Convergences and divergences of the kinetic equation.

Neglecting the effects of the rotation of the Earth yields a scale-invariant system with dispersion relation given by Eq. (19) and matrix element given only by the $I_{\bm{p},\bm{p}_1,\bm{p}_2}$ in Eq. (15). This is the kinetic equation of Refs. gm_lvov,lvov-2004-92, describing internal waves in hydrostatic balance in a non-rotating environment.

Proposing a self-similar separable spectrum of the form (2), it is clear from the bi-homogeneous nature of the azimuthally-integrated kinetic equation (18) that

In order to find a steady scale-invariant solution for all values of $k$ and $m$, it is therefore sufficient to find exponents that give zero collision integral for one wavenumber. One can fix $k$ and $m$, adopting for instance $k=m=1$, and seek zeros of the collision integral as a function of $a$ and $b$:

Figure 2: Divergence/convergence due to IR wavenumbers (top) and due to UV wavenumbers (bottom). The integral converge for the exponents in the shaded regions or on the segments $(b=0, 7/2 < a < 4$) and $(b=1, 3 < a < 7/2)$. Dashed lines distinguish the domains where the indicated named triads dominate the singularity.

Before embarking on numerical or analytical integration of the kinetic equation (18) with scale-invariant solutions (2), it is necessary to check whether or not the collision integral converges. Appendix A outlines these calculations. The condition for the scale-invariant collision integral (32) to converge at the IR end, $k_1 \;\mathrm{or}\; k_2 \to 0$ is given by

Similarly, UV convergence as $k_1 \;\mathrm{and}\; k_2 \to \infty$ implies that

The domains of divergence and convergence are shown in Fig. 2.

Figure 2 also displays the classes of triads dominating the interactions. Knowing the classes of interactions that lead to the divergences of the kinetic equation allows us to find possible physical scenarios of the convergent solutions or to find a possible physical regularization of the divergences.

Note that in addition to the two-dimensional domain of IR convergence [the regions (33a, 33d, 33e)] there are two additional IR convergent line segments given by Eqs. (33b) and (33c). These two special line segments appear because of the $b(b-1)$ prefactor to the divergent contributions to the collision integral (A3). Similarly, for the UV limit, in addition to the two-dimensional region of convergence (34a, 34c, 34d) there is an additional special line segment of $b=0$ (34b).

We see that these domains of convergence overlap only on the segment

Note that $b=0$ corresponds to wave action independent of vertical wavenumbers, $\partial n / \partial m = 0$. Existence of the $b=0$ line will allow us to find novel convergent solution in Sec. IV. We also note that the IR segment on $b=0$ coincides with one of the ID solution determined in Sec. VI. The other segment on $b=1$ do not coincides with the ID solution in Sec. VI since the scale-invariant system has higher symmetry than the system with Coriolis effect.