EULER EQUATIONS AND TURBULENCE: ANALYTICAL APPROACH ...
EULER EQUATIONS AND TURBULENCE: ANALYTICAL APPROACH TO INTERMITTENCY A. CHESKIDOV AND R. SHVYDKOY Abstract. In this note we introduce in precise mathematical terms some of the empirical concepts used to describe intermittency in a fully developed turbulence. We will give definitions of the active turbulent region, volume, eddies, energy dissipation set, and derive rigorously some power laws of turbulence. In particular, the formula for the Hausdorff dimension of the energy dissipation set will be justified, and upper/lower bounds on the energy spectrum will be obtained.
1. Introduction Intermittency, if viewed as a measure of non-uniformity of the energy cascade, bears its signature on many statistical laws of a fully developed turbulence. A typical example is the power law for the energy density function, 2 ε 3 κ0 1− d3 (1) E(κ) ∼ 5 , κ3 κ where ε is the energy dissipation rate per unit mass, κ0 the integral scale, and d is the dimension of a set that carries turbulent energy dissipation. One observes similar corrections for longitudinal structure functions or skewness and flatness factors of velocity gradients (see the texts [15, 19, 18] for detailed discussions). Due to intermittency the deterministic counterparts of the power laws appear in a form of bounds to account for possible corrections as in Constantin et al [9, 6, 7], even though the intermittency itself has not been mathematically defined. As Foias remarked [13], “The rigorous mathematical reformulation of the heuristic estimates provided by some widely accepted turbulence theories always seem to represent the upper bound of the rigorous mathematical estimates.” At present there is no compelling analytical evidence that the fluid equations are consistent with the full range Date: February 7, 2012. 2000 Mathematics Subject Classification. Primary: 76F02; Secondary: 35Q31. The work of A. Cheskidov is partially supported by NSF grant DMS–1108864. R. Shvydkoy acknowledges the support of NSF grant DMS–0907812. 1
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A. CHESKIDOV AND R. SHVYDKOY
of intermittent dimensions 0 ≤ d ≤ 3 (or with d < 0). On the other hand, experimental evidence points only to moderate deviations from the classical Kolmogorov theory. Based on several subjective factors, such as the overwhelming success of widely used hot wire anemometry, or high incidence rates of linear cross-sections with (2+)D fractals (see [17]), one is lead to believe that the values of d above 2 are more likely to occur in real turbulence. For Leray-Hopf solutions of the Navier-Stokes equation this question is directly related to the regularity problem for the following reason. If d is defined as a saturation level of Bernstein’s inequality at the upper end of the inertial range (in a proper sense) then, as shown in [3], a solution with dimension d above 3/2 is automatically regular. We note that such an interpretation of intermittency also appeared in a derivation of the dyadic [4] and continuous [5] models of the energy cascade. Without implementation of viscous dissipation the statistical theory offers examples of models where the dimension is introduced as a rarefaction exponent of Richardson’s cascade (see [14, 20, 17]). Let us briefly recall the β-model of Frisch, Sulem, and Nelkin [14]. Assume that a fluid is in a state of turbulent motion confined to the periodic box T3L = [0, L]3 and observed on a long interval of time [0, T ]. On each stage of the cascade the kinetic energy is carried through by a set of active eddies {eq }. Let the eddied fill a volume Vq . If one assumes that Vq is a fraction of the preceding volume, i.e. Vq ∼ βVq−1 , for some 0 < β ≤ 1, then going up the scales, Vq ∼ β q L3 . Letting h = − log2 β, we have Vq ∼ 2−qh L3 and h ≥ 0. A heuristic incidence argument shows that the ultimate accumulation of the cascade occurs on a set that crosses infinitely many generations of eddies, i.e. A = lim supq→∞ ∪eq , and the dimension of the accumulant A does not exceed d = 3 − h. A simple dimensional analysis recovers the energy spectrum law (1). Further hypothesis of a statistical uni-scaling of velocity displacements δu(`) on A made the model unsuitable to capture experimentally observed exponents for higher order structure functions. This lead to subsequent multi-fractal ramifications of Frisch and Parisi (see [22, 12, 15]). Our objective in this paper is to define the above mentioned empirical concepts for deterministic fields, and use them to justify intermittency corrections to statistical laws without the use of viscous dissipation. We outline our basic definitions and results. Let u ∈ L2 ([0, T ] × T3L ) be a divergence free time dependent vector field. P We will use the LittlewoodPaley decomposition of u given by u = q≥−1 uq (see Section 2.1). We interpret uq as the collective velocity field of all eddies eq of dyadic size
ANALYTICAL APPROACH TO INTERMITTENCY
3
`q = L/2q . By bridging relations between the Euleran and Lagrangian description of the energy flux through scales `q we define the active volume as h|uq |2 i3 (2) Vq = L3 . h|uq |3 i2 Here and throughout the bracket h·i denotes the average over the spacetime domain ΩT = [0, T ] × T3L . The dimension d is then introduced as the following exponential co-type of the sequence {Vq }: log2 (L3 /Vq ) . q→∞ q Note that d is a number ”attached” to the field u without any assumptions on u. By the H¨older inequality we have one obvious bound Vq ≤ L3 , and in the stationary case Bernstein’s inequality implies Vq ≥ c`3q , where c is a-dimensional. The latter implies d ≥ 0, which is a natural bound. However, in the time-dependent case, Vq can scale down faster due to possible temporal rarefaction of active regions giving negative values of the dimension d (see also a discussion in Frisch [15]). To define active regions that fill volume Vq we find that for an eddie to qualify as “active” it has to have speed at least proportional to (3)
d = 3 − lim inf
sq ∼
h|uq |3 i . h|uq |2 i
The speeds will be interpreted as magnitudes of coefficients in the atomic decomposition of uq , and the supports of atoms will be viewed as eddies. The collection of those eddies will constitute what we call an “active” region Aq . The number of active eddies is represented by Vq /`3q . Our Theorem 3.2 shows that the Hausdorff dimension of A = lim sup Aq q→∞
is not to exceed d as predicted by the model, and Aq supports most of the “active” part of the field uq in terms of L3 -averages. We will recover the energy law (1) in the form of an upper and lower bounds. The scaling of the second order structure function is as predicted by the β-model " d 43 # 1− 3 −δ 2 2 ` ` (4) S2 (`) . ε 3 ` 3 + Cδ . L L for all ` < L, and any δ > 0. Here and throughout, . denotes inequality up to a universal adimensional constant. Note that the intermittency
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A. CHESKIDOV AND R. SHVYDKOY
correction exponent 1 − d3 is the same as in the energy law (1), however (4) shows that it is prone to deviations into smaller range. This phenomenon has been observed in direct numerical simulations as well by Kaneda et al in [16]. In Section 4 we stipulate on the role of A as an accumulant of the energy cascade for weak solutions to the Euler equations. We define the energy flux through scales `q by Πq = hπq i where density πq is the trilinear term that contains all relevant interactions involved in the energy transfer. Due to a localization of Πq observed in [2] all appreciable interactions are frequency-local near the `q scales. We show in Theorem 4.3 that the complement of A takes passive part in the cascade process in the sense that there is a nested sequence of sets Gp → A, in fact Gp = ∪k>p Ak , such that for all p > 0 lim h|πq |ΩT \Gp |i = 0.
q→∞
In other words, the turbulent cascade tends to dump the energy on the carrier A. The results of this section are established under the natural Onsager regularity condition ε = lim supq→∞ h|uq |3 i/`q < ∞ which is known to be suitable for the turbulent interpretation of the field (see [2, 8, 24, 10, 21, 11]). Any better regularity of u, namely simply requiring ε = 0, erases turbulent energy dissipation. This lack of dissipation may also occur locally in a proper sense. Therefore it is necessary to further confine the cascade to the smaller set A ∩ S, where S is the Onsager singular set of u, i.e. S is the smallest set on the complement of which ε = 0. This reduction is performed in Section 4.3. We will demonstrate by several examples that the regularity-related intermittency due to S and the cascade-related intermittency due to A are essentially independent. Therefore one has to consider both to have a more complete description of the accumulant. The active regions Aq can be used to give a more detailed description of a measure-theoretic support of the anomalous energy dissipation introduced by Duchon and Robert in [10]. Recall that for every weak solution u ∈ L3 (ΩT ) to the Euler equation one can associate a distribution D(u) such that 1 2 1 2 (5) ∂t |u| + div u |u| + p + D(u) = 0. 2 2 In our notation, and under the assumption ε < ∞ , D(u) is obtained as a weak limit of the sequence {πq }q which is uniformly bounded in L1 (ΩT ). Thus, D(u) is a measure of bounded variation. One can easily show that supp D(u) ⊂ S, but it is unclear whether the topological support of D(u) is contained in the accumulant A or is at all relevant
ANALYTICAL APPROACH TO INTERMITTENCY
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to the intermittent cascade. Instead, we will show that the measuretheoretic support of D(u), in the sense of Hahn decomposition, can in fact be described by limits of algebraic progressions of the active regions Aq in a sense defined in Section 6. However, due to a possible slow convergence rate of such a limit, D(u) may not enjoy the same dimensional bound as A itself. 2. Phenomenology of active volumes 2.1. Dimensional Fourier analysis on a periodic box. Let B(0, r) denote the ball centered at 0 of radius r in R3 . We fix a nonnegative radial function χ belonging to C0∞ (B(0, 1)) such that χ(ξ) = 1 for |ξ| ≤ 1/2. We define φ(ξ) = χ(ξ/2) − χ(ξ) and let Z φ(ξ)e2πiξ·x dx, h(x) = R3
the usual inverse Fourier transform of φ on R3 . We now introduce a dimensional dyadic decomposition. Let L > 0 be fixed length scale q and P let ϕ−1 (ξ) = χ(Lξ) and ϕq (ξ) = φ(Lξ/2 ) for q ≥ 0. We have q≥−1 ϕq (ξ) = 1, and (6)
supp ϕq ∩ supp ϕp = ∅ for all |p − q| ≥ 2.
Let T3L denote the torus of side length L. Let us denote by F the Fourier transform on T3L : Z 1 1 u(x)e−2πik·x dx, k ∈ Z3 , F(u)(k) = 3 L T3L L then F −1 (f )(x) =
X
f (k)e2πik·x
1 3 k∈ L Z
determines the inverse transform. We define uq = F −1 (ϕq F(u)). Here ϕq must be understood as restricted on the lattice L1 Z3 . In particR ular, u−1 (x) = L13 T3 u(y)dy. It is informative to note the formula for L hq = F −1 ϕq , q ≥ 0, which can be derived via the standard transference argument: X 2q x 3q q (7) hq (x) = 2 h + 2 n , x ∈ T3L . L 3 n∈Z
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A. CHESKIDOV AND R. SHVYDKOY
Then for all r ∈ [1, ∞] we have !1/r Z 1 (8) khq kr = |hq (x)|r dx ∼ 23q(r−1)/r 3 L T3L which is independent of the scale. From (8) follow the classical Bernstein’s inequalities, which are also scaling invariant: (9)
1
1
kuq kr00 ≤ c23q( r0 − r00 ) kuq kr0
for all 1 ≤ r0 < r00 ≤ ∞; and the differential Bernsteins’s inequalities: (10)
k∂ β uq kr ∼ (2q /L)|β| kuq kr ,
q ≥ 0.
We will use the special notation λq = 2q /L for the dimensional wavenumbers in order to distinguish them from the a-dimensional ones 2q . The corresponding dyadic length scale is denoted `q = λ−1 q . 2.2. Active volumes. We assume that a turbulent fluid fills a periodic box of linear dimension L. Let us denote it by T3L . The motion of the fluid is driven by an external stirring force f , which we assume to be time independent and of scale ηf ∼ L. More specifically, supp F(f ) ⊂ B(0, c/L) for some a-dimensional c > 1. We observe the fluid on a time interval [0, T ] long enough to capture the necessary statistics. Let us envision that the turbulent motion of the fluid at scale λq consists of actively interacting eddies and these eddies fill a region of volume Vq , which we also call active. Let us now proceed using the classical phenomenological argument, giving every concept its Littlewood-Paley analogue. Our immediate goal will be to derive the following explicit formula for Vq : (11)
Vq = L3
h|uq |2 i3 . h|uq |3 i2
Here the bracket h·i denote the average in space-time on the domain ΩT = [0, T ] × T3L . Let us start by noticing that since ϕq has mean zero, for q ≥ 0, we can prescribe the turn-over velocity of an `q -eddy at location x to be uq (x). Let Uq be a characteristic velocity of an `q -eddy. In order to single out active eddies from passive ones, one will use an Lp -average such as Uq ∼ h|uq |p i1/p . The minimal value of p proved to be suitable for studying intermittent cascade turns out to be 3 (see Section 4), while higher values can be adopted to formulate multi-fractal hypotheses as shown in Section 4.5. So, we define L (12) Uq = 1/3 h|uq |3 i1/3 , Vq
ANALYTICAL APPROACH TO INTERMITTENCY
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or more explicitly, (13)
Uq =
1 T Vq
Z
3
1/3
|uq (x, t)| dxdt
,
ΩT
Even though the integration in (13) is performed over the entire domain, we will prove later that there exists indeed an active region Aq ⊂ ΩT with |Aq | . T Vq , and Z Z 3 (14) |uq | dxdt ∼ |uq |3 dxdt. Aq
ΩT
The input energy produced by the force f is passing from larger to smaller scales. The energy flux per unit volume carried through the scales of order `q is given by (15)
Uq3 L3 = h|uq |3 i. εq = `q `q Vq
On the other hand, (16)
εq =
Kq tq
where Kq is the average kinetic energy of an active `q -eddy given by (17)
Kq =
L3 h|uq |2 i, Vq
(here again we use the active proportion of the volume) and tq is the typical turnover time given by 1/3
(18)
`q `q Vq tq = = . Uq Lh|uq |3 i1/3
Putting together (17) and (18) we obtain another expression for εq : (19)
εq =
L4 4/3 `q Vq
h|uq |2 ih|uq |3 i1/3 .
Equating (15) and (19) we finally obtain (11). 2.3. Intermittent dimension. We define the intermittency dimension as follows log2 (L3 /Vq ) dq = 3 − , q (20) d = lim sup dq . q→∞
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A. CHESKIDOV AND R. SHVYDKOY
Observe that all volumes are bounded by L3 . Therefore we have dq ≤ 3 for all q ≥ 0. The dimension d is defined without any scaling hypothesis like in the β-model and therefore is a property of the field. 3. Active regions Let us assume as before that u(x, t) is the velocity field of a turbulent fluid in the periodic box T3L on a time interval [0, T ], and Vq ’s are defined by (11). In this section we will introduce regions that capture most of the active eddies, i.e. eddies with at least a certain turn-over speed sq . In order to properly define the speed and the regions we will make use of atomic decompositions. Let us briefly recall the definition (for more, see [1]). For q ≥ 0 and k ∈ [0, 2q − 1]3 ∩ Z3 we define the dyadic cubes Qqk = [0, `q )3 + `q k ⊂ T3L , and dilated cubes [
Q∗qk =
Qqk0 .
|k0 −k| 1 one can find a spacial decomposition of u(t, ·) into M -atoms (21)
u(t, x) =
∞ X
X
sqk (t)aqk (t, x),
q=0 k∈[0,2q −1]3 ∩Z3
where supp(aqk (t, ·)) ⊂ Q∗qk compactly, and max λ−|β| k∂xβ aqk kL∞ (ΩT ) ≤ 1. q |β|≤M
For every r < ∞ one has X 1 (22) λ3q q
1/r |sqk (t)|r
∼ kuq (t, ·)kr .
k∈[0,2 −1]3
Let us recall how the coefficients sqk are constructed. Let χqk be the characteristic function of Qqk and let η be an approximative kernel supported on B(0, 1). Let ηq (x) = 23q η(λq x), χ0qk = χqk ? ηq and bqk = uq χ0qk . Then let sqk (t) = max λ−|β| k∂xβ bqk (t, ·)k∞ q |β|≤M
aqk = bqk /sqk .
ANALYTICAL APPROACH TO INTERMITTENCY
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Let us now fix any positive a-dimensional decreasing sequence σq → 0 with zero exponential type log σq = 0. q→∞ q lim
Definition 3.1. An active region occupied by eddies of size `p at time t is defined by [ (23) Aq (t) = Q∗qk . k:|sqk (t)|>σq
h|uq |3 i h|uq |2 i
Furthermore, let Aq = {(t, x) : x ∈ Aq (t)}.
(24)
3
q| i Roughly, the threshold speed σq h|u is determined by comparing h|uq |2 i the energy flux of an eddy with the total flux through scales `q . The flux of an eddie averaged over its size is given by εqk ∼ |sqk |3 /`q . Then the selection criterion εqk > σq εq , where εq is defined in (15), gives the desired formula. In the following theorem we collect several properties of active regions that are independent of particular nature of the flow or even the evolution law of the field u. For a set A ⊂ ΩT , let A(t) = {x ∈ T3L : (x, t) ∈ A}.
Theorem 3.2. Let Aq be the active region defined by (23), and let A = lim sup Aq = ∩∞ p=1 ∪q>p Aq .
(25)
q→∞
Then for some absolute constant c > 0 we have |Aq | ≤ cσq−3 Vq T, Z Z 3 3 |uq | dxdt ≤ |u| dxdt ≤
(26) Z (27)
(1 − cσq ) ΩT
(28)
Aq
|uq |3 dxdt,
ΩT
dimH A(t) ≤ d, for a.e. t ∈ [0, T ].
In particular, if d < 0 then A(t) = ∅ for a.e. t. We thus see that the active regions are regions where most of the characteristic velocity Uq is concentrated, yet the measure of Aq on average does not exceed Vq up to an algebraic multiple. Inequality (28) confirms the dimensional prediction of the β-model, and we will argue in the next section that A indeed represents the set where the energy flux accumulates.
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A. CHESKIDOV AND R. SHVYDKOY
Proof. Let Nq (t) be the number of cubes in the union (23). Then trivially 27 |Aq (t)| ≤ 3 Nq (t). λq On the other hand, [ 1 Qqk ≤ |Aq (t)|. Nq (t) = 3 λq q |3 i k:|sqk (t)|>σq h|u h|uq |2 i Integrating in time we obtain Z T Z 1 27 T (29) Nq (t)dt ≤ |Aq | ≤ 3 Nq (t)dt. λ3q 0 λq 0 In view of (22) we have Z Z TX Z σq3 h|uq |3 i3 T 1 3 3 |u| dxdt ∼ 3 |sqk (t)| dt ≥ 3 Nq (t) dt. λq 0 k λq h|uq |2 i3 0 ΩT Thus, in view of (29), h|uq |2 i3 cσq−3 h|uq |3 i3
|Aq | ≤
Z
|u|3 dxdt = cσq−3 Vq T,
ΩT
as claimed. Now let us observe that on the complement Acq = ΩT \A, we have 3
h|uq | i |u| ≤ cσq h|u 2 . We therefore obtain q| i
Z
3
R
Z
|u| dxdt = Ac
ΩT
2
Ac
|u| |u|dxdt ≤ σq R
ZΩT ≤ σq
|u|3 dxdt
Z
|u|2 dxdt
|u|2 dxdt
Ac
|u|3 dxdt,
ΩT
which proves (27). Let us first recall that the Hausdorff dimension of a set A ⊂ T3L , dimH (A) is the smallest d for which the Hausdorff measure vanishes Hd+δ (A) = lim Hd+δ,ε (A) = 0, ε→0
for all δ > 0, where Hd,ε (A) = inf
(∞ X i=1
) (diam Ai )d : A ⊂ ∪i Ai , diam Ai < ε .
ANALYTICAL APPROACH TO INTERMITTENCY
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Directly from (26) we have Z 1 T (30) Nq (t) dt . σq−3 λ3q Vq . T 0 Given δ > 0, we note that Vq (31) ≤ 2−q(h−δ/2) , L3 for all q large enough. Let us now assume that d ≥ 0. Then X 1 Hd+δ,`q (A(t)) ≤ Hd+δ,`q (∪p>q Ap (t)) ≤ Np (t) d+δ . λp p>q Integrating in time, using (30), (31) and the fact that σq → 0 algebraically, we obtain Z T X Ld 2δ/2 X Vp ≤ T → 0, Hd+δ,`q (A(t))dt ≤ T 3 −h+δ σp3 λδp 0 p>q p>q σp λp as q → ∞. So, in the limit we obtain Z T Hd+δ (A(t))dt = 0. 0
Hence, dimH A(t) ≤ d + δ for a.e. t ∈ [0, T ], which concludes the proof. If d < 0, then (30) and (31) imply Z TX Np (t)dt . σq−3 2−qδ/2 , 0
p>q
for q large enough. Thus the measure ( ) X Np (t) ≥ 1 ≤ σq−3 2−qδ/2 , t: p>q
and on the complement the set Aq (t) is empty. Passing to the limit we conclude that A(t) is empty a.e. 4. Energy dissipation set Let us consider the Euler equations in the periodic box T3L : (32) (33)
∂u + (u · ∇)u = −∇p + f, ∂t ∇ · u = 0.
A vector field u ∈ Cw ([0, T ]; L2 (T3L )), (the space of weakly continuous functions), is a weak solution of the Euler equations with initial data
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A. CHESKIDOV AND R. SHVYDKOY
u0 ∈ L2 (T3L ) if for every ψ ∈ C0∞ ([0, T ] × T3L ) with ∇x · ψ = 0 and 0 ≤ t ≤ T , we have Z Z Z Z Z (34) u·ψ− u0 ·ψ− u·∂s ψ = (u⊗u) : ∇ψ+ f ·ψ, T3L ×{t}
T3L ×{0}
Ωt
Ωt
Ωt
and ∇x · u(t) = 0 in the sense of distributions. We define the operation : by A : B = Tr[AB]. 4.1. Energy flux and density. In order to properly define the flux of kinetic energy across the scales, let us fix a q ≥ 1 and test (32) against the filtered field u
1. Introduction Intermittency, if viewed as a measure of non-uniformity of the energy cascade, bears its signature on many statistical laws of a fully developed turbulence. A typical example is the power law for the energy density function, 2 ε 3 κ0 1− d3 (1) E(κ) ∼ 5 , κ3 κ where ε is the energy dissipation rate per unit mass, κ0 the integral scale, and d is the dimension of a set that carries turbulent energy dissipation. One observes similar corrections for longitudinal structure functions or skewness and flatness factors of velocity gradients (see the texts [15, 19, 18] for detailed discussions). Due to intermittency the deterministic counterparts of the power laws appear in a form of bounds to account for possible corrections as in Constantin et al [9, 6, 7], even though the intermittency itself has not been mathematically defined. As Foias remarked [13], “The rigorous mathematical reformulation of the heuristic estimates provided by some widely accepted turbulence theories always seem to represent the upper bound of the rigorous mathematical estimates.” At present there is no compelling analytical evidence that the fluid equations are consistent with the full range Date: February 7, 2012. 2000 Mathematics Subject Classification. Primary: 76F02; Secondary: 35Q31. The work of A. Cheskidov is partially supported by NSF grant DMS–1108864. R. Shvydkoy acknowledges the support of NSF grant DMS–0907812. 1
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A. CHESKIDOV AND R. SHVYDKOY
of intermittent dimensions 0 ≤ d ≤ 3 (or with d < 0). On the other hand, experimental evidence points only to moderate deviations from the classical Kolmogorov theory. Based on several subjective factors, such as the overwhelming success of widely used hot wire anemometry, or high incidence rates of linear cross-sections with (2+)D fractals (see [17]), one is lead to believe that the values of d above 2 are more likely to occur in real turbulence. For Leray-Hopf solutions of the Navier-Stokes equation this question is directly related to the regularity problem for the following reason. If d is defined as a saturation level of Bernstein’s inequality at the upper end of the inertial range (in a proper sense) then, as shown in [3], a solution with dimension d above 3/2 is automatically regular. We note that such an interpretation of intermittency also appeared in a derivation of the dyadic [4] and continuous [5] models of the energy cascade. Without implementation of viscous dissipation the statistical theory offers examples of models where the dimension is introduced as a rarefaction exponent of Richardson’s cascade (see [14, 20, 17]). Let us briefly recall the β-model of Frisch, Sulem, and Nelkin [14]. Assume that a fluid is in a state of turbulent motion confined to the periodic box T3L = [0, L]3 and observed on a long interval of time [0, T ]. On each stage of the cascade the kinetic energy is carried through by a set of active eddies {eq }. Let the eddied fill a volume Vq . If one assumes that Vq is a fraction of the preceding volume, i.e. Vq ∼ βVq−1 , for some 0 < β ≤ 1, then going up the scales, Vq ∼ β q L3 . Letting h = − log2 β, we have Vq ∼ 2−qh L3 and h ≥ 0. A heuristic incidence argument shows that the ultimate accumulation of the cascade occurs on a set that crosses infinitely many generations of eddies, i.e. A = lim supq→∞ ∪eq , and the dimension of the accumulant A does not exceed d = 3 − h. A simple dimensional analysis recovers the energy spectrum law (1). Further hypothesis of a statistical uni-scaling of velocity displacements δu(`) on A made the model unsuitable to capture experimentally observed exponents for higher order structure functions. This lead to subsequent multi-fractal ramifications of Frisch and Parisi (see [22, 12, 15]). Our objective in this paper is to define the above mentioned empirical concepts for deterministic fields, and use them to justify intermittency corrections to statistical laws without the use of viscous dissipation. We outline our basic definitions and results. Let u ∈ L2 ([0, T ] × T3L ) be a divergence free time dependent vector field. P We will use the LittlewoodPaley decomposition of u given by u = q≥−1 uq (see Section 2.1). We interpret uq as the collective velocity field of all eddies eq of dyadic size
ANALYTICAL APPROACH TO INTERMITTENCY
3
`q = L/2q . By bridging relations between the Euleran and Lagrangian description of the energy flux through scales `q we define the active volume as h|uq |2 i3 (2) Vq = L3 . h|uq |3 i2 Here and throughout the bracket h·i denotes the average over the spacetime domain ΩT = [0, T ] × T3L . The dimension d is then introduced as the following exponential co-type of the sequence {Vq }: log2 (L3 /Vq ) . q→∞ q Note that d is a number ”attached” to the field u without any assumptions on u. By the H¨older inequality we have one obvious bound Vq ≤ L3 , and in the stationary case Bernstein’s inequality implies Vq ≥ c`3q , where c is a-dimensional. The latter implies d ≥ 0, which is a natural bound. However, in the time-dependent case, Vq can scale down faster due to possible temporal rarefaction of active regions giving negative values of the dimension d (see also a discussion in Frisch [15]). To define active regions that fill volume Vq we find that for an eddie to qualify as “active” it has to have speed at least proportional to (3)
d = 3 − lim inf
sq ∼
h|uq |3 i . h|uq |2 i
The speeds will be interpreted as magnitudes of coefficients in the atomic decomposition of uq , and the supports of atoms will be viewed as eddies. The collection of those eddies will constitute what we call an “active” region Aq . The number of active eddies is represented by Vq /`3q . Our Theorem 3.2 shows that the Hausdorff dimension of A = lim sup Aq q→∞
is not to exceed d as predicted by the model, and Aq supports most of the “active” part of the field uq in terms of L3 -averages. We will recover the energy law (1) in the form of an upper and lower bounds. The scaling of the second order structure function is as predicted by the β-model " d 43 # 1− 3 −δ 2 2 ` ` (4) S2 (`) . ε 3 ` 3 + Cδ . L L for all ` < L, and any δ > 0. Here and throughout, . denotes inequality up to a universal adimensional constant. Note that the intermittency
4
A. CHESKIDOV AND R. SHVYDKOY
correction exponent 1 − d3 is the same as in the energy law (1), however (4) shows that it is prone to deviations into smaller range. This phenomenon has been observed in direct numerical simulations as well by Kaneda et al in [16]. In Section 4 we stipulate on the role of A as an accumulant of the energy cascade for weak solutions to the Euler equations. We define the energy flux through scales `q by Πq = hπq i where density πq is the trilinear term that contains all relevant interactions involved in the energy transfer. Due to a localization of Πq observed in [2] all appreciable interactions are frequency-local near the `q scales. We show in Theorem 4.3 that the complement of A takes passive part in the cascade process in the sense that there is a nested sequence of sets Gp → A, in fact Gp = ∪k>p Ak , such that for all p > 0 lim h|πq |ΩT \Gp |i = 0.
q→∞
In other words, the turbulent cascade tends to dump the energy on the carrier A. The results of this section are established under the natural Onsager regularity condition ε = lim supq→∞ h|uq |3 i/`q < ∞ which is known to be suitable for the turbulent interpretation of the field (see [2, 8, 24, 10, 21, 11]). Any better regularity of u, namely simply requiring ε = 0, erases turbulent energy dissipation. This lack of dissipation may also occur locally in a proper sense. Therefore it is necessary to further confine the cascade to the smaller set A ∩ S, where S is the Onsager singular set of u, i.e. S is the smallest set on the complement of which ε = 0. This reduction is performed in Section 4.3. We will demonstrate by several examples that the regularity-related intermittency due to S and the cascade-related intermittency due to A are essentially independent. Therefore one has to consider both to have a more complete description of the accumulant. The active regions Aq can be used to give a more detailed description of a measure-theoretic support of the anomalous energy dissipation introduced by Duchon and Robert in [10]. Recall that for every weak solution u ∈ L3 (ΩT ) to the Euler equation one can associate a distribution D(u) such that 1 2 1 2 (5) ∂t |u| + div u |u| + p + D(u) = 0. 2 2 In our notation, and under the assumption ε < ∞ , D(u) is obtained as a weak limit of the sequence {πq }q which is uniformly bounded in L1 (ΩT ). Thus, D(u) is a measure of bounded variation. One can easily show that supp D(u) ⊂ S, but it is unclear whether the topological support of D(u) is contained in the accumulant A or is at all relevant
ANALYTICAL APPROACH TO INTERMITTENCY
5
to the intermittent cascade. Instead, we will show that the measuretheoretic support of D(u), in the sense of Hahn decomposition, can in fact be described by limits of algebraic progressions of the active regions Aq in a sense defined in Section 6. However, due to a possible slow convergence rate of such a limit, D(u) may not enjoy the same dimensional bound as A itself. 2. Phenomenology of active volumes 2.1. Dimensional Fourier analysis on a periodic box. Let B(0, r) denote the ball centered at 0 of radius r in R3 . We fix a nonnegative radial function χ belonging to C0∞ (B(0, 1)) such that χ(ξ) = 1 for |ξ| ≤ 1/2. We define φ(ξ) = χ(ξ/2) − χ(ξ) and let Z φ(ξ)e2πiξ·x dx, h(x) = R3
the usual inverse Fourier transform of φ on R3 . We now introduce a dimensional dyadic decomposition. Let L > 0 be fixed length scale q and P let ϕ−1 (ξ) = χ(Lξ) and ϕq (ξ) = φ(Lξ/2 ) for q ≥ 0. We have q≥−1 ϕq (ξ) = 1, and (6)
supp ϕq ∩ supp ϕp = ∅ for all |p − q| ≥ 2.
Let T3L denote the torus of side length L. Let us denote by F the Fourier transform on T3L : Z 1 1 u(x)e−2πik·x dx, k ∈ Z3 , F(u)(k) = 3 L T3L L then F −1 (f )(x) =
X
f (k)e2πik·x
1 3 k∈ L Z
determines the inverse transform. We define uq = F −1 (ϕq F(u)). Here ϕq must be understood as restricted on the lattice L1 Z3 . In particR ular, u−1 (x) = L13 T3 u(y)dy. It is informative to note the formula for L hq = F −1 ϕq , q ≥ 0, which can be derived via the standard transference argument: X 2q x 3q q (7) hq (x) = 2 h + 2 n , x ∈ T3L . L 3 n∈Z
6
A. CHESKIDOV AND R. SHVYDKOY
Then for all r ∈ [1, ∞] we have !1/r Z 1 (8) khq kr = |hq (x)|r dx ∼ 23q(r−1)/r 3 L T3L which is independent of the scale. From (8) follow the classical Bernstein’s inequalities, which are also scaling invariant: (9)
1
1
kuq kr00 ≤ c23q( r0 − r00 ) kuq kr0
for all 1 ≤ r0 < r00 ≤ ∞; and the differential Bernsteins’s inequalities: (10)
k∂ β uq kr ∼ (2q /L)|β| kuq kr ,
q ≥ 0.
We will use the special notation λq = 2q /L for the dimensional wavenumbers in order to distinguish them from the a-dimensional ones 2q . The corresponding dyadic length scale is denoted `q = λ−1 q . 2.2. Active volumes. We assume that a turbulent fluid fills a periodic box of linear dimension L. Let us denote it by T3L . The motion of the fluid is driven by an external stirring force f , which we assume to be time independent and of scale ηf ∼ L. More specifically, supp F(f ) ⊂ B(0, c/L) for some a-dimensional c > 1. We observe the fluid on a time interval [0, T ] long enough to capture the necessary statistics. Let us envision that the turbulent motion of the fluid at scale λq consists of actively interacting eddies and these eddies fill a region of volume Vq , which we also call active. Let us now proceed using the classical phenomenological argument, giving every concept its Littlewood-Paley analogue. Our immediate goal will be to derive the following explicit formula for Vq : (11)
Vq = L3
h|uq |2 i3 . h|uq |3 i2
Here the bracket h·i denote the average in space-time on the domain ΩT = [0, T ] × T3L . Let us start by noticing that since ϕq has mean zero, for q ≥ 0, we can prescribe the turn-over velocity of an `q -eddy at location x to be uq (x). Let Uq be a characteristic velocity of an `q -eddy. In order to single out active eddies from passive ones, one will use an Lp -average such as Uq ∼ h|uq |p i1/p . The minimal value of p proved to be suitable for studying intermittent cascade turns out to be 3 (see Section 4), while higher values can be adopted to formulate multi-fractal hypotheses as shown in Section 4.5. So, we define L (12) Uq = 1/3 h|uq |3 i1/3 , Vq
ANALYTICAL APPROACH TO INTERMITTENCY
7
or more explicitly, (13)
Uq =
1 T Vq
Z
3
1/3
|uq (x, t)| dxdt
,
ΩT
Even though the integration in (13) is performed over the entire domain, we will prove later that there exists indeed an active region Aq ⊂ ΩT with |Aq | . T Vq , and Z Z 3 (14) |uq | dxdt ∼ |uq |3 dxdt. Aq
ΩT
The input energy produced by the force f is passing from larger to smaller scales. The energy flux per unit volume carried through the scales of order `q is given by (15)
Uq3 L3 = h|uq |3 i. εq = `q `q Vq
On the other hand, (16)
εq =
Kq tq
where Kq is the average kinetic energy of an active `q -eddy given by (17)
Kq =
L3 h|uq |2 i, Vq
(here again we use the active proportion of the volume) and tq is the typical turnover time given by 1/3
(18)
`q `q Vq tq = = . Uq Lh|uq |3 i1/3
Putting together (17) and (18) we obtain another expression for εq : (19)
εq =
L4 4/3 `q Vq
h|uq |2 ih|uq |3 i1/3 .
Equating (15) and (19) we finally obtain (11). 2.3. Intermittent dimension. We define the intermittency dimension as follows log2 (L3 /Vq ) dq = 3 − , q (20) d = lim sup dq . q→∞
8
A. CHESKIDOV AND R. SHVYDKOY
Observe that all volumes are bounded by L3 . Therefore we have dq ≤ 3 for all q ≥ 0. The dimension d is defined without any scaling hypothesis like in the β-model and therefore is a property of the field. 3. Active regions Let us assume as before that u(x, t) is the velocity field of a turbulent fluid in the periodic box T3L on a time interval [0, T ], and Vq ’s are defined by (11). In this section we will introduce regions that capture most of the active eddies, i.e. eddies with at least a certain turn-over speed sq . In order to properly define the speed and the regions we will make use of atomic decompositions. Let us briefly recall the definition (for more, see [1]). For q ≥ 0 and k ∈ [0, 2q − 1]3 ∩ Z3 we define the dyadic cubes Qqk = [0, `q )3 + `q k ⊂ T3L , and dilated cubes [
Q∗qk =
Qqk0 .
|k0 −k| 1 one can find a spacial decomposition of u(t, ·) into M -atoms (21)
u(t, x) =
∞ X
X
sqk (t)aqk (t, x),
q=0 k∈[0,2q −1]3 ∩Z3
where supp(aqk (t, ·)) ⊂ Q∗qk compactly, and max λ−|β| k∂xβ aqk kL∞ (ΩT ) ≤ 1. q |β|≤M
For every r < ∞ one has X 1 (22) λ3q q
1/r |sqk (t)|r
∼ kuq (t, ·)kr .
k∈[0,2 −1]3
Let us recall how the coefficients sqk are constructed. Let χqk be the characteristic function of Qqk and let η be an approximative kernel supported on B(0, 1). Let ηq (x) = 23q η(λq x), χ0qk = χqk ? ηq and bqk = uq χ0qk . Then let sqk (t) = max λ−|β| k∂xβ bqk (t, ·)k∞ q |β|≤M
aqk = bqk /sqk .
ANALYTICAL APPROACH TO INTERMITTENCY
9
Let us now fix any positive a-dimensional decreasing sequence σq → 0 with zero exponential type log σq = 0. q→∞ q lim
Definition 3.1. An active region occupied by eddies of size `p at time t is defined by [ (23) Aq (t) = Q∗qk . k:|sqk (t)|>σq
h|uq |3 i h|uq |2 i
Furthermore, let Aq = {(t, x) : x ∈ Aq (t)}.
(24)
3
q| i Roughly, the threshold speed σq h|u is determined by comparing h|uq |2 i the energy flux of an eddy with the total flux through scales `q . The flux of an eddie averaged over its size is given by εqk ∼ |sqk |3 /`q . Then the selection criterion εqk > σq εq , where εq is defined in (15), gives the desired formula. In the following theorem we collect several properties of active regions that are independent of particular nature of the flow or even the evolution law of the field u. For a set A ⊂ ΩT , let A(t) = {x ∈ T3L : (x, t) ∈ A}.
Theorem 3.2. Let Aq be the active region defined by (23), and let A = lim sup Aq = ∩∞ p=1 ∪q>p Aq .
(25)
q→∞
Then for some absolute constant c > 0 we have |Aq | ≤ cσq−3 Vq T, Z Z 3 3 |uq | dxdt ≤ |u| dxdt ≤
(26) Z (27)
(1 − cσq ) ΩT
(28)
Aq
|uq |3 dxdt,
ΩT
dimH A(t) ≤ d, for a.e. t ∈ [0, T ].
In particular, if d < 0 then A(t) = ∅ for a.e. t. We thus see that the active regions are regions where most of the characteristic velocity Uq is concentrated, yet the measure of Aq on average does not exceed Vq up to an algebraic multiple. Inequality (28) confirms the dimensional prediction of the β-model, and we will argue in the next section that A indeed represents the set where the energy flux accumulates.
10
A. CHESKIDOV AND R. SHVYDKOY
Proof. Let Nq (t) be the number of cubes in the union (23). Then trivially 27 |Aq (t)| ≤ 3 Nq (t). λq On the other hand, [ 1 Qqk ≤ |Aq (t)|. Nq (t) = 3 λq q |3 i k:|sqk (t)|>σq h|u h|uq |2 i Integrating in time we obtain Z T Z 1 27 T (29) Nq (t)dt ≤ |Aq | ≤ 3 Nq (t)dt. λ3q 0 λq 0 In view of (22) we have Z Z TX Z σq3 h|uq |3 i3 T 1 3 3 |u| dxdt ∼ 3 |sqk (t)| dt ≥ 3 Nq (t) dt. λq 0 k λq h|uq |2 i3 0 ΩT Thus, in view of (29), h|uq |2 i3 cσq−3 h|uq |3 i3
|Aq | ≤
Z
|u|3 dxdt = cσq−3 Vq T,
ΩT
as claimed. Now let us observe that on the complement Acq = ΩT \A, we have 3
h|uq | i |u| ≤ cσq h|u 2 . We therefore obtain q| i
Z
3
R
Z
|u| dxdt = Ac
ΩT
2
Ac
|u| |u|dxdt ≤ σq R
ZΩT ≤ σq
|u|3 dxdt
Z
|u|2 dxdt
|u|2 dxdt
Ac
|u|3 dxdt,
ΩT
which proves (27). Let us first recall that the Hausdorff dimension of a set A ⊂ T3L , dimH (A) is the smallest d for which the Hausdorff measure vanishes Hd+δ (A) = lim Hd+δ,ε (A) = 0, ε→0
for all δ > 0, where Hd,ε (A) = inf
(∞ X i=1
) (diam Ai )d : A ⊂ ∪i Ai , diam Ai < ε .
ANALYTICAL APPROACH TO INTERMITTENCY
11
Directly from (26) we have Z 1 T (30) Nq (t) dt . σq−3 λ3q Vq . T 0 Given δ > 0, we note that Vq (31) ≤ 2−q(h−δ/2) , L3 for all q large enough. Let us now assume that d ≥ 0. Then X 1 Hd+δ,`q (A(t)) ≤ Hd+δ,`q (∪p>q Ap (t)) ≤ Np (t) d+δ . λp p>q Integrating in time, using (30), (31) and the fact that σq → 0 algebraically, we obtain Z T X Ld 2δ/2 X Vp ≤ T → 0, Hd+δ,`q (A(t))dt ≤ T 3 −h+δ σp3 λδp 0 p>q p>q σp λp as q → ∞. So, in the limit we obtain Z T Hd+δ (A(t))dt = 0. 0
Hence, dimH A(t) ≤ d + δ for a.e. t ∈ [0, T ], which concludes the proof. If d < 0, then (30) and (31) imply Z TX Np (t)dt . σq−3 2−qδ/2 , 0
p>q
for q large enough. Thus the measure ( ) X Np (t) ≥ 1 ≤ σq−3 2−qδ/2 , t: p>q
and on the complement the set Aq (t) is empty. Passing to the limit we conclude that A(t) is empty a.e. 4. Energy dissipation set Let us consider the Euler equations in the periodic box T3L : (32) (33)
∂u + (u · ∇)u = −∇p + f, ∂t ∇ · u = 0.
A vector field u ∈ Cw ([0, T ]; L2 (T3L )), (the space of weakly continuous functions), is a weak solution of the Euler equations with initial data
12
A. CHESKIDOV AND R. SHVYDKOY
u0 ∈ L2 (T3L ) if for every ψ ∈ C0∞ ([0, T ] × T3L ) with ∇x · ψ = 0 and 0 ≤ t ≤ T , we have Z Z Z Z Z (34) u·ψ− u0 ·ψ− u·∂s ψ = (u⊗u) : ∇ψ+ f ·ψ, T3L ×{t}
T3L ×{0}
Ωt
Ωt
Ωt
and ∇x · u(t) = 0 in the sense of distributions. We define the operation : by A : B = Tr[AB]. 4.1. Energy flux and density. In order to properly define the flux of kinetic energy across the scales, let us fix a q ≥ 1 and test (32) against the filtered field u
