Yuri V. Lvov, Kurt L. Polzin and Esteban G. Tabak,
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy NY 12180
Introduction. Internal waves are an important piece of energy and momentum budgets for the earth's atmosphere and ocean. The drag associated with internal wave breaking needs to be included in order to obtain accurate simulations of the atmospheric Jet Stream  and it has been argued that the ocean's Meridional Overturning Circulation  is forced by the diffusion of mass  associated with internal wave breaking  rather than by the production of cold, dense water by convection at high latitudes. Both circulations represent important pieces of the earth's climate system.
In a classical work , Garrett and Munk demonstrated how observations from various sensor types could be synthesized into a combined wavenumber-frequency spectrum, now called the Garrett-and-Munk (GM) spectrum of internal waves. Consistent only with linear internal wave kinematics, the GM spectrum was developed as an empirical curve fit to available data. Even though deviations have been noted near boundaries , and at the equator , the last significant model revision [8,9] has surprisingly stood the test of time. However, a review of open ocean data sets reveals subtle variability in spectral power laws. We show in this letter that predictions based upon a weakly nonlinear wave turbulence theory are consistent with both the high frequency-high wave number limit of GM spectrum and the observed variability.
In this letter, we will consider only the high frequency-high wave
number limit of GM; for brevity, we shall denote this henceforth as
GM. The GM is given by
The possibility that the internal wavefield might exhibit a universal character represents an attractive theoretical target, and much effort (as reviewed by ) was devoted to studying the issue of nonlinearity in the context of resonant wave interactions. That line of work is based on a Lagrangian description of the flow, with two main approximations: that fluid particles undergo small displacements, and that nonlinear interactions take place on a much longer time-scale than the underlying linear dynamics. An approximate kinetic equation describing the time evolution of spectral wave energy was derived, and it was shown  that the GM spectrum (1) was close to being a stationary solution.
An alternative to the Lagrangian formulation, based on a Hamiltonian description in isopycnal (density) coordinates, was recently proposed . This approach does not invoke a small-displacement assumption and yields a comparatively simple kinetic equation with an exact steady power-law solution in the high frequency limit. That steady state solution, [see (5) below] is close to the GM spectrum (1), yet there is a noticeable difference. Motivated by this difference, we tried to estimate the accuracy of the GM power laws and thus reviewed extant observations from the literature. In the process of analyzing the data, we found that there was subtle variability in the high wavenumber, high frequency spectrum, forming a distinct pattern.
We then reexamined the kinetic equation of  and found its full family of steady state solutions, of which the solution reported in  is just one member. This family of solutions compares well with the variability found in the observations. Moreover, the GM spectrum (1) is a member of this family, thus describing the GM spectrum simply as an exact steady-state solution to the kinetic equation derived in .
Hence, in this article we present evidence for variability in the high frequency-high wavenumber open-ocean internal wavefield, and find that a wave turbulence approach predicts that both GM itself and the observed variability are stationary states of the kinetic equation. The variability itself, and its likely roots in variable forcing, Coriolis effects, underlying stratification and currents, as well as the low frequency range of the energy spectrum, are fundamental problems posing exciting challenges for future research.
Overview of observations: a family of spectra. Below we
present a summary of historical oceanic internal wave
energy spectra. These observations are re-analyzed to
study whether the high-frequency, high-wavenumber spectra may form a
pattern. We review seven data sets available in the
literature. We shall present a detailed analysis of these data sets
elsewhere; here we just list them along with their high-frequency,
asymptotics. Let us assume that, in this limit,
the three dimensional wave action can be approximated by
horizontally isotropic power laws of the form
Using the linear dispersion relation of internal waves,
, this action spectrum can be
transformed from the wavenumber space to the vertical
wave-number-frequency space . Multiplication by the
frequency yields the corresponding energy spectrum,
A wave turbulence formulation for the internal wave field. In this section we assume that the internal wave field can be viewed as a field of weakly interacting waves, thus falling into the class of systems describable by wave turbulence. Wave turbulence is a universal statistical theory for the description of an ensemble of weakly interacting particles, or waves. This theory has contributed to our understanding of spectral energy transfer in complex systems , and has been used for describing surface water waves since pioneering works by Hasselmann , Benney and Newell  and Zakharov [25,26].
The dynamics of oceanic internal waves can be most easily described in isopycnal (i.e. density) coordinates, which allow for a simple and intuitive Hamiltonian description . To describe the wave field, we introduce two variables: a velocity potential , and an isopycnal straining . The horizontal velocity is given by the isopycnal gradient of the velocity potential, . The straining can also be interpreted as the fluid density in isopycnal coordinates.
These two variables form a canonically conjugated Hamiltonian pair, so
that the primitive equations of motion (i.e. conservation of
horizontal momentum, hydrostatic balance, mass conservation and the
incompressibility constraint) can be written as a pair of canonical
Performing the Fourier transform, and introducing a complex field
These field equations are equivalent to the primitive equations of motion for internal waves (up to the hydrostatic balance and Boussinesq approximation); the work reviewed in  instead resorted to a small displacement approximation to arrive at similar equations. We will argue elsewhere that this extra assumption does not provide an internally consistent description of interactions between extremely scale separated waves. For the purposes of this letter, it suffices to note that the two kinetic equations are different and yield different steady solutions.
We shall characterize the field of interacting internal waves by its wave action
Under the assumption of weak nonlinear interaction, one derives a
closed equation for the evolution of the wave action, the kinetic
equation. Assuming horizontal isotropy, the kinetic equation can be
reduced further by averaging over all horizontal angles, obtaining
A family of steady state power-law solutions to the kinetic
equation. In wave turbulence theory, three-wave kinetic equations
admit two classes of exact stationary solutions: thermodynamic
equilibrium and Kolmogorov flux solutions, with the latter
corresponding to a direct cascade of energy -or other conserved
quantities- toward the higher modes. The fact that the thermodynamic
equilibrium -or equipartition of energy-
is a stationary solution of (4) can be seen by inspection,
whereas in order to find Kolmogorov spectra one needs to be more
elaborate. In  we used the Zakharov-Kuznetsov conformal
mapping [25,26,27] to show analytically that the following
wave action spectrum constitutes an exact steady state solution of
(4) [note the difference with (1)]:
Since this family of steady state solutions is not all apparently
amenable to a closed form, we sought the zeros of by
numerical integration. This involves a certain amount of work.
First, the delta-functions in (4) restrict
contributions to the resonant set. Consider, for example, the resonant
The resulting family of zeros is depicted in Figure 1. Notice that the curve passes through the exact solution (5). More importantly, it also passes through the point , corresponding to the GM spectrum (1). Hence this classical spectrum is for the first time shown to correspond to an exact steady solution to a kinetic equation based on first fluid principles.
Finally, we note the integrals converge in the parameter regime occupied by the observations. In regions of tightly spaced contour lines ( and , and ) (4) is nonintegrable.
The other points marked on the figure correspond to the observational sets discussed above. Notice that, with the exception of NATRE, they all lie very close to the zeros of . Therefore the predictions of wave turbulence are consistent with the observed deviations from GM.
In fact, the NATRE point lies in an area of space where and are nearly tangential, thus making the line of zeros effectively ``thicker'' (in other words, the collision integral is not zero at the observed points, but it is very small, possibly allowing other, typically smaller effects to take over.)
- Conclusions We have shown that the wave turbulence formalism captures much of the variability apparent in the oceanic internal wave field. This includes the characterization of the spectral curve put together by Garrett and Munk as an exact steady solution to a kinetic equation for the evolution of the wave field, derived from first principles. In addition, the curve of steady solutions to this kinetic equation is consistent with much of the observed variability in the energy spectra. We conjecture that the placement along this curve of individual observations depends on the nature of the forcing (for instance, by tides and atmospheric winds), the local degree of stratification, vorticity and shear, and the variable magnitude of the Coriolis parameter. This is the subject of ongoing research.
Acknowledgments YL is supported by NSF CAREER grant DMS 0134955 and by ONR YIP grant N000140210528; KP is supported by NSF grant OCE 9906731; ET is supported by NSF grant DMS 0306729.