Experimental motivation

Power laws provide a simple and intuitive physical description of complicated wave fields. Therefore we assumed that the spectral energy density can be represented as Eq. (2), and undertook a systematic study of published observational programs. In doing so we were fitting the experimental data available to us by power-law spectra. We were presuming that the power laws offer a good fit of the data. We were not assuming that spectra are given by Garrett and Munk spectrum. That effort is reported in detail on elsewhere(19), here we just give a brief synopsis to motivate present theoretical study.

We reviewed the following observational programs:

• The Internal Wave Experiment (IWEX)(20);
• The Arctic Internal Wave Experiment (AIWEX)(22,21);
• The Frontal Air-Sea Interaction Experiment (FASINEX) (24,23);
• The Patches Experiment (PATCHEX)(25);
• The Surface Wave Process Program (SWAPP) experiment(26);
• North Atlantic Tracer Release Experiment (NATRE)(27) / Subduction(28);
• Salt Finger Tracer Release Experiment (SFTRE)(29) / Polymode IIIc (PMIII)(30);
• Site-D(31);
• Storm Transfer and Response Experiment (STREX)(32) / Ocean Storms Experiment (OS)(33).

Here we extract the summary of that study in the form of Fig. 1, in which estimates of power-law exponents $(a,b)$ are plotted against each other.

That study comes with many caveats. The most important here is that the power-law exponents, $(a,b)$, are not typically derived from two-dimensional horizontal wavenumber, vertical wavenumber spectra. The power laws were fit to one-dimensional frequency and one-dimensional vertical wavenumber spectra of energy, $e(\omega) \propto \omega^{-c}$ and $e(m) \propto m^{-d}$, with conversion to $(a,b)$ requiring that:

This procedure explicitly assumes that the two-dimensional spectra are separable. That is to say that the $\omega$-$m$ spectrum can be represented as a product of function of $\omega$ and function of $m$ alone. It is clear that this assumption is not warranted in all cases. This is particularly true of the NATRE data, for which two power-law estimates are provided. Nevertheless, this is the best data that is available to us at a present time. We see that these points are not randomly scattered, but have some pattern. Explaining the location of the experimental points and making sense out of this pattern is the main physical motivation for this study.

Figure 1: The observational points. The filled circles represent the Pelinovsky-Raevsky (PR) spectrum, the convergent numerical solution determined in Sec. IV (C) and the GM spectrum. Circles with stars represent estimates based upon one-dimensional spectra from the western North Atlantic south of the Gulf Stream (IWEX, FASINEX and SFTRE/PMIII), the eastern North Pacific (STREX/OS and PATCHEX), the western North Atlantic north of the Gulf Stream (Site-D), the Arctic (AIWEX) and the eastern North Atlantic (NATRE and NATRE$^2$). There are two estimates obtained from two-dimensional data sets from the eastern North Pacific (SWAPP and PATCHEX$^2$) represented as circles with cross hairs. NATRE represents a fit over frequencies of $1 < \omega < 6$cpd and NATRE$^2$ a fit over higher frequencies.