Hamiltonian Structure and Wave Turbulence Theory

This subsection briefly summarizes the derivation in Ref. gm_lvov; it is included here only for completeness and to allow references from the core of the paper.

The equations of motion satisfied by an incompressible stratified rotating flow in hydrostatic balance under the Boussinesq approximation are:

$$\frac{\partial}{\partial t}\frac{\partial z}{\partial \rho} + \nabla \cdot \left(\frac{\partial z}{\partial \rho} \bm{u} \right) = 0 ,$$

$$\frac{\partial \bm{u}}{\partial t} +f \bm{u}^\perp+ \bm{u} \cdot \nabla \bm{u} + \frac{\nabla M}{\rho_0} = 0 ,$$

$$\frac{\partial M}{\partial \rho} - g z = 0 .$$ (6)

These equations result from mass convervation, horizontal momentum convervation and hydrostatic balance. The equations are written in isopycnal coordinates with the density $\rho$ replacing the height $z$ in its role as independent vertical variable. Here $\bm{u} = (u,v)$ is the horizontal component of the velocity field, $\bm{u}^{\perp} = (-v, u)$, $\nabla = (\partial/\partial x, \partial/\partial y)$ is the gradient operator along isopycnals, $M$ is the Montgomery potential

$f$ is the Coriolis parameter, and $\rho_0$ is a reference density in its role as inertia, considered constant under the Boussinesq approximation.

The potential vorticity is given by

where $\Pi = \rho / g \partial^2 M/\partial \rho^2 = \rho \partial z/\partial \rho$ is a normalized differential layer thickness. Since both the potential vorticity and the fluid density are conserved along particle trajectories, an initial profile of the potential vorticity that is a function of the density will be preserved by the flow. Hence it is self-consistent to assume that

where $\Pi_0(\rho) = -g / N(\rho)^2$ is a reference stratification profile with the constant background buoyancy frequency, $N = (-g/(\rho \partial z/\partial\rho\vert _{\mathrm{bg}}))^{1/2}$. This assumption is not unrealistic: it represents a pancake-like distribution of potential vorticity, the result of its comparatively faster homogenization along than across isopycnal surfaces.

It is shown in Ref. gm_lvov that the primitive equations of motion (6) under the assumption (8) can be written as a pair of canonical Hamiltonian equations,

where $\phi$ is the isopycnal velocity potential, and the Hamiltonian is the sum of kinetic and potential energies,

Here, $\nabla^{\perp}=(-\partial / \partial y, \partial / \partial x)$ and $\Delta^{-1}$ is the inverse Laplacian.

Switching to Fourier space, and introducing a complex field variable $a_{\bm{p}}$ through the transformation

$$\phi_{\bm{p}} = \frac{i N \sqrt{\omega_{\bm{p}}}}{\sqrt{2 g} \vert\bm{k}\vert} \left(a_{\bm{p}}- a^{\ast}_{-{\bm{p}}}\right)\, ,$$

$$\Pi_{\bm{p}} = \Pi_0-\frac{N\, \Pi_0\, \vert\bm{k}\vert}{\sqrt{2\, g \omega_{\bm{p}}}}\left(a_{\bm{p}}+a^{\ast}_{-{\bm{p}}}\right)\, ,$$ (10)

where the frequency $\omega$ satisfies the linear dispersion relation

$$\omega_{\bm{p}}=\sqrt{ f^2 + \frac{g^2}{\rho_0^2 N^2} \frac{\vert\bm{k}\vert^2 }{m^2}},$$ (11)

the equations of motion (6) adopt the canonical form

with Hamiltonian:

$${\cal H} = \int d\bm{p} \, \omega_{\bm{p}} \vert a_{\bm{p}}\vert^2$$

$$\quad + \int d\bm{p}_{012} \left( \delta_{\bm{p}+\bm{p}_1+\bm{p}_... ...\bm{p}}^{\ast} a_{\bm{p}_1}^{\ast} a_{\bm{p}_2}^{\ast} + \mathrm{c.c.}) \right.$$

$$\quad \left. + \delta_{-\bm{p}+\bm{p}_1+\bm{p}_2} (V_{\bm{p}_1,\b... ...{\bm{p}} a_{\bm{p}}^{\ast} a_{\bm{p}_1} a_{\bm{p}_2} + \mathrm{c.c.}) \right) .$$ (13)

This is the standard form of the Hamiltonian of a system dominated by three-wave interactions(14). Calculations of interaction coefficients are tedious but straightforward task, completed in Ref. gm_lvov. These coefficients are given by

where $((0,1,2) \to (1,2,0))$ and $((0,1,2) \to (2,0,1))$ denote exchanges of suffixes. We stress that the field equation (13) with the three-wave Hamiltonian (12, 14, 15) is equivalent to the primitive equations of motion for internal waves (6). Other approaches, in particular the Lagrangian approach, is based on small-amplitude expansion to arrive to this type of equations.

In wave turbulence theory, one proposes a perturbation expansion in the amplitude of the nonlinearity, yielding linear waves at the leading order. Wave amplitudes are modulated by the nonlinear interactions, and the modulation is statistically described by an kinetic equation(14) for the wave action $n_{\bm{p}}$ defined by

$$n_{\bm{p}} \delta(\bm{p} - \bm{p}^{\prime}) = \langle a_{\bm{p}}^{\ast} a_{\bm{p}^{\prime}}\rangle.$$ (13)

The derivation of this kinetic equation is well studied and understood. For the three-wave Hamiltonian (14), the kinetic equation is the one in Eq. (1), describing general internal waves interacting in both rotating and non-rotating environments.

The delta functions in the kinetic equation ensures that spectral transfer happens on the resonant manifold, defined as

Now let us assume that the wave action is independent of the direction of the horizontal wavenumber,

Note that value of the interaction matrix element is independent of horizontal azimuth as it depends only on the magnitude of interacting wavenumbers. Therefore one can integrate the kinetic equation (1) over horizontal azimuth (14), yielding

$$\frac{\partial n_{\bm{p}}}{\partial t} = \frac{2}{k}\int \left(R^{0}_{12} - R^{1}_{20} - R^{2}_{01} \right) \, d k_1 d k_2 d m_1 d m_2 \, ,$$

$$R^{0}_{12}= f^{\bm{p}}_{\bm{p}_1\bm{p}_2} \, \vert V^{\bm{p}}_{\b... ...\omega_{\bm{p}}-\omega_{\bm{p}_1}-\omega_{\bm{p}_2}} k k_1 k_2 / S^0_{1,2} \, .$$

Here $S^0_{1,2}$ appears as the result of integration of the horizontal-momentum conservative delta function over all possible orientations and is equal to the area of the triangle with sides with the length of the horizontal wavenumbers $k = \vert\bm{k}\vert$, $k_1 = \vert\bm{k}_1\vert$ and $k_2 = \vert\bm{k}_2\vert$. This is the form of the kinetic equation which will be used to find scale-invariant solutions in the next section.