Reduction of Kinetic Equation to the Resonant Manifold

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Reduction of Kinetic Equation to the Resonant Manifold

In the high-frequency limit $\omega\gg f$ , one could conceivably neglect the effects of the rotation of the Earth. The dispersion relation (12) then becomes

(13)

and, to the leading order, the matrix element (15) retains only its first term, $I_{\bm{p},\bm{p}_1,\bm{p}_2}$ .

The azimuthally-integrated kinetic equation (18) includes integration over and since the integrations over and can be done by using delta functions. To use delta functions, we need to perform what is called reduction to the resonant manifold. Consider, for example, resonances of type (17a). Given , , and , one can find and satisfying the resonant condition by solving simultaneous equations

$\displaystyle m = m_1 + m_2 , \qquad \frac{k}{\vert m\vert}=\frac{k_1}{\vert m_1\vert} +\frac{k_2}{\vert m-m_1\vert} .$

(14)

The solutions of this quadratic equation are given by
$\begin{subequations} \begin{eqnarray} m_1 &=& \displaystyle\frac{m}{2 k} \left(... ...4 k k_1}\right) \nonumber\\ m_2 &=& m - m_1. . \end{eqnarray}\end{subequations}$
Note that Eq. (21a) translates into Eq. (21b) if the indices

and

are exchanged. Similarly, resonances of type (17b) yield
$\begin{subequations} \begin{eqnarray} m_2 &=& - \displaystyle\frac{m}{2 k} \lef... ...4 k k_2}\right) \nonumber\\ m_1 &=& m + m_2 . \end{eqnarray}\end{subequations}$
and resonances of type (17c) yield
$\begin{subequations} \begin{eqnarray} m_1 &=& - \displaystyle\frac{m}{2k} \left... ... k k_1}\right) \nonumber\\ m_2 &=& m + m_1. . \end{eqnarray}\end{subequations}$

After this reduction, a double integral over and is left. The domain of integration is further restricted by the triangle inequalities

(12)

These conditions ensure that one can construct a triangle out of the wavenumbers with lengths

and

and determine the domain in the

plane called the kinematic box in the oceanographic literature.

Numerical evaluation of the collision integral is a complicated yet straightforward task. Interpretation of the results, though, is more difficult, mostly due to the complexity of the interaction matrix element and the nontrivial nature of the resonant set. Starting with mccomas-1977-82, therefore, predictions were made based on a further simplification. This simplification is based on the assertion that it is interactions between wavenumbers with extreme scale separation that contribute mostly to the nonlinear dynamics. Three main classes of such resonant triads appear, characterized by extreme scale separation. These three main classes are

the vertical backscattering of a high-frequency wave by a low-frequency wave of twice the vertical wavenumber into a second high-frequency wave of oppositely signed vertical wavenumber. This type of scattering, as in Eqs. (25a, 27b, 29a, 30a) below, is called elastic scattering (ES).
The scattering of a high-frequency wave by a low-frequency, small-wavenumber wave into a second, nearly identical, high-frequency large-wavenumber wave. This type of scattering, as in Eqs. (25b, 27a, 29b, 30b) below, is called induced diffusion (ID).
The decay of a small-wavenumber wave into two large vertical-wavenumber waves of approximately one-half the frequency. This type of scattering, as in Eqs. (26a, 26b, 28a, 28b) below, is called parametric subharmonic instability (PSI).

To see how this classification appears analytically, we perform the limit of $k_1 \to 0$ and the limit $k_1 \to \infty$ in Eqs (21-23). We will refer to the or $k_2 \to 0$ limits as IR limits, while the and $k_2 \to \infty$ limit will be referred as an UV limit. Since the integrals in the kinetic equation for power-law solutions will be dominated by the scale-separated interaction, this will help us analyze possible solutions to the kinetic equation.

The results of the $k_1 \to 0$ limit of Eqs. (21-23) are given by
$\begin{subequations} \begin{eqnarray} & \left\{ \begin{array}{l} m_1 \to 2m, m_... ...ega, \omega_2 \sim \omega \end{array}\right. , \end{eqnarray}\end{subequations}$

$\begin{subequations} \begin{eqnarray} & \left\{ \begin{array}{l} m_1 \ll m, m_2... ...mega, \omega_2 \sim \omega \end{array}\right. , \end{eqnarray}\end{subequations}$

$\begin{subequations} \begin{eqnarray} & \left\{ \begin{array}{l} -m_1 \ll m, m_... ...mega, \omega_2 \sim \omega \end{array}. \right. \end{eqnarray}\end{subequations}$
We now see that the interactions (25a, 27b) correspond to the elastic scattering (ES) mechanism, the interactions (25b, 27a, correspond to the induced diffusion (ID). The interactions (26a, 26b), correspond to the parametric subharmonic instability (PSI).

Similarly, taking the and $k_2 \to \infty$ limits, of Eqs. (21-23) we obtain
$\begin{subequations} \begin{eqnarray} & \left\{ \begin{array}{l} m_1 \gg m, -m_... ...1, \omega_2 \sim \omega/2 \end{array}\right. , \end{eqnarray}\end{subequations}$

$\begin{subequations} \begin{eqnarray} & \left\{ \begin{array}{l} m_1 \sim m/2, ... ...mega_1,\omega_2 \gg \omega \end{array}\right. , \end{eqnarray}\end{subequations}$

$\begin{subequations} \begin{eqnarray} & \left\{ \begin{array}{l} m_1 \sim -m/2,... ...ega_1, \omega_2 \gg \omega \end{array}. \right. \end{eqnarray}\end{subequations}$
We now can identify the interactions (28a, 28b) to be PSI, the interactions (29a, 30a) to be ES, and finally the interactions (29b, 30b) as being ID.

This classification provides an easy and intuitive tool for describing extremely scale-separated interactions. We will see below that one of these interactions, namely ID, explain reasonably well the experimental data that is available to us.

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Dr Yuri V Lvov 2008-07-08