The Dynamic Response of the Hurricane Wind Field to Spiral ...
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The Dynamic Response of the Hurricane Wind Field to Spiral Rainband Heating YUMIN MOON AND DAVID S. NOLAN Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida (Manuscript received 6 April 2009, in final form 28 January 2010) ABSTRACT The response of the hurricane wind field to spiral rainband heating is examined by using a three-dimensional, nonhydrostatic, linear model of the vortex–anelastic equations. Diabatic heat sources, which are designed in accordance with previous observations of spiral rainbands, are made to rotate with the flow around the hurricane-like wind field of a balanced, axisymmetric vortex. Common kinematic features are recovered, such as the overturning secondary circulation, descending midlevel radial inflow, and cyclonically accelerated tangential flow on the radially outward side of spiral rainbands. Comparison of the responses to the purely convective and stratiform rainbands indicates that the overturning secondary circulation is mostly due to the convective part of the rainband and is stronger in the upwind region, while midlevel radial inflow descending to the surface is due to the stratiform characteristics of the rainband and is stronger in the downwind region. The secondary horizontal wind maximum is exhibited in both convective and stratiform parts of the rainband, but it tends to be stronger in the downwind region. The results indicate that the primary effects of rainbands on the hurricane wind field are caused by the direct response to diabatic heating in convection embedded in them and that the structure of the diabatic heating is primarily responsible for their unique kinematic structures. Sensitivity tests confirm the robustness of the results. In addition, the response of the hurricane wind field to the rainband heating is, in the linear limit, the sum of the asymmetric potential vorticity and symmetric transverse circulations.
1. Introduction One of the most prominent features of tropical cyclones is their banded structures of convection in circular or spiral shapes called spiral rainbands. The importance of spiral rainbands on tropical cyclones was recognized early (e.g., Wexler 1947; Senn and Hiser 1959; Atlas et al. 1963), and their cloud characteristics are often used to help diagnose the current intensity and forecast the future evolution of tropical cyclones (Dvorak 1975). There are several ways to distinguish spiral rainbands. They can be classified as inner or outer rainbands. Inner rainbands are located near the vortex core, usually within 100 km from the center of the storm. They are visible on radar images but are not always apparent on satellite images because of the dense overcast clouds near the center. In comparison, outer rainbands often do not show much spiral curvature and can be quite long. They are typically visible on both radar and satellite images and located far from the center, on the order of
Corresponding author address: Yumin Moon, 4600 Rickenbacker Causeway, University of Miami, Miami, FL 33149. E-mail: [email protected] DOI: 10.1175/2010JAS3171.1 Ó 2010 American Meteorological Society
several hundred kilometers (e.g., Willoughby et al. 1984; Guinn and Schubert 1993). The observed radial distribution of lightning within tropical cyclones agrees with this categorization (Molinari et al. 1999; Cecil et al. 2002) in which inner rainbands are associated with the region of relative minimum in lightning frequency while outer rainbands are associated with the region of larger lightning frequency. Wang (2008) suggested the use of the rapid filamentation zone (Rozoff et al. 2006) as a criterion to separate inner rainbands from outer rainbands. The rapid filamentation zone is a strain-dominated flow region just outside the radius of maximum wind (RMW), where convection and its associated vorticity are expected to be quickly filamented into thin strips, which are ultimately lost to diffusion. Other classification methods exist as well. According to Willoughby et al. (1984), spiral rainbands can also be classified as moving or stationary, based on their propagation relative to the mean tangential circulation. They are primary rainbands if they merge into the eyewall of the storm or secondary rainbands if they are disconnected from the eyewall. In addition, recent observational studies (e.g., Gall et al. 1998; Kusunoki and Mashiko 2006) have identified numerous smaller fine-scale bands with
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cross-band scales of 2–10 km that are found near the eyewall and have very little propagation relative to the mean tangential circulation of the storm. There exist contrasting views of the effect of spiral rainbands on the intensity of tropical cyclones. Observational studies have indicated that from a thermodynamic viewpoint spiral rainbands have a negative impact on intensity because they intercept some of the moist low-level radial inflow into the eyewall and replace it with cooler, dry air originating from downdrafts (e.g., Barnes et al. 1983, hereafter BZJM83; Powell 1990, hereafter P90; Cione et al. 2000). At the same time, however, spiral rainbands can also be considered to have a positive impact on intensity if they are viewed as barriers to protect the storm core from dry air intrusion and environmental wind shear (Kimball 2006). From a fluid dynamical point of view, spiral rainbands are viewed as potential vorticity (PV) bands, and they are believed to have a positive impact on intensity as they transport angular momentum inward through the axisymmetrization process (e.g., Carr and Williams 1989; Montgomery and Kallenbach 1997; Nolan and Farrell 1999). Full-physics numerical simulations of tropical cyclones have shown that PV anomalies are generated in the middle troposphere within spiral rainbands (Chen and Yau 2001) and that the PV generation is greater in the stratiform portion of the rainbands (May and Holland 1999; Franklin et al. 2006). Recently, Wang (2009) examined the effects of outer spiral rainbands on tropical cyclone structure and intensity by modifying the diabatic heating rate in outer spiral rainbands formed within numerically simulated tropical cyclones. Wang (2009) found that the effect of the increased (decreased) diabatic heating rate in rainbands is to weaken (strengthen) the intensity of hurricanes but make them larger (smaller) in size. To understand the effects of spiral rainbands on tropical cyclones, it is necessary to fully incorporate their dynamic and thermodynamic aspects; however, their dynamics and thermodynamics must be driven in large part by latent heat release in convection embedded within them. Here, we propose that the effects of spiral rainbands on the tropical cyclone wind field are caused by the direct response to diabatic heating in their convection. This approach is similar to that of Pandya and Durran (1996) and Pandya et al. (2000) in which the mesoscale circulation around squall lines was approximated by the direct response to steady thermal forcing that resembles the latent heat release in a convective leading line. In this paper, it is also proposed that spiral rainbands can be regarded as quasi-steady, asymmetric heat sources that may rotate around the storm. It is the goal of this study to evaluate these approximations and
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the overall view that the diabatic heating structure in spiral rainbands is primarily responsible for their kinematic structures. We first review previous observational studies of spiral rainbands in section 2. Section 3 introduces the numerical model and the basic-state vortex. Section 4 describes how the structures of asymmetric heat sources used to represent spiral rainbands are represented in the numerical model. Section 5 describes the response of the tropical cyclone wind field to the rainband heating and compares the results to observations. Section 6 presents sensitivity tests. Summary and discussions are provided in section 7.
2. Observed structures of spiral rainbands a. Previous observational studies A large portion of previous observational studies of spiral rainbands examined the reflectivity of a single ground-based or airborne Doppler radar (e.g., Senn and Hiser 1959; Atlas et al. 1963; BZJM83). Composites of pseudodual Doppler radar data have also been used before (e.g., P90; Barnes et al. 1991; Barnes and Powell 1995). More recent studies (e.g., May et al. 1994; Kim et al. 2009) analyzed wind profiler observations to infer the structures of the spiral rainbands. Previous studies have shown that spiral rainbands are usually 10–40 km in width and often spiral radially inward in a cyclonic direction. Their precipitation structures show that the upwind portion is mostly convective while the downwind side is mostly stratiform (Atlas et al. 1963; BZJM83), although there is a large variation in the degree of organization (e.g., Ryan et al. 1992; May 1996). A prominent rainband that extends outward toward one side of the vortex is often called a principal rainband that sometimes joins the eyewall through a connecting rainband (Willoughby et al. 1984). BZJM83 hypothesized that the convective-scale circulation within spiral rainbands is made up of two flows: an updraft accompanied by a low-level radial inflow and an upper-level radial outflow, and a midlevel radial inflow that passes through the rainband and descends to the surface. This circulation was illustrated in Fig. 18 of BZJM83, reproduced here as Fig. 1a. Also noted in the BZJM83 model is that the reflectivity tower is tilted radially outward with height. Another kinematic feature of the spiral rainband circulation is a midlevel jet of tangential wind located on the radially outward side of the rainbands (P90; Samsury and Zipser 1995), as shown in the horizontal cross-sectional view of a spiral rainband circulation presented by P90 (Fig. 16a of P90, reproduced here as Fig. 1b). This midlevel jet is also
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FIG. 1. Schematic diagrams of kinematic structures within spiral rainbands, originally presented as (a) Fig. 18 in BZJM83 and (b) Fig. 16 in P90. Contours are for radar reflectivity (dBZ), and the 25-dBZ contour marks the boundary of spiral rainbands. In (a), arrows represent convective-scale circulations, and potential temperature (K) is marked at the beginning and ending of each circulation.
called a secondary horizontal wind maximum (SHWM). Samsury and Zipser (1995) found that the location of a SHWM is highly correlated with the location of spiral rainbands and speculated on the possibility that the SHWM is generated and maintained by spiral rainband processes, which were not clearly defined at that time.
b. Recent dual-Doppler radar observations from the Hurricane Rainband and Intensity Change Experiment program Hence and Houze (2008, hereafter HH08) analyzed high-resolution dual-Doppler radar observations of spiral rainbands in Hurricanes Katrina and Rita (2005) collected during the Hurricane Rainband and Intensity
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Change Experiment field program (Houze et al. 2006). One of the goals of HH08 was to evaluate the main features of the BZJM83 and P90 conceptual spiral rainband models. HH08 examined the middle portion of a spiral rainband in Hurricanes Katrina and Rita (2005). The vertical cross sections (Figs. 6 and 7 of HH08) were taken through an intense updraft located on the radially inward edge of the rainband. They illustrate that the reflectivity of the convective cell slopes radially outward with height. In addition, there is an overturning secondary circulation that has radial inflow at lower levels and radial outflow at upper levels. The cross sections also reveal layers of convergence and divergence at the base and top of the reflectivity core, respectively, with the largest upward vertical motion found in between. A SHWM is located on the radially outward side of the spiral rainband. In addition, HH08 examined the vertical cross sections (Figs. 8 and 9 of HH08) through a downdraft found in the middle portion of the same rainband of Katrina and Rita. The cross sections show two downdraft circulations: a radial inflow (near height z 5 2 km) enters from the radially outward side of the rainband and descends to surface, while the other downdraft originates from upper levels (near z 5 8 km) but stays on the radially inward side of the reflectivity core. From examining the structures of the updrafts and downdrafts found in the middle portion of a spiral rainband (both Hurricanes Katrina and Rita), HH08 were able to find supporting evidence for the hypothesized features of the BZJM83 and P90 models: an overturning secondary circulation associated with a convective updraft, a midlevel radial inflow that descends to surface, and a midlevel SHWM. By using the retrieved three-dimensional velocity fields, HH08 also computed and examined the vertical profile of mass transport. Their analysis showed that the net vertical mass transport in the upwind and middle regions is positive everywhere throughout the troposphere, with a maximum value at middle levels, while the downwind region has a nearly zero net vertical mass transport. The magnitude of the upward vertical mass transport is greatest in the middle region of the spiral rainband. Figure 2 shows the conceptualized model of a spiral rainband circulation as presented in HH08. The horizontal view (Fig. 2a) shows that the rainband is convective upwind and through the middle portion of the rainband before becoming stratiform downwind. As the rainband becomes more stratiform, it also becomes wider. A SHWM is located near the rainband and also on its radially outward side. A vertical cross section through
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FIG. 2. Schematic diagram of precipitation and kinematic structures within spiral rainbands, originally presented as Fig. 13 in HH08. (a) Horizontal cross-sectional view of the conceptual spiral rainband model. (b) Radius–height cross section along the thin solid line in (a).
the middle part of the rainband (Fig. 2b) shows that the convective cell leans radially outward with altitude and the SHWM is located on its radially outward edge. There is an overturning secondary circulation coupled with an updraft located on the radially inward edge of the cell, with convergence and divergence at the base and top of the cell, respectively. Another radial inflow can be found slightly above the low-level radial inflow, and it descends to the surface while passing through the rainband. A downdraft that originates in the upper levels stays radially inward of the rainband. The boundary of the rainband is marked by the reflectivity value of 25 dBZ.
3. The model and basic-state vortex a. The numerical model: Three-Dimensional Vortex Perturbation Analysis and Simulation To simulate the effects of spiral rainband heating on the tropical cyclone wind field, we use a three-dimensional,
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nonhydrostatic, linear numerical model of vortex dynamics, now known as the Three-Dimensional Vortex Perturbation Analysis and Simulation (3DVPAS). It is based on the vortex–anelastic equations in cylindrical coordinates on the f plane (Hodyss and Nolan 2007), and its dynamics are linear and nonhydrostatic. It can simulate both asymmetric and symmetric motions on balanced axisymmetric vortices. Perturbations can have arbitrary radial and vertical structures but have harmonic variation in the azimuthal (tangential) direction. The symmetric response to purely asymmetric motions can also be computed by using the eddy flux divergences arising from asymmetric motions as forcing for symmetric motions. Free-slip, solid-wall boundary conditions are enforced on all boundaries, and Rayleigh damping regions (sponges) are placed at the upper and outer boundaries to suppress the reflection of gravity waves. The domain size of all simulations in this study is 300 km in the radial direction and 20 km in the vertical direction. The grids are stretched in the radial direction but equally spaced in the vertical direction. In the radial direction, the grid size is approximately 2 km in the inner-core region (0 km , r , 95 km) but larger (approximately 6 km) in the far outer-core region with a smooth transition in between. The grid size in the vertical direction is 1/ 3 km. More complete details of the 3DVPAS model can be found in Nolan and Montgomery (2002a), Nolan and Grasso (2003), and Nolan et al. (2007).
b. Basic-state vortex There are a number of idealized profiles that can provide a reasonably realistic representation of the tangential wind field of tropical cyclones (e.g., Willoughby et al. 2006). As the control case in this study, we use the modified Rankine (MR) profile because it has a very simple form yet provides a realistic representation of tangential wind, as shown by Mallen et al. (2005). The MR profile has a tangential velocity field defined by r 8 > y < max RMW , a ~y(r) 5 RMW , > :y max r
r # RMW (1) r . RMW,
where y max is the prescribed maximum tangential velocity and a is the decay parameter. The radial structure of tangential wind field of tropical cyclones is well documented because of the relative abundance of observations obtained from United States Air Force and National Oceanic and Atmospheric Administration aircrafts investigating a large number of hurricanes. The same cannot be said, however, about the vertical structure of the tangential wind field. From
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early studies (e.g., Frank 1977; Jorgensen 1984; Shea and Gray 1973), it has been known qualitatively for some time that the tangential wind decays away from the surface, and the RMW generally slopes radially outward with height. Some arbitrary vertical structures (e.g., Nolan and Montgomery 2002a; Nolan and Grasso 2003; Nolan et al. 2007) have been used before; however, the RMW of the wind fields used in these previous studies did not in fact slope outward and is thus unrealistic. A realistic radially outward slope of the RMW with height and a vertical decay rate of tangential velocity at the RMW can be quantitatively obtained by utilizing the steady-state axisymmetric hurricane model by Emanuel (1986, hereafter E86), which assumes slantwise moist neutrality. The E86 model has two components: an inviscid, free atmosphere in hydrostatic and gradient wind balance and a boundary layer in which viscous and surface processes play an important role. Only the free atmosphere portion of the E86 model is used in our approach, which utilizes the fact that the RMW is very closely approximated by a surface of constant angular momentum (Stern and Nolan 2009). Details of this approach are outlined in the appendix. By using the MR profile from Eq. (1) with ymax 5 45 m s21, RMW 5 31 km, and a 5 0.5 to prescribe the tangential velocity field at the lowest level (Fig. 3a), we obtain the vertical slope of the RMW and decay rate of tangential velocity along the RMW (Figs. 3b,c). The level at which tangential velocity becomes zero at the RMW is z 5 15.9 km. Then, the radial profile of tangential velocity at each vertical level is constructed by using Eq. (1). It turns out that the tangential wind field constructed this way decays too rapidly near the surface, such that the linear model supports unstable modes in that region. To remedy this problem, the lowest level of the tangential wind field is extended 0.5 km downward barotropically with the same radial structure, and the tangential wind field is smoothed in the vertical direction. Figure 4a shows the tangential wind field of this MR basic-state vortex. Although frictionally induced secondary circulations are important, they are neglected here, because above the boundary layer tropical cyclones can be approximated as vortices in hydrostatic and gradient wind balance (Willoughby 1990). After constructing the tangential wind field of the basic-state vortex, pressure and temperature fields that hold the vortex wind field in hydrostatic and gradient wind balance are computed, which is achieved with an iterative procedure described in Nolan et al. (2001) using the Jordan (1958) mean hurricane season sounding as the far-field environment. Figure 4b shows the potential temperature anomalies required to keep the vortex in balance. This MR vortex is stable to static and symmetric
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FIG. 3. (a) MR radial profile of tangential velocity (m s21) with ymax 5 45 m s21 and RMW 5 31 km. (b) Vertical profile of the RMW of tangential velocity field constructed by using the free atmosphere component of the E86 model. (c) Vertical profile of tangential velocity at the RMW from the same model.
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FIG. 4. Radius–height cross sections of (a) tangential velocity (m s21), (b) anomalous potential temperature (K), (c) square of stratification frequency (s22), and (d) PV fields [potential vorticity units (PVU)] of the basic-state MR vortex. Contour intervals are 5.0 m s21, 2.0 K, 5.0 3 1025 s22, and 2.5 PVU from the zero line, respectively.
instabilities by ensuring ›u/›z . 0 and PV . 0 over the whole domain (Figs. 4c,d). Stability analysis using the methods of Nolan and Montgomery (2002a) and Nolan and Grasso (2003) confirms that there are no exponentially growing modes in this MR vortex.
4. Designing a spiral rainband heat source a. The structure of an idealized spiral rainband heating Spiral rainbands are banded structures of convection and can be considered as a cluster of individual updrafts and downdrafts where most of the strong vertical motions and associated latent heat release occur. Steady improvements in observational and computational
technology have made it possible to observe and simulate tropical cyclones with convective features on the same length scale as individual updrafts; however, it is beyond the ability of 3DVPAS to simulate the effect of individual updrafts and downdrafts in a spiral rainband. Rather, the goal of this study is to understand the response of the tropical cyclone wind field to the overall rainband heating in a simple framework that nonetheless can capture the time-dependent, nonhydrostatic dynamics of the response. The observational studies discussed in section 2 have shown that the upwind portion of spiral rainband heating is convective and the downwind end is stratiform, with a transition between them. In addition, the rainband heating spirals radially inward in a cyclonic direction and tilts radially outward with height. One simple way
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to represent the overall diabatic heating structure of such rainbands is to overlap multiple heat sources similar to the individual heat sources previously used in
Nolan et al. (2007). First, we define the diabatic heating structures of purely convective and stratiform heat sources as
8 0, for z # > bc > " z > # < r rbc 2 z zbc _ _ sin p exp(inl invt), Q con (r, l, z, t) 5 Qcon max exp > src src > > : 0, for z $ zbc 1 szc
for
8 0, for z # zbs szs > > " ! > 2# < r r z z bs bs sin p exp(inl invt), for Q_ str (r, l, z, t) 5 Q_ str max exp > srs szs > > : 0, for z $ zbs 1 szs
In Eqs. (2) and (3), rb(z) is the radial center position and sr and sz are the radial and vertical half-widths of the heating, respectively. Subscripts c and s refer to purely convective and stratiform heat sources. For convective heat sources (Fig. 5a), zbc is chosen to be the lowest altitude where positive diabatic heating is present, and the vertical depth of the heating is controlled by taking the half vertical wavelength szc only upward from zbc. For stratiform heat sources (Fig. 5b), zbs is the level of zero heating, and its vertical depth is controlled by taking the half vertical wavelength szs both upward and downward from zbs. Observations (e.g., BZJM83; P90) indicate that rainband heating tilts radially outward with height, which can be achieved by linearly varying rb with altitude, such that rb(z) 5 rbsfc 1 z, where rbsfc is the radial center position at the surface. The complex exponential term in Eqs. (2) and (3) rotates the heat source around the center of the vortex at an angular velocity v 5 yrot/rc, where rc is the radial location of the center point of the rainband heating at the surface and y rot is the rotation speed of the heat source. Now we create an idealized but realistic-looking spiral rainband by overlapping 40 heat sources. By ‘‘overlapping’’ heat sources, we mean that heat sources are placed close to each other so that most points in the rainband receive a heating contribution from multiple heat sources. The sum of these overlapped heat sources is the rainband heating. Figure 6a shows the horizontal cross section of this rainband at z 5 4.5 km. This rainband covers approximately one quarter of the circumference in the lower right quadrant, but its upwind end is at r 5 80 km, while its downwind end is at r 5 60 km. The radial center location of each heat source is equal to rbc in Eq. (2) and rbs in Eq. (3), which varies linearly between the ends of the rainband (from r 5 80 km to 60 km). The heat sources at the upwind (r 5 80 km)
zbc , z , zbc 1 szc ,
zbs szs , z , zbs 1 szs
(2)
(3)
and downwind (r 5 60 km) ends are purely convective (Fig. 5a) and purely stratiform (Fig. 5b), respectively. The transition from the convective to stratiform diabatic heating is defined to be linear. As the rainband makes a transition from convective to stratiform, the diabatic heating structure becomes shallower and wider, and an area of significant cooling develops in the lower part of the rainband. Figure 5c shows the vertical cross section in the middle part of the rainband, illustrating the change in the vertical structure of diabatic heating during this linear transition. At this location (r 5 70 km), the heat source is a combination of 50% convective and 50% stratiform heat sources. The maximum heating rate of each individual convective heat source is chosen to be twice the maximum heating rate of the stratiform heat source, so Q_ con max 5 3.0 K h21 and Q_ str max 5 1.5 K h21.
b. How 3DVPAS sees the spiral rainband diabatic heating To compute the impact of this idealized spiral rainband in 3DVPAS, the idealized spiral rainband heating in Fig. 6a is separated into different azimuthal wavenumber components by Fourier series decomposition. The magnitude of the purely asymmetric components decreases as the azimuthal wavenumber increases (not shown). Figure 6b shows the sum of its Fourier decompositions from n 5 0 to 4, and it is a bit different from the original rainband heating (Fig. 6a) because of the truncation of the series at n 5 4. Each wavenumber component of the heating is simulated separately in 3DVPAS. To obtain the full response of the vortex to a rainband heating, it is necessary to consider the asymmetric and symmetric responses to each asymmetric component plus the symmetric response to the symmetric part. Therefore, the full response of the vortex to
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the rainband heating can be approximated in the linear limit by the sum of the asymmetric and symmetric responses to its first four asymmetric components plus the symmetric response to its symmetric part. Hereafter, the response refers to the sum of the asymmetric and symmetric responses to the purely asymmetric components (n 5 1–4) and the symmetric response to the symmetric part (n 5 0). In addition, all responses are rotated such that the rainband heating at the time the response is shown would always be in the same location as in Fig. 6a.
5. The response of the hurricane wind field to rainband heating a. A purely convective spiral rainband heating
FIG. 5. (a),(b) Vertical cross sections of normalized purely convective and stratiform diabatic heat sources centered at r 5 80 and 60 km, respectively. (c) Vertical cross section of normalized mixed (50% convective, 50% stratiform) diabatic heat source centered at r 5 70 km. For (a), rbsfc 5 80 km [so rbc(z) 5 80 km 1 z in Eq. (2)], zbc 5 1 km, src 5 2 km, and szc 5 7 km are used; for (b), rbsfc 5 60 km [so rbs(z) 5 60 km 1 z in Eq. (3)], zbs 5 4 km, srs 5 6 km, and szs 5 2 km are used. For (c), rbsfc 5 70 km.
A typical spiral rainband heating contains both convective and stratiform components; however, to better understand the combined response to these two different heating profiles, it is helpful to first consider the response to a rainband heating that is entirely convective or stratiform. Figures 6c and 6d show the horizontal cross sections at z 5 4.50 km of the idealized purely convective rainband heating and its sum of Fourier decompositions from n 5 0 to 4. Just like the mixed rainband described in the previous section, a purely convective rainband spirals inward (from r 5 80 to 60 km), but the structure of its diabatic heating remains the same along the band (see Fig. 5a). The first four purely asymmetric components plus the symmetric part are rotated around the MR vortex for 24 h. Since rainbands are observed to move more slowly than the mean circulation speed (Anthes 1982), the rotation speed is set to be 70% of the maximum tangential wind at the middle part of the rainband at r 5 70 km, so y rot 5 21.1 m s21. Figure 7 shows time evolution of kinetic energy (KE) of the sum of the asymmetric responses to the asymmetric components (n 5 1–4, hereafter KEaa), the symmetric response to the symmetric part (n 5 0, hereafter KEss), and the sum of the symmetric responses to the asymmetric components (n 5 1–4, hereafter KEsa). It shows that KEaa increases rapidly in the first 8 h and then gradually levels off although it never quite reaches a steady state, which is due to the excitation of a weakly damped n 5 1 inner-core mode, as KE for n $ 2 reaches a nearly steady state by approximately t 5 15 h (not shown). After t 5 16 h, the change in KE only results from the accumulation of KE in the n 5 1 inner-core mode. Unlike KEaa, however, after a period of adjustments in the first several hours KEss increases at a linear rate after t 5 5 h. The period of initial adjustment in KEsa is longer but it increases linearly after t 5 15 h.
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FIG. 6. (a) The horizontal cross section at z 5 4.50 km of the mixed spiral rainband heating (K h21). (b) The sum of its Fourier decompositions from n 5 0 to 4. (c),(d) As in (a),(b), but for the purely convective spiral rainband heating (K h21). (e),(f) As in (a),(b), but for the purely stratiform spiral rainband heating (K h21). Forty heat bubbles are used to construct each rainband heating, which spirals cyclonically inward from r 5 80 to 60 km. The zero line is not plotted. Maximum and minimum values can be found above each panel. See section 4a for explanations.
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FIG. 7. Time evolution of normalized KE of the asymmetric responses to the asymmetric components (KEaa; solid), the symmetric response to the symmetric part (KEss; dashed), and the symmetric responses to the asymmetric components (KEsa; dashed–dotted) of the convective rainband. KE values are normalized by their respective maximum value during the 24-h period, which are 7.90 3 1012 J for KEaa, 4.44 3 1014 J for KEss, and 8.30 3 1012 J for KEsa.
KEsa decreases1 for the first 6 h as the asymmetric components of the rainband heating interact with the basicstate vortex and extract KE from it (see section 4b of Nolan et al. 2007). At t 5 16 h and thereafter, the response to the rainband heating is expected to change only quantitatively. Unless noted otherwise, the response hereafter is evaluated at t 5 16 h for all cases considered. Figure 8 shows horizontal cross sections of the response to the convective rainband heating at z 5 1.83 (lower level), 4.50 (middle level), and 7.17 km (upper level); z 5 1.83 and 7.17 km are located slightly above the bottom of the heating and slightly below the top of the heating respectively, while z 5 4.50 km is at the middle level of the heating. Arrows and contours represent horizontal and vertical velocity components of the response. All cross sections show that an inner-core mode is excited by the rainband heating, and this mode has a strong n 5 1 structure. Despite the presence of an inner-core mode, the response to the rainband heating can be identified in all cross sections. They show that there is upward vertical velocity at the location of the
1 KEsa can in fact be negative, as it is defined as the difference in total KE between the initial symmetric vortex and the symmetric vortex at later times. If interactions with the evolving asymmetries change the symmetric vortex to a flow with less KE, then KEsa can be negative.
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heating. Comparing the location of vertical motion at each cross section with the location of the rainband heating (Figs. 6c,d) indicates that vertical motion is well coupled to the heating, and this is found to be true for all subsequent cases. Hereafter, the location of vertical motion of the response is used interchangeably with the location of the rainband heating. Examining the horizontal velocity components of the response at the lower levels (Fig. 8c) shows an accelerated tangential circulation on the radially outward side of the rainband heating. However, this feature becomes weak at the middle levels (Fig. 8b), and at the upper levels (Fig. 8c) there is instead a decelerated tangential circulation on the radially outward side of the rainband heating. Figure 9 shows vertical cross sections through the downwind, middle, and upwind regions (see white lines in Fig. 6a) of the response to the convective rainband heating. Arrows depict radial and vertical velocity components of the response, while contours show the tangential velocity component of the response. All cross sections show that tangential circulation around the location of the heating and on its radially outward side is accelerated at the lower levels (from the surface to z 5 4 km) but decelerated at the upper levels between z 5 5 and 10 km. In addition, there is radial inflow toward the rainband from its radially outward side at the lower levels but radial outflow away from the rainband at the upper levels. Upward vertical motion is found between radial inflow and accelerated tangential velocity at the lower levels and between radial outflow and decelerated tangential velocity at the upper levels. The strongest rising motion is found between z 5 4 and 6 km. The axis of the largest upward velocity moves radially inward in the downstream direction, in accordance with the inwardly spiraling structure of the convective rainband heating. Although full understanding of the response to the rainband heating requires examining both asymmetric and symmetric responses to the asymmetric and symmetric components of the heating, the symmetric responses alone capture a large part of the full response. Figure 10a shows the vertical cross section of the sum of the symmetric responses to both asymmetric and symmetric parts (n 5 0–4) of the convective rainband, while Fig. 10b shows the vertical cross section of the symmetric response to the symmetric part (n 5 0) alone. Both cross sections clearly show that around the rainband location and on its radially outward side there is low-level radial inflow and upper-level radial outflow, which are connected by upward vertical motion through the rainband heating. On its radially inward side, there is weak lowlevel radial outflow that enters the updraft and some of the updraft gets diverted into weak radial inflow at the
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upper levels. Clearly, there is convergence and divergence at the base and top of the updraft, respectively. These vertical cross sections also indicate that tangential circulation at the lower levels is accelerated at the location of the heating, as well as on its radially outward side, but decelerated at the upper levels. This symmetric response is qualitatively similar to transverse circulations presented in Eliassen (1951), Shapiro and Willoughby (1982), and Hack and Schubert (1986). All vertical cross sections from Fig. 9 exhibit these similar features of the symmetric responses shown in Fig. 10. There are, however, some notable differences among different parts of the rainband. The low-level radial inflow toward the rainband is deeper and stronger in the downwind region in comparison to the upwind region. This radial inflow after passing the rainband continues to travel toward the vortex center in the downwind region, while there is actually a radial outflow toward the rainband in the upwind region. At the upper levels, the radial outflow from the rainband is weaker and shallower in the upwind region than in the downwind region. In addition, on the radially inward side of the rainband, there is radial inflow in the upwind region but radial outflow in the downwind region. These differences may be understood conceptually by considering the PV of the asymmetric response. In a linearized system such as 3DVPAS, the perturbation PV is h ›u 1 1 ›u ›u 1 n 1 zn qn 5 (vn $u 1 v $un ) 5 jn ›r ›z r r r ›l ›un ›un h ›un , (4) 1j 1 1 ( f 1 z) r ›l ›r ›z
FIG. 8. Horizontal cross sections at z 5 (a) 7.17, (b) 4.50, and (c) 1.83 km of the response to the convective rainband heating (Fig. 6c). Contours represent vertical velocity (m s21) of the response with a contour interval of 0.01 m s21 from the zero line, which is not plotted. In this and all subsequent grayscale figures, dark colors are more positive than light colors, and positive values are indicated by solid contours, while negative values are indicated by dashed contours. Arrows show horizontal velocity (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
where j, h, and z are, respectively, the radial, azimuthal, and vertical components of the three-dimensional vorticity vector v; r is density; u is potential temperature; the overbar and subscript n (azimuthal wavenumber) denote the axisymmetric (basic state) and asymmetric perturbation quantities, respectively; and f is a Coriolis parameter of 5.0 3 1025 s21. In a baroclinic vortex without secondary circulations, such as the MR vortex here, a thermal perturbation such as the rainband heating produces a PV perturbation that has contributions from the fourth and sixth terms in Eq. (4). For the convective rainband, the fourth term creates positive PV on its radially inner side and negative PV on its radially outer side, while the sixth term produces positive PV at the lower levels and negative PV at its upper levels. The sum of these four PV components results in the PV forcing that has positive PV at the low levels and negative PV at the upper levels (Fig. 11a).
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FIG. 10. Vertical cross sections of (a) the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0– 4) and (b) the symmetric response to the symmetric component (n 5 0) of the convective rainband heating (Fig. 6c). Contours represent tangential velocity (m s21) of the response with a contour interval of 0.10 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
FIG. 9. Vertical cross sections through the (a) downwind, (b) middle, and (c) upwind regions of the response to the convective rainband heating (Fig. 6c). Contours represent tangential velocity (m s21) of the response with a contour interval of 0.20 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
In response to this PV forcing, positive PV and cyclonic circulations are expected to develop at the lower levels, but negative PV and anticyclonic circulations should develop at the upper levels. Figure 12 shows the sum of the asymmetric responses to the asymmetric components of the convective rainband, and it clearly illustrates these features; the full response also hints at them (Figs. 8a,c). Figure 11b also shows the vertical cross section of PV of the sum of the asymmetric responses to the asymmetric components (n 5 1–4) to the convective rainband. Careful examination of Figs. 10
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FIG. 11. Vertical cross sections through the middle region of (a) PV forcing (PVU h21) associated with the convective rainband heating (Fig. 6c) and (b) PV (PVU) associated with the response to the convective rainband heating. (c),(d) As in (a),(b), but for the stratiform rainband (Fig. 6e) case. Note that different contour interval values are used, and the zero line is not plotted. Maximum, minimum, and contour interval can be found above each panel.
and 12 indicates that the asymmetric PV and symmetric transverse responses cooperate in some places but act in opposite directions in other places. For example, at the lower levels in the upwind region, the asymmetric and symmetric responses are in the same direction on the radially inward side of the rainband, producing radial outflow into the rainband, but in the opposite direction on its radially outward side, resulting in weak radial inflow into the rainband. At the lower levels in the downwind region, the asymmetric and symmetric responses are in the same direction on the outward side, but they are in the opposite direction on the inward side and result in radial inflow toward the center because the asymmetric response is stronger. Similarly, the differences in the strength and direction of the radial flow at the
upper levels between the upwind and downwind regions can be attributed to the different superimpositions of the asymmetric response with the symmetric response. The collocation of radial inflow with increased tangential velocity at the lower levels and radial outflow with decreased tangential velocity at the upper levels suggests that the change in tangential wind field is simply the result of an angular momentum-conserving radial flow. The rainband heating is located outside the RMW where the tangential wind decays as r2a (a 5 0.5 in the MR vortex); thus, angular momentum associated with tropical cyclone tangential wind field increases radially outward. Therefore, radial flow from a region of higher angular momentum to a region of lower angular momentum (e.g., low-level radial inflow) will increase
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height, more outward than the updraft. The change in tangential wind field due to angular momentumconserving radial flow, however, is larger in comparison to that due to angular momentum-conserving vertical flow, because the angular momentum transport by radial flow is larger than that by vertical flow by a factor of 3 or higher. Comparing Figs. 6d and 6b suggests that the error from truncating the Fourier series at n 5 4 is larger in the convective rainband forcing (Fig. 6c) than in the mixed rainband forcing (Fig. 6a). The convective rainband case was repeated with the truncation at n 5 8, but the response of the hurricane wind field to the n 5 0–8 convective rainband heating remained essentially the same (not shown).
b. A purely stratiform spiral rainband heating
FIG. 12. Horizontal cross sections at z 5 (a) 7.17 and (b) 1.83 km of the sum of the asymmetric responses to the asymmetric components of the convective rainband (Fig. 6c). Contours show PV (PVU) and arrows show horizontal velocity (m s21). Note that different contour interval values are used, and the zero line is not plotted. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
tangential wind, while radial flow from a region of lower angular momentum to a region of higher angular momentum (e.g., upper-level radial outflow) will decrease tangential velocity. In addition, the updraft component of the overturning secondary circulation flows from a region of higher angular momentum to a region of lower angular momentum, leading to positive tangential velocity change at the upper levels but negative tangential velocity change at the lower levels (not shown). This process occurs because the tangential velocity field of the MR vortex decreases with height, so the angular momentum surfaces are sloped radially outward with
Figures 6e and 6f show the horizontal cross sections at z 5 4.50 km of the idealized purely stratiform rainband heating and its sum of Fourier decompositions from n 5 0 to 4. The purely stratiform rainband has heating above and cooling below the level of zero heating at z 5 4.0 km (see Fig. 5b). Figure 13 shows horizontal cross sections of the response at z 5 2.17 (lower level), 3.83 (middle level), and 5.83 km (upper level). Note that z 5 2.17 and 5.83 km are located slightly above the bottom of the cooling component and slightly below the top of the heating component of the stratiform rainband heating, respectively, while z 5 3.83 km is slightly below the level of zero heating. All cross sections show that there are inner-core modes excited by the heating. While the modes from the convective rainband case have a strong wavenumber-1 signature at all levels (Fig. 8), different wavenumber structures tend to dominate at different levels in the stratiform case. Despite the excitation of the inner-core modes, it is easy to identify the response to the stratiform rainband heating. Since this stratiform rainband heating has heating above and cooling below z 5 4.0 km (see Fig. 5b), there are rising and sinking motions above and below this level, indicating vertical divergence at this level. Also present near this level (Fig. 13b) is accelerated tangential circulation on the radially outward side of the rainband heating. Near the top of the heating component (Fig. 13a), there is anticyclonic radial outflow emanating away from the rainband where rising motion is present. At the base of the cooling component (Fig. 13c) there is weak anticyclonic radial outflow away from the rainband where sinking motion exists. It is interesting to note that the low-level horizontal circulation is substantially weaker than that at middle and upper levels. Figure 14 shows vertical cross sections through the downwind, middle, and upwind regions (see white lines
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in Fig. 6a) of the response to the stratiform rainband heating. All cross sections show that there is radial inflow toward the rainband at the middle levels (between z 5 3 and 5 km). After passing through the rainband, this inflow splits into two branches: one descends to the surface and continues to travel toward the vortex center, and the other ascends to the upper levels (near z 5 6 km). This midlevel radial inflow pattern is collocated with the positive tangential velocity component of the response, indicating that angular momentum conservation leads to the accelerated tangential circulation. Also present at these levels is radial outflow that enters the rainband from its radially inward side and decreased tangential velocity response. In addition, on the radially outward side of the rainband, there is negative tangential velocity response above and below the midlevel radial inflow layer, respectively. Figure 15 shows vertical cross sections of the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0–4) of the stratiform rainband heating and the symmetric response to its symmetric part (n 5 0) alone. Both cross sections clearly show that at the location of the rainband heating there are rising and sinking motions above and below z 5 4 km, respectively. In addition, on the radially outward side there is radial inflow toward the rainband at this altitude. It appears that the symmetric responses are very similar to the full response through the middle portion (Fig. 14b). The collocation of positive and negative tangential velocity responses with radial inflow and outflow indicates that angular momentum conservation leads to these tangential velocity responses. The angular momentum transport by radial flow is larger than that by vertical flow by a factor of 4 or higher. Examining Fig. 14 in detail reveals that the response to the stratiform rainband heating does not show as much variation along the band as the response to the convective rainband (Fig. 9), which indicates that the symmetric responses to the stratiform rainband capture an even larger portion of the full response in comparison to the convective case. There are, however, still some notable differences, especially in the radial flow pattern. For example, there is radial inflow toward the rainband from its radially outward side at the middle levels in the downwind region but weak radial outflow in the upwind region. In addition, the radial flow on the radially outward side at the upper levels is oppositely directed between the downwind and upwind regions. These differences may be attributed to the asymmetric PV responses. Figure 11c shows the vertical cross section of PV forcing associated with the stratiform rainband. There is negative PV at the lower and upper levels but positive PV at the middle levels. In response,
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FIG. 13. As in Fig. 8, but for the response to the stratiform rainband heating (Fig. 6e) at z 5 (a) 5.83, (b) 3.83, and (c) 2.17 km, with a contour interval of 0.03 m s21.
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FIG. 15. As in Fig. 10, but for the stratiform rainband heating (Fig. 6e), with a contour interval of 0.50 m s21.
FIG. 14. As in Fig. 9, but for the response to the stratiform rainband heating (Fig. 6e), with a contour interval of 0.50 m s21.
positive PV and cyclonic circulations develop at the middle levels and negative PV and anticyclonic circulations prevail at the lower and upper levels, as illustrated in the horizontal cross sections of the sum of the asymmetric responses to the asymmetric components to the stratiform rainband (Fig. 16). Figure 11d also shows the vertical cross section of PV of the sum of the asymmetric responses to the asymmetric components to the stratiform rainband. Comparing Figs. 15 and 16 suggests that the asymmetric PV and symmetric transverse responses are in the same direction in some regions but in the opposite direction in other places. For example, on the radially outward side of the rainband, the asymmetric and symmetric responses are in the same direction in the downwind region but in the opposite direction in the upwind region. Because the radial velocity of the asymmetric response is stronger than that of
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the symmetric response, there is weak midlevel radial outflow in the upwind region. A more pronounced midlevel tangential velocity response in the downwind region is due to the fact that the cyclonic asymmetric response strongly projects into the azimuthal direction in the downwind region but more into the radial direction in the upwind region.
c. A mixed spiral rainband heating
FIG. 16. Horizontal cross sections at z 5 (a) 5.83, (b) 3.83, and (c) 2.17 km of the sum of the asymmetric responses to the asymmetric components of the stratiform rainband (Fig. 6e). Contours show PV (PVU) with a contour interval of 0.60 PVU from the zero line, which is not plotted. Arrows show horizontal velocity (m s21). Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
With the previous results in mind, we now present the response to a mixed spiral rainband heating. The detailed configurations of the mixed rainband were described in section 4a. The mixed rainband heating spirals radially inward in a cyclonic direction from r 5 80 to 60 km, and the upwind and downwind regions of the heating are convective and stratiform, respectively, with a linear transition between them. Figure 17 shows horizontal cross sections of the response at z 5 2.50 (lower level), 4.17 (middle level), and 6.83 km (upper level). Despite the usual excitation of an inner-core mode, it is not difficult to isolate the response to the rainband heating. At the lower levels (Fig. 17c), the rainband heating is positive upwind and negative downwind, and accordingly there are rising and sinking motions in those respective regions. On the radially outward side of the rainband, there are some signs of accelerated tangential circulation. This positive tangential velocity response becomes more evident at the middle levels (Fig. 17b), especially on the downwind region. At these altitudes, the heating is positive everywhere along the rainband, and accordingly there is rising motion everywhere. At the upper levels (Fig. 17a), there is instead a negative tangential velocity response. Also present at these altitudes is radial flow that diverges away from the rainband in an anticyclonic direction. It seems evident from these cross sections that the asymmetric PV response is cyclonic at the lower and middle levels but anticyclonic at the upper levels. Figure 18 shows vertical cross sections through the upwind, middle, and downwind regions (see white lines in Fig. 6a) of the mixed rainband. All cross sections show that on the radially outward side of the rainband there is accelerated tangential velocity at the lower levels, especially between z 5 3 and 4 km. Above the positive tangential velocity response, there is a decelerated tangential circulation, centered near z 5 7 km. In the upwind region, there is weak radial inflow toward the rainband from its outward side at the lower levels and radial outflow from the radially inward side. They converge at the base of the rainband and then rise through it at approximately r 5 80 km, and most of it becomes
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radial inflow at the upper levels. In the downwind region, there is deep (from the surface to z 5 4 km) radial inflow toward the rainband from its outward side, and most of it continues to flow toward the center after passing the rainband. Between z 5 3 and 4 km, some of the radial inflow that enters the rainband descends to lower levels between r 5 80 and 60 km. Radial inflow between z 5 4 and 5 km meets radial outflow that enters the rainband from its inward side and rises to upper levels, where there is mostly radial outflow. The tangential velocity response in the downwind region is greater than in the upwind region. The middle region appears to be a mix of the upwind and downwind regions. Figure 19 shows vertical cross sections of the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0–4) of the heating and the symmetric response to the symmetric part (n 5 0) alone. Both cross sections show that there is deep radial inflow toward the rainband from its radially outward side, and this inflow is stronger at z 5 4 km than at the surface. Most of this inflow rises through the rainband and becomes radial outflow away from the rainband between z 5 5 and 9 km; however, some of this inflow descends from z 5 3 to 2 km and continues to travel radially inward. Near z 5 6 km, some of the updraft diverts toward the vortex center. On the radially inward side, there is radial outflow at z 5 4 km toward the rainband. Clearly, there is convergence at the location of the rainband heating at this altitude, and divergence above it. This symmetric response to the mixed rainband also appears to be a mix of the convective and stratiform symmetric responses presented earlier, and it explains a large portion of the full response. As in the previous cases, the asymmetric and symmetric responses act in the same direction in some locations, such as on the radially outward side at the lower and middle levels in the downwind region, but in the opposite direction in other locations, such as on the radially outward side at the upper levels in the upwind region.
d. Comparison to observations Previous observational studies of spiral rainbands indicated that there are an overturning secondary circulation, an accelerated tangential circulation on the radially outward side of the rainband, and a midlevel radial inflow that descends to the surface, intertwining with the lower branch of the overturning circulation. Figure 2 illustrates a conceptualized model of a spiral rainband circulation as presented in HH08. The idealized spiral rainband heating structures considered so far are designed to represent only the gross effects of latent heat release within the spiral rainband. Nonetheless, the
FIG. 17. As in Fig. 8, but for the response to the mixed rainband heating (Fig. 6a) at z 5 (a) 6.83, (b) 4.17, and (c) 2.50 km, with a contour interval of 0.03 m s21.
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FIG. 19. As in Fig. 10, but for the mixed rainband heating (Fig. 6a), with a contour interval of 0.20 m s21.
FIG. 18. As in Fig. 9, but for the response to the mixed rainband heating (Fig. 6a), with a contour interval of 0.40 m s21.
response of the MR vortex wind field to the mixed spiral rainband heating captures all of the aforementioned features with the contributions from the responses to both purely convective and stratiform rainband heating. It appears that the midlevel radial inflow and its descending motion to the surface closely resemble the stratiform response and are stronger in the downwind region. The overturning secondary circulation feature belongs to the convective response and is stronger in the upwind region. An accelerated tangential circulation on the radially outward side of the rainband shares the characteristics of both the convective and stratiform responses, but it tends to be stronger in the downwind region where stratiform response prevails. In addition, this accelerated tangential circulation feature extends further radially away from the rainband than what observations indicate (see Figs. 1b and 2).
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e. The effects of the inner-core modes All cross sections through the responses indicate that the response to the rainband heating occurs not only at the location of the heating but also around the RMW. In the convective rainband case, horizontal cross sections (Fig. 8) indicate that there are strong azimuthal wavenumber-1 signatures in the inner-core response. Vertical cross sections (Fig. 9) show that the tangential wind field at the lower levels (from the surface to z 5 3 km) is decelerated inside the RMW but accelerated just outside the RMW. In addition, there is increased tangential velocity response near the RMW at the upper levels (between z 5 6 and 10 km). In the linear limit, this process would suggest that the effect of the inner-core modes excited in response to the purely convective rainband is to vertically stretch the tangential circulation on the rainband side and pull the whole tangential circulation toward the rainband. The inner-core modes in the response to the purely stratiform rainband are not as severe as in the convective case. The inner-core modes in the mixed rainband case show similar low-level structures as in the convective case but are disorganized in the upper levels. The overall effects of the asymmetric inner-core modes in all cases, however, are small because the differences in the wind speed change between the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0–4) of the rainband and the sum of the symmetric response to the symmetric component (n 5 0) of the rainband are less than 5% (see Figs. 10, 15, and 19).
6. Sensitivity tests We must consider whether the responses described in the previous section are highly sensitive to the particular parameters that define the structure and movement of the rainband heating. As described in section 4a, there are a large number of parameters involved in controlling the evolution of imposed rainband heating. Tests are performed for only parameters that appear to have a large variation based on observations.
a. Depth of convection in the upwind region Cecil et al. (2002) created the composites of radar reflectivity over tropical cyclones as observed by the Tropical Rainfall Measuring Mission satellite (see their Fig. 3). Their statistics indicate that the convective region of spiral rainbands sometimes has convection penetrating deeper than z 5 8 km, which is the upper boundary of the convective heat forcings used in the previous section (see section 4a and Fig. 5a). By using 20 dBZ to define the boundary for a spiral rainband
FIG. 20. Vertical cross sections through the (a) downwind and (b) upwind regions of the response to the mixed rainband heating whose convective heat bubbles are 10 km deep in the vertical direction. Contours represent tangential velocity (m s21) of the response with a contour interval of 0.30 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
heating, the upper bound of z 5 8 km is very close to the 75th percentile of the cumulative density function of radar reflectivity. To evaluate the effects of the different depths of upwind convection, the mixed rainband case is repeated with sz 5 9 and 11 km so the upper bound of upwind convection is z 5 10 and 12 km, respectively. Figure 20 shows vertical cross sections of the response through the upwind and downwind region (see white lines in Fig. 6a) for the sz 5 9 km case (thus 10 km deep). It is evident that the effect of deeper upwind convection is to elevate
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FIG. 21. Vertical cross sections through the (top) downwind and (bottom) upwind regions of the response to the mixed rainbands that are (left) 50% convective and (right) 50% stratiform. See section 6b for details. Contours represent tangential velocity (m s21) of the response with a contour interval of 0.50 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
the radial outflow of the overturning secondary circulation feature; however, the midlevel radial inflow and its descending motion, as well as the accelerated tangential circulation on the radially outward side of the rainband, remain unaffected by the presence of deeper upwind convection. This point further supports the conclusion from the previous section that the overturning circulation is due to the response to purely convective parts of the rainband.
b. Organization of convection within rainbands All rainband cases examined so far have the linear transition occurring over the whole rainband, which may not be representative for all rainbands. The mixed rainband case is repeated but with two different transition
schemes. One scheme (hereafter 50% convective) keeps the rainband purely convective over the first half of the rainband from its upwind end and then makes a linear transition to purely stratiform regime over the second half, while the other scheme (hereafter 50% stratiform) makes a linear transition from purely convective to purely stratiform over the first half and then remains purely stratiform over the second half. Three common rainband circulation features are present in the response to both cases (Fig. 21). Notable differences from the mixed rainband case are that the overturning circulation is strengthened in the 50% convective case while the 50% stratiform case shows enhanced descending midlevel radial inflow. Both cases show increased tangential circulation on the radially outward side of the rainband,
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but the altitude of the largest positive tangential velocity response is higher for the 50% stratiform case than the 50% convective case. While a large number of observed rainbands are convective upwind and stratiform downwind, it is possible to have highly disorganized rainbands, which can be represented by constructing a rainband only with heat sources that have equal convective and stratiform contributions at all locations along the band (see Fig. 5c). Other parameters for this disorganized rainband are the same as in the mixed rainband. The response to the disorganized rainband (Fig. 22) recovers the overturning circulation and accelerated tangential circulation on its radially outward side. The descending midlevel radial inflow, however, is not present in the downwind region response, likely due to the reduced area of cooling in the downwind region of this disorganized rainband in comparison to the mixed rainband (cf. Figs. 5c and 5b). We also considered another way to change the transition between the upwind and downwind ends, which is to modify the values of Q_ con max in Eq. (2) and Q_ str max in Eq. (3). Their ratio represents the different contributions of purely convective and purely stratiform heat sources. When Q_ con max : Q_ str max is increased, the overturning secondary circulation is strengthened as in the 50% convective case. Decreasing the ratio leads to enhanced descending midlevel radial inflow as in the 50% stratiform case. It also appears that the effect of stronger convective heat sources in the rainband in the upwind region is to vertically stretch the positive tangential velocity response near the rainband because of the stronger updraft. In the downwind region, stronger stratiform heat sources lead to enhanced tangential circulation at the middle levels on the radially outward side of the rainband.
c. Transient rainband So far we have examined the response of the hurricane wind field to rainband heating that is persistent over time. However, observations (e.g., Anthes 1982) indicate that the duration of individual convective cells embedded within rainband heating is typically on the order of several hours. The rainbands themselves are transient, but the precise mechanisms controlling their initiation, duration, and decline remain unknown. To represent the transient effects of the rainband heating, we now multiply Eqs. (2) and (3) by " # t tforc 4 . (5) F(t) 5 exp ptforc In Eq. (5), tforc is the time at which the rainband heating is at its maximum. Setting the coefficient p 5 0.55163
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FIG. 22. Vertical cross sections through the (a) downwind and (b) upwind regions of the response to the disorganized rainband (see section 6b). Contours represent tangential velocity (m s21) of the response with a contour interval of 0.50 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
shapes the rainband heating to approximately last for 2tforc h. The structure of this exponential function is very similar to that used in Nolan et al. (2007; see their Fig. 5b). Another simple function to represent the transient effects is to use a piecewise linear function such that the rainband heating rate reaches its peak at t 5 tpeak and then decays until it dissipates at t 5 tdiss. The mixed rainband case from the previous section is repeated with the heating now following the exponential and linear functions described previously. For the exponential function, tforc 5 6 h is used, and for the linear function, tpeak 5 4 h and tdiss 5 3tpeak 5 12 h are used.
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Figure 23 shows horizontal cross sections at z 5 4.17 km of the response to the transient mixed rainband heating using Eq. (5) at different times. Comparison with the response to the persistent mixed rainband heating (Fig. 17) indicates that cyclonic tangential acceleration on the radially outward side of the rainband can be seen locally around the rainband heating as early as t 5 4 h (Fig. 23a). The overturning secondary circulation and descending midlevel radial inflow are also present in the response by this time (not shown). As the heating rate becomes larger, the response becomes more easily discernable (Fig. 23b, t 5 7 h), and this tangential acceleration starts to propagate in the downstream direction relative to the rainband heating. As the heating rate ramps down (Fig. 23c, t 5 10 h), the response starts to weaken, especially in vertical velocity response; however, the cyclonic tangential acceleration feature remains coherent long after the heating is turned off (Figs. 23e,f). The evolution of the transient mixed rainband that follows the piecewise linear forcing is qualitatively similar (not shown).
d. Rotation speed and radial location of rainband heating Previous observational studies suggested that spiral rainbands move around the center more slowly than the mean circulation. There are observations that some spiral rainbands (e.g., principal bands) stay at the same region for a long period of time. The mixed rainband case was repeated but with y rot prescribed to be 60% and 80% of the maximum tangential wind at the middle part of the rainband (r 5 70 km). The corresponding values of y rot are 18.1 and 24.1 m s21, respectively. A stationary mixed rainband case (y rot 5 0.0 m s21) was also considered. All responses (not shown) recovered the aforementioned main features of rainband circulation, indicating that the sensitivity of the response to the rotation speed is not significantly large. The sensitivity of the response to the radial location of the rainband was examined by considering the mixed rainband case with the convective upwind end at r 5 120 km and the stratiform downwind end at r 5 100 km. Sixty heat sources were used to construct the rainband, which still makes a linear transition in the downstream direction. The response was qualitatively similar to the mixed rainband except that the features are shifted 20 km radially outward (not shown).
7. Summary and discussions The response of the hurricane wind field to rainband heating has been examined by using a three-dimensional, nonhydrostatic, linear model of vortex dynamics. Diabatic
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heat sources were designed in accordance with previous observations of spiral rainbands, and they were embedded in the hurricane wind field and rotated around the vortex. The response to a typical mixed rainband captures many common features of spiral rainband kinematic structures, such as an overturning secondary circulation, an enhanced tangential circulation on the radially outward side of the rainband (or secondary horizontal wind maximum), and a midlevel radial inflow that descends to the surface. Comparison of the responses to the purely convective and stratiform rainbands indicates that the overturning secondary circulation mostly results from the convective part of the rainband and is stronger in the upwind region, while midlevel radial inflow descending to surface results from the stratiform characteristics of the rainband and is stronger in the downwind region. The secondary horizontal wind maximum is exhibited in both convective and stratiform parts of the rainband, but it tends to be stronger in the downwind region. Sensitivity tests confirm that the response is robust and that the latent heating structure of convection embedded in rainbands is primarily responsible for their kinematic structures. In addition, the response of the hurricane wind field to the rainband heating is, in the linear limit, the sum of the asymmetric PV and symmetric transverse circulations. Frictionally induced secondary circulations are an important part of the tropical cyclone wind field. The basicstate MR vortex used in this study, however, lacks them, which is justified by the work by Willoughby (1990) that showed in the free atmosphere above the boundary layer tropical cyclones are approximately cyclonic vortices in hydrostatic and gradient wind balance. The secondary circulations are closely related with the distribution of convection around tropical cyclones, which are highly asymmetric and transient; however, the responses to the rainband in all cases considered develop their own secondary circulations, so the qualitative aspects of the response appear not to be significantly affected by the lack of the secondary circulation. In addition, the location of the rainband secondary circulation above the boundary layer is different from the location of the mean secondary circulation. It is possible that incorporating the secondary circulation into the basic-state vortex would lead to a more vigorous response to rainband heating because the mean anticyclonic circulation at the upper levels reduces inertial stability. The excitation of inner-core modes could be less severe because of the upward and outward advection of the PV anomalies (Nolan and Montgomery 2002b). This study examines the response of the hurricane wind field to imposed rainband heating whose structures are already prescribed. A fundamental question that still remains unanswered concerns the factors or processes
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FIG. 23. Horizontal cross sections at z 5 4.17 km of the response to the transient mixed rainband heating at t 5 (a) 4, (b) 7, (c) 10, (d) 12, (e) 14, and (f) 16 h. Contours show vertical velocity (m s21) of the response, with a contour interval shown above each panel from the zero line, which is not plotted. Note that different contour interval values are used for different plots. Positive and negative values are marked by solid and dashed contour lines, respectively. Arrows show horizontal velocity (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
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that control the evolution of these diabatic heating structures of convection in tropical cyclones. The answer appears to be closely related to the evolution of the boundary layer structures of hurricanes, which are not represented in 3DVPAS. Comparisons to spiral rainbands that are seen to develop ubiquitously during full-physics simulations of tropical cyclones should be made as a next step. Attempts to isolate the impact of spiral rainbands on the surrounding wind field in high-resolution simulations should also be made. The response of the hurricane wind field to transient mixed rainbands is intriguing. It suggests that the accelerated cyclonic tangential circulation can wrap around the entire vortex if the heating lasts long enough and that the change in the tangential circulation would remain coherent for a long time after the rainband heating dissipates. This process may help us better understand secondary eyewall formation, a phenomenon not well understood at the moment. Kuo et al. (2004) and Wang (2009) also suggested that the presence of strong rainbands may favor the formation of secondary eyewalls. Our efforts to understand the role of spiral rainbands in the formation of secondary eyewalls will continue with numerical models of increasing complexity. Acknowledgments. This research was supported by the National Science Foundation Grants ATM-0432717 and ATM-0756308. We also would like to sincerely thank Drs. Brian E. Mapes, Robert F. Rogers, and Hugh E. Willoughby for their helpful comments and feedback on this study. Detailed and constructive reviews by Drs. Yuqing Wang, Shigeo Yoden, and two anonymous reviewers are greatly appreciated. Thanks are due to Mr. Daniel P. Stern for sharing his version of the code for the free atmospheric portion of the E86 model and also to Mr. David Painemal for help with figures.
APPENDIX Obtaining the Slope of the RMW and Decay Rate of Tangential Velocity at the RMW from E86 E86 assumes that the primary tangential circulation of tropical cyclones in the free atmosphere is in hydrostatic and gradient wind balance and that saturation moist entropy is conserved along angular momentum surfaces, which slope radially outward with height, thus implying a state of slantwise moist neutrality. By using pseudoadiabatic thermodynamic conditions, as clarified by Bryan and Rotunno (2009), E86 derived expressions [Eqs. (51) and (56) in E86] for how an angular momentum surface slopes outward with height in the free atmosphere. Since the RMW of the tropical cyclone
tangential wind field is closely approximated by an angular momentum surface (Stern and Nolan 2009), the rates at which the RMW slopes radially outward with altitude and the tangential velocity decays along the RMW may be obtained from these expressions. First, the tangential wind field at the lowest level of the domain, which should be considered to be the top of the boundary layer, is prescribed by a profile such as the MR vortex in Eq. (1), and the pressure and temperature fields are initialized with constant values p0 and TB, respectively. Then, we compute the pressure field that would hold the prescribed wind field in gradient wind balance 1 ›p y2 5 fy 1 , r ›r r
(A1)
where r is density, p is pressure, r is radius, f is the Coriolis parameter, and y is tangential velocity. Equation (A1) is combined with the equation of state— r5
p , Rd T
(A2)
where Rd is the dry gas constant and T is temperature— to eliminate r and then is integrated inward from the outer boundary using a Crank–Nicholson scheme while holding T fixed. The density field is computed by enforcing Eq. (A2) with fixed T. Next, we compute the environmental thermodynamic properties consistent with the prescribed tangential wind field by following an iterative procedure outlined in E86 (597–598). Then, Eq. (51) of E86 is evaluated at the RMW to obtain the radius of an outward sloping angular momentum surface that is congruent to the RMW. The altitude of this angular momentum surface can be found from Eq. (56) of E86. Vertical decay of tangential velocity along the RMW can be found by inverting the definition of absolute angular momentum M [ ry 1 fr2/2 at each altitude. We use values of p0 5 1015.1 mb, TB 5 26.38C, and T0 5 270.08C. REFERENCES Anthes, R. A., 1982: Tropical Cyclones: Their Evolution, Structure and Effects. Meteor. Monogr., No. 41, Amer. Meteor. Soc., 208 pp. Atlas, D., K. R. Hardy, R. Wexler, and R. J. Boucher, 1963: On the origin of hurricane spiral bands. Geofis. Int., 3, 123–132. Barnes, G. M., and M. D. Powell, 1995: Evolution of the inflow boundary layer of Hurricane Gilbert (1988). Mon. Wea. Rev., 123, 2348–2368. ——, E. J. Zipser, D. Jorgensen, and F. Marks Jr., 1983: Mesoscale and convective structure of a hurricane rainband. J. Atmos. Sci., 40, 2125–2137.
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——, J. F. Gamache, M. A. LeMone, and G. J. Stossmeister, 1991: A convective cell in a hurricane rainband. Mon. Wea. Rev., 119, 776–794. Bryan, G. H., and R. Rotunno, 2009: Evaluation of an analytical model for the maximum intensity of tropical cyclones. J. Atmos. Sci., 66, 3042–3060. Carr, L. E., and R. T. Williams, 1989: Barotropic vortex stability to perturbations from axisymmetry. J. Atmos. Sci., 46, 3177–3191. Cecil, D. J., E. J. Zipser, and S. W. Nesbitt, 2002: Reflectivity, ice scattering, and lightning characteristics of hurricane eyewalls and rainbands. Part I: Quantitative description. Mon. Wea. Rev., 130, 769–784. Chen, Y., and M. K. Yau, 2001: Spiral bands in a simulated hurricane. Part I: Vortex Rossby wave verification. J. Atmos. Sci., 58, 2128–2145. Cione, J. J., P. G. Black, and S. H. Houston, 2000: Surface observations in the hurricane environment. Mon. Wea. Rev., 128, 1550–1561. Dvorak, V. F., 1975: Tropical cyclone intensity analysis and forecasting from satellite imagery. Mon. Wea. Rev., 103, 420–430. Eliassen, A., 1951: Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astrophys. Norv., 5, 19–60. Emanuel, K. A., 1986: An air–sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., 43, 585–605. Frank, W. M., 1977: The structure and energetics of the tropical cyclone. Part I: Storm structure. Mon. Wea. Rev., 105, 1119– 1135. Franklin, C. N., G. J. Holland, and P. T. May, 2006: Mechanisms for the generation of mesoscale vorticity features in tropical cyclone rainbands. Mon. Wea. Rev., 134, 2649–2669. Gall, R., J. Tuttle, and P. Hildebrand, 1998: Small-scale spiral bands observed in hurricanes Andrew, Hugo, and Erin. Mon. Wea. Rev., 126, 1749–1766. Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands. J. Atmos. Sci., 50, 3380–3403. Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of atmospheric vortices to heating by organized cumulus convection. J. Atmos. Sci., 43, 1559–1573. Hence, D. A., and R. A. Houze Jr., 2008: Kinematic structure of convective-scale elements in the rainbands of Hurricanes Katrina and Rita (2005). J. Geophys. Res., 113, D15108, doi:10.1029/2007JD009429. Hodyss, D., and D. S. Nolan, 2007: Anelastic equations for atmospheric vortices. J. Atmos. Sci., 64, 2947–2959. Houze, R. A., Jr., and Coauthors, 2006: The Hurricane Rainband and Intensity Change Experiment. Observations and modeling of Hurricanes Katrina, Ophelia, and Rita. Bull. Amer. Meteor. Soc., 87, 1503–1521. Jordan, C. L., 1958: Mean soundings for the West Indies area. J. Meteor., 15, 91–97. Jorgensen, D. P., 1984: Mesoscale and convective-scale characteristics of mature hurricanes. Part I: General observations by research aircraft. J. Atmos. Sci., 41, 1268–1285. Kim, D.-K., K. R. Knupp, and C. R. Williams, 2009: Airflow and precipitation properties within the stratiform region of Tropical Storm Gabrielle during landfall. Mon. Wea. Rev., 137, 1954–1971. Kimball, S., 2006: A modeling study of hurricane landfall in a dry environment. Mon. Wea. Rev., 134, 1901–1918. Kuo, H.-C., L.-Y. Lin, C.-P. Chang, and R. T. Williams, 2004: The formation of concentric vorticity structures in typhoons. J. Atmos. Sci., 61, 2722–2734.
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Kusunoki, K., and W. Mashiko, 2006: Doppler radar investigations of the inner core of Typhoon Songda (2004): Polygonal/elliptical eyewalls, eye contraction, and small-scale spiral bands. Extended Abstracts, 27th Conf. on Hurricanes and Tropical Meteorology, Monterey, CA, Amer. Meteor. Soc., P4.10. [Available online at http://ams.confex.com/ams/27Hurricanes/ techprogram/paper_107480.htm.] Mallen, K. J., M. T. Montgomery, and B. Wang, 2005: Reexamining the near-core radial structure of the tropical cyclone primary circulation: Implications for vortex resiliency. J. Atmos. Sci., 62, 408–425. May, P. T., 1996: The organization of convection in the rainbands of Tropical Cyclone Laurence. Mon. Wea. Rev., 124, 807–815. ——, and G. J. Holland, 1999: The role of potential vorticity generation in tropical cyclone rainbands. J. Atmos. Sci., 56, 1224– 1228. ——, ——, and W. L. Ecklund, 1994: Wind profiler observations of Tropical Storm Flo at Saipan. Wea. Forecasting, 9, 410–426. Molinari, J., P. Moore, and V. Idone, 1999: Convective structure of hurricanes as revealed by lightning locations. Mon. Wea. Rev., 127, 520–534. Montgomery, M. T., and R. J. Kallenbach, 1997: A theory for vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes. Quart. J. Roy. Meteor. Soc., 123, 435–465. Nolan, D. S., and B. F. Farrell, 1999: The intensification of twodimensional swirling flows by stochastic asymmetric forcing. J. Atmos. Sci., 56, 3937–3962. ——, and M. T. Montgomery, 2002a: Nonhydrostatic, threedimensional perturbations to hurricane-like vortices. Part I: Linearized formulation, stability, and evolution. J. Atmos. Sci., 59, 2989–3020. ——, and ——, 2002b: Three-dimensional stability analyses of tornado-like vortices with secondary circulations. Preprints, 21st Conf. on Severe Local Storms, San Antonio, TX, Amer. Meteor. Soc., 477–480. ——, and L. D. Grasso, 2003: Nonhydrostatic, three-dimensional perturbations to hurricane-like vortices. Part II: Symmetric response and nonlinear simulations. J. Atmos. Sci., 60, 2717– 2745. ——, M. T. Montgomery, and L. D. Grasso, 2001: The wavenumberone instability and trochoidal motion of hurricane-like vortices. J. Atmos. Sci., 58, 3243–3270. ——, Y. Moon, and D. P. Stern, 2007: Tropical cyclone intensification from asymmetric convection: Energetics and efficiency. J. Atmos. Sci., 64, 3377–3405. Pandya, R. E., and D. R. Durran, 1996: The influence of convectively generated thermal forcing on the mesoscale circulation around squall lines. J. Atmos. Sci., 53, 2924–2951. ——, ——, and M. L. Weisman, 2000: The influence of convective thermal forcing on the three-dimensional circulation around squall lines. J. Atmos. Sci., 57, 29–45. Powell, M. D., 1990: Boundary layer structure and dynamics in outer hurricane rainbands. Part I: Mesoscale rainfall and kinematic structure. Mon. Wea. Rev., 118, 891–917. Rozoff, C. M., W. H. Schubert, B. D. McNoldy, and J. P. Kossin, 2006: Rapid filamentation zones in intense tropical cyclones. J. Atmos. Sci., 63, 325–340. Ryan, B. F., G. M. Barnes, and E. J. Zipser, 1992: A wide rainband in a developing tropical cyclone. Mon. Wea. Rev., 120, 431–447. Samsury, C. E., and E. J. Zipser, 1995: Secondary wind maxima in hurricanes: Airflow and relationship to rainbands. Mon. Wea. Rev., 123, 3502–3517.
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Senn, H. V., and H. W. Hiser, 1959: On the origin of hurricane spiral rain bands. J. Meteor., 16, 419–426. Shapiro, L. J., and H. E. Willoughby, 1982: The response of balanced hurricanes to local sources of heat and momentum. J. Atmos. Sci., 39, 378–394. Shea, D. J., and W. M. Gray, 1973: The hurricane’s inner core region. I. Symmetric and asymmetric structure. J. Atmos. Sci., 30, 1544–1564. Stern, D. P., and D. S. Nolan, 2009: Reexamining the vertical structure of tangential winds in tropical cyclones: Observations and theory. J. Atmos. Sci., 66, 3579–3600. Wang, Y., 2008: Rapid filamentation zone in a numerically simulated tropical cyclone. J. Atmos. Sci., 65, 1158–1181.
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——, 2009: How do outer spiral rainbands affect tropical cyclone structure and intensity? J. Atmos. Sci., 66, 1250–1273. Wexler, H., 1947: Structure of hurricanes as determined by radar. Ann. N. Y. Acad. Sci., 48, 821–844. Willoughby, H. E., 1990: Gradient balance in tropical cyclones. J. Atmos. Sci., 47, 265–274. ——, F. D. Marks Jr., and R. J. Feinberg, 1984: Stationary and moving convective bands in hurricanes. J. Atmos. Sci., 41, 3189– 3211. ——, R. W. R. Darling, and M. E. Rahn, 2006: Parametric representation of the primary hurricane vortex. Part II: A new family of sectionally continuous profiles. J. Atmos. Sci., 134, 1102–1120.
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The Dynamic Response of the Hurricane Wind Field to Spiral Rainband Heating YUMIN MOON AND DAVID S. NOLAN Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida (Manuscript received 6 April 2009, in final form 28 January 2010) ABSTRACT The response of the hurricane wind field to spiral rainband heating is examined by using a three-dimensional, nonhydrostatic, linear model of the vortex–anelastic equations. Diabatic heat sources, which are designed in accordance with previous observations of spiral rainbands, are made to rotate with the flow around the hurricane-like wind field of a balanced, axisymmetric vortex. Common kinematic features are recovered, such as the overturning secondary circulation, descending midlevel radial inflow, and cyclonically accelerated tangential flow on the radially outward side of spiral rainbands. Comparison of the responses to the purely convective and stratiform rainbands indicates that the overturning secondary circulation is mostly due to the convective part of the rainband and is stronger in the upwind region, while midlevel radial inflow descending to the surface is due to the stratiform characteristics of the rainband and is stronger in the downwind region. The secondary horizontal wind maximum is exhibited in both convective and stratiform parts of the rainband, but it tends to be stronger in the downwind region. The results indicate that the primary effects of rainbands on the hurricane wind field are caused by the direct response to diabatic heating in convection embedded in them and that the structure of the diabatic heating is primarily responsible for their unique kinematic structures. Sensitivity tests confirm the robustness of the results. In addition, the response of the hurricane wind field to the rainband heating is, in the linear limit, the sum of the asymmetric potential vorticity and symmetric transverse circulations.
1. Introduction One of the most prominent features of tropical cyclones is their banded structures of convection in circular or spiral shapes called spiral rainbands. The importance of spiral rainbands on tropical cyclones was recognized early (e.g., Wexler 1947; Senn and Hiser 1959; Atlas et al. 1963), and their cloud characteristics are often used to help diagnose the current intensity and forecast the future evolution of tropical cyclones (Dvorak 1975). There are several ways to distinguish spiral rainbands. They can be classified as inner or outer rainbands. Inner rainbands are located near the vortex core, usually within 100 km from the center of the storm. They are visible on radar images but are not always apparent on satellite images because of the dense overcast clouds near the center. In comparison, outer rainbands often do not show much spiral curvature and can be quite long. They are typically visible on both radar and satellite images and located far from the center, on the order of
Corresponding author address: Yumin Moon, 4600 Rickenbacker Causeway, University of Miami, Miami, FL 33149. E-mail: [email protected] DOI: 10.1175/2010JAS3171.1 Ó 2010 American Meteorological Society
several hundred kilometers (e.g., Willoughby et al. 1984; Guinn and Schubert 1993). The observed radial distribution of lightning within tropical cyclones agrees with this categorization (Molinari et al. 1999; Cecil et al. 2002) in which inner rainbands are associated with the region of relative minimum in lightning frequency while outer rainbands are associated with the region of larger lightning frequency. Wang (2008) suggested the use of the rapid filamentation zone (Rozoff et al. 2006) as a criterion to separate inner rainbands from outer rainbands. The rapid filamentation zone is a strain-dominated flow region just outside the radius of maximum wind (RMW), where convection and its associated vorticity are expected to be quickly filamented into thin strips, which are ultimately lost to diffusion. Other classification methods exist as well. According to Willoughby et al. (1984), spiral rainbands can also be classified as moving or stationary, based on their propagation relative to the mean tangential circulation. They are primary rainbands if they merge into the eyewall of the storm or secondary rainbands if they are disconnected from the eyewall. In addition, recent observational studies (e.g., Gall et al. 1998; Kusunoki and Mashiko 2006) have identified numerous smaller fine-scale bands with
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cross-band scales of 2–10 km that are found near the eyewall and have very little propagation relative to the mean tangential circulation of the storm. There exist contrasting views of the effect of spiral rainbands on the intensity of tropical cyclones. Observational studies have indicated that from a thermodynamic viewpoint spiral rainbands have a negative impact on intensity because they intercept some of the moist low-level radial inflow into the eyewall and replace it with cooler, dry air originating from downdrafts (e.g., Barnes et al. 1983, hereafter BZJM83; Powell 1990, hereafter P90; Cione et al. 2000). At the same time, however, spiral rainbands can also be considered to have a positive impact on intensity if they are viewed as barriers to protect the storm core from dry air intrusion and environmental wind shear (Kimball 2006). From a fluid dynamical point of view, spiral rainbands are viewed as potential vorticity (PV) bands, and they are believed to have a positive impact on intensity as they transport angular momentum inward through the axisymmetrization process (e.g., Carr and Williams 1989; Montgomery and Kallenbach 1997; Nolan and Farrell 1999). Full-physics numerical simulations of tropical cyclones have shown that PV anomalies are generated in the middle troposphere within spiral rainbands (Chen and Yau 2001) and that the PV generation is greater in the stratiform portion of the rainbands (May and Holland 1999; Franklin et al. 2006). Recently, Wang (2009) examined the effects of outer spiral rainbands on tropical cyclone structure and intensity by modifying the diabatic heating rate in outer spiral rainbands formed within numerically simulated tropical cyclones. Wang (2009) found that the effect of the increased (decreased) diabatic heating rate in rainbands is to weaken (strengthen) the intensity of hurricanes but make them larger (smaller) in size. To understand the effects of spiral rainbands on tropical cyclones, it is necessary to fully incorporate their dynamic and thermodynamic aspects; however, their dynamics and thermodynamics must be driven in large part by latent heat release in convection embedded within them. Here, we propose that the effects of spiral rainbands on the tropical cyclone wind field are caused by the direct response to diabatic heating in their convection. This approach is similar to that of Pandya and Durran (1996) and Pandya et al. (2000) in which the mesoscale circulation around squall lines was approximated by the direct response to steady thermal forcing that resembles the latent heat release in a convective leading line. In this paper, it is also proposed that spiral rainbands can be regarded as quasi-steady, asymmetric heat sources that may rotate around the storm. It is the goal of this study to evaluate these approximations and
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the overall view that the diabatic heating structure in spiral rainbands is primarily responsible for their kinematic structures. We first review previous observational studies of spiral rainbands in section 2. Section 3 introduces the numerical model and the basic-state vortex. Section 4 describes how the structures of asymmetric heat sources used to represent spiral rainbands are represented in the numerical model. Section 5 describes the response of the tropical cyclone wind field to the rainband heating and compares the results to observations. Section 6 presents sensitivity tests. Summary and discussions are provided in section 7.
2. Observed structures of spiral rainbands a. Previous observational studies A large portion of previous observational studies of spiral rainbands examined the reflectivity of a single ground-based or airborne Doppler radar (e.g., Senn and Hiser 1959; Atlas et al. 1963; BZJM83). Composites of pseudodual Doppler radar data have also been used before (e.g., P90; Barnes et al. 1991; Barnes and Powell 1995). More recent studies (e.g., May et al. 1994; Kim et al. 2009) analyzed wind profiler observations to infer the structures of the spiral rainbands. Previous studies have shown that spiral rainbands are usually 10–40 km in width and often spiral radially inward in a cyclonic direction. Their precipitation structures show that the upwind portion is mostly convective while the downwind side is mostly stratiform (Atlas et al. 1963; BZJM83), although there is a large variation in the degree of organization (e.g., Ryan et al. 1992; May 1996). A prominent rainband that extends outward toward one side of the vortex is often called a principal rainband that sometimes joins the eyewall through a connecting rainband (Willoughby et al. 1984). BZJM83 hypothesized that the convective-scale circulation within spiral rainbands is made up of two flows: an updraft accompanied by a low-level radial inflow and an upper-level radial outflow, and a midlevel radial inflow that passes through the rainband and descends to the surface. This circulation was illustrated in Fig. 18 of BZJM83, reproduced here as Fig. 1a. Also noted in the BZJM83 model is that the reflectivity tower is tilted radially outward with height. Another kinematic feature of the spiral rainband circulation is a midlevel jet of tangential wind located on the radially outward side of the rainbands (P90; Samsury and Zipser 1995), as shown in the horizontal cross-sectional view of a spiral rainband circulation presented by P90 (Fig. 16a of P90, reproduced here as Fig. 1b). This midlevel jet is also
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FIG. 1. Schematic diagrams of kinematic structures within spiral rainbands, originally presented as (a) Fig. 18 in BZJM83 and (b) Fig. 16 in P90. Contours are for radar reflectivity (dBZ), and the 25-dBZ contour marks the boundary of spiral rainbands. In (a), arrows represent convective-scale circulations, and potential temperature (K) is marked at the beginning and ending of each circulation.
called a secondary horizontal wind maximum (SHWM). Samsury and Zipser (1995) found that the location of a SHWM is highly correlated with the location of spiral rainbands and speculated on the possibility that the SHWM is generated and maintained by spiral rainband processes, which were not clearly defined at that time.
b. Recent dual-Doppler radar observations from the Hurricane Rainband and Intensity Change Experiment program Hence and Houze (2008, hereafter HH08) analyzed high-resolution dual-Doppler radar observations of spiral rainbands in Hurricanes Katrina and Rita (2005) collected during the Hurricane Rainband and Intensity
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Change Experiment field program (Houze et al. 2006). One of the goals of HH08 was to evaluate the main features of the BZJM83 and P90 conceptual spiral rainband models. HH08 examined the middle portion of a spiral rainband in Hurricanes Katrina and Rita (2005). The vertical cross sections (Figs. 6 and 7 of HH08) were taken through an intense updraft located on the radially inward edge of the rainband. They illustrate that the reflectivity of the convective cell slopes radially outward with height. In addition, there is an overturning secondary circulation that has radial inflow at lower levels and radial outflow at upper levels. The cross sections also reveal layers of convergence and divergence at the base and top of the reflectivity core, respectively, with the largest upward vertical motion found in between. A SHWM is located on the radially outward side of the spiral rainband. In addition, HH08 examined the vertical cross sections (Figs. 8 and 9 of HH08) through a downdraft found in the middle portion of the same rainband of Katrina and Rita. The cross sections show two downdraft circulations: a radial inflow (near height z 5 2 km) enters from the radially outward side of the rainband and descends to surface, while the other downdraft originates from upper levels (near z 5 8 km) but stays on the radially inward side of the reflectivity core. From examining the structures of the updrafts and downdrafts found in the middle portion of a spiral rainband (both Hurricanes Katrina and Rita), HH08 were able to find supporting evidence for the hypothesized features of the BZJM83 and P90 models: an overturning secondary circulation associated with a convective updraft, a midlevel radial inflow that descends to surface, and a midlevel SHWM. By using the retrieved three-dimensional velocity fields, HH08 also computed and examined the vertical profile of mass transport. Their analysis showed that the net vertical mass transport in the upwind and middle regions is positive everywhere throughout the troposphere, with a maximum value at middle levels, while the downwind region has a nearly zero net vertical mass transport. The magnitude of the upward vertical mass transport is greatest in the middle region of the spiral rainband. Figure 2 shows the conceptualized model of a spiral rainband circulation as presented in HH08. The horizontal view (Fig. 2a) shows that the rainband is convective upwind and through the middle portion of the rainband before becoming stratiform downwind. As the rainband becomes more stratiform, it also becomes wider. A SHWM is located near the rainband and also on its radially outward side. A vertical cross section through
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FIG. 2. Schematic diagram of precipitation and kinematic structures within spiral rainbands, originally presented as Fig. 13 in HH08. (a) Horizontal cross-sectional view of the conceptual spiral rainband model. (b) Radius–height cross section along the thin solid line in (a).
the middle part of the rainband (Fig. 2b) shows that the convective cell leans radially outward with altitude and the SHWM is located on its radially outward edge. There is an overturning secondary circulation coupled with an updraft located on the radially inward edge of the cell, with convergence and divergence at the base and top of the cell, respectively. Another radial inflow can be found slightly above the low-level radial inflow, and it descends to the surface while passing through the rainband. A downdraft that originates in the upper levels stays radially inward of the rainband. The boundary of the rainband is marked by the reflectivity value of 25 dBZ.
3. The model and basic-state vortex a. The numerical model: Three-Dimensional Vortex Perturbation Analysis and Simulation To simulate the effects of spiral rainband heating on the tropical cyclone wind field, we use a three-dimensional,
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nonhydrostatic, linear numerical model of vortex dynamics, now known as the Three-Dimensional Vortex Perturbation Analysis and Simulation (3DVPAS). It is based on the vortex–anelastic equations in cylindrical coordinates on the f plane (Hodyss and Nolan 2007), and its dynamics are linear and nonhydrostatic. It can simulate both asymmetric and symmetric motions on balanced axisymmetric vortices. Perturbations can have arbitrary radial and vertical structures but have harmonic variation in the azimuthal (tangential) direction. The symmetric response to purely asymmetric motions can also be computed by using the eddy flux divergences arising from asymmetric motions as forcing for symmetric motions. Free-slip, solid-wall boundary conditions are enforced on all boundaries, and Rayleigh damping regions (sponges) are placed at the upper and outer boundaries to suppress the reflection of gravity waves. The domain size of all simulations in this study is 300 km in the radial direction and 20 km in the vertical direction. The grids are stretched in the radial direction but equally spaced in the vertical direction. In the radial direction, the grid size is approximately 2 km in the inner-core region (0 km , r , 95 km) but larger (approximately 6 km) in the far outer-core region with a smooth transition in between. The grid size in the vertical direction is 1/ 3 km. More complete details of the 3DVPAS model can be found in Nolan and Montgomery (2002a), Nolan and Grasso (2003), and Nolan et al. (2007).
b. Basic-state vortex There are a number of idealized profiles that can provide a reasonably realistic representation of the tangential wind field of tropical cyclones (e.g., Willoughby et al. 2006). As the control case in this study, we use the modified Rankine (MR) profile because it has a very simple form yet provides a realistic representation of tangential wind, as shown by Mallen et al. (2005). The MR profile has a tangential velocity field defined by r 8 > y < max RMW , a ~y(r) 5 RMW , > :y max r
r # RMW (1) r . RMW,
where y max is the prescribed maximum tangential velocity and a is the decay parameter. The radial structure of tangential wind field of tropical cyclones is well documented because of the relative abundance of observations obtained from United States Air Force and National Oceanic and Atmospheric Administration aircrafts investigating a large number of hurricanes. The same cannot be said, however, about the vertical structure of the tangential wind field. From
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early studies (e.g., Frank 1977; Jorgensen 1984; Shea and Gray 1973), it has been known qualitatively for some time that the tangential wind decays away from the surface, and the RMW generally slopes radially outward with height. Some arbitrary vertical structures (e.g., Nolan and Montgomery 2002a; Nolan and Grasso 2003; Nolan et al. 2007) have been used before; however, the RMW of the wind fields used in these previous studies did not in fact slope outward and is thus unrealistic. A realistic radially outward slope of the RMW with height and a vertical decay rate of tangential velocity at the RMW can be quantitatively obtained by utilizing the steady-state axisymmetric hurricane model by Emanuel (1986, hereafter E86), which assumes slantwise moist neutrality. The E86 model has two components: an inviscid, free atmosphere in hydrostatic and gradient wind balance and a boundary layer in which viscous and surface processes play an important role. Only the free atmosphere portion of the E86 model is used in our approach, which utilizes the fact that the RMW is very closely approximated by a surface of constant angular momentum (Stern and Nolan 2009). Details of this approach are outlined in the appendix. By using the MR profile from Eq. (1) with ymax 5 45 m s21, RMW 5 31 km, and a 5 0.5 to prescribe the tangential velocity field at the lowest level (Fig. 3a), we obtain the vertical slope of the RMW and decay rate of tangential velocity along the RMW (Figs. 3b,c). The level at which tangential velocity becomes zero at the RMW is z 5 15.9 km. Then, the radial profile of tangential velocity at each vertical level is constructed by using Eq. (1). It turns out that the tangential wind field constructed this way decays too rapidly near the surface, such that the linear model supports unstable modes in that region. To remedy this problem, the lowest level of the tangential wind field is extended 0.5 km downward barotropically with the same radial structure, and the tangential wind field is smoothed in the vertical direction. Figure 4a shows the tangential wind field of this MR basic-state vortex. Although frictionally induced secondary circulations are important, they are neglected here, because above the boundary layer tropical cyclones can be approximated as vortices in hydrostatic and gradient wind balance (Willoughby 1990). After constructing the tangential wind field of the basic-state vortex, pressure and temperature fields that hold the vortex wind field in hydrostatic and gradient wind balance are computed, which is achieved with an iterative procedure described in Nolan et al. (2001) using the Jordan (1958) mean hurricane season sounding as the far-field environment. Figure 4b shows the potential temperature anomalies required to keep the vortex in balance. This MR vortex is stable to static and symmetric
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FIG. 3. (a) MR radial profile of tangential velocity (m s21) with ymax 5 45 m s21 and RMW 5 31 km. (b) Vertical profile of the RMW of tangential velocity field constructed by using the free atmosphere component of the E86 model. (c) Vertical profile of tangential velocity at the RMW from the same model.
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FIG. 4. Radius–height cross sections of (a) tangential velocity (m s21), (b) anomalous potential temperature (K), (c) square of stratification frequency (s22), and (d) PV fields [potential vorticity units (PVU)] of the basic-state MR vortex. Contour intervals are 5.0 m s21, 2.0 K, 5.0 3 1025 s22, and 2.5 PVU from the zero line, respectively.
instabilities by ensuring ›u/›z . 0 and PV . 0 over the whole domain (Figs. 4c,d). Stability analysis using the methods of Nolan and Montgomery (2002a) and Nolan and Grasso (2003) confirms that there are no exponentially growing modes in this MR vortex.
4. Designing a spiral rainband heat source a. The structure of an idealized spiral rainband heating Spiral rainbands are banded structures of convection and can be considered as a cluster of individual updrafts and downdrafts where most of the strong vertical motions and associated latent heat release occur. Steady improvements in observational and computational
technology have made it possible to observe and simulate tropical cyclones with convective features on the same length scale as individual updrafts; however, it is beyond the ability of 3DVPAS to simulate the effect of individual updrafts and downdrafts in a spiral rainband. Rather, the goal of this study is to understand the response of the tropical cyclone wind field to the overall rainband heating in a simple framework that nonetheless can capture the time-dependent, nonhydrostatic dynamics of the response. The observational studies discussed in section 2 have shown that the upwind portion of spiral rainband heating is convective and the downwind end is stratiform, with a transition between them. In addition, the rainband heating spirals radially inward in a cyclonic direction and tilts radially outward with height. One simple way
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to represent the overall diabatic heating structure of such rainbands is to overlap multiple heat sources similar to the individual heat sources previously used in
Nolan et al. (2007). First, we define the diabatic heating structures of purely convective and stratiform heat sources as
8 0, for z # > bc > " z > # < r rbc 2 z zbc _ _ sin p exp(inl invt), Q con (r, l, z, t) 5 Qcon max exp > src src > > : 0, for z $ zbc 1 szc
for
8 0, for z # zbs szs > > " ! > 2# < r r z z bs bs sin p exp(inl invt), for Q_ str (r, l, z, t) 5 Q_ str max exp > srs szs > > : 0, for z $ zbs 1 szs
In Eqs. (2) and (3), rb(z) is the radial center position and sr and sz are the radial and vertical half-widths of the heating, respectively. Subscripts c and s refer to purely convective and stratiform heat sources. For convective heat sources (Fig. 5a), zbc is chosen to be the lowest altitude where positive diabatic heating is present, and the vertical depth of the heating is controlled by taking the half vertical wavelength szc only upward from zbc. For stratiform heat sources (Fig. 5b), zbs is the level of zero heating, and its vertical depth is controlled by taking the half vertical wavelength szs both upward and downward from zbs. Observations (e.g., BZJM83; P90) indicate that rainband heating tilts radially outward with height, which can be achieved by linearly varying rb with altitude, such that rb(z) 5 rbsfc 1 z, where rbsfc is the radial center position at the surface. The complex exponential term in Eqs. (2) and (3) rotates the heat source around the center of the vortex at an angular velocity v 5 yrot/rc, where rc is the radial location of the center point of the rainband heating at the surface and y rot is the rotation speed of the heat source. Now we create an idealized but realistic-looking spiral rainband by overlapping 40 heat sources. By ‘‘overlapping’’ heat sources, we mean that heat sources are placed close to each other so that most points in the rainband receive a heating contribution from multiple heat sources. The sum of these overlapped heat sources is the rainband heating. Figure 6a shows the horizontal cross section of this rainband at z 5 4.5 km. This rainband covers approximately one quarter of the circumference in the lower right quadrant, but its upwind end is at r 5 80 km, while its downwind end is at r 5 60 km. The radial center location of each heat source is equal to rbc in Eq. (2) and rbs in Eq. (3), which varies linearly between the ends of the rainband (from r 5 80 km to 60 km). The heat sources at the upwind (r 5 80 km)
zbc , z , zbc 1 szc ,
zbs szs , z , zbs 1 szs
(2)
(3)
and downwind (r 5 60 km) ends are purely convective (Fig. 5a) and purely stratiform (Fig. 5b), respectively. The transition from the convective to stratiform diabatic heating is defined to be linear. As the rainband makes a transition from convective to stratiform, the diabatic heating structure becomes shallower and wider, and an area of significant cooling develops in the lower part of the rainband. Figure 5c shows the vertical cross section in the middle part of the rainband, illustrating the change in the vertical structure of diabatic heating during this linear transition. At this location (r 5 70 km), the heat source is a combination of 50% convective and 50% stratiform heat sources. The maximum heating rate of each individual convective heat source is chosen to be twice the maximum heating rate of the stratiform heat source, so Q_ con max 5 3.0 K h21 and Q_ str max 5 1.5 K h21.
b. How 3DVPAS sees the spiral rainband diabatic heating To compute the impact of this idealized spiral rainband in 3DVPAS, the idealized spiral rainband heating in Fig. 6a is separated into different azimuthal wavenumber components by Fourier series decomposition. The magnitude of the purely asymmetric components decreases as the azimuthal wavenumber increases (not shown). Figure 6b shows the sum of its Fourier decompositions from n 5 0 to 4, and it is a bit different from the original rainband heating (Fig. 6a) because of the truncation of the series at n 5 4. Each wavenumber component of the heating is simulated separately in 3DVPAS. To obtain the full response of the vortex to a rainband heating, it is necessary to consider the asymmetric and symmetric responses to each asymmetric component plus the symmetric response to the symmetric part. Therefore, the full response of the vortex to
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the rainband heating can be approximated in the linear limit by the sum of the asymmetric and symmetric responses to its first four asymmetric components plus the symmetric response to its symmetric part. Hereafter, the response refers to the sum of the asymmetric and symmetric responses to the purely asymmetric components (n 5 1–4) and the symmetric response to the symmetric part (n 5 0). In addition, all responses are rotated such that the rainband heating at the time the response is shown would always be in the same location as in Fig. 6a.
5. The response of the hurricane wind field to rainband heating a. A purely convective spiral rainband heating
FIG. 5. (a),(b) Vertical cross sections of normalized purely convective and stratiform diabatic heat sources centered at r 5 80 and 60 km, respectively. (c) Vertical cross section of normalized mixed (50% convective, 50% stratiform) diabatic heat source centered at r 5 70 km. For (a), rbsfc 5 80 km [so rbc(z) 5 80 km 1 z in Eq. (2)], zbc 5 1 km, src 5 2 km, and szc 5 7 km are used; for (b), rbsfc 5 60 km [so rbs(z) 5 60 km 1 z in Eq. (3)], zbs 5 4 km, srs 5 6 km, and szs 5 2 km are used. For (c), rbsfc 5 70 km.
A typical spiral rainband heating contains both convective and stratiform components; however, to better understand the combined response to these two different heating profiles, it is helpful to first consider the response to a rainband heating that is entirely convective or stratiform. Figures 6c and 6d show the horizontal cross sections at z 5 4.50 km of the idealized purely convective rainband heating and its sum of Fourier decompositions from n 5 0 to 4. Just like the mixed rainband described in the previous section, a purely convective rainband spirals inward (from r 5 80 to 60 km), but the structure of its diabatic heating remains the same along the band (see Fig. 5a). The first four purely asymmetric components plus the symmetric part are rotated around the MR vortex for 24 h. Since rainbands are observed to move more slowly than the mean circulation speed (Anthes 1982), the rotation speed is set to be 70% of the maximum tangential wind at the middle part of the rainband at r 5 70 km, so y rot 5 21.1 m s21. Figure 7 shows time evolution of kinetic energy (KE) of the sum of the asymmetric responses to the asymmetric components (n 5 1–4, hereafter KEaa), the symmetric response to the symmetric part (n 5 0, hereafter KEss), and the sum of the symmetric responses to the asymmetric components (n 5 1–4, hereafter KEsa). It shows that KEaa increases rapidly in the first 8 h and then gradually levels off although it never quite reaches a steady state, which is due to the excitation of a weakly damped n 5 1 inner-core mode, as KE for n $ 2 reaches a nearly steady state by approximately t 5 15 h (not shown). After t 5 16 h, the change in KE only results from the accumulation of KE in the n 5 1 inner-core mode. Unlike KEaa, however, after a period of adjustments in the first several hours KEss increases at a linear rate after t 5 5 h. The period of initial adjustment in KEsa is longer but it increases linearly after t 5 15 h.
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FIG. 6. (a) The horizontal cross section at z 5 4.50 km of the mixed spiral rainband heating (K h21). (b) The sum of its Fourier decompositions from n 5 0 to 4. (c),(d) As in (a),(b), but for the purely convective spiral rainband heating (K h21). (e),(f) As in (a),(b), but for the purely stratiform spiral rainband heating (K h21). Forty heat bubbles are used to construct each rainband heating, which spirals cyclonically inward from r 5 80 to 60 km. The zero line is not plotted. Maximum and minimum values can be found above each panel. See section 4a for explanations.
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FIG. 7. Time evolution of normalized KE of the asymmetric responses to the asymmetric components (KEaa; solid), the symmetric response to the symmetric part (KEss; dashed), and the symmetric responses to the asymmetric components (KEsa; dashed–dotted) of the convective rainband. KE values are normalized by their respective maximum value during the 24-h period, which are 7.90 3 1012 J for KEaa, 4.44 3 1014 J for KEss, and 8.30 3 1012 J for KEsa.
KEsa decreases1 for the first 6 h as the asymmetric components of the rainband heating interact with the basicstate vortex and extract KE from it (see section 4b of Nolan et al. 2007). At t 5 16 h and thereafter, the response to the rainband heating is expected to change only quantitatively. Unless noted otherwise, the response hereafter is evaluated at t 5 16 h for all cases considered. Figure 8 shows horizontal cross sections of the response to the convective rainband heating at z 5 1.83 (lower level), 4.50 (middle level), and 7.17 km (upper level); z 5 1.83 and 7.17 km are located slightly above the bottom of the heating and slightly below the top of the heating respectively, while z 5 4.50 km is at the middle level of the heating. Arrows and contours represent horizontal and vertical velocity components of the response. All cross sections show that an inner-core mode is excited by the rainband heating, and this mode has a strong n 5 1 structure. Despite the presence of an inner-core mode, the response to the rainband heating can be identified in all cross sections. They show that there is upward vertical velocity at the location of the
1 KEsa can in fact be negative, as it is defined as the difference in total KE between the initial symmetric vortex and the symmetric vortex at later times. If interactions with the evolving asymmetries change the symmetric vortex to a flow with less KE, then KEsa can be negative.
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heating. Comparing the location of vertical motion at each cross section with the location of the rainband heating (Figs. 6c,d) indicates that vertical motion is well coupled to the heating, and this is found to be true for all subsequent cases. Hereafter, the location of vertical motion of the response is used interchangeably with the location of the rainband heating. Examining the horizontal velocity components of the response at the lower levels (Fig. 8c) shows an accelerated tangential circulation on the radially outward side of the rainband heating. However, this feature becomes weak at the middle levels (Fig. 8b), and at the upper levels (Fig. 8c) there is instead a decelerated tangential circulation on the radially outward side of the rainband heating. Figure 9 shows vertical cross sections through the downwind, middle, and upwind regions (see white lines in Fig. 6a) of the response to the convective rainband heating. Arrows depict radial and vertical velocity components of the response, while contours show the tangential velocity component of the response. All cross sections show that tangential circulation around the location of the heating and on its radially outward side is accelerated at the lower levels (from the surface to z 5 4 km) but decelerated at the upper levels between z 5 5 and 10 km. In addition, there is radial inflow toward the rainband from its radially outward side at the lower levels but radial outflow away from the rainband at the upper levels. Upward vertical motion is found between radial inflow and accelerated tangential velocity at the lower levels and between radial outflow and decelerated tangential velocity at the upper levels. The strongest rising motion is found between z 5 4 and 6 km. The axis of the largest upward velocity moves radially inward in the downstream direction, in accordance with the inwardly spiraling structure of the convective rainband heating. Although full understanding of the response to the rainband heating requires examining both asymmetric and symmetric responses to the asymmetric and symmetric components of the heating, the symmetric responses alone capture a large part of the full response. Figure 10a shows the vertical cross section of the sum of the symmetric responses to both asymmetric and symmetric parts (n 5 0–4) of the convective rainband, while Fig. 10b shows the vertical cross section of the symmetric response to the symmetric part (n 5 0) alone. Both cross sections clearly show that around the rainband location and on its radially outward side there is low-level radial inflow and upper-level radial outflow, which are connected by upward vertical motion through the rainband heating. On its radially inward side, there is weak lowlevel radial outflow that enters the updraft and some of the updraft gets diverted into weak radial inflow at the
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upper levels. Clearly, there is convergence and divergence at the base and top of the updraft, respectively. These vertical cross sections also indicate that tangential circulation at the lower levels is accelerated at the location of the heating, as well as on its radially outward side, but decelerated at the upper levels. This symmetric response is qualitatively similar to transverse circulations presented in Eliassen (1951), Shapiro and Willoughby (1982), and Hack and Schubert (1986). All vertical cross sections from Fig. 9 exhibit these similar features of the symmetric responses shown in Fig. 10. There are, however, some notable differences among different parts of the rainband. The low-level radial inflow toward the rainband is deeper and stronger in the downwind region in comparison to the upwind region. This radial inflow after passing the rainband continues to travel toward the vortex center in the downwind region, while there is actually a radial outflow toward the rainband in the upwind region. At the upper levels, the radial outflow from the rainband is weaker and shallower in the upwind region than in the downwind region. In addition, on the radially inward side of the rainband, there is radial inflow in the upwind region but radial outflow in the downwind region. These differences may be understood conceptually by considering the PV of the asymmetric response. In a linearized system such as 3DVPAS, the perturbation PV is h ›u 1 1 ›u ›u 1 n 1 zn qn 5 (vn $u 1 v $un ) 5 jn ›r ›z r r r ›l ›un ›un h ›un , (4) 1j 1 1 ( f 1 z) r ›l ›r ›z
FIG. 8. Horizontal cross sections at z 5 (a) 7.17, (b) 4.50, and (c) 1.83 km of the response to the convective rainband heating (Fig. 6c). Contours represent vertical velocity (m s21) of the response with a contour interval of 0.01 m s21 from the zero line, which is not plotted. In this and all subsequent grayscale figures, dark colors are more positive than light colors, and positive values are indicated by solid contours, while negative values are indicated by dashed contours. Arrows show horizontal velocity (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
where j, h, and z are, respectively, the radial, azimuthal, and vertical components of the three-dimensional vorticity vector v; r is density; u is potential temperature; the overbar and subscript n (azimuthal wavenumber) denote the axisymmetric (basic state) and asymmetric perturbation quantities, respectively; and f is a Coriolis parameter of 5.0 3 1025 s21. In a baroclinic vortex without secondary circulations, such as the MR vortex here, a thermal perturbation such as the rainband heating produces a PV perturbation that has contributions from the fourth and sixth terms in Eq. (4). For the convective rainband, the fourth term creates positive PV on its radially inner side and negative PV on its radially outer side, while the sixth term produces positive PV at the lower levels and negative PV at its upper levels. The sum of these four PV components results in the PV forcing that has positive PV at the low levels and negative PV at the upper levels (Fig. 11a).
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FIG. 10. Vertical cross sections of (a) the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0– 4) and (b) the symmetric response to the symmetric component (n 5 0) of the convective rainband heating (Fig. 6c). Contours represent tangential velocity (m s21) of the response with a contour interval of 0.10 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
FIG. 9. Vertical cross sections through the (a) downwind, (b) middle, and (c) upwind regions of the response to the convective rainband heating (Fig. 6c). Contours represent tangential velocity (m s21) of the response with a contour interval of 0.20 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
In response to this PV forcing, positive PV and cyclonic circulations are expected to develop at the lower levels, but negative PV and anticyclonic circulations should develop at the upper levels. Figure 12 shows the sum of the asymmetric responses to the asymmetric components of the convective rainband, and it clearly illustrates these features; the full response also hints at them (Figs. 8a,c). Figure 11b also shows the vertical cross section of PV of the sum of the asymmetric responses to the asymmetric components (n 5 1–4) to the convective rainband. Careful examination of Figs. 10
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FIG. 11. Vertical cross sections through the middle region of (a) PV forcing (PVU h21) associated with the convective rainband heating (Fig. 6c) and (b) PV (PVU) associated with the response to the convective rainband heating. (c),(d) As in (a),(b), but for the stratiform rainband (Fig. 6e) case. Note that different contour interval values are used, and the zero line is not plotted. Maximum, minimum, and contour interval can be found above each panel.
and 12 indicates that the asymmetric PV and symmetric transverse responses cooperate in some places but act in opposite directions in other places. For example, at the lower levels in the upwind region, the asymmetric and symmetric responses are in the same direction on the radially inward side of the rainband, producing radial outflow into the rainband, but in the opposite direction on its radially outward side, resulting in weak radial inflow into the rainband. At the lower levels in the downwind region, the asymmetric and symmetric responses are in the same direction on the outward side, but they are in the opposite direction on the inward side and result in radial inflow toward the center because the asymmetric response is stronger. Similarly, the differences in the strength and direction of the radial flow at the
upper levels between the upwind and downwind regions can be attributed to the different superimpositions of the asymmetric response with the symmetric response. The collocation of radial inflow with increased tangential velocity at the lower levels and radial outflow with decreased tangential velocity at the upper levels suggests that the change in tangential wind field is simply the result of an angular momentum-conserving radial flow. The rainband heating is located outside the RMW where the tangential wind decays as r2a (a 5 0.5 in the MR vortex); thus, angular momentum associated with tropical cyclone tangential wind field increases radially outward. Therefore, radial flow from a region of higher angular momentum to a region of lower angular momentum (e.g., low-level radial inflow) will increase
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height, more outward than the updraft. The change in tangential wind field due to angular momentumconserving radial flow, however, is larger in comparison to that due to angular momentum-conserving vertical flow, because the angular momentum transport by radial flow is larger than that by vertical flow by a factor of 3 or higher. Comparing Figs. 6d and 6b suggests that the error from truncating the Fourier series at n 5 4 is larger in the convective rainband forcing (Fig. 6c) than in the mixed rainband forcing (Fig. 6a). The convective rainband case was repeated with the truncation at n 5 8, but the response of the hurricane wind field to the n 5 0–8 convective rainband heating remained essentially the same (not shown).
b. A purely stratiform spiral rainband heating
FIG. 12. Horizontal cross sections at z 5 (a) 7.17 and (b) 1.83 km of the sum of the asymmetric responses to the asymmetric components of the convective rainband (Fig. 6c). Contours show PV (PVU) and arrows show horizontal velocity (m s21). Note that different contour interval values are used, and the zero line is not plotted. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
tangential wind, while radial flow from a region of lower angular momentum to a region of higher angular momentum (e.g., upper-level radial outflow) will decrease tangential velocity. In addition, the updraft component of the overturning secondary circulation flows from a region of higher angular momentum to a region of lower angular momentum, leading to positive tangential velocity change at the upper levels but negative tangential velocity change at the lower levels (not shown). This process occurs because the tangential velocity field of the MR vortex decreases with height, so the angular momentum surfaces are sloped radially outward with
Figures 6e and 6f show the horizontal cross sections at z 5 4.50 km of the idealized purely stratiform rainband heating and its sum of Fourier decompositions from n 5 0 to 4. The purely stratiform rainband has heating above and cooling below the level of zero heating at z 5 4.0 km (see Fig. 5b). Figure 13 shows horizontal cross sections of the response at z 5 2.17 (lower level), 3.83 (middle level), and 5.83 km (upper level). Note that z 5 2.17 and 5.83 km are located slightly above the bottom of the cooling component and slightly below the top of the heating component of the stratiform rainband heating, respectively, while z 5 3.83 km is slightly below the level of zero heating. All cross sections show that there are inner-core modes excited by the heating. While the modes from the convective rainband case have a strong wavenumber-1 signature at all levels (Fig. 8), different wavenumber structures tend to dominate at different levels in the stratiform case. Despite the excitation of the inner-core modes, it is easy to identify the response to the stratiform rainband heating. Since this stratiform rainband heating has heating above and cooling below z 5 4.0 km (see Fig. 5b), there are rising and sinking motions above and below this level, indicating vertical divergence at this level. Also present near this level (Fig. 13b) is accelerated tangential circulation on the radially outward side of the rainband heating. Near the top of the heating component (Fig. 13a), there is anticyclonic radial outflow emanating away from the rainband where rising motion is present. At the base of the cooling component (Fig. 13c) there is weak anticyclonic radial outflow away from the rainband where sinking motion exists. It is interesting to note that the low-level horizontal circulation is substantially weaker than that at middle and upper levels. Figure 14 shows vertical cross sections through the downwind, middle, and upwind regions (see white lines
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in Fig. 6a) of the response to the stratiform rainband heating. All cross sections show that there is radial inflow toward the rainband at the middle levels (between z 5 3 and 5 km). After passing through the rainband, this inflow splits into two branches: one descends to the surface and continues to travel toward the vortex center, and the other ascends to the upper levels (near z 5 6 km). This midlevel radial inflow pattern is collocated with the positive tangential velocity component of the response, indicating that angular momentum conservation leads to the accelerated tangential circulation. Also present at these levels is radial outflow that enters the rainband from its radially inward side and decreased tangential velocity response. In addition, on the radially outward side of the rainband, there is negative tangential velocity response above and below the midlevel radial inflow layer, respectively. Figure 15 shows vertical cross sections of the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0–4) of the stratiform rainband heating and the symmetric response to its symmetric part (n 5 0) alone. Both cross sections clearly show that at the location of the rainband heating there are rising and sinking motions above and below z 5 4 km, respectively. In addition, on the radially outward side there is radial inflow toward the rainband at this altitude. It appears that the symmetric responses are very similar to the full response through the middle portion (Fig. 14b). The collocation of positive and negative tangential velocity responses with radial inflow and outflow indicates that angular momentum conservation leads to these tangential velocity responses. The angular momentum transport by radial flow is larger than that by vertical flow by a factor of 4 or higher. Examining Fig. 14 in detail reveals that the response to the stratiform rainband heating does not show as much variation along the band as the response to the convective rainband (Fig. 9), which indicates that the symmetric responses to the stratiform rainband capture an even larger portion of the full response in comparison to the convective case. There are, however, still some notable differences, especially in the radial flow pattern. For example, there is radial inflow toward the rainband from its radially outward side at the middle levels in the downwind region but weak radial outflow in the upwind region. In addition, the radial flow on the radially outward side at the upper levels is oppositely directed between the downwind and upwind regions. These differences may be attributed to the asymmetric PV responses. Figure 11c shows the vertical cross section of PV forcing associated with the stratiform rainband. There is negative PV at the lower and upper levels but positive PV at the middle levels. In response,
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FIG. 13. As in Fig. 8, but for the response to the stratiform rainband heating (Fig. 6e) at z 5 (a) 5.83, (b) 3.83, and (c) 2.17 km, with a contour interval of 0.03 m s21.
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FIG. 15. As in Fig. 10, but for the stratiform rainband heating (Fig. 6e), with a contour interval of 0.50 m s21.
FIG. 14. As in Fig. 9, but for the response to the stratiform rainband heating (Fig. 6e), with a contour interval of 0.50 m s21.
positive PV and cyclonic circulations develop at the middle levels and negative PV and anticyclonic circulations prevail at the lower and upper levels, as illustrated in the horizontal cross sections of the sum of the asymmetric responses to the asymmetric components to the stratiform rainband (Fig. 16). Figure 11d also shows the vertical cross section of PV of the sum of the asymmetric responses to the asymmetric components to the stratiform rainband. Comparing Figs. 15 and 16 suggests that the asymmetric PV and symmetric transverse responses are in the same direction in some regions but in the opposite direction in other places. For example, on the radially outward side of the rainband, the asymmetric and symmetric responses are in the same direction in the downwind region but in the opposite direction in the upwind region. Because the radial velocity of the asymmetric response is stronger than that of
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the symmetric response, there is weak midlevel radial outflow in the upwind region. A more pronounced midlevel tangential velocity response in the downwind region is due to the fact that the cyclonic asymmetric response strongly projects into the azimuthal direction in the downwind region but more into the radial direction in the upwind region.
c. A mixed spiral rainband heating
FIG. 16. Horizontal cross sections at z 5 (a) 5.83, (b) 3.83, and (c) 2.17 km of the sum of the asymmetric responses to the asymmetric components of the stratiform rainband (Fig. 6e). Contours show PV (PVU) with a contour interval of 0.60 PVU from the zero line, which is not plotted. Arrows show horizontal velocity (m s21). Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
With the previous results in mind, we now present the response to a mixed spiral rainband heating. The detailed configurations of the mixed rainband were described in section 4a. The mixed rainband heating spirals radially inward in a cyclonic direction from r 5 80 to 60 km, and the upwind and downwind regions of the heating are convective and stratiform, respectively, with a linear transition between them. Figure 17 shows horizontal cross sections of the response at z 5 2.50 (lower level), 4.17 (middle level), and 6.83 km (upper level). Despite the usual excitation of an inner-core mode, it is not difficult to isolate the response to the rainband heating. At the lower levels (Fig. 17c), the rainband heating is positive upwind and negative downwind, and accordingly there are rising and sinking motions in those respective regions. On the radially outward side of the rainband, there are some signs of accelerated tangential circulation. This positive tangential velocity response becomes more evident at the middle levels (Fig. 17b), especially on the downwind region. At these altitudes, the heating is positive everywhere along the rainband, and accordingly there is rising motion everywhere. At the upper levels (Fig. 17a), there is instead a negative tangential velocity response. Also present at these altitudes is radial flow that diverges away from the rainband in an anticyclonic direction. It seems evident from these cross sections that the asymmetric PV response is cyclonic at the lower and middle levels but anticyclonic at the upper levels. Figure 18 shows vertical cross sections through the upwind, middle, and downwind regions (see white lines in Fig. 6a) of the mixed rainband. All cross sections show that on the radially outward side of the rainband there is accelerated tangential velocity at the lower levels, especially between z 5 3 and 4 km. Above the positive tangential velocity response, there is a decelerated tangential circulation, centered near z 5 7 km. In the upwind region, there is weak radial inflow toward the rainband from its outward side at the lower levels and radial outflow from the radially inward side. They converge at the base of the rainband and then rise through it at approximately r 5 80 km, and most of it becomes
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radial inflow at the upper levels. In the downwind region, there is deep (from the surface to z 5 4 km) radial inflow toward the rainband from its outward side, and most of it continues to flow toward the center after passing the rainband. Between z 5 3 and 4 km, some of the radial inflow that enters the rainband descends to lower levels between r 5 80 and 60 km. Radial inflow between z 5 4 and 5 km meets radial outflow that enters the rainband from its inward side and rises to upper levels, where there is mostly radial outflow. The tangential velocity response in the downwind region is greater than in the upwind region. The middle region appears to be a mix of the upwind and downwind regions. Figure 19 shows vertical cross sections of the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0–4) of the heating and the symmetric response to the symmetric part (n 5 0) alone. Both cross sections show that there is deep radial inflow toward the rainband from its radially outward side, and this inflow is stronger at z 5 4 km than at the surface. Most of this inflow rises through the rainband and becomes radial outflow away from the rainband between z 5 5 and 9 km; however, some of this inflow descends from z 5 3 to 2 km and continues to travel radially inward. Near z 5 6 km, some of the updraft diverts toward the vortex center. On the radially inward side, there is radial outflow at z 5 4 km toward the rainband. Clearly, there is convergence at the location of the rainband heating at this altitude, and divergence above it. This symmetric response to the mixed rainband also appears to be a mix of the convective and stratiform symmetric responses presented earlier, and it explains a large portion of the full response. As in the previous cases, the asymmetric and symmetric responses act in the same direction in some locations, such as on the radially outward side at the lower and middle levels in the downwind region, but in the opposite direction in other locations, such as on the radially outward side at the upper levels in the upwind region.
d. Comparison to observations Previous observational studies of spiral rainbands indicated that there are an overturning secondary circulation, an accelerated tangential circulation on the radially outward side of the rainband, and a midlevel radial inflow that descends to the surface, intertwining with the lower branch of the overturning circulation. Figure 2 illustrates a conceptualized model of a spiral rainband circulation as presented in HH08. The idealized spiral rainband heating structures considered so far are designed to represent only the gross effects of latent heat release within the spiral rainband. Nonetheless, the
FIG. 17. As in Fig. 8, but for the response to the mixed rainband heating (Fig. 6a) at z 5 (a) 6.83, (b) 4.17, and (c) 2.50 km, with a contour interval of 0.03 m s21.
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FIG. 19. As in Fig. 10, but for the mixed rainband heating (Fig. 6a), with a contour interval of 0.20 m s21.
FIG. 18. As in Fig. 9, but for the response to the mixed rainband heating (Fig. 6a), with a contour interval of 0.40 m s21.
response of the MR vortex wind field to the mixed spiral rainband heating captures all of the aforementioned features with the contributions from the responses to both purely convective and stratiform rainband heating. It appears that the midlevel radial inflow and its descending motion to the surface closely resemble the stratiform response and are stronger in the downwind region. The overturning secondary circulation feature belongs to the convective response and is stronger in the upwind region. An accelerated tangential circulation on the radially outward side of the rainband shares the characteristics of both the convective and stratiform responses, but it tends to be stronger in the downwind region where stratiform response prevails. In addition, this accelerated tangential circulation feature extends further radially away from the rainband than what observations indicate (see Figs. 1b and 2).
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e. The effects of the inner-core modes All cross sections through the responses indicate that the response to the rainband heating occurs not only at the location of the heating but also around the RMW. In the convective rainband case, horizontal cross sections (Fig. 8) indicate that there are strong azimuthal wavenumber-1 signatures in the inner-core response. Vertical cross sections (Fig. 9) show that the tangential wind field at the lower levels (from the surface to z 5 3 km) is decelerated inside the RMW but accelerated just outside the RMW. In addition, there is increased tangential velocity response near the RMW at the upper levels (between z 5 6 and 10 km). In the linear limit, this process would suggest that the effect of the inner-core modes excited in response to the purely convective rainband is to vertically stretch the tangential circulation on the rainband side and pull the whole tangential circulation toward the rainband. The inner-core modes in the response to the purely stratiform rainband are not as severe as in the convective case. The inner-core modes in the mixed rainband case show similar low-level structures as in the convective case but are disorganized in the upper levels. The overall effects of the asymmetric inner-core modes in all cases, however, are small because the differences in the wind speed change between the sum of the symmetric responses to both asymmetric and symmetric components (n 5 0–4) of the rainband and the sum of the symmetric response to the symmetric component (n 5 0) of the rainband are less than 5% (see Figs. 10, 15, and 19).
6. Sensitivity tests We must consider whether the responses described in the previous section are highly sensitive to the particular parameters that define the structure and movement of the rainband heating. As described in section 4a, there are a large number of parameters involved in controlling the evolution of imposed rainband heating. Tests are performed for only parameters that appear to have a large variation based on observations.
a. Depth of convection in the upwind region Cecil et al. (2002) created the composites of radar reflectivity over tropical cyclones as observed by the Tropical Rainfall Measuring Mission satellite (see their Fig. 3). Their statistics indicate that the convective region of spiral rainbands sometimes has convection penetrating deeper than z 5 8 km, which is the upper boundary of the convective heat forcings used in the previous section (see section 4a and Fig. 5a). By using 20 dBZ to define the boundary for a spiral rainband
FIG. 20. Vertical cross sections through the (a) downwind and (b) upwind regions of the response to the mixed rainband heating whose convective heat bubbles are 10 km deep in the vertical direction. Contours represent tangential velocity (m s21) of the response with a contour interval of 0.30 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
heating, the upper bound of z 5 8 km is very close to the 75th percentile of the cumulative density function of radar reflectivity. To evaluate the effects of the different depths of upwind convection, the mixed rainband case is repeated with sz 5 9 and 11 km so the upper bound of upwind convection is z 5 10 and 12 km, respectively. Figure 20 shows vertical cross sections of the response through the upwind and downwind region (see white lines in Fig. 6a) for the sz 5 9 km case (thus 10 km deep). It is evident that the effect of deeper upwind convection is to elevate
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FIG. 21. Vertical cross sections through the (top) downwind and (bottom) upwind regions of the response to the mixed rainbands that are (left) 50% convective and (right) 50% stratiform. See section 6b for details. Contours represent tangential velocity (m s21) of the response with a contour interval of 0.50 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
the radial outflow of the overturning secondary circulation feature; however, the midlevel radial inflow and its descending motion, as well as the accelerated tangential circulation on the radially outward side of the rainband, remain unaffected by the presence of deeper upwind convection. This point further supports the conclusion from the previous section that the overturning circulation is due to the response to purely convective parts of the rainband.
b. Organization of convection within rainbands All rainband cases examined so far have the linear transition occurring over the whole rainband, which may not be representative for all rainbands. The mixed rainband case is repeated but with two different transition
schemes. One scheme (hereafter 50% convective) keeps the rainband purely convective over the first half of the rainband from its upwind end and then makes a linear transition to purely stratiform regime over the second half, while the other scheme (hereafter 50% stratiform) makes a linear transition from purely convective to purely stratiform over the first half and then remains purely stratiform over the second half. Three common rainband circulation features are present in the response to both cases (Fig. 21). Notable differences from the mixed rainband case are that the overturning circulation is strengthened in the 50% convective case while the 50% stratiform case shows enhanced descending midlevel radial inflow. Both cases show increased tangential circulation on the radially outward side of the rainband,
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but the altitude of the largest positive tangential velocity response is higher for the 50% stratiform case than the 50% convective case. While a large number of observed rainbands are convective upwind and stratiform downwind, it is possible to have highly disorganized rainbands, which can be represented by constructing a rainband only with heat sources that have equal convective and stratiform contributions at all locations along the band (see Fig. 5c). Other parameters for this disorganized rainband are the same as in the mixed rainband. The response to the disorganized rainband (Fig. 22) recovers the overturning circulation and accelerated tangential circulation on its radially outward side. The descending midlevel radial inflow, however, is not present in the downwind region response, likely due to the reduced area of cooling in the downwind region of this disorganized rainband in comparison to the mixed rainband (cf. Figs. 5c and 5b). We also considered another way to change the transition between the upwind and downwind ends, which is to modify the values of Q_ con max in Eq. (2) and Q_ str max in Eq. (3). Their ratio represents the different contributions of purely convective and purely stratiform heat sources. When Q_ con max : Q_ str max is increased, the overturning secondary circulation is strengthened as in the 50% convective case. Decreasing the ratio leads to enhanced descending midlevel radial inflow as in the 50% stratiform case. It also appears that the effect of stronger convective heat sources in the rainband in the upwind region is to vertically stretch the positive tangential velocity response near the rainband because of the stronger updraft. In the downwind region, stronger stratiform heat sources lead to enhanced tangential circulation at the middle levels on the radially outward side of the rainband.
c. Transient rainband So far we have examined the response of the hurricane wind field to rainband heating that is persistent over time. However, observations (e.g., Anthes 1982) indicate that the duration of individual convective cells embedded within rainband heating is typically on the order of several hours. The rainbands themselves are transient, but the precise mechanisms controlling their initiation, duration, and decline remain unknown. To represent the transient effects of the rainband heating, we now multiply Eqs. (2) and (3) by " # t tforc 4 . (5) F(t) 5 exp ptforc In Eq. (5), tforc is the time at which the rainband heating is at its maximum. Setting the coefficient p 5 0.55163
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FIG. 22. Vertical cross sections through the (a) downwind and (b) upwind regions of the response to the disorganized rainband (see section 6b). Contours represent tangential velocity (m s21) of the response with a contour interval of 0.50 m s21 from the zero line, which is not plotted. Arrows show radial and vertical velocity components (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
shapes the rainband heating to approximately last for 2tforc h. The structure of this exponential function is very similar to that used in Nolan et al. (2007; see their Fig. 5b). Another simple function to represent the transient effects is to use a piecewise linear function such that the rainband heating rate reaches its peak at t 5 tpeak and then decays until it dissipates at t 5 tdiss. The mixed rainband case from the previous section is repeated with the heating now following the exponential and linear functions described previously. For the exponential function, tforc 5 6 h is used, and for the linear function, tpeak 5 4 h and tdiss 5 3tpeak 5 12 h are used.
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Figure 23 shows horizontal cross sections at z 5 4.17 km of the response to the transient mixed rainband heating using Eq. (5) at different times. Comparison with the response to the persistent mixed rainband heating (Fig. 17) indicates that cyclonic tangential acceleration on the radially outward side of the rainband can be seen locally around the rainband heating as early as t 5 4 h (Fig. 23a). The overturning secondary circulation and descending midlevel radial inflow are also present in the response by this time (not shown). As the heating rate becomes larger, the response becomes more easily discernable (Fig. 23b, t 5 7 h), and this tangential acceleration starts to propagate in the downstream direction relative to the rainband heating. As the heating rate ramps down (Fig. 23c, t 5 10 h), the response starts to weaken, especially in vertical velocity response; however, the cyclonic tangential acceleration feature remains coherent long after the heating is turned off (Figs. 23e,f). The evolution of the transient mixed rainband that follows the piecewise linear forcing is qualitatively similar (not shown).
d. Rotation speed and radial location of rainband heating Previous observational studies suggested that spiral rainbands move around the center more slowly than the mean circulation. There are observations that some spiral rainbands (e.g., principal bands) stay at the same region for a long period of time. The mixed rainband case was repeated but with y rot prescribed to be 60% and 80% of the maximum tangential wind at the middle part of the rainband (r 5 70 km). The corresponding values of y rot are 18.1 and 24.1 m s21, respectively. A stationary mixed rainband case (y rot 5 0.0 m s21) was also considered. All responses (not shown) recovered the aforementioned main features of rainband circulation, indicating that the sensitivity of the response to the rotation speed is not significantly large. The sensitivity of the response to the radial location of the rainband was examined by considering the mixed rainband case with the convective upwind end at r 5 120 km and the stratiform downwind end at r 5 100 km. Sixty heat sources were used to construct the rainband, which still makes a linear transition in the downstream direction. The response was qualitatively similar to the mixed rainband except that the features are shifted 20 km radially outward (not shown).
7. Summary and discussions The response of the hurricane wind field to rainband heating has been examined by using a three-dimensional, nonhydrostatic, linear model of vortex dynamics. Diabatic
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heat sources were designed in accordance with previous observations of spiral rainbands, and they were embedded in the hurricane wind field and rotated around the vortex. The response to a typical mixed rainband captures many common features of spiral rainband kinematic structures, such as an overturning secondary circulation, an enhanced tangential circulation on the radially outward side of the rainband (or secondary horizontal wind maximum), and a midlevel radial inflow that descends to the surface. Comparison of the responses to the purely convective and stratiform rainbands indicates that the overturning secondary circulation mostly results from the convective part of the rainband and is stronger in the upwind region, while midlevel radial inflow descending to surface results from the stratiform characteristics of the rainband and is stronger in the downwind region. The secondary horizontal wind maximum is exhibited in both convective and stratiform parts of the rainband, but it tends to be stronger in the downwind region. Sensitivity tests confirm that the response is robust and that the latent heating structure of convection embedded in rainbands is primarily responsible for their kinematic structures. In addition, the response of the hurricane wind field to the rainband heating is, in the linear limit, the sum of the asymmetric PV and symmetric transverse circulations. Frictionally induced secondary circulations are an important part of the tropical cyclone wind field. The basicstate MR vortex used in this study, however, lacks them, which is justified by the work by Willoughby (1990) that showed in the free atmosphere above the boundary layer tropical cyclones are approximately cyclonic vortices in hydrostatic and gradient wind balance. The secondary circulations are closely related with the distribution of convection around tropical cyclones, which are highly asymmetric and transient; however, the responses to the rainband in all cases considered develop their own secondary circulations, so the qualitative aspects of the response appear not to be significantly affected by the lack of the secondary circulation. In addition, the location of the rainband secondary circulation above the boundary layer is different from the location of the mean secondary circulation. It is possible that incorporating the secondary circulation into the basic-state vortex would lead to a more vigorous response to rainband heating because the mean anticyclonic circulation at the upper levels reduces inertial stability. The excitation of inner-core modes could be less severe because of the upward and outward advection of the PV anomalies (Nolan and Montgomery 2002b). This study examines the response of the hurricane wind field to imposed rainband heating whose structures are already prescribed. A fundamental question that still remains unanswered concerns the factors or processes
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FIG. 23. Horizontal cross sections at z 5 4.17 km of the response to the transient mixed rainband heating at t 5 (a) 4, (b) 7, (c) 10, (d) 12, (e) 14, and (f) 16 h. Contours show vertical velocity (m s21) of the response, with a contour interval shown above each panel from the zero line, which is not plotted. Note that different contour interval values are used for different plots. Positive and negative values are marked by solid and dashed contour lines, respectively. Arrows show horizontal velocity (m s21) of the response. Maximum and minimum values of the contoured field as well as the magnitude of the largest arrow (jmaxj) can be found above each panel.
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that control the evolution of these diabatic heating structures of convection in tropical cyclones. The answer appears to be closely related to the evolution of the boundary layer structures of hurricanes, which are not represented in 3DVPAS. Comparisons to spiral rainbands that are seen to develop ubiquitously during full-physics simulations of tropical cyclones should be made as a next step. Attempts to isolate the impact of spiral rainbands on the surrounding wind field in high-resolution simulations should also be made. The response of the hurricane wind field to transient mixed rainbands is intriguing. It suggests that the accelerated cyclonic tangential circulation can wrap around the entire vortex if the heating lasts long enough and that the change in the tangential circulation would remain coherent for a long time after the rainband heating dissipates. This process may help us better understand secondary eyewall formation, a phenomenon not well understood at the moment. Kuo et al. (2004) and Wang (2009) also suggested that the presence of strong rainbands may favor the formation of secondary eyewalls. Our efforts to understand the role of spiral rainbands in the formation of secondary eyewalls will continue with numerical models of increasing complexity. Acknowledgments. This research was supported by the National Science Foundation Grants ATM-0432717 and ATM-0756308. We also would like to sincerely thank Drs. Brian E. Mapes, Robert F. Rogers, and Hugh E. Willoughby for their helpful comments and feedback on this study. Detailed and constructive reviews by Drs. Yuqing Wang, Shigeo Yoden, and two anonymous reviewers are greatly appreciated. Thanks are due to Mr. Daniel P. Stern for sharing his version of the code for the free atmospheric portion of the E86 model and also to Mr. David Painemal for help with figures.
APPENDIX Obtaining the Slope of the RMW and Decay Rate of Tangential Velocity at the RMW from E86 E86 assumes that the primary tangential circulation of tropical cyclones in the free atmosphere is in hydrostatic and gradient wind balance and that saturation moist entropy is conserved along angular momentum surfaces, which slope radially outward with height, thus implying a state of slantwise moist neutrality. By using pseudoadiabatic thermodynamic conditions, as clarified by Bryan and Rotunno (2009), E86 derived expressions [Eqs. (51) and (56) in E86] for how an angular momentum surface slopes outward with height in the free atmosphere. Since the RMW of the tropical cyclone
tangential wind field is closely approximated by an angular momentum surface (Stern and Nolan 2009), the rates at which the RMW slopes radially outward with altitude and the tangential velocity decays along the RMW may be obtained from these expressions. First, the tangential wind field at the lowest level of the domain, which should be considered to be the top of the boundary layer, is prescribed by a profile such as the MR vortex in Eq. (1), and the pressure and temperature fields are initialized with constant values p0 and TB, respectively. Then, we compute the pressure field that would hold the prescribed wind field in gradient wind balance 1 ›p y2 5 fy 1 , r ›r r
(A1)
where r is density, p is pressure, r is radius, f is the Coriolis parameter, and y is tangential velocity. Equation (A1) is combined with the equation of state— r5
p , Rd T
(A2)
where Rd is the dry gas constant and T is temperature— to eliminate r and then is integrated inward from the outer boundary using a Crank–Nicholson scheme while holding T fixed. The density field is computed by enforcing Eq. (A2) with fixed T. Next, we compute the environmental thermodynamic properties consistent with the prescribed tangential wind field by following an iterative procedure outlined in E86 (597–598). Then, Eq. (51) of E86 is evaluated at the RMW to obtain the radius of an outward sloping angular momentum surface that is congruent to the RMW. The altitude of this angular momentum surface can be found from Eq. (56) of E86. Vertical decay of tangential velocity along the RMW can be found by inverting the definition of absolute angular momentum M [ ry 1 fr2/2 at each altitude. We use values of p0 5 1015.1 mb, TB 5 26.38C, and T0 5 270.08C. REFERENCES Anthes, R. A., 1982: Tropical Cyclones: Their Evolution, Structure and Effects. Meteor. Monogr., No. 41, Amer. Meteor. Soc., 208 pp. Atlas, D., K. R. Hardy, R. Wexler, and R. J. Boucher, 1963: On the origin of hurricane spiral bands. Geofis. Int., 3, 123–132. Barnes, G. M., and M. D. Powell, 1995: Evolution of the inflow boundary layer of Hurricane Gilbert (1988). Mon. Wea. Rev., 123, 2348–2368. ——, E. J. Zipser, D. Jorgensen, and F. Marks Jr., 1983: Mesoscale and convective structure of a hurricane rainband. J. Atmos. Sci., 40, 2125–2137.
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——, J. F. Gamache, M. A. LeMone, and G. J. Stossmeister, 1991: A convective cell in a hurricane rainband. Mon. Wea. Rev., 119, 776–794. Bryan, G. H., and R. Rotunno, 2009: Evaluation of an analytical model for the maximum intensity of tropical cyclones. J. Atmos. Sci., 66, 3042–3060. Carr, L. E., and R. T. Williams, 1989: Barotropic vortex stability to perturbations from axisymmetry. J. Atmos. Sci., 46, 3177–3191. Cecil, D. J., E. J. Zipser, and S. W. Nesbitt, 2002: Reflectivity, ice scattering, and lightning characteristics of hurricane eyewalls and rainbands. Part I: Quantitative description. Mon. Wea. Rev., 130, 769–784. Chen, Y., and M. K. Yau, 2001: Spiral bands in a simulated hurricane. Part I: Vortex Rossby wave verification. J. Atmos. Sci., 58, 2128–2145. Cione, J. J., P. G. Black, and S. H. Houston, 2000: Surface observations in the hurricane environment. Mon. Wea. Rev., 128, 1550–1561. Dvorak, V. F., 1975: Tropical cyclone intensity analysis and forecasting from satellite imagery. Mon. Wea. Rev., 103, 420–430. Eliassen, A., 1951: Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astrophys. Norv., 5, 19–60. Emanuel, K. A., 1986: An air–sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., 43, 585–605. Frank, W. M., 1977: The structure and energetics of the tropical cyclone. Part I: Storm structure. Mon. Wea. Rev., 105, 1119– 1135. Franklin, C. N., G. J. Holland, and P. T. May, 2006: Mechanisms for the generation of mesoscale vorticity features in tropical cyclone rainbands. Mon. Wea. Rev., 134, 2649–2669. Gall, R., J. Tuttle, and P. Hildebrand, 1998: Small-scale spiral bands observed in hurricanes Andrew, Hugo, and Erin. Mon. Wea. Rev., 126, 1749–1766. Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands. J. Atmos. Sci., 50, 3380–3403. Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of atmospheric vortices to heating by organized cumulus convection. J. Atmos. Sci., 43, 1559–1573. Hence, D. A., and R. A. Houze Jr., 2008: Kinematic structure of convective-scale elements in the rainbands of Hurricanes Katrina and Rita (2005). J. Geophys. Res., 113, D15108, doi:10.1029/2007JD009429. Hodyss, D., and D. S. Nolan, 2007: Anelastic equations for atmospheric vortices. J. Atmos. Sci., 64, 2947–2959. Houze, R. A., Jr., and Coauthors, 2006: The Hurricane Rainband and Intensity Change Experiment. Observations and modeling of Hurricanes Katrina, Ophelia, and Rita. Bull. Amer. Meteor. Soc., 87, 1503–1521. Jordan, C. L., 1958: Mean soundings for the West Indies area. J. Meteor., 15, 91–97. Jorgensen, D. P., 1984: Mesoscale and convective-scale characteristics of mature hurricanes. Part I: General observations by research aircraft. J. Atmos. Sci., 41, 1268–1285. Kim, D.-K., K. R. Knupp, and C. R. Williams, 2009: Airflow and precipitation properties within the stratiform region of Tropical Storm Gabrielle during landfall. Mon. Wea. Rev., 137, 1954–1971. Kimball, S., 2006: A modeling study of hurricane landfall in a dry environment. Mon. Wea. Rev., 134, 1901–1918. Kuo, H.-C., L.-Y. Lin, C.-P. Chang, and R. T. Williams, 2004: The formation of concentric vorticity structures in typhoons. J. Atmos. Sci., 61, 2722–2734.
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Kusunoki, K., and W. Mashiko, 2006: Doppler radar investigations of the inner core of Typhoon Songda (2004): Polygonal/elliptical eyewalls, eye contraction, and small-scale spiral bands. Extended Abstracts, 27th Conf. on Hurricanes and Tropical Meteorology, Monterey, CA, Amer. Meteor. Soc., P4.10. [Available online at http://ams.confex.com/ams/27Hurricanes/ techprogram/paper_107480.htm.] Mallen, K. J., M. T. Montgomery, and B. Wang, 2005: Reexamining the near-core radial structure of the tropical cyclone primary circulation: Implications for vortex resiliency. J. Atmos. Sci., 62, 408–425. May, P. T., 1996: The organization of convection in the rainbands of Tropical Cyclone Laurence. Mon. Wea. Rev., 124, 807–815. ——, and G. J. Holland, 1999: The role of potential vorticity generation in tropical cyclone rainbands. J. Atmos. Sci., 56, 1224– 1228. ——, ——, and W. L. Ecklund, 1994: Wind profiler observations of Tropical Storm Flo at Saipan. Wea. Forecasting, 9, 410–426. Molinari, J., P. Moore, and V. Idone, 1999: Convective structure of hurricanes as revealed by lightning locations. Mon. Wea. Rev., 127, 520–534. Montgomery, M. T., and R. J. Kallenbach, 1997: A theory for vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes. Quart. J. Roy. Meteor. Soc., 123, 435–465. Nolan, D. S., and B. F. Farrell, 1999: The intensification of twodimensional swirling flows by stochastic asymmetric forcing. J. Atmos. Sci., 56, 3937–3962. ——, and M. T. Montgomery, 2002a: Nonhydrostatic, threedimensional perturbations to hurricane-like vortices. Part I: Linearized formulation, stability, and evolution. J. Atmos. Sci., 59, 2989–3020. ——, and ——, 2002b: Three-dimensional stability analyses of tornado-like vortices with secondary circulations. Preprints, 21st Conf. on Severe Local Storms, San Antonio, TX, Amer. Meteor. Soc., 477–480. ——, and L. D. Grasso, 2003: Nonhydrostatic, three-dimensional perturbations to hurricane-like vortices. Part II: Symmetric response and nonlinear simulations. J. Atmos. Sci., 60, 2717– 2745. ——, M. T. Montgomery, and L. D. Grasso, 2001: The wavenumberone instability and trochoidal motion of hurricane-like vortices. J. Atmos. Sci., 58, 3243–3270. ——, Y. Moon, and D. P. Stern, 2007: Tropical cyclone intensification from asymmetric convection: Energetics and efficiency. J. Atmos. Sci., 64, 3377–3405. Pandya, R. E., and D. R. Durran, 1996: The influence of convectively generated thermal forcing on the mesoscale circulation around squall lines. J. Atmos. Sci., 53, 2924–2951. ——, ——, and M. L. Weisman, 2000: The influence of convective thermal forcing on the three-dimensional circulation around squall lines. J. Atmos. Sci., 57, 29–45. Powell, M. D., 1990: Boundary layer structure and dynamics in outer hurricane rainbands. Part I: Mesoscale rainfall and kinematic structure. Mon. Wea. Rev., 118, 891–917. Rozoff, C. M., W. H. Schubert, B. D. McNoldy, and J. P. Kossin, 2006: Rapid filamentation zones in intense tropical cyclones. J. Atmos. Sci., 63, 325–340. Ryan, B. F., G. M. Barnes, and E. J. Zipser, 1992: A wide rainband in a developing tropical cyclone. Mon. Wea. Rev., 120, 431–447. Samsury, C. E., and E. J. Zipser, 1995: Secondary wind maxima in hurricanes: Airflow and relationship to rainbands. Mon. Wea. Rev., 123, 3502–3517.
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