Wave Turbulence and the Phillips Spectrum

Transcription

1 Wave Turbulence and the Phillips Spectrum Alan Newell Department of Mathematics University of Arizona September 29, 2008

2 The role of the generalized Phillips spectrum in wave turbulence 1 What is wave turbulence? 2 Why is it generally believed to be a solved problem? 3 Why in fact the theory is incomplete and the story is far from over. 4 The role that the generalized Phillips spectrum might play in completing the theory.

3 The role of the generalized Phillips spectrum in wave turbulence 1 What is wave turbulence? 2 Why is it generally believed to be a solved problem? 3 Why in fact the theory is incomplete and the story is far from over. 4 The role that the generalized Phillips spectrum might play in completing the theory.

4 The role of the generalized Phillips spectrum in wave turbulence 1 What is wave turbulence? 2 Why is it generally believed to be a solved problem? 3 Why in fact the theory is incomplete and the story is far from over. 4 The role that the generalized Phillips spectrum might play in completing the theory.

5 The role of the generalized Phillips spectrum in wave turbulence 1 What is wave turbulence? 2 Why is it generally believed to be a solved problem? 3 Why in fact the theory is incomplete and the story is far from over. 4 The role that the generalized Phillips spectrum might play in completing the theory.

6 The role of the generalized Phillips spectrum in wave turbulence 1 What is wave turbulence? 2 Why is it generally believed to be a solved problem? 3 Why in fact the theory is incomplete and the story is far from over. 4 The role that the generalized Phillips spectrum might play in completing the theory.

7 What is wave turbulence? Wave Turbulence is the study of the long time statistical behaviour of solutions of nonlinear field equations, usually conservative and Hamiltonian, describing a sea of weakly nonlinear interacting dispersive waves. In most interesting contexts, the system is nonisolated having both sources and sinks of energy and other conserved densities.

8 Why is it believed to be a solved problem? A natural asymptotic closure A closed kinetic equation for the particle or wave action density n k from which all other quantities of interest such as nonlinear frequency modification, higher order cumulants, spatial structure functions can be calculated. In addition to the usual thermodynamic stationary solutions, the kinetic equation has finite flux (Kolmogorov-Zakharov) solutions which capture the flow of conserved densities (energy, waveaction) from sources to sinks.

9 Why is it believed to be a solved problem? A natural asymptotic closure A closed kinetic equation for the particle or wave action density n k from which all other quantities of interest such as nonlinear frequency modification, higher order cumulants, spatial structure functions can be calculated. In addition to the usual thermodynamic stationary solutions, the kinetic equation has finite flux (Kolmogorov-Zakharov) solutions which capture the flow of conserved densities (energy, waveaction) from sources to sinks.

10 Why is it believed to be a solved problem? A natural asymptotic closure A closed kinetic equation for the particle or wave action density n k from which all other quantities of interest such as nonlinear frequency modification, higher order cumulants, spatial structure functions can be calculated. In addition to the usual thermodynamic stationary solutions, the kinetic equation has finite flux (Kolmogorov-Zakharov) solutions which capture the flow of conserved densities (energy, waveaction) from sources to sinks.

11 The theory in outline The variables u s (x, t)(e.g. η(x, t), ϕ(x, z = η(x, t), t)) A s k; dim k = d. The governing equations da s k isω k A s k = dt r=2 εr 1 L ss 1...s r kk 1...k r A s 1 k 1... A sr k r δ(k k r k)dk 1... dk r.

12 The theory in outline The variables u s (x, t)(e.g. η(x, t), ϕ(x, z = η(x, t), t)) A s k; dim k = d. The governing equations da s k isω k A s k = dt r=2 εr 1 L ss 1...s r kk 1...k r A s 1 k 1... A sr k r δ(k k r k)dk 1... dk r.

13 The theory in outline The statistics < A s ka s k >= δ(k + k )N s k 2 point correlation {A s ka s k... } higher order cumulants

14 The theory in outline 2 The strategy 1 Form BBGKY hierarchy. 2 Solve iteratively in power series in ε. 3 Choose slow variations of leading approximations to make 2. uniformly valid for long times. The outcome (N k (t) = n k (t) + corrections + ) dn k dt = ε 2 T 2 [n k ] + ε 4 T 4 [n k ] + ε 6 T 6 [n k ] + sω k sω k + ε 2 Ω s 2[n k ] +

15 The theory in outline 2 The strategy 1 Form BBGKY hierarchy. 2 Solve iteratively in power series in ε. 3 Choose slow variations of leading approximations to make 2. uniformly valid for long times. The outcome (N k (t) = n k (t) + corrections + ) dn k dt = ε 2 T 2 [n k ] + ε 4 T 4 [n k ] + ε 6 T 6 [n k ] + sω k sω k + ε 2 Ω s 2[n k ] +

16 The theory in outline 2 The strategy 1 Form BBGKY hierarchy. 2 Solve iteratively in power series in ε. 3 Choose slow variations of leading approximations to make 2. uniformly valid for long times. The outcome (N k (t) = n k (t) + corrections + ) dn k dt = ε 2 T 2 [n k ] + ε 4 T 4 [n k ] + ε 6 T 6 [n k ] + sω k sω k + ε 2 Ω s 2[n k ] +

17 The theory in outline 2 The strategy 1 Form BBGKY hierarchy. 2 Solve iteratively in power series in ε. 3 Choose slow variations of leading approximations to make 2. uniformly valid for long times. The outcome (N k (t) = n k (t) + corrections + ) dn k dt = ε 2 T 2 [n k ] + ε 4 T 4 [n k ] + ε 6 T 6 [n k ] + sω k sω k + ε 2 Ω s 2[n k ] +

18 The theory in outline 2 The strategy 1 Form BBGKY hierarchy. 2 Solve iteratively in power series in ε. 3 Choose slow variations of leading approximations to make 2. uniformly valid for long times. The outcome (N k (t) = n k (t) + corrections + ) dn k dt = ε 2 T 2 [n k ] + ε 4 T 4 [n k ] + ε 6 T 6 [n k ] + sω k sω k + ε 2 Ω s 2[n k ] +

19 Ocean gravity waves Hasselmann (1962) T 4 [n k ] = L kk1 k 2 k 3 2 n k n k1 n k2 n k3 ( ) n k n k1 n k2 n k3 δ(k + k 1 k 2 k 3 )δ(ω + ω 1 ω 2 ω 3 )dk 1 dk 2 dk 3

20 Ocean gravity waves

21 Breakdown! The markers of a successful closure are t L t NL = 1 1 dn k ω k n k c 2 P 2/3 k 1 dt g T 6 [n k ], Ω k [n k ] 1/3 k 1/2 T 4 [n k ] w k,..., cp 1 g 1/2 S 4 3S2 2,..., cp 1/3 1 1 S2 2 g 1/2 r 1/2

22 Breakdown! The finite flux solutions of wave turbulence are almost never uniformly valid for all wavenumbers. They almost always fail at very high or very low wavenumbers.

23 The generalized Phillips spectrum (GPS) We (ACN VEZ, Phys. Lett. A 372, (2008)) argue that the GPS can play a central role. It has four important properties. It is the only spectrum for which symmetries of the original governing equations are inherited by the asymptotic statistics. This in fact will serve as the definition of the GPS. It is the unique spectrum on which wave turbulence is uniformly valid at all wavenumbers.

24 The generalized Phillips spectrum (GPS) We (ACN VEZ, Phys. Lett. A 372, (2008)) argue that the GPS can play a central role. It has four important properties. It is the only spectrum for which symmetries of the original governing equations are inherited by the asymptotic statistics. This in fact will serve as the definition of the GPS. It is the unique spectrum on which wave turbulence is uniformly valid at all wavenumbers.

25 The generalized Phillips spectrum (GPS) We (ACN VEZ, Phys. Lett. A 372, (2008)) argue that the GPS can play a central role. It has four important properties. It is the only spectrum for which symmetries of the original governing equations are inherited by the asymptotic statistics. This in fact will serve as the definition of the GPS. It is the unique spectrum on which wave turbulence is uniformly valid at all wavenumbers.

26 The generalized Phillips spectrum (GPS) We (ACN VEZ, Phys. Lett. A 372, (2008)) argue that the GPS can play a central role. It has four important properties. It is the only spectrum for which symmetries of the original governing equations are inherited by the asymptotic statistics. This in fact will serve as the definition of the GPS. It is the unique spectrum on which wave turbulence is uniformly valid at all wavenumbers.

27 The generalized Phillips spectrum (GPS) It is a solution of T 4 [n k ] γ k [n k ] = 0, i.e. a balance of nonlinear transfer and dissipation and independent of energy flux.(i.e. can absorb any excess flux) It is connected with whitecapping and Phillips picture.

28 The generalized Phillips spectrum (GPS) It is a solution of T 4 [n k ] γ k [n k ] = 0, i.e. a balance of nonlinear transfer and dissipation and independent of energy flux.(i.e. can absorb any excess flux) It is connected with whitecapping and Phillips picture.

29 The generalized Phillips spectrum (GPS) It is a solution of T 4 [n k ] γ k [n k ] = 0, i.e. a balance of nonlinear transfer and dissipation and independent of energy flux.(i.e. can absorb any excess flux) It is connected with whitecapping and Phillips picture.

30 Inheriting symmetries

31 Inheriting symmetries Gravity waves: α = 1 2, γ 2 = 7 4, γ 3 = 3, d = 2 n k k 9/2, < η 2 k > k 4. Capillary waves: α = 3 2, γ 2 = 9 4, γ 3 = 3, d = 2 n k k 7/2, < η 2 k > k 4.

32 WT U valid all k: n k k αx, αx =? dn k dt = T 2 [n k ] + T 4 [n k ] + T 6 [n k ] + sω k sω k + Ω s k [n k] + U valid in k? t L t NL = 1 1 dn k ω k n k dt = s. k ω k 1 all k? S 4 3S 2 2 S all r? t L t NL n k = ck αx T 4 ω k n k k 2γ r c 3 k 3αx k α k 2d k α ck αx c 2 k 2(γ 3+d α αx)

33 WT U valid all k: n k k αx, αx =? On GPS, αx = γ 3 + d α, t L t NL k independent On KZ, αx = 2γ d, t L t NL cp 2/3 k 2( γ 3 3 α) P 2/3 k g

34 GPS solves T [n k ] γ[n k ] = 0 From dn k dt = T [n k ] γ[n k ], angle average to obtain de k dt = p(k) k γ k where ω k n k dk = Let RHS=0 in integral sense E k dk, p(k) k =< ω k T [n k ] > k 1 p(k) k dk = p(k 1) = breakdown wavenumber k 1 γ k dk = P all dissipation between k 1 and absorbs flux P

35 GPS solves T [n k ] γ[n k ] = 0 But on n k = Ck αx, p(k 1 ) = C 3 I(x)k 3α(x KZ x) 1 C 3 I(x)k 3α(x KZ x) 1 = P = k γ 3+3α 1 αx = γ 3 + d α, GPS.

36 Connection with whitecaps Consider sea surface dominated by a series of wedge shapes of finite length. < η 2 k > η(x, y) 1 2s e s x e y 2 L 2 1 k 2 L 2 sin 2 ϕ (s 2 + k 2 cos 2 ϕ) 2 e 2

37 Connection with whitecaps For s fixed, average over ϕ k 5, not Phillips!!! But averaging over s, s > k 1,..., π L 2 e k2 L 2 sin 2 ϕ 1 k cos 3 ϕ [π 4 1 k 1 2 tan 1 k cos ϕ 1 2 for kl 1, < η 2 k > k 4, Phillips!!! kk 1 cos ϕ k1 2 + k 2 cos 2 ϕ ]dϕ

38 Connection with whitecaps For s fixed, average over ϕ k 5, not Phillips!!! But averaging over s, s > k 1,..., π L 2 e k2 L 2 sin 2 ϕ 1 k cos 3 ϕ [π 4 1 k 1 2 tan 1 k cos ϕ 1 2 for kl 1, < η 2 k > k 4, Phillips!!! kk 1 cos ϕ k1 2 + k 2 cos 2 ϕ ]dϕ

39 Inverse and direct cascades in ocean waves turbulence... Qualitative picture: whitecapping University of New Mexico Albuquerque,

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43 Inverse and direct cascades in ocean waves turbulence... Surface. With condensate. University of New Mexico Albuquerque,

44 Inverse and direct cascades in ocean waves turbulence... Surface. Without condensate. University of New Mexico Albuquerque,

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