A novel convergent solution

To find out whether there is a steady solution of the kinetic equation along the convergent segment (35), we substitute the power-law ansatz (2) with $b=0$ into the azimuthally-integrated kinetic equation (18). We then compute numerically the collision integral as a function of $a$ for $b=0$. To this end, we fix $k=m=1$ and perform a numerical integration over the kinematic box (24), reducing the integral to the resonant manifold as described in Sec. III+.1667emA.

The result of this numerical integration is shown in Fig. 3. The figure clearly shows the existence of a steady solution of the kinetic equation (18) near $a \cong 3.7$ and $b=0$.

Figure 3: Value of the collision integral as a function of $a$ on the convergent segment, $b=0$.

This is, therefore, the only convergent steady solution to the scale-invariant kinetic equation for the internal wave field. It is highly suggestive that it should exist so close to the GM spectrum, $a=4$ and $b=0$ for large wavenumbers, the most agreed-upon fit to the spectra observed throughout the ocean. It remains to be seen whether how this solution is modified by inclusion of background rotation.