Chapter 10: On the wave turbulence theory for stratified flows in the ocean

This post is based on my joint paper with Leslie M. Smith and Irene M. Gamba.

After the pioneering work of Garrett and Munk, the statistics of oceanic internal gravity waves has become a central subject of research in oceanography. The time evolution of the spectral energy of internal waves in the ocean can be described by a near-resonance wave turbulence equation, of quantum Boltzmann type. In this work, we provide the first rigorous mathematical study for the equation by showing the global existence and uniqueness of strong solutions.

UnderwaterWave1

The study of wave turbulence has obtained spectacular success in the understanding of spectral energy transfer processes in plasmas, oceans, and planetary atmospheres. Wave-wave interactions in continuously stratified fluids have been a fascinating subject of intensive research in the last few decades. In particular, the observation of a nearly universal internal-wave energy spectrum in the ocean, first described by Garrett and Munk plays a very important role in understanding such wave-wave interactions. The existence of a universal spectrum is generally perceived to be the result of nonlinear interactions of waves with different wavenumbers. As the nonlinearity of the underlying primitive equations is quadratic, waves interact in triads. Furthermore, since the linear internal wave dispersion relation can satisfy a three-wave resonance condition, resonant triads are expected to dominate the dynamics for weak nonlinearity. Resonant wave interactions can be characterized by Zakharov kinetic equations. The equations describe, under the assumption of weak nonlinearity, the resonant spectral energy transfer on the resonant manifold, which is a set of wave vectors k, k_1, k_2 satisfying

   k=k_1+k_2,  \omega_k=\omega_{k_1}+\omega_{k_2},

where the frequency \omega is given by a linear the dispersion relation between the wave frequency  \omega and the wavenumber  k. However, it is known that exact resonances defined by  \omega_k=\omega_{k_1}+\omega_{k_2} do not capture some important physical effects, such as energy transfer to non-propagating wave modes with zero frequency, corresponding to generation of anisotropic coherent structures. Some authors have included more physics by allowing near-resonant interactions defined as

  k=k_1+k_2, |\omega_k-\omega_{k_1}-\omega_{k_2}|<\theta(f,k),

where  \theta accounts for broadening of the resonant surfaces and depends on the wave density   f and the wave number  k. When near resonances are included in the dynamics, numerical studies have demonstrated the formation of the anisotropic, non-propagating wave modes in dispersive wave systems relevant to geophysical flows. We consider in this work the following near-resonance turbulence kinetic equation for internal wave interactions in the open ocean

\partial_tf(t,k) + \mu_k f(t,k)  =  Q[f](t,k),  f(0,k)=f_0(k),

in which  f(t,k) is the nonnegative wave density at wavenumber k \in R^d,  d \ge 2. We add   \mu_kf=2\nu|k|^2 f as the viscous damping term, where  \nu is the viscosity coefficient. The equation is a three-wave kinetic one, in which the collision operator is of the form

 Q[f](k) \ = \iint_{\mathbb{R}^{2d}} \Big[ R_{k,k_1,k_2}[f] - R_{k_1,k,k_2}[f] - R_{k_2,k,k_1}[f] \Big] dk_1dk_2

with

R_{k,k_1,k_2} [f]:= |V_{k,k_1,k_2}|^2\delta(k-k_1-k_2)\mathcal{L}_f(\omega_k -\omega_{k_1}-\omega_{k_2})(f_1f_2-ff_1-ff_2),

where we have used the short-hand notation  f = f(t,k) and  f_j = f(t,k_j). The Dirac delta function  \delta(\cdot) ensures the following resonant condition for the wavenumbers: that interactions are between triads with  k = k_1 + k_2. The collision kernel  V_{k,k_1,k_2} we consider in this work is of the form  V_{k,k_1,k_2} = \mathfrak{C}\left({|k||k_1||k_2|}\right)^\frac12, where  \mathfrak{C} is some physical constant. The dispersion law is linear

\omega_k=\sqrt{F^2+\frac{g^2}{\rho_0^2N^2}\frac{|k|^2}{m^2}},

where  F is the Coriolis parameter,  N is the buoyancy frequency,  m is the reference vertical wave number determined from observations, g is the gravitational constant,  \rho_0 is the constant reference value for the density.  The operator   \mathcal{L}_f is defined as

\mathcal{L}_f(\zeta)=\frac{\Gamma_{k,k_1,k_2}}{\zeta^2+\Gamma_{k,k_1,k_2}^2}, with the condition that

 \lim_{\Gamma_{k,k_1,k_2}\to 0}\mathcal{L}_f(\zeta)=\pi\delta(\zeta).

Thus when  \Gamma_{k,k_1,k_2} tends to  0, the equation becomes the following exact resonance collision operator

Q_e[f](k) \ = \ \iint_{\mathbb{R}^{2d}} \Big[ \tilde{R}_{k,k_1,k_2}[f] - \tilde{R}_{k_1,k,k_2}[f] - \tilde{R}_{k_2,k,k_1}[f] \Big] dk_1dk_2

with

 \tilde{R}_{k,k_1,k_2} [f]:= |V_{k,k_1,k_2}|^2\delta(k-k_1-k_2)\delta(\omega_k -\omega_{k_1}-\omega_{k_2})(f_1f_2-ff_1-ff_2).

Moreover, the resonance broadening frequency  \Gamma_{k,k_1,k_2} may be written

 \Gamma_{k,k_1,k_2}=\gamma_k+\gamma_{k_1}+\gamma_{k_2},

where  \gamma_k is computed using a one-loop approximation:

 \gamma_k\backsim \mathfrak{c}|k|^2\int_{\mathbb{R}_+}|k|^2|f(t,|k|)|d|k|,

and  \mathfrak{c} is a physical constant, which can be normalized to be  1. Approximating the integral

 \int_{\mathbb{R}_+}|k|^2|f(t,|k|)|d|k| \approx \int_{\mathbb{R}^3}f(t,k)dk,

we obtain a formula for  \gamma_k that will be used throughout the paper

 \gamma_k=|k|^2\int_{\mathbb{R}^3}f(t,k)dk

Note that the formulation of  \Gamma_{k,k_1,k_2} is given

\Gamma_{k,k_1,k_2}=(|k|^2+|k_1|^2+|k_2|^2)\int_{\mathbb{R}^3}f(t,k)dk.

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