Asymptotic expansion for small $f$ values.

In this section we perform the small $f$ calculations of Sec. VI. We start from the kinetic equation written as Eq. (40). There we change variables in the first line of Eq. (40) as

and in the second line of Eq. (40) as

Then the Eq. (40) becomes the following form:

$$\frac{\partial}{\partial t} n(k(\omega,m),m) = \frac{2}{k} \int _f^{f+\omega_{\mathrm{s}}} d \omega_1 \int_0^1 d\mu {\cal P}_1$$

$$- {\frac{2}{k}} \int_{\omega-f-\omega_{\mathrm{s}}}^{\omega-f} d \omega_1 \int_0^1 d \mu {\cal P}_2 .$$ (13)

Here we introduced integrand ${\cal P}_1$ and ${\cal P}_2$ to be

$${\cal P}_1 = J\frac{\vert V^0_{1,2}\vert^2}{S^0_{1,2}} n_1 (n_2 - n) (E_1(\omega_1) - E_3(\omega_1)),$$

$${\cal P}_2 = J\frac{\vert V^0_{1,2}\vert^2}{S^0_{1,2}} n_2 (n - n_1) (E_2(\omega_1)-E_3(\omega_1)).$$ (14)

Before proceeding, note the following symmetry:

and

To quantify the contribution of near-inertial waves to a $(\omega,m)$ mode, we write

Subsequently, in the domain (a) we write

in ${\cal P}_1$, and

in ${\cal P}_2$. Furthermore, we expand ${\cal P}_1$ and ${\cal P}_2$ in powers of $\epsilon$ and $f$ without making any assumptions of the relative smallness of $f$ and $\epsilon$. We use the facts that

Define

and

This allows us to expand $P_1$, $P_2$, $P_3$ and $P_4$ in powers of $f$ and $\epsilon$. We perform these calculations on Mathematica using Series command, and extensively using Assumptions field in the FullSimplify command.

Mathematica was then able to perform the integrals of ${\cal P}_1$ and ${\cal P}_2$ over $\mu$ from $0$ to $1$ in (C1) analytically. The result is given by Eq. (41).