Balance between divergences

The fact that the collision integral ${\cal C}$ diverges for almost all values of $a$ and $b$ can be viewed both as a challenge and as a blessing. On the one hand, it makes the prediction of steady spectral slopes more difficult, since it now depends on the details of the truncation of the domain of the integration. Fortunately, it provides a powerful tool for quantifying the effects of fundamental players in ocean dynamics, most of which live on the fringes of the inertial subrange of the internal wave field: the Coriolis effect, as well as tides and storms at the IR end of the spectrum, and wave breaking and dissipation at the UV end. The sensitive response of the inertial subrange to the detailed modeling of these scale-separated mechanisms permits, in principle, building simple models in which these are the only players, bypassing the need to consider the long range of wave scales in between.

Figure 4: Signs of the divergences where the both IR and UV contributions diverge. The left symbols show the signs due to IR wavenumbers and the right symbols show the signs due to UV wavenumbers.

At the IR end, the resonant interactions are dominated by the ID singularity for $-3<b<3$, Fig. 2. The sign of the divergences is given by $-b(b-1)$ (Eq. (A3)). Similarly, at the UV end resonant interactions are dominated by the ID singularity for $-2<b<2$ (Fig. 2). The sign of the divergence is given by $b$ (Eq. (A4)). At the UV end for $b>2$, where ES determines the divergences, the sign of the singularity is given by $-b$, i.e. the sign is negative (Eq. (A5)). Figure 4 shows the signs of the divergences where both the IR and UV contributions diverge: the left sign corresponds to the IR contribution, and the right sign to the UV contribution.

Hence, in the regions,

the divergences of the collision integral at the IR and UV ends have opposite signs. Then formal solutions can be constructed by having these two divergences cancel each other out.

This observation justifies the existence of the PR solution (5). Indeed, the PR spectrum has divergent power-law exponents at the both ends. One can prove that the PR spectrum is an exact steady solution of the kinetic equation (1) by applying the Zakharov-Kuznetsov conformal mapping for systems with cylindrical symmetry (36,35,37). This Zakharov-Kolmogorov conformal mapping effectively establishes a map between the neighborhoods of zero and infinity. Making these two contributions cancel point-wisely yields the solution (5). For this transformation to be mathematically applicable the integrals have to converge. This transformation leads only to a formal solution for divergent integrals.

The PR spectrum was first found by pelinovsky1977wti. However, they realized that it was only a formal solution. The solution was found again in Ref. caillol2000kea through a renormalization argument, and in Ref. gm_lvov within an isopycnal formulation of the wave field.

The idea of a formal solution, such as PR, can be generalized quite widely: in fact, any point in the regions with opposite-signed divergences can be made into a steady solution under a suitable mapping that makes the divergences at zero and infinity cancel each other, as does the Kuznetsov-Zakharov transformation. Such generalized Kuznetsov-Zakharov transformation is an extension of the idea of principle value for a divergent integral, whereby two divergent contributions are made to cancel each other through a specific relation between their respective contributions.

This scenario also suggests an possible explanation for the variability of the power-law exponents of the quasi-steady spectra: inertial subrange spectral variability is to be expected when it is driven by the nonlocal interactions. The natural local variability of players outside the inertial range, such as tides and storms, translates into a certain degree of non-universality.