Ultra-violet asymptotics

Table III: Asymptotics as $k_1 \to \infty$. PSI (21a, 21b) gives $\epsilon ^{a+b-3}$. ES (22a, 23a) gives $\epsilon ^{a-3}$. ID (22b, 23b) gives $\epsilon ^{a+b/2-4}$ ( $\epsilon ^{a-7/2}$). The asymptotics for $b=0$ appear in parentheses.

Next, we consider the limit $k_1 \to \infty$. In this case, $k_2$ also approaches to infinity. We employ the independent variables $x$ and $y$ as $k_1 = k/2 (1 + 1/x + y)$ and $k_2 = k/2 (1 + 1/x - y)$, where $x = O(\epsilon)$ and $-1 < y < 1$. Again, $m>0$ is assumed.

The leading orders are obtained by the similar manner used in the IR asymptotic and are summarized in Table III. The leading order of the integral is given by ID, whose wavenumbers are given by Eqs. (22b, 23b), when $-2<b<2$. In this limit, no second cancellation is made.

As the result of the perturbation theory, we get the leading order,

$$\frac{\partial n_{\bm{p}}}{\partial t} \propto k^{4-2a} m^{1-2b} b \int_0 x^{a+b/2-5} d x .$$ (11)

It has $O(\epsilon^{a+b/2-4})$. Therefore, the integral converges if

The integral for the PR spectrum, which gives $O(\epsilon^{-1/4})$, diverges as $k_1 \to \infty$. and that for the Garrett-Munk spectrum, which gives $O(\epsilon^0)$, converges owing that $b=0$.

Similarly,

$$\frac{\partial n_{\bm{p}}}{\partial t} \propto - k^{4-2a} m^{1-2b} b \int_0 x^{a-2} d x$$ (12)

for ES, which is dominant for $b>2$. Consequently, the integral converges also if

In the same manner, the convergent domain of the integral for PSI is given by