Effects of Ambient Velocity Shear on Internal ... - Semantic Scholar
1 2 3
Effects of Ambient Velocity Shear on Nonlinear Internal Waves
4
and Associated Mixing at the Columbia River Plume Front
5 6 7
Jiayi Pan and David A. Jay
8 9 10 11 12
Department of Civil and Environmental Engineering
13
Portland State University, Portland, OR 97201 USA
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
Submitted to Journal of Geophysical Research-Oceans June 2008
1
Abstract
2
Large-amplitude nonlinear internal waves (NLIWs) are frequently observed propagat-
3
ing away from the Columbia River tidal plume front. They are generated during the de-
4
celeration of the frontal bulge. Cruise vessel observations indicate that in the presence of
5
strong ambient velocity shear, the maximum amplitude of the NLIW structure function
6
appear well below the density interface on which the NLIWs is traveling, and at a deeper
7
depth than in the absence of shear. The Observations of NLIW properties are based on
8
density profiles, ADCP velocities and ADCP beam echo intensity, all obtained by a
9
towed sled during the RISE (River Influences on Shelf Ecosystems) project. The effects
10
of ambient shear on NLIW dynamic characteristics can be analyzed using a high-order
11
KdV model forced by an ambient velocity field described by three parameters: the sur-
12
face layer ambient velocity (U0), the depth of the maximum ambient velocity shear (hU),
13
and the shear depth range (thickness) δhU.
14
RISE vessel data show there are two depth ranges associated with density overturns
15
and vertical mixing. Dynamic analysis reveals that the overturns correspond to the depths
16
with the gradient Richardson number Rig less than a critical value of ~0.25. The nonlinear
17
interaction between ambient shear and the NLIWs is responsible for velocity shear inten-
18
sification and causes Rig to decrease below the critical value. In addition, due to the pres-
19
ence of ambient velocity shear, the maximum NLIW velocity shear occurs at a depth be-
20
low the interface where there is less stratification, which enhances the NLIW-induced
21
turbulent mixing. The Rig calculated from the in-situ measurements is consistent with the
22
results of the theoretical analysis.
23
Key words: River plume front; nonlinear internal waves; turbulent mixing.
1
1
1. Introduction
2
The tidal outflow from the Columbia River (CR) forms a coastal plume that is impor-
3
tant to the coastal ecosystem [Barnes et al., 1972; Grimes and Kingsford, 1996; Hickey et
4
al, 1998]. Observations indicate that the CR plume consists of four distinct water masses:
5
(a) source water at the lift-off point, and (b) the tidal, (c) re-circulating, and (d) far-field
6
plumes [Horner-Devine et al, 2008]. The tidal plume is the water from the most recent
7
ebb, with a radius of ~10-30 km. It is bounded by a distinct front that often generates
8
nonlinear internal waves (NLIWs; [Nash and Moum, 2005]). The plume front is the most
9
energetic part of the plume and exhibits vigorous turbulent mixing, which disturbs the
10
seabed, dissipates plume energy and mixes upwelled nutrients and iron (Fe) from re-
11
suspended river sediments into the surface layer [Orton and Jay, 2005; Zaron and Jay,
12
2008, this volume].
13
The strong stratification and high energy levels of the CR plume lead to the frequent
14
occurrence of NLIWs. NLIWs often appear by interaction between shoreward propagat-
15
ing tides and the sharp topography of the continental slope [Moum et al., 2003, Stanton
16
and Ostrovsky, 1998]. NLIWs are also generated at the plume front, propagating off-
17
shore, and the generation occurs as the tidal plume front transitions from supercritical to
18
subcritical conditions [Nash and Moum, 2005; Jay et al., 2008]. These NLIWs are
19
nonlinear solitary waves, or internal solitons. Pan et al. [2007] analyzed a group of inter-
20
nal solitons generated at and traveling off the CR plume front using a satellite synthetic
21
aperture radar (SAR) image and extracted the internal soliton dynamic parameters based
22
on a SAR internal soliton imaging model. For the CR plume frontal internal solitons, Pan
23
and Jay [2008a] found that high-order KdV theory is preferable for description of the the
2
1
dynamic properties of the solitons. They demonstrated that the plume front-generated in-
2
ternal solitons can expand the plume area up to 20%, and carry ~75% frontal energy out
3
of the frontal region. Jay et al. [2008] reported that there are obvious differences in inter-
4
nal wave generation at upstream fronts between upwelling and downwelling conditions.
5
Under typical downwelling conditions, the tidal plume front is usually broad (up to 5 km)
6
and diffuse on its upstream southern side. However, under summer upwelling conditions,
7
the upstream front remains sharp and narrow (only ~50-200 m wide on its upwind or
8
northern side). They found that NLIW generation occurs regularly under upwelling con-
9
ditions. The generation is first seen on the southern side, and the front proceeds to “un-
10
zip” from south to north. Jay et al. suggested that potential vorticity conservation causes
11
northerly fronts to thin and remain supercritical for an extended period, up to 12 hrs.
12
Frontal NLIWs influence the interaction between the CR plume and ambient coastal
13
water and have a major impact on the coastal ecosystem because the frontal internal
14
waves can influence vertical mixing near the plume front and affects interaction of the
15
tidal plume with plume near- and far-field waters. Many investigators documented mix-
16
ing caused by internal waves. Sandstrom et al. [1989] argued that the shear intensified by
17
internal waves could reduce the gradient Richardson number Rig. Sandstrom and Oakey
18
[1995] found enhanced turbulent mixing on the Scotian Shelf occurring in a region with
19
strong shear caused internal waves. Observations of internal waves off the Oregon coast
20
by Moum et al. [2003] showed that high acoustic backscatter beginning in the vicinity of
21
the internal wave trough and continuing through its trailing edge and wake. The acoustic
22
backscatter coincided with overturning, high-density microstructure, and turbulence at the
23
interface. Nevertheless, the calculated Rig was larger than critical value of 0.25, suggest-
3
1
ing a stable condition. They speculated that the acoustic Doppler current profiler (ADCP)
2
time and space resolution did not allow capture of velocity shear caused by the internal
3
waves. Based on the high-resolution density measurements, they deduced existence of a
4
velocity layer with strong shear in the interface, which could produce the observed turbu-
5
lence. Working in the same area, Avicola et al. [2007] reported that there are three prin-
6
cipal components contributing to the velocity shear for the mixing process, the slowly
7
varying thermal wind shear, the M2 internal tide, and near f waves, all of which were of
8
similar magnitude. When the three dominant shear constituents interfere constructively,
9
enhanced turbulent mixing occurs.
10
Clearly, it is common in coastal seas for NLIWs to co-exist with ambient currents.
11
This is particularly pertinent in the CR plume region, because NLIWs are frequently gen-
12
erated at the tidal plume front, an environment that often exhibits strong shear. It is still,
13
however, under investigation of how the ambient shear modifies dynamic structures of
14
the frontal NLIWs and how the interaction of the ambient shear with NLIWs affects the
15
mixing status in the frontal area. In this study, we analyze the influence of the ambient
16
current shear on NLIW dynamic properties and associated turbulent mixing using theory
17
and in-situ measurement data collected by the River Influence on the Shelf Ecosystem
18
(RISE) project. This investigation deepens our knowledge of frontal mixing processes
19
and the plume ecosystem as a whole.
20 21
2. RISE cruises and observations
22
The RISE study hypothesizes that waters influenced by the CR plume are more pro-
23
ductive than adjacent coastal waters, especially off Washington. Understanding the im-
4
1
pact of the highly mobile CR plume on coastal production and transport patterns requires
2
that measurements be made on a variety of scales from turbulence to internal waves and
3
fronts, to those of the plume and the underlying shelf circulation. RISE emphasizes,
4
therefore, rapid surveys and detailed process investigations.
5
The RISE field program consists of 4 cruises in July 2004, spring 2005 and 2006, and
6
August 2005. Each cruise was carried out by two vessels, the R/V Wecoma (biological
7
and chemical studies) and the R/V Pt Sur (plume surveys, mixing processes and zoo-
8
plankton dynamics). The R/V Pt Sur carried out rapid surveys using a towed body (TRI-
9
AXUS, steerable in 3D) and the vessel’s near-surface underway data acquisition system
10
or UDAS (normally at 3 m depth). The high mobility of TRIAXUS was used to sample
11
surface waters (from 60 m up to within 0.5-2 m of the surface, depending on sea state)
12
outside of the ship wake.
13
The R/V Pt Sur carried a side-mounted ADCP: 300 kHz in 2004 (RISE1) and 1200
14
kHz in 2005 and 2006 (RISE2 and RISE4). The R/V Pt Sur UDAS acquired position, me-
15
teorological data, salinity (S), temperature (T), and fluorescence at 3 m. TRIAXUS car-
16
ried an upward looking (REMUS configuration) 1200 kHz ADCP with mode 12 firm-
17
ware, a 911 Seabird conductivity-temperature-depth (CTD) profiler equipped with sen-
18
sors for nitrate (N), C, T, pressure, transmissivity and fluorescence. The UDAS, TRI-
19
AXUS ADCP, and ship-mounted ADCP data sets are used in the analyses described
20
herein, along with scalar data from TRIAXUS. With the GPS data of the cruise ship, the
21
ship-mounted ADCP measured ocean current velocities can be converted into those in the
22
Earth coordinate system, and the converted velocities can be used as references for the
23
TRIAXUS ADCP velocity measurements [Pan and Jay, 2008b]. In addition, X band
5
1
shipboard radar images are collected every minute in 2006 RISE cruise. The ship radar
2
images show sea surface roughness, and often reveal the presence and properties of inter-
3
nal waves and plume fronts.
4
We seek here to contrast NLIW processes in situations with significant vs. weak am-
5
bient shear as a means to understand the interaction of NLIW with ambient shear. The
6
first situation pertained when we observed an NLIW packet on June 10, 2006 around
7
0230 UTC. Figures 1a, 1b, and 1c are three shipboard radar images, taken on June 10,
8
2006 at 0229, 0249, and 0256 UTC, respectively. The images show a group of three
9
NLIWs traveling off the CR front northwestward in the direction of 298°. The cruise
10
ship navigation direction is 314°. Therefore, the angle between the ship navigation and
11
the NLIW traveling is 16°. The locations of the NLIW lines and fronts at these three
12
times are illustrated in Figure 2 together with the bathymetry contours.
13
The density profiles and the ADCP data are shown in Figure 3. The upper panel (Fig-
14
ure 3a) is density profiles observed by the Seabird CTD onboard the TRIAXUS towfish.
15
The figure suggests that there are density undulations between longitude 124.292°W and
16
124.286°W. However, the TRIAXUS CTD cannot fully resolve NLIW structures, be-
17
cause the NLIW wavelength (~70m) is considerable less than the cycle distance of the
18
TRIAXUS (~300m). The middle panels (Figures 3b and 3c) show the horizontal crest-
19
normal and vertical velocities, respectively. There are positive horizontal velocity anoma-
20
lies relative to the background induced by the NLIW (Figure 3b) suggesting that wave-
21
induced water movement are in the direction of the wave travel. On the leading edge of
22
the NLIW, the water particles move downward, and on the other side, upward, which
23
causes a convergence zone on the leading edge and a divergence zone on the other for
6
1
each of the NLIWs (Figure 3c). Figure 3d shows the ADCP beam echo intensity, which
2
better resolves the NLIW features, indicating intensified ADCP echo backscattering be-
3
low the interface of upper and lower layers and suggesting that the NLIW are stronger
4
than those at the interface of density layers and the maximum wave amplitude is below
5
the normal interface depth.
6
The NLIWs in Figures 2 and 3 differ greatly from the NLIWs observed around 1400
7
UTC on June 8, 2006, analyzed by Pan and Jay [2008a] and shown in Figure 4. Compar-
8
ing the two cases, we notice a difference in ambient current conditions between the two
9
cases. In the first case (Figures 2 and 3), there is significant ambient current, whereas for
10
the second (Figure 4), the current vertical shear is weak (Figure 5). The presence of am-
11
bient velocity shear is a likely reason for the different behavior of the NLIWs, a hypothe-
12
sis we investigate here. Pan and Jay [2008a] suggested the high-order KdV model is bet-
13
ter for predicting the plume front-generated NLIWs than other weakly nonlinear NLIW
14
theories. In the next section, based on that model, we analyze the effects of the ambient
15
shear on the plume frontal NLIW dynamic behavior.
16 17
3. Effects of ambient shear on dynamics of frontal NLIWs
18
Two-dimensional internal waves may be described by a stream function ψ. The
19
stream function is a product of linear wave speed (C0-U) and the vertical displacement
20
A(x,z,t): ψ = (c0 − U ) A( χ , z ) , where χ = x − ct ; c0 and c are the linear wave and NLIW
21
speeds, respectively, and U is the ambient velocity. The vertical displacement has a sepa-
22
rable form A( x, z , t ) = η ( χ )φ ( z ) , in which, φ(z) is the internal wave vertical structure
23
function, satisfying the following eigenvalue equation
7
d dφ [(U − c0 ) 2 ] + N 2φ = 0 , dz dz
1
φ ( 0) = φ ( − H ) = 0
(1)
2
where c0 is determined from eigenvalues, and N is the buoyancy frequency of undisturbed
3
density ( N = −
4
ity for the first case are shown in Figure 5. The normalization condition is φ(zmax)=1,
5
where zmax is the depth corresponding to the maximum value of φ. The internal NLIW
6
amplitude η (ξ ) is governed by
∂η ∂η ∂ 3η + (c0 + αη + α 1η 2 ) + β 3 = 0, ∂t ∂x ∂x
7 8
g ∂ρ ). The profile of density, buoyancy frequency, and ambient velocρ ∂z
(2)
where 0
2 3 3 ∫− H (c0 − U ) (dφ / dz ) dz , α= 2 0 (c − U )(dφ / dz ) 2 dz ∫ 0
9
(3)
−H
1 β= 2
10
∫
0
−H
∫
0
−H
(c 0 −U ) 2 φ 2 dz
(c0 − U )(dφ / dz ) 2 dz
.
(4)
11
The role of shear on coefficients α and β is evident from (3) and (4), but the coefficient
12
α1 in equation (2) is more complex – [Holloway, 2002] α1 =
13
1
∫
0
−H
(c0 − U )(dφ / dz) dz 2
∫
0
−H
{3(c0 − U ) 2 [3dξ / dz − 2(dφ / dz) 2 ]
,
(5)
(dφ / dz) 2 − α 2 (dφ / dz) 2 + α (c0 − U )(dφ / dz)[5(dφ / dz) 2 − 4dξ / dz]}dz
14
where ξ (z ) is the first correction function to the NLIW mode; ξ (z ) is a solution of the
15
following equation:
16
3 d d dξ d dφ dφ [(c 0 − U ) 2 ] + N 2ξ = −α [c 0 − U ) ] + [(c 0 − U ) 2 ( ) 2 ] , 2 dz dz dz dz dz dz
8
(6)
1
with the boundary condition ξ(0)= ξ(-H)=0 and normalized condition ξ ( z max ) = 0 .
2
For the high-order KdV equation (2), there is a soliton solution,
η=−
3
4
7 8 9
(7)
where ν and δ are parameters satisfying
1 4
1 +ν ) , 1 −ν
δ (ν ) = ln(
5 6
αν x − ct x − ct [tanh( + δ ) − tanh( − δ )] , ∆ ∆ α1 2
(8)
and ∆=
− 24α 1 β
α 2ν 2
.
(9)
The NLIW phase speed is given by c = c0 −
α 2ν 2 . 6α 1
(10)
10
The internal wave vertical structure function (φ) is obtained by solving the eigenvalue
11
equation (1) with the boundary conditions. We employ the MATLAB partial differential
12
equation (PDE) toolbox to obtain the numerical solution to the eigenvalue equation. Only
13
the first mode is considered because the phase speeds of higher-order modes are much
14
slower than the first mode and even slower than the frontal speed, so that the higher-order
15
modes cannot escape from the front. To determine first correction function (ξ) to the
16
NLIW mode from (6), with two boundary conditions (ξ(0)= ξ(-H)=0) and the normaliza-
17
tion condition ( ξ ( z max ) = 0 ), we break the solution into two portions with three boundary
18
conditions. The first segment is on [-H to zmax] with boundary conditions of ξ(-
19
H)=ξ(zmax)= 0. The second is on [zmax to 0] with the boundary condition of
9
1
ξ(zmax)=ξ(0)=0. The final solution is the combination of the two portions, which satisfies
2
the normalized condition, ξ ( z max ) = 0 .
3
The solutions for φ and ξ from (1) and (6) are shown in Figures 6a and 6b, with and
4
without shear. The value of maximum φ (zmax) with ambient current shear is deeper in the
5
water column than for the case without shear, suggesting that the ambient current shear
6
modifies the vertical structures of NLIWs. Without shear, zmax is -10.7 m, whereas with
7
shear, the zmax is changed to -13.5 m. There is a little difference in the position of the ex-
8
trema in ξ between the two cases. Because zmax is deeper with shear than without, the
9
depth range for ξ in the upper level with shear is broader and the magnitude is larger than
10
that without shear.
11
The vertical structure function φ and the correction function ξ are used to determine
12
the parameters α, β, and α1 from equations (3)-(5). These parameters are listed in Table
13
2. Larger differences are found between shear and no-shear conditions for β and α1 than
14
for the α. The absolute values of β and α1 are one and five times smaller under the shear
15
than under no-shear condition, whereas α shows a 10% difference.
16
The above solutions can be compared to observations. Using the ADCP echo intensity
17
data (Figure 2d) and ship radar images (Figure 3), we extract the NLIW amplitude and
18
phase speed data, as per Pan and Jay [2008a]. The predicted relationships between the
19
NLIW phase speed and amplitude for the shear and no-shear cases are displayed in Fig-
20
ure 7, along with observations for the case with shear. The observations fit the model re-
21
lationship closely. Compared with the no-shear condition, NLIW amplitudes are larger
22
for the same phase speed and density stratification when shear is present.
10
1 2 3
In order to determine the response of the solution to variations in the ambient shear we use a three-parameter model to parameterize the ambient velocity 2( z + hU ) 1 1 U m ( z ) = U 0 { arctan[ ]+ }, π δhU 2
(11)
4
where hU is the depth of the maximum ambient velocity shear; δhU is the shear depth
5
range; and U0 is approximately the surface layer ambient velocity. The model is dis-
6
played in Figure 5c. Model parameters are listed in Table 1.We judge that the models
7
adequately represent velocity profiles.
8
Using (1)-(6) and Um, we re-calculate the parameters, α, β, and α1 for a variety of
9
shear conditions through adjustment of the velocity shear model parameters, U0, hU, and
10
δhU. Figure 8 illustrates variations of α, β, and α1 in response to changes in the parame-
11
ters, U0, hU, and δhU. Note that α decreases as U0 varies from -0.4 to 0.4 (Figure 8a),
12
whereas it exhibits less variation with hU and δhU (Figures 8b and 8c). An interesting fea-
13
ture in the relationship between α and hU is that as hU approaches -10 m, α increases and
14
then drops dramatically as hU reaches -7.5 m (Figures 8b). In contrast, increases of U0,
15
hU, and δhU result in larger β (Figures 8d, 8e, and 8f). Parameter α1 has the same re-
16
sponses to U0, hU, and δhU as α. With an increase in U0, α1 decreases (Figure 8g), but the
17
changes of α1 caused by hU and δhU are relatively small (Figures 8h and 8i).
18
Figure 9 illustrates the relationships between amplitude and phase speed in different
19
ambient velocity profiles. The direction of the ambient current is important. We describe
20
the sense of the ambient as “counter-NLIW” (U0 0) when the two
22
move in the same direction. The stronger the counter-NLIW velocity, the larger the
11
1
NLIW amplitude for any given phase speed. However, when U0
Effects of Ambient Velocity Shear on Nonlinear Internal Waves
4
and Associated Mixing at the Columbia River Plume Front
5 6 7
Jiayi Pan and David A. Jay
8 9 10 11 12
Department of Civil and Environmental Engineering
13
Portland State University, Portland, OR 97201 USA
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
Submitted to Journal of Geophysical Research-Oceans June 2008
1
Abstract
2
Large-amplitude nonlinear internal waves (NLIWs) are frequently observed propagat-
3
ing away from the Columbia River tidal plume front. They are generated during the de-
4
celeration of the frontal bulge. Cruise vessel observations indicate that in the presence of
5
strong ambient velocity shear, the maximum amplitude of the NLIW structure function
6
appear well below the density interface on which the NLIWs is traveling, and at a deeper
7
depth than in the absence of shear. The Observations of NLIW properties are based on
8
density profiles, ADCP velocities and ADCP beam echo intensity, all obtained by a
9
towed sled during the RISE (River Influences on Shelf Ecosystems) project. The effects
10
of ambient shear on NLIW dynamic characteristics can be analyzed using a high-order
11
KdV model forced by an ambient velocity field described by three parameters: the sur-
12
face layer ambient velocity (U0), the depth of the maximum ambient velocity shear (hU),
13
and the shear depth range (thickness) δhU.
14
RISE vessel data show there are two depth ranges associated with density overturns
15
and vertical mixing. Dynamic analysis reveals that the overturns correspond to the depths
16
with the gradient Richardson number Rig less than a critical value of ~0.25. The nonlinear
17
interaction between ambient shear and the NLIWs is responsible for velocity shear inten-
18
sification and causes Rig to decrease below the critical value. In addition, due to the pres-
19
ence of ambient velocity shear, the maximum NLIW velocity shear occurs at a depth be-
20
low the interface where there is less stratification, which enhances the NLIW-induced
21
turbulent mixing. The Rig calculated from the in-situ measurements is consistent with the
22
results of the theoretical analysis.
23
Key words: River plume front; nonlinear internal waves; turbulent mixing.
1
1
1. Introduction
2
The tidal outflow from the Columbia River (CR) forms a coastal plume that is impor-
3
tant to the coastal ecosystem [Barnes et al., 1972; Grimes and Kingsford, 1996; Hickey et
4
al, 1998]. Observations indicate that the CR plume consists of four distinct water masses:
5
(a) source water at the lift-off point, and (b) the tidal, (c) re-circulating, and (d) far-field
6
plumes [Horner-Devine et al, 2008]. The tidal plume is the water from the most recent
7
ebb, with a radius of ~10-30 km. It is bounded by a distinct front that often generates
8
nonlinear internal waves (NLIWs; [Nash and Moum, 2005]). The plume front is the most
9
energetic part of the plume and exhibits vigorous turbulent mixing, which disturbs the
10
seabed, dissipates plume energy and mixes upwelled nutrients and iron (Fe) from re-
11
suspended river sediments into the surface layer [Orton and Jay, 2005; Zaron and Jay,
12
2008, this volume].
13
The strong stratification and high energy levels of the CR plume lead to the frequent
14
occurrence of NLIWs. NLIWs often appear by interaction between shoreward propagat-
15
ing tides and the sharp topography of the continental slope [Moum et al., 2003, Stanton
16
and Ostrovsky, 1998]. NLIWs are also generated at the plume front, propagating off-
17
shore, and the generation occurs as the tidal plume front transitions from supercritical to
18
subcritical conditions [Nash and Moum, 2005; Jay et al., 2008]. These NLIWs are
19
nonlinear solitary waves, or internal solitons. Pan et al. [2007] analyzed a group of inter-
20
nal solitons generated at and traveling off the CR plume front using a satellite synthetic
21
aperture radar (SAR) image and extracted the internal soliton dynamic parameters based
22
on a SAR internal soliton imaging model. For the CR plume frontal internal solitons, Pan
23
and Jay [2008a] found that high-order KdV theory is preferable for description of the the
2
1
dynamic properties of the solitons. They demonstrated that the plume front-generated in-
2
ternal solitons can expand the plume area up to 20%, and carry ~75% frontal energy out
3
of the frontal region. Jay et al. [2008] reported that there are obvious differences in inter-
4
nal wave generation at upstream fronts between upwelling and downwelling conditions.
5
Under typical downwelling conditions, the tidal plume front is usually broad (up to 5 km)
6
and diffuse on its upstream southern side. However, under summer upwelling conditions,
7
the upstream front remains sharp and narrow (only ~50-200 m wide on its upwind or
8
northern side). They found that NLIW generation occurs regularly under upwelling con-
9
ditions. The generation is first seen on the southern side, and the front proceeds to “un-
10
zip” from south to north. Jay et al. suggested that potential vorticity conservation causes
11
northerly fronts to thin and remain supercritical for an extended period, up to 12 hrs.
12
Frontal NLIWs influence the interaction between the CR plume and ambient coastal
13
water and have a major impact on the coastal ecosystem because the frontal internal
14
waves can influence vertical mixing near the plume front and affects interaction of the
15
tidal plume with plume near- and far-field waters. Many investigators documented mix-
16
ing caused by internal waves. Sandstrom et al. [1989] argued that the shear intensified by
17
internal waves could reduce the gradient Richardson number Rig. Sandstrom and Oakey
18
[1995] found enhanced turbulent mixing on the Scotian Shelf occurring in a region with
19
strong shear caused internal waves. Observations of internal waves off the Oregon coast
20
by Moum et al. [2003] showed that high acoustic backscatter beginning in the vicinity of
21
the internal wave trough and continuing through its trailing edge and wake. The acoustic
22
backscatter coincided with overturning, high-density microstructure, and turbulence at the
23
interface. Nevertheless, the calculated Rig was larger than critical value of 0.25, suggest-
3
1
ing a stable condition. They speculated that the acoustic Doppler current profiler (ADCP)
2
time and space resolution did not allow capture of velocity shear caused by the internal
3
waves. Based on the high-resolution density measurements, they deduced existence of a
4
velocity layer with strong shear in the interface, which could produce the observed turbu-
5
lence. Working in the same area, Avicola et al. [2007] reported that there are three prin-
6
cipal components contributing to the velocity shear for the mixing process, the slowly
7
varying thermal wind shear, the M2 internal tide, and near f waves, all of which were of
8
similar magnitude. When the three dominant shear constituents interfere constructively,
9
enhanced turbulent mixing occurs.
10
Clearly, it is common in coastal seas for NLIWs to co-exist with ambient currents.
11
This is particularly pertinent in the CR plume region, because NLIWs are frequently gen-
12
erated at the tidal plume front, an environment that often exhibits strong shear. It is still,
13
however, under investigation of how the ambient shear modifies dynamic structures of
14
the frontal NLIWs and how the interaction of the ambient shear with NLIWs affects the
15
mixing status in the frontal area. In this study, we analyze the influence of the ambient
16
current shear on NLIW dynamic properties and associated turbulent mixing using theory
17
and in-situ measurement data collected by the River Influence on the Shelf Ecosystem
18
(RISE) project. This investigation deepens our knowledge of frontal mixing processes
19
and the plume ecosystem as a whole.
20 21
2. RISE cruises and observations
22
The RISE study hypothesizes that waters influenced by the CR plume are more pro-
23
ductive than adjacent coastal waters, especially off Washington. Understanding the im-
4
1
pact of the highly mobile CR plume on coastal production and transport patterns requires
2
that measurements be made on a variety of scales from turbulence to internal waves and
3
fronts, to those of the plume and the underlying shelf circulation. RISE emphasizes,
4
therefore, rapid surveys and detailed process investigations.
5
The RISE field program consists of 4 cruises in July 2004, spring 2005 and 2006, and
6
August 2005. Each cruise was carried out by two vessels, the R/V Wecoma (biological
7
and chemical studies) and the R/V Pt Sur (plume surveys, mixing processes and zoo-
8
plankton dynamics). The R/V Pt Sur carried out rapid surveys using a towed body (TRI-
9
AXUS, steerable in 3D) and the vessel’s near-surface underway data acquisition system
10
or UDAS (normally at 3 m depth). The high mobility of TRIAXUS was used to sample
11
surface waters (from 60 m up to within 0.5-2 m of the surface, depending on sea state)
12
outside of the ship wake.
13
The R/V Pt Sur carried a side-mounted ADCP: 300 kHz in 2004 (RISE1) and 1200
14
kHz in 2005 and 2006 (RISE2 and RISE4). The R/V Pt Sur UDAS acquired position, me-
15
teorological data, salinity (S), temperature (T), and fluorescence at 3 m. TRIAXUS car-
16
ried an upward looking (REMUS configuration) 1200 kHz ADCP with mode 12 firm-
17
ware, a 911 Seabird conductivity-temperature-depth (CTD) profiler equipped with sen-
18
sors for nitrate (N), C, T, pressure, transmissivity and fluorescence. The UDAS, TRI-
19
AXUS ADCP, and ship-mounted ADCP data sets are used in the analyses described
20
herein, along with scalar data from TRIAXUS. With the GPS data of the cruise ship, the
21
ship-mounted ADCP measured ocean current velocities can be converted into those in the
22
Earth coordinate system, and the converted velocities can be used as references for the
23
TRIAXUS ADCP velocity measurements [Pan and Jay, 2008b]. In addition, X band
5
1
shipboard radar images are collected every minute in 2006 RISE cruise. The ship radar
2
images show sea surface roughness, and often reveal the presence and properties of inter-
3
nal waves and plume fronts.
4
We seek here to contrast NLIW processes in situations with significant vs. weak am-
5
bient shear as a means to understand the interaction of NLIW with ambient shear. The
6
first situation pertained when we observed an NLIW packet on June 10, 2006 around
7
0230 UTC. Figures 1a, 1b, and 1c are three shipboard radar images, taken on June 10,
8
2006 at 0229, 0249, and 0256 UTC, respectively. The images show a group of three
9
NLIWs traveling off the CR front northwestward in the direction of 298°. The cruise
10
ship navigation direction is 314°. Therefore, the angle between the ship navigation and
11
the NLIW traveling is 16°. The locations of the NLIW lines and fronts at these three
12
times are illustrated in Figure 2 together with the bathymetry contours.
13
The density profiles and the ADCP data are shown in Figure 3. The upper panel (Fig-
14
ure 3a) is density profiles observed by the Seabird CTD onboard the TRIAXUS towfish.
15
The figure suggests that there are density undulations between longitude 124.292°W and
16
124.286°W. However, the TRIAXUS CTD cannot fully resolve NLIW structures, be-
17
cause the NLIW wavelength (~70m) is considerable less than the cycle distance of the
18
TRIAXUS (~300m). The middle panels (Figures 3b and 3c) show the horizontal crest-
19
normal and vertical velocities, respectively. There are positive horizontal velocity anoma-
20
lies relative to the background induced by the NLIW (Figure 3b) suggesting that wave-
21
induced water movement are in the direction of the wave travel. On the leading edge of
22
the NLIW, the water particles move downward, and on the other side, upward, which
23
causes a convergence zone on the leading edge and a divergence zone on the other for
6
1
each of the NLIWs (Figure 3c). Figure 3d shows the ADCP beam echo intensity, which
2
better resolves the NLIW features, indicating intensified ADCP echo backscattering be-
3
low the interface of upper and lower layers and suggesting that the NLIW are stronger
4
than those at the interface of density layers and the maximum wave amplitude is below
5
the normal interface depth.
6
The NLIWs in Figures 2 and 3 differ greatly from the NLIWs observed around 1400
7
UTC on June 8, 2006, analyzed by Pan and Jay [2008a] and shown in Figure 4. Compar-
8
ing the two cases, we notice a difference in ambient current conditions between the two
9
cases. In the first case (Figures 2 and 3), there is significant ambient current, whereas for
10
the second (Figure 4), the current vertical shear is weak (Figure 5). The presence of am-
11
bient velocity shear is a likely reason for the different behavior of the NLIWs, a hypothe-
12
sis we investigate here. Pan and Jay [2008a] suggested the high-order KdV model is bet-
13
ter for predicting the plume front-generated NLIWs than other weakly nonlinear NLIW
14
theories. In the next section, based on that model, we analyze the effects of the ambient
15
shear on the plume frontal NLIW dynamic behavior.
16 17
3. Effects of ambient shear on dynamics of frontal NLIWs
18
Two-dimensional internal waves may be described by a stream function ψ. The
19
stream function is a product of linear wave speed (C0-U) and the vertical displacement
20
A(x,z,t): ψ = (c0 − U ) A( χ , z ) , where χ = x − ct ; c0 and c are the linear wave and NLIW
21
speeds, respectively, and U is the ambient velocity. The vertical displacement has a sepa-
22
rable form A( x, z , t ) = η ( χ )φ ( z ) , in which, φ(z) is the internal wave vertical structure
23
function, satisfying the following eigenvalue equation
7
d dφ [(U − c0 ) 2 ] + N 2φ = 0 , dz dz
1
φ ( 0) = φ ( − H ) = 0
(1)
2
where c0 is determined from eigenvalues, and N is the buoyancy frequency of undisturbed
3
density ( N = −
4
ity for the first case are shown in Figure 5. The normalization condition is φ(zmax)=1,
5
where zmax is the depth corresponding to the maximum value of φ. The internal NLIW
6
amplitude η (ξ ) is governed by
∂η ∂η ∂ 3η + (c0 + αη + α 1η 2 ) + β 3 = 0, ∂t ∂x ∂x
7 8
g ∂ρ ). The profile of density, buoyancy frequency, and ambient velocρ ∂z
(2)
where 0
2 3 3 ∫− H (c0 − U ) (dφ / dz ) dz , α= 2 0 (c − U )(dφ / dz ) 2 dz ∫ 0
9
(3)
−H
1 β= 2
10
∫
0
−H
∫
0
−H
(c 0 −U ) 2 φ 2 dz
(c0 − U )(dφ / dz ) 2 dz
.
(4)
11
The role of shear on coefficients α and β is evident from (3) and (4), but the coefficient
12
α1 in equation (2) is more complex – [Holloway, 2002] α1 =
13
1
∫
0
−H
(c0 − U )(dφ / dz) dz 2
∫
0
−H
{3(c0 − U ) 2 [3dξ / dz − 2(dφ / dz) 2 ]
,
(5)
(dφ / dz) 2 − α 2 (dφ / dz) 2 + α (c0 − U )(dφ / dz)[5(dφ / dz) 2 − 4dξ / dz]}dz
14
where ξ (z ) is the first correction function to the NLIW mode; ξ (z ) is a solution of the
15
following equation:
16
3 d d dξ d dφ dφ [(c 0 − U ) 2 ] + N 2ξ = −α [c 0 − U ) ] + [(c 0 − U ) 2 ( ) 2 ] , 2 dz dz dz dz dz dz
8
(6)
1
with the boundary condition ξ(0)= ξ(-H)=0 and normalized condition ξ ( z max ) = 0 .
2
For the high-order KdV equation (2), there is a soliton solution,
η=−
3
4
7 8 9
(7)
where ν and δ are parameters satisfying
1 4
1 +ν ) , 1 −ν
δ (ν ) = ln(
5 6
αν x − ct x − ct [tanh( + δ ) − tanh( − δ )] , ∆ ∆ α1 2
(8)
and ∆=
− 24α 1 β
α 2ν 2
.
(9)
The NLIW phase speed is given by c = c0 −
α 2ν 2 . 6α 1
(10)
10
The internal wave vertical structure function (φ) is obtained by solving the eigenvalue
11
equation (1) with the boundary conditions. We employ the MATLAB partial differential
12
equation (PDE) toolbox to obtain the numerical solution to the eigenvalue equation. Only
13
the first mode is considered because the phase speeds of higher-order modes are much
14
slower than the first mode and even slower than the frontal speed, so that the higher-order
15
modes cannot escape from the front. To determine first correction function (ξ) to the
16
NLIW mode from (6), with two boundary conditions (ξ(0)= ξ(-H)=0) and the normaliza-
17
tion condition ( ξ ( z max ) = 0 ), we break the solution into two portions with three boundary
18
conditions. The first segment is on [-H to zmax] with boundary conditions of ξ(-
19
H)=ξ(zmax)= 0. The second is on [zmax to 0] with the boundary condition of
9
1
ξ(zmax)=ξ(0)=0. The final solution is the combination of the two portions, which satisfies
2
the normalized condition, ξ ( z max ) = 0 .
3
The solutions for φ and ξ from (1) and (6) are shown in Figures 6a and 6b, with and
4
without shear. The value of maximum φ (zmax) with ambient current shear is deeper in the
5
water column than for the case without shear, suggesting that the ambient current shear
6
modifies the vertical structures of NLIWs. Without shear, zmax is -10.7 m, whereas with
7
shear, the zmax is changed to -13.5 m. There is a little difference in the position of the ex-
8
trema in ξ between the two cases. Because zmax is deeper with shear than without, the
9
depth range for ξ in the upper level with shear is broader and the magnitude is larger than
10
that without shear.
11
The vertical structure function φ and the correction function ξ are used to determine
12
the parameters α, β, and α1 from equations (3)-(5). These parameters are listed in Table
13
2. Larger differences are found between shear and no-shear conditions for β and α1 than
14
for the α. The absolute values of β and α1 are one and five times smaller under the shear
15
than under no-shear condition, whereas α shows a 10% difference.
16
The above solutions can be compared to observations. Using the ADCP echo intensity
17
data (Figure 2d) and ship radar images (Figure 3), we extract the NLIW amplitude and
18
phase speed data, as per Pan and Jay [2008a]. The predicted relationships between the
19
NLIW phase speed and amplitude for the shear and no-shear cases are displayed in Fig-
20
ure 7, along with observations for the case with shear. The observations fit the model re-
21
lationship closely. Compared with the no-shear condition, NLIW amplitudes are larger
22
for the same phase speed and density stratification when shear is present.
10
1 2 3
In order to determine the response of the solution to variations in the ambient shear we use a three-parameter model to parameterize the ambient velocity 2( z + hU ) 1 1 U m ( z ) = U 0 { arctan[ ]+ }, π δhU 2
(11)
4
where hU is the depth of the maximum ambient velocity shear; δhU is the shear depth
5
range; and U0 is approximately the surface layer ambient velocity. The model is dis-
6
played in Figure 5c. Model parameters are listed in Table 1.We judge that the models
7
adequately represent velocity profiles.
8
Using (1)-(6) and Um, we re-calculate the parameters, α, β, and α1 for a variety of
9
shear conditions through adjustment of the velocity shear model parameters, U0, hU, and
10
δhU. Figure 8 illustrates variations of α, β, and α1 in response to changes in the parame-
11
ters, U0, hU, and δhU. Note that α decreases as U0 varies from -0.4 to 0.4 (Figure 8a),
12
whereas it exhibits less variation with hU and δhU (Figures 8b and 8c). An interesting fea-
13
ture in the relationship between α and hU is that as hU approaches -10 m, α increases and
14
then drops dramatically as hU reaches -7.5 m (Figures 8b). In contrast, increases of U0,
15
hU, and δhU result in larger β (Figures 8d, 8e, and 8f). Parameter α1 has the same re-
16
sponses to U0, hU, and δhU as α. With an increase in U0, α1 decreases (Figure 8g), but the
17
changes of α1 caused by hU and δhU are relatively small (Figures 8h and 8i).
18
Figure 9 illustrates the relationships between amplitude and phase speed in different
19
ambient velocity profiles. The direction of the ambient current is important. We describe
20
the sense of the ambient as “counter-NLIW” (U0 0) when the two
22
move in the same direction. The stronger the counter-NLIW velocity, the larger the
11
1
NLIW amplitude for any given phase speed. However, when U0
