Conclusions

The results in this paper provide an interpretation of the variability in the observed spectral power laws. Combining Figs. 1, 2 and 4 with Eqs. (4), (5) and (43), produces the results shown in Fig 6.

Notice that most of the observational points are UV convergent and IR divergent and that regularization of the collision integral by $f$, as in Sec. VI, produces a family of stationary states that collapses much of the variability. Four points land in the region where the spectra are both IR and UV divergent, with all but one point close to the ID lines. None of the observations lies in a region where both IR and UV divergences have the same sign. Furthermore, the novel convergent (non-rotating) solution is in close proximity to the experimental points determined from the two-dimensional spectra.

Figure 6: The observational points and the theories. The filled circles represent the Pelinovsky-Raevsky (PR) spectrum, the convergent numerical solution determined in Section 4 and the GM spectrum. Circles with stars represent power-law estimates based upon one-dimensional spectra. Circles with cross hairs represent estimates based upon two-dimensional data sets. See Fig. 1 for the identification of the field programs. Light grey shading represents regions of the power-law domain for which the collision integral converges in either the IR or UV limit. The dark grey shading represents the region of the power-law domain for which neither the IR or UV limits converge. The region of black shading represents the sub-domain for which the IR and UV divergences have the same sign. Overlain as solid white lines are the induced diffusion stationary states.

Summarizing the paper, we have analyzed the scale-invariant kinetic equation for internal gravity waves, and shown that its collision integral diverges for almost all spectral exponents. Figure 6 shows that the integral nearly always diverges, either at zero or at infinity. This means that, in the wave turbulence kinetic equation framework, the energy transfer is dominated by the scale-separated interactions with either large or small scales.

The only exception where the integral converges is a segment of a line, $7/2 < a < 4$, with $b=0$. On this convergent segment, we found a special solution, $(a,b) = (3.7, 0)$. This new solution is not far from the large-wavenumber asymptotic form of the Garrett-Munk spectrum, $(a,b)=(4,0)$.

We have argued that there exist two regions of power-law exponents which can yield quasi-steady solutions of the kinetic equation. For these ranges of exponents, the contribution of the scale-separated interactions due to the IR and UV wavenumbers can be made to balance each other. The Pelinovsky-Raevsky spectrum is a special case of this scenario. None of the observational power-law exponents lie within the region of same-signed UV and IR divergences.

This scenario, in which the energy spectrum in the inertial subrange is determined by the nonlocal interactions, provides an explanation for the variability of the power-law exponents of the observed spectra: they are a reflection of the variability of dominant players outside of the inertial range, such as the Coriolis effect, tides and storms. can not be universal.

We also rederive ID steady-solution lines, which are also shown as white lines in Fig. 6.

All of theory, experimental data, and the results of numerical simulation in Ref. numericsLY hint at the importance of the IR contribution to the collision integral. The nonlocal interactions with large scales will therefore play a dominant role in forming the internal-wave spectrum. To the degree that the large scales are location dependent and not universal, the high-frequency, high vertical-wavenumber internal-wave spectrum ought to be affected by this variability. Consequently, the internal-wave spectrum should be strongly dependent on the regional characteristics of the ocean.