Introduction

Wave-wave interactions in continuously stratified flows have been a subject of intensive research in the last few decades. Of particular importance is the observation of a nearly universal internal-wave energy spectrum in the ocean, first described by Garrett and Munk. It is generally thought that the existence of a universal spectrum is at least partly the result of nonlinear interactions among internal waves. Due to the quadratic nonlinearity of the underlying fluid equations and dispersion relation allowing three-wave resonances, internal waves interact through triads. In the weakly nonlinear regime, the nonlinear interactions among internal waves concentrate on their resonant set, and can be described by a kinetic equation, which assumes the familiar form (3,14,5,7,12,8,6,9,13,1,10,2,4,11):

$$\frac{\partial n_{\bm{p}}}{\partial t} \! = 4\pi \int d \bm{p}_{12} \left( \vert V_{\bm{p}_1,\bm{p}_2}^{\bm{p... ...}} \, \delta_{\omega_{\bm{p}} -\omega_{{\bm{p}_1}}-\omega_{{\bm{p}_2}}} \right.$$

$$\left. - \vert V_{\bm{p}_2,\bm{p}}^{\bm{p}_1}\vert^2\, f^{\bm{p}_... ...p}}} \, \delta_{{\omega_{\bm{p}_1} -\omega_{\bm{p}_2}-\omega_{\bm{p}}}} \right.$$

$$\left. - \, \vert V_{\bm{p},\bm{p}_1}^{\bm{p}_2}\vert^2\, f^{\bm{... ...\, \delta_{{\omega_{\bm{p}_2} -\omega_{\bm{p}}-\omega_{\bm{p}_1}}} \right) \, ,$$

$$\mathrm{with} \quad f^{\bm{p}}_{\bm{p}_1\bm{p}_2} = n_{\bm{p}_1}n_{\bm{p}_2} - n_{\bm{p}}(n_{\bm{p}_1}+n_{\bm{p}_2}) \, .$$ (1)

Here $n_{\bm{p}} = n(\bm{p})$ is a three-dimensional action spectrum (see Eq. (16)) with wavenumber $\bm{p} = (\bm{k}, m),$ i.e. $\bm{k}$ and $m$ are the horizontal and vertical components of $\bm{p}$. The frequency $\omega$ is given by a linear dispersion relation. The wavenumbers are signed variables, while the wave frequencies are always positive. The factor $V^{\bm{p}}_{\bm{p}_1\bm{p}_2}$ is the interaction matrix describing the transfer of wave action among the members of a resonant triad.

Following Kolmogorov's viewpoint of energy cascades in isotropic Navier-Stokes turbulence, one may look for statistically stationary states using scale-invariant solutions to the kinetic equation (1). The solution may occur in an inertial subrange of wavenumbers and frequencies that are far from those where forcing and dissipation act, and also far from characteristic scales of the system, including the Coriolis frequency resulting from the rotation of the Earth, the buoyancy frequency due to stratification and the ocean depth. Under these assumptions, the dispersion relation and the interaction matrix elements are locally scale-invariant. It is natural, therefore, in this restricted domain, to look for self-similar solutions of Eq. (1), which take the form

$$n(\bm{k}, m)= \vert\bm{k}\vert^{-a} \vert m\vert^{-b}.$$ (2)

Values of $a$ and $b$ in the right-hand side of Eq. (1) vanishes identically correspond to steady solutions of the kinetic equation, and hopefully also to statistically steady states of the ocean's wave field. Unlike Kolmogorov turbulence, the exponents which give steady solutions can not be determined by the dimensional analysis alone (see, for example, Ref. polzin-2004). This is the case owing to multiple characteristic length scales in anisotropic systems.

Before seeking steady solutions, however, one should find out whether the improper integrals in the kinetic equation (1) converge. This is related to the question of locality of the interactions: a convergent integral characterizes the physical scenario where interactions of neighboring wavenumbers dominate the evolution of the wave spectrum, while a divergent one implies that distant, nonlocal interactions in the wavenumber space dominate.

It turns out that the internal-wave collision integral diverges for almost all values of $a$ and $b$. In particular, the collision integral has an infra-red (IR) divergence at zero, i.e. $\vert\bm{k}_1\vert$ or $\vert\bm{k}_2\vert$ $\to 0$ and an ultra-violet (UV) divergence at infinity, i.e. $\vert\bm{k}_1\vert$ and $\vert\bm{k}_2\vert$ $\to \infty$. There is only one exception where the integral converges: the segment with $b=0$ and $7/2 < a < 4$. This segment corresponds to wave action independent of vertical wavenumbers, $\partial n / \partial m = 0$. Within this segment we numerically determine a new steady convergent solution to Eq. (1), with

$$n(\bm{k},m) \propto \vert\bm{k}\vert^{-3.7}.$$ (3)

This solution is not far from the large-wavenumber form of the Garrett-Munk (GM) spectrum:

Alternatively, one can explore the physical interpretation of divergent solutions. We find a region in $(a,b)$ space where there are both IR and UV divergences having opposite signs. This suggests a possible scenario where the two divergent contributions may cancel each other, yielding a steady state. An example of such a case is provided by the Pelinovsky-Raevsky (PR) spectrum,

$$n_{\bm{k},m} \propto \vert\bm{k}\vert^{-7/2} \vert m\vert^{-1/2} .$$ (5)

This solution, however, is only one among infinitely many. The problem at hand is a generalization of the concept of principal value integrals: for $a$ and $b$ which give opposite signs of the divergences at zero and infinity, one can regularize the integral by cutting out small neighborhoods of the two singularities in such a way that the divergences cancel each other and the remaining contributions are small. Hence all the exponents which yield opposite-signed divergence at the both ends can be steady solutions of Eq. (1). As we will see below this general statement helps to describe the experimental oceanographic data which are available to us. The nature of such steady solutions depends on the particular truncation of the divergent integrals.

So far, we have kept the formalism at the level of the self-similar limit of the kinetic equation (1). However, once one considers energy transfer mechanisms dominated by interactions with extreme modes of the system, one can no longer neglect the deviations from self-similarity near the spectral boundaries: the inertial frequency due to the rotation of the Earth at the IR end, and the buoyancy frequency and/or dissipative cut-offs at the UV end.

For example, we may consider a scenario in which interactions with the smallest horizontal wavenumbers dominate the energy transfer within the inertial subrange, either because the collision integral at infinity converges or because the system is more heavily truncated at the large wavenumbers by wave breaking or dissipation. We will demonstrate that the IR divergence of the collision integral has a simple physical interpretation: the evolution of each wave is dominated by the interaction with its nearest neighboring vertical wavenumbers, mediated by the smallest horizontal wavenumbers of the system, a mechanism denoted Induced Diffusion in the oceanographic literature.

To bring back the effects of the rotation of the Earth in Eq. (1), one introduces the Coriolis parameter $f$ there and in the linear dispersion relation. Since we are considering the evolution of waves with frequency $\omega$ much larger than $f$, $f$ can be considered to be small. However, since the interaction with waves near $f$ dominates the energy transfer, one needs to invert the order in which the limits are taken, postponing making $f$ zero to the end. This gives rise to an integral that diverges like $f$ raised to a negative power smaller than $-1$, but multiplied by a prefactor that vanishes if either $9-2a-3b=0$ or $b=0$. These are the induced diffusion lines of steady state solutions, found originally in Ref. mccomas-1981. This family of stationary states does a reasonable job of explaining the gamut of observed variability.

The paper is organized as follows. Wave turbulence theory for the internal wave field and the corresponding kinetic equation are briefly summarized in Sec. II along with the motivating observations. We analyze the divergence of the kinetic equation in Sec. III. Section IV includes a special, convergent power-law solution that may account for the GM spectrum. In Sec. V we introduce possible quasi-steady solutions of the kinetic equation which are based on cancellations of two singularities. Section VI shows that the IR divergence is dominated by induced diffusion, and computes the family of power-law solutions which arises from taking it into account. We conclude in Sec. VII.