Hamiltonian formalism for internal waves

Equations of motion for incompressible stratified fluid (i.e. conservation of horizontal momentum, hydrostatic balance, mass conservation and the incompressibility constraint) under the assumption of zero potential vorticity in the isopycnal coordinates, $(\bldx, \rho)$, can be written as a pair of canonical Hamiltonian equations,

$$\frac{\partial \Pi}{\partial t} = - \frac{\delta {\cal H}}{... ... \phi}{\partial t} = \frac{\delta {\cal H}}{\delta \Pi} \, ,$$ (1)

where $\Pi = \rho dz/d\rho$ is the fluctuation of stratification profile around the mean, $-g / N_0^2$, and $\phi$ is the isopycnal velocity potential. The Hamiltonian is the sum of kinetic and potential energies,

$${\cal H} = \int \mathrm{d} \bldx \mathrm{d} \rho \left( - \frac{1}{2} \left( -\frac{g}{N_0^2}+\Pi(\bldx, \rho) \right)$$ (2)

$$end{tex2html_deferred}, \left\vert\bnabla \phi(\bldx, \rho) - \fr... ...N_0^2 }{g} \bnabla^{{\perp}}\Delta^{-1} \Pi(\bldx, \rho) \right\vert^2 \right.$$

$$end{tex2html_deferred}$$ (3)

$$end{tex2html_deferred} \quad \left. + \frac{g}{2} \left\vert\int^{\rho} \mathrm{d}\rho^{\prime} \frac{\Pi(\bldx, \rho^{\prime})}{\rho^{\prime}} \right\vert^2 \right)$$ (4)

$$end{tex2html_deferred}, ,$$ (5)

(see Lvov & Tabak (2004) for complete details). The differential operators on the isopycnal surfaces, $\bnabla=(\upartial/\upartial x, \upartial/\upartial y)$ and $\bnabla^{\perp}=(-\upartial/\upartial y, \upartial/\upartial x)$, are the horizontal gradient operator and the rotation operator, respectively. Also, $g$ is the acceleration of gravity, $N_0$ is the buoyancy (Brunt-Väisälä) frequency, $f$ is the inertial frequency due to the rotation of the Earth, and $\rho_0$ is the mean density.

We then perform Fourier transformation and canonical transformation to the field variable, $a(\bldp)$, as

$$a(\bldp) = \sqrt{\frac{\omega}{2 g}} \frac{N_0}{\vert\bldk\vert} ... ... \sqrt{\frac{g}{2 \omega}}\frac{\vert\bldk\vert}{N_0} {\widetilde \phi}(\bldp)$$ (6)

$$end{tex2html_deferred}, ,$$ (7)

with linear coupling of the Fourier components of the stratification profile, ${\widetilde \Pi}$, and the horizontal velocity potential, ${\widetilde \phi}$. The three-dimensional wavenumber, $\bldp$, consists of a two-dimensional horizontal wavenumber in the isopycnal surface, $\bldk$, and a vertical density wavenumber, $m$. The linear frequency is given by the dispersion relation,

$$\omega(\bldp) = \sqrt{f^2 + \frac{g^2}{\rho_0^2 N_0^2} \frac{\vert\bldk\vert^2}{m^2}}$$ (8)

$$end{tex2html_deferred}, .$$ (9)

The usual vertical wavenumber, $k_z$, and the density wavenumber, $m$, are related as $m = - g/(\rho_0 N_0^2) k_z$. The buoyancy frequency, $N_0$, and the inertial frequency, $f$, are assumed to be constants.

Then, the equations of motion (1) are rewritten as a canonical equation,

$$\mathrm{i} \frac{\upartial a(\bldp)}{\upartial t} = \frac{\delta {\cal H}}{\delta a^{\ast}(\bldp)}$$ (10)

with the standard Hamiltonian of three-wave interactive system,

$${\cal H} = \int \mathrm{d}\bldp$$ (11)

$$end{tex2html_deferred}: \omega(\bldp) \vert a(\bldp)\vert^2$$

$$end{tex2html_deferred}$$ (12)

$$end{tex2html_deferred} + \int$$ (13)

$$end{tex2html_deferred}! \mathrm{d}\bldp \mathrm{d}\bldp_1 \mathrm... ...p} a(\bldp) a^{\ast}(\bldp_1) a^{\ast}(\bldp_2) + \mathrm{c.c.}\right) \right.$$

$$end{tex2html_deferred}$$ (14)

$$end{tex2html_deferred} \left. \qquad\qquad\qquad\qquad + \left( U_{\bldp,\bldp_1,\bldp_2} a(\bldp) a(\bldp_1) a(\bldp_2) + \mathrm{c.c.}\right) \right)$$ (15)

$$end{tex2html_deferred}, .$$ (16)

Here, $\delta/\delta a^{\ast}$ is the functional derivative with respect to $a^{\ast}(\bldp)$ that is the complex conjugate of $a(\bldp)$ and the abbreviation c.c. denotes complex conjugates. Matrix elements, $V_{\bldp_1,\bldp_2}^{\bldp}$ and $U_{\bldp,\bldp_1,\bldp_2}$, have exchange symmetries such that $V_{\bldp_1,\bldp_2}^{\bldp} = V_{\bldp_2,\bldp_1}^{\bldp}$ and $U_{\bldp,\bldp_1,\bldp_2} = U_{\bldp,\bldp_2,\bldp_1} = U_{\bldp_1,\bldp,\bldp_2}$.