Oceanic Internal-Wave Field: Theory of Scale-Invariant Spectra

1. Introduction

Wave–wave interactions in continuously stratified fluids have been a subject of intensive research in the last few decades. Of particular importance is the observation of a nearly universal internal-wave energy spectrum in the ocean, first described by Garrett and Munk (Garrett and Munk 1972, 1975; Cairns and Williams 1976; Garrett and Munk 1979). However, it appears that ocean is too complex to be described by one universal model. Accumulating evidence suggests that there is measurable variability of observed experimental spectra (Polzin and Lvov 2010, manuscript submitted to Rev. Geophys., hereafter PL). In particular, we have analyzed the decades of observational programs, and we have come to the conclusion that the high-frequency–high-wavenumber part of the spectrum can be characterized through simple power-law fits with variable exponents.

It is generally thought (Müller et al. 1986; Olbers 1974, 1976; PL) that nonlinear interactions significantly contribute to determining the background oceanic spectrum, and this belief motivates the investigation of spectral evolution equations for steady balances. A particularly important study in this regard is the demonstration that the Garrett–Munk (GM) vertical wavenumber spectrum is stationary and supports a constant downscale energy flux (McComas and Müller 1981) associated with resonant interactions. The accumulating evidence alluded to above suggests there is more to the story.

The purpose of the present study is to lay down a firm theoretical framework that allows a detailed analysis of power-law spectra of internal waves in the ocean. We investigate the parameter space of the possible power laws with a specific focus on extreme scale-separated interactions and their role in dominating spectral transfers. We then use this theoretical framework to interpret the observed oceanic variability.

Because of the quadratic nonlinearity of the underlying fluid equations and dispersion relation allowing three-wave resonances, internal waves interact through triads. In the weakly nonlinear regime, the nonlinear interactions among internal waves concentrate on their resonant set and can be described by a kinetic equation, which assumes the familiar form (Caillol and Zeitlin 2000; Hasselmann 1966; Kenyon 1966, 1968; Lvov and Tabak 2001, 2004; McComas and Bretherton 1977; Milder 1990; Müller and Olbers 1975; Olbers 1974, 1976; Pelinovsky and Raevsky 1977; Pomphrey et al. 1980; Voronovich 1979; Zakharov et al. 1992)

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Here n_p = n(p) is a three-dimensional (3D) action spectrum [see Eq. (16)] with wavenumber p = (k, m): that is, k and m are the horizontal and vertical components of p. Action or wave action can be viewed as “number” of waves with a given wavenumber. The frequency ω_p is given by a linear dispersion relation (12) below. Consequently, wave action multiplied by frequency ω_pn_p can be seen as quadratic spectral energy density of internal waves. Note that the wavenumbers are oppositely signed variables, whereas the wave frequencies are always positive. The factor is the interaction matrix element describing the transfer of wave action among the members of a triad composed of three wave vectors p = p₁ + p₂.

Following Kolmogorov’s viewpoint of energy cascades in isotropic Navier–Stokes turbulence, one may look for statistically stationary states using scale-invariant solutions to the kinetic Eq. (1). The solution may occur in an inertial subrange of wavenumbers and frequencies that are far from those where forcing and dissipation act and also are far from characteristic scales of the system, including the Coriolis frequency resulting from the rotation of the earth, the buoyancy frequency due to stratification and the ocean depth. Under these assumptions, the dispersion relation and the interaction matrix elements are locally scale invariant. It is natural, therefore, in this restricted domain, to look for self-similar solutions of Eq. (1), which take the form

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Values of a and b for which the right-hand side of Eq. (1) vanishes identically correspond to steady solutions of the kinetic equation and we hope also to statistically steady states of the ocean’s wave field. Unlike Kolmogorov turbulence, the exponents that give steady solutions cannot be determined by the dimensional analysis alone (see, e.g., Polzin 2004). This is the case because of multiple characteristic length scales in anisotropic systems.

Before seeking steady solutions, however, one should find out whether the improper integrals¹ in the kinetic Eq. (1) converge. This is related to the question of locality of the interactions: a convergent integral characterizes the physical scenario where interactions of neighboring wavenumbers dominate the evolution of the wave spectrum, whereas a divergent one implies that distant, nonlocal interactions in the wavenumber space dominate.

In the present paper, we demonstrate analytically that the internal-wave collision integral diverges for almost all values of a and b. In particular, the collision integral has an infrared (IR) divergence at 0 (i.e., |k₁| or |k₂| → 0) and an ultraviolet (UV) divergence at infinity (i.e., |k₁| and |k₂| → ∞). Thus, IR divergence comes from interactions with the smallest wavenumber, and UV divergence comes from interactions with the largest horizontal wavenumber. There is only one exception where the integral converges: the segment with b = 0 and . The b = 0 line corresponds to wave action independent of vertical wavenumbers, ∂n/∂m = 0. Within this segment, we numerically determine a new steady convergent solution to Eq. (1), with

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This solution is not far from the large-wavenumber form of the GM spectrum (Garrett and Munk 1972, 1975; Cairns and Williams 1976; Garrett and Munk 1979),

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Alternatively, one can explore the physical interpretation of divergent solutions. We find a region in (a, b) space where there are both IR and UV divergences having opposite signs. This suggests a possible scenario where the two divergent contributions may cancel each other, yielding a steady state. An example of such a case is provided by the Pelinovsky–Raevsky (PR) spectrum (Pelinovsky and Raevsky 1977),

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This solution, however, is only one among infinitely many. The problem at hand is a generalization of the concept of principal value integrals: for a and b, which give opposite signs of the divergences at zero and infinity, one can regularize the integral by cutting out small neighborhoods of the two singularities in such a way that the divergences cancel each other and the remaining contributions are small. Hence, all the exponents that yield opposite-signed divergence at both ends can be steady solutions of Eq. (1). As we will see below this general statement helps to describe the experimental oceanographic data that are available to us. The nature of such steady solutions depends on the particular truncation of the divergent integrals.

So far, we have kept the formalism at the level of the self-similar limit of the kinetic Eq. (1). However, once one considers energy transfer mechanisms dominated by interactions with extreme modes of the system, one can no longer neglect the deviations from self-similarity near the spectral boundaries: the inertial frequency due to the rotation of the earth at the IR end and the buoyancy frequency and/or dissipative cutoffs at the UV end.

For example, we may consider a scenario in which interactions with the smallest horizontal wavenumbers dominate the energy transfer within the inertial subrange, either because the collision integral at infinity converges or because the system is more heavily truncated at the large wavenumbers by wave breaking or dissipation. We will demonstrate that the IR divergence of the collision integral has a simple physical interpretation: the evolution of each wave is dominated by the interaction with its nearest neighboring vertical wavenumbers, mediated by the smallest horizontal wavenumbers of the system. Such a mechanism is called induced diffusion (ID) in the oceanographic literature.

To bring back the effects of the rotation of the earth in Eq. (1), one introduces the Coriolis parameter f there and in the linear dispersion relation. Because we are considering the evolution of waves with frequency ω much larger than f, f can be considered to be small. However, because the interaction with waves near f dominates the energy transfer, one needs to invert the order in which the limits are taken, postponing making f = 0 to the end. This procedure gives rise to an integral that diverges like f raised to a negative power smaller than −1 but multiplied by a prefactor that vanishes if either 9 − 2a − 3b = 0 or b = 0. These are the induced diffusion lines of steady-state solutions, found originally in McComas and Müller (1981) as a diffusive approximation to the kinetic equation. This family of stationary states does a reasonable job of explaining the gamut of observed variability. The rigorous asymptotic analysis presented here clearly implies the induced diffusion family of stationary states makes sense only in the IR divergent subdomain of (a, b) space, and we find that the data are located in this subdomain.

The present paper investigates in detail the parameter space (a, b) of a general power-law spectrum (2), compares this parameter space to the ocean observations, and gives possible interpretation. Furthermore, the present study places the previously obtained ID curves (McComas and Müller 1981) and Pelinovsky–Raevsky spectrum (Pelinovsky and Raevsky 1977) into a much wider context. Finally, we present a general theoretical background that we are going to exploit for future studies.

The paper is organized as follows: Wave turbulence theory for the internal wave field and the corresponding kinetic equation are briefly summarized in section 2 along with the motivating observations. We analyze the divergence of the kinetic equation in section 3. Section 4 includes a special convergent power-law solution that may account for the GM spectrum. In section 5, we introduce possible quasi-steady solutions of the kinetic equation that are based on cancellations of two singularities. Section 6 shows that the IR divergence is dominated by induced diffusion, and we compute the family of power-law solutions that arises from taking it into account. We conclude in section 7.

2. Wave turbulence theory for internal waves

b. 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 assume that the power laws offer a good fit of the data in the high-frequency, high-wavenumber parts of the spectrum. We do not assume that spectra are given by Garrett–Munk spectrum.

In most instances, vertical wavenumber and frequency power laws were estimated by superimposing best-fit lines on top of one-dimensional spectra. The quoted power laws are the asymptotic relations of these best-fit lines. Fits in the frequency domain included only periods smaller than 10 h, thereby eliminating the inertial peak and semidiurnal tides from consideration. For one-dimensional spectra, there is an implicit assumption that the high-frequency–high-wavenumber spectra are separable. Because both 1D spectra are red, frequency spectra are typically dominated by low-vertical-wavenumber motions, and vertical wavenumber spectra are dominated by low frequencies. Care should be taken to distinguish these results from the high-wavenumber asymptotics of truly two-dimensional spectra. We have included two realizations of two-dimensional displacement spectra in isopycnal coordinates [the Patches Experiment (PATCHEX²) and the Surface Wave Process Program (SWAPP)]. The quoted power laws in these instances were estimated using a straight edge and χ-by-eye procedure.

Below we list extant datasets with concurrent vertical profile and current meter observations and some major experiments utilizing moored arrays, along with our best estimate of their high-wavenumber, high-frequency asymptotics:

• Site-D (Foffonoff 1969; Silverthorne and Toole 2009): energy spectra are m^−2.0 and ω^−2.0;

• the Frontal Air-Sea Interaction Experiment (FASINEX; Weller et al. 1991; Eriksen et al. 1991): energy spectra are m^−2.3 and ω^−1.85;

• the Internal Wave Experiment (IWEX; Müller et al. 1978): energy spectrum is −k^−2.4±0.4ω^−1.75;

• the Salt Finger Tracer Release Experiment (SFTRE; Schmitt et al. 2005)/Polymode IIIc (PMIII; Keffer 1983): energy spectra are m^−2.4 and ω^−1.9;

• the North Atlantic Tracer Release Experiment (NATRE; Polzin et al. 2003)–subduction (Weller et al. 2004): energy spectra are m^−2.55 (observed: NATRE¹) or m^−2.75 (minus vortical contribution: NATRE²) and ω^−1.35;

• PATCHEX¹ (Gregg et al. 1993; Chereskin et al. 2000): energy spectra are m^−1.75 and ω^−1.65 − ω^−2.0;

• PATCHEX² (Sherman and Pinkel 1991): energy spectrum is m^−1.75ω^−1.65 − m^−1.75ω^−2.0;

• the SWAPP experiment (Anderson 1992): energy spectrum is m^−1.9ω^−2.0.

• the Storm Transfer and Response Experiment (STREX; D’Asaro 1984)/Ocean Storms Experiment (OS; D’Asaro 1995): energy spectra are m^−2.3 and ω^−2.2;

• the Midocean Acoustic Transmission Experiment (MATE; Levine et al. 1986): energy spectra are −m^−2.1 and ω^−1.7; and

• the Arctic Internal Wave Experiment (AIWEX; Levine et al. 1987; D’Asaro and Morehead 1991): energy spectra are m^−2.25 and ω^−1.2.

Two estimates of the NATRE spectrum are provided: NATRE¹ represents the observed spectrum, and NATRE² represents the observed spectrum minus the quasi-permanent finestructure spectrum identified in Polzin et al. (2003). The residual (NATRE²) represents our best estimate of the internal-wave spectrum. Two estimates of the PATCHEX spectrum are provided: PATCHEX¹ combines free-fall vertical profiler data from Gregg et al. (1993) and long-term current meter data from Chereskin et al. (2000), and PATCHEX² is an estimate from a two-dimensional displacement spectrum appearing in Sherman and Pinkel (1991). Further details and a regional characterization of these data appear in PL. Finally, power laws of a two-dimensional vertical wavenumber–frequency spectrum, e(m, ω) ∝ ω^−cm^−d, correspond to the power laws of a three-dimensional vertical wavenumber–horizontal wavenumber action spectrum n(k, m) ∝ k^−am^−b with the mapping,

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Figure 1 suggests that the data points are not randomly distributed 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.

c. Hamiltonian structure and wave turbulence theory

This subsection briefly summarizes the derivation in Lvov and Tabak (2001, 2004); it is included here only for completeness and to allow references from the core of the paper.

The equations of motion satisfied by an incompressible stratified rotating flow in hydrostatic balance under the Boussinesq approximation are (Cushman-Roisin 1994)

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These equations result from mass conservation, horizontal-momentum conservation, and hydrostatic balance. The equations are written in isopycnal coordinates with the density ρ replacing the height z in its role as independent vertical variable. Here u = (u, υ) is the horizontal component of the velocity field; u^⊥ = (−υ, u); ∇ = (∂/∂x, ∂/∂y) is the gradient operator along isopycnals; M is the Montgomery potential,

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f is the Coriolis parameter; and ρ₀ is a reference density in its role as inertia, considered constant under the Boussinesq approximation.

The potential vorticity is given by

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where ∏ = ρ/g∂²M/∂ρ² = ρ∂z/∂ρ is a normalized differential layer thickness. Because both the potential vorticity and the fluid density are conserved along particle trajectories, an initial profile of the potential vorticity that is a function of the density will be preserved by the flow. Hence, it is self-consistent to assume that

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where ∏₀(ρ) = −g/N(ρ)² is a reference stratification profile with constant background buoyancy frequency, N = (−g/(ρ∂z/∂ρ|_bg))^1/2, independent of x and y. This assumption is not unrealistic: it represents a pancake-like distribution of potential vorticity, the result of its comparatively faster homogenization along than across isopycnal surfaces.

It is shown in Lvov and Tabak (2001, 2004) that the primitive equations of motion (6) under the assumption (8) can be written as a pair of canonical Hamiltonian equations,

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where ϕ is the isopycnal velocity potential and the Hamiltonian is the sum of kinetic and potential energies,

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Here, ∇^⊥ = (−∂/∂y, ∂/∂x), Δ⁻¹ is the inverse Laplacian, and ρ′ represents a variable of integration rather than perturbation.

Switching to Fourier space and introducing a complex field variable c_p through the transformation

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where the frequency ω satisfies the linear dispersion relation²

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the equations of motion (6) adopt the canonical form

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with Hamiltonian

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This is the standard form of the Hamiltonian of a system dominated by three-wave interactions (Zakharov et al. 1992). Calculations of interaction coefficients are tedious but straightforward tasks, completed in Lvov and Tabak (2001, 2004). These coefficients are given by

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where [(0, 1, 2) → (1, 2, 0)] and [(0, 1, 2) → (2, 0, 1)] denote exchanges of suffixes and, for two dimensional vector k = (k_x, k_y), k^⊥ = (−k_y, k_x).³ We stress that the field Eq. (13) with the three-wave Hamiltonian [(12), (14), and (15)] is equivalent to the primitive equations of motion for internal waves (6). The approach using a Lagrangian coordinate system is based on small-amplitude expansion to arrive at this type of equation.

In wave turbulence theory, one proposes a perturbation expansion in the amplitude of the nonlinearity, yielding linear waves at the leading order. Wave amplitudes are modulated by the nonlinear interactions, and the modulation is statistically described by a kinetic equation (Zakharov et al. 1992) for the wave action n_p defined by

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Here 〈·〉 denotes an ensemble averaging: that is, averaging over many realizations of the random wave field. The derivation of this kinetic equation is well studied and understood (Zakharov et al. 1992; Lvov and Nazarenko 2004). For the three-wave Hamiltonian (14), the kinetic equation is the one in Eq. (1), describing general internal waves interacting in both rotating and nonrotating environments.

The delta functions in the kinetic equation ensures that spectral transfer happens on the resonant manifold, which is defined as

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Now let us assume that the wave action n is independent of the direction of the horizontal wavenumber and is symmetric with respect to m → −m change,

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Note that value of the interaction matrix element is independent of horizontal azimuth because it depends only on the magnitude of interacting wavenumbers. Therefore, one can integrate the kinetic Eq. (1) over horizontal azimuth (Zakharov et al. 1992), yielding

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Here, appears as the result of integration of the horizontal-momentum conservative delta function over all possible orientations and is equal to the area of the triangle with sides with the length of the horizontal wavenumbers k = |k|, k₁ = |k₁| and k₂ = |k₂|. This is the form of the kinetic equation that will be used to find scale-invariant solutions in the next section.

3. Scale-invariant kinetic equation

a. Reduction of kinetic equation to the resonant manifold

In the high-frequency limit ω ≫ f, one could conceivably neglect the effects of the rotation of the earth. The dispersion relation (12) then becomes (Lvov and Tabak 2001)

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and, in this limit, the matrix element (15) retains only its first term, I_p,p1,p2.

The azimuthally integrated kinetic Eq. (18) includes integration over k₁ and k₂, because the integrations over m₁ and m₂ can be done by using delta functions. To use delta functions, we need to perform what is called reduction to the resonant manifold. Consider, for example, resonances of type (17a). Given k, k₁, k₂, and m, one can find m₁ and m₂ satisfying the resonant condition by solving simultaneous equations

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The solutions of this quadratic equation are given by

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Note that Eq. (21a) translates into Eq. (21b) if the indices 1 and 2 are exchanged. Indeed, exchanging indices 1 and 2 in Eq. (21a) we obtain

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which simplifies then to m₁ of Eq. (21b).

Similarly, resonances of type (17b) yield

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and resonances of type (17c) yield

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After this reduction, a double integral over k₁ and k₂ is left. The domain of integration is further restricted by the triangle inequalities

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These conditions ensure that one can construct a triangle out of the wavenumbers with lengths k, k₁, and k₂ and determine the domain in the (k₁, k₂) plane called the kinematic box in the oceanographic literature.

Numerical evaluation of the collision integral is a complicated but straightforward task. Interpretation of the results, though, is more difficult, mostly because of the complexity of the interaction matrix element and the nontrivial nature of the resonant set. Starting with McComas and Bretherton (1977), therefore, predictions were made based on a further simplification. This simplification is based on the assertion that interactions between wavenumbers with extreme scale separation contribute mostly to the nonlinear dynamics. Three main classes of such resonant triads appear, characterized by extreme scale separation. These three main classes are as follows:

• The vertical backscattering of a high-frequency wave by a low-frequency wave of twice the vertical wavenumber into a second high-frequency wave of oppositely signed vertical wavenumber: this type of scattering, as in Eqs. (25a), (27b), (29a), and (30a) below, is called elastic scattering (ES).

• The scattering of a high-frequency wave by a low-frequency, small-wavenumber wave into a second, nearly identical, high-frequency, large-wavenumber wave: this type of scattering, as in Eqs. (25b), (27a), (29b), and (30b) below, is called induced diffusion.

• The decay of a small-wavenumber wave into two large-vertical-wavenumber waves of approximately one-half the frequency: this type of scattering, as in Eqs. (26a), (26b), (28a), (28b) below, is called parametric subharmonic instability (PSI).

To see how this classification appears analytically, we perform the limit of k₁ → 0 and the limit k₁ → ∞ in Eqs. (21)–(23). We will refer to the k₁ or k₂ → 0 limits as IR limits, whereas the k₁ and k₂ → ∞ limit will be referred as a UV limit. Because the integrals in the kinetic equation for power-law solutions will be dominated by the scale-separated interaction, this will help us analyze possible solutions to the kinetic equation.

The results of the k₁ → 0 limit of Eqs. (21)–(23) are given by

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We now see that the interactions (25a) and (27b) correspond to the ES mechanism, that the interactions (25b) and (27a) correspond to the ID, and that the interactions (26a) and (26b) correspond to the PSI.

Similarly, taking the k₁ and k₂ → ∞ limits of Eqs. (21)–(23), we obtain

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We now can identify the interactions (28a) and (28b) as being PSI, the interactions (29a) and (30a) as being ES, and the interactions (29b) and (30b) as being ID.

This classification provides an easy and intuitive tool for describing extremely scale-separated interactions. We will see below that one of these interactions, namely ID, explains reasonably well the experimental data that is available to us.

b. Convergences and divergences of the kinetic equation

Neglecting the effects of the rotation of the earth yields a scale-invariant system with dispersion relation given by Eq. (19) and matrix element given only by the I_p,p1,p2 in Eq. (15). This is the kinetic equation of Lvov and Tabak (2001), Lvov et al. (2004), and Lvov and Tabak (2004), describing internal waves in hydrostatic balance in a nonrotating environment.

Proposing a self-similar separable spectrum of the form (2), one can show from the azimuthally integrated kinetic Eq. (18) that (Zakharov et al. 1992)

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for constants α and β. To find a steady scale-invariant solution for all values of k and m, it is therefore sufficient to find exponents that give a 0 collision integral for one wavenumber. One can fix k and m, adopting for instance k = m = 1, and seek 0 of the collision integral (represented as below) as a function of a and b,

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Before embarking on numerical or analytical integration of the kinetic Eq. (18) with scale-invariant solutions (2), it is necessary to check whether the collision integral converges. Appendix A outlines these calculations. The condition for the scale-invariant collision integral (32) to converge at the IR end, k₁ or k₂ → 0, is given by

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Similarly, UV convergence as k₁ and k₂ → ∞ implies that

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The domains of divergence and convergence are shown in Fig. 2.

Figure 2 also displays the classes of triads dominating the interactions. Knowing the classes of interactions that lead to the divergences of the kinetic equation allows us to find possible physical scenarios of the convergent solutions or to find a possible physical regularization of the divergences.

Note that, in addition to the two-dimensional domain of IR convergence [the regions (33a), (33d), and (33e)], there are two additional IR convergent line segments given by Eqs. (33b) and (33c). These two special line segments appear because of the b(b − 1) prefactor to the divergent contributions to the collision integral (A3). Similarly, for the UV limit, in addition to the two-dimensional region of convergence [(33a), (34c), and (34d)], there is an additional special line segment of b = 0 [(34b)].

We see that these domains of convergence overlap only on the segment

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This result contradicts the one in Lvov et al. (2004), where a coding error led us to believe that the integrals converge and that GM is an exact steady-state solution to the scale-invariant kinetic equation. Note that b = 0 corresponds to wave action independent of vertical wavenumbers, ∂n/∂m = 0. Existence of the b = 0 line will allow us to find novel convergent solution in section 4. We also note that the IR segment on b = 0 coincides with one of the ID solutions determined in section 6. The other segment on b = 1 does not coincide with the ID solution in section 6, because the scale-invariant system has higher symmetry than the system with Coriolis effect.

4. 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 Eq. (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 section 3a.

The result of this numerical integration is shown in Fig. 3. The figure clearly shows the existence of a steady solution of the kinetic Eq. (18) near a ≅ 3.7 and 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 and how this solution is modified by inclusion of background rotation.

5. Balance between divergences

The fact that the collision integral diverges for almost all values of a and b can be viewed both as a challenge and as a blessing. On one hand, it makes the prediction of steady spectral slopes more difficult, because 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.

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: that is, 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 corresponds to the UV contribution.

Hence, in the regions,

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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 Eq. (1) by applying the Zakharov–Kuznetsov conformal mapping for systems with cylindrical symmetry (Zakharov 1968, 1967; Kuznetsov 1972). This Zakharov–Kuznetsov conformal mapping effectively establishes a map between the neighborhoods of zero and infinity. Making these two contributions cancel pointwise 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. Some other transformation, as we explain below, may lead to completely different solutions.

The PR spectrum was first found by Pelinovsky and Raevsky (1977). However, they realized that it was only a formal solution. The solution was found again in Caillol and Zeitlin (2000) through a renormalization argument and in Lvov and Tabak (2001, 2004) 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 conformal mapping that makes the divergences at zero and infinity cancel each other, as does the Zakharov–Kuznetsov transformation. Such generalized Zakharov–Kuznetsov transformation is an extension of the idea of principal value for a divergent integral, whereby two divergent contributions are made to cancel each other through a specific relation between their respective contributions.

Indeed, in the ocean internal waves cannot have zero horizontal wavenumber. Rather, a smallest horizontal wavenumber exists, which corresponds to largest horizontal scales of internal waves. The largest wavenumber that was observed is on the order of thousands of kilometers. On larger scales, β effects become important, and they prevent wavenumbers to achieve even smaller scales. Other effects possibly affecting small wavenumbers include ocean storms, interactions with large scale vortices and shear, and ocean boundaries. Similarly, there is no infinitely large wavenumber for internal waves, but rather there is possibly large horizontal wavenumber that is affected by wave breaking, interaction with the turbulence, and other processes.

The idea of a generalized Zakharov–Kuznetsov transformation leading to an infinite number of steady states provides a 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 translates into a certain degree of nonuniversality. Such players include storms and tides, as well as possible geometrical constrains and interactions with large-scale shear and vortices. Investigation of the nature of possible balances between IR and UV divergences is outside of scope of the present paper.

6. Regularization by the Coriolis effect

Physically, the ocean does not perform generalized Zakharov–Kuznetsov transformation. However, in the ocean there are finite boundaries in the frequency domain. In particular, the inertial frequency f provides a truncation for the IR part of the spectrum, whereas the UV truncation is provided by the buoyancy frequency N. These two frequencies vary from place to place, giving grounds for spectral variability. Consequently, the integrals are not truly divergent, but rather they have a large numerical value dictated by the location of the IR cutoff.

Note that the f → 0 calculations presented in this section describe intermediate frequencies ω such that ω ≫ f > 0. Consequently, these intermediate frequencies feel inertial frequency as being small. These calculations will not be applicable for equatorial regions, where f is identically zero.

Observe that all the experimental points are located in regions of the (a, b) domain a > (−b + 7)/2 for which the integral diverges in the IR region. Also note that 5 out of 12 experimental points are located in the region where collision integrals are UV convergent: that is, in a > −b/2 + 4. The UV region is therefore assumed to be either subdominant or convergent in this section, where we study the regularization resulting from a finite value of f.

We note that, because we consider scale-invariant case only in the present paper, the IR cutoff can equivalently be considered as k approaching smallest possible value or equivalently when ω approaches its smallest value. Because the IR cutoff is given by f, a frequency, it is easier to analyze the resulting integral in (ω₁, m₁) rather than in the more traditional (k₁, m₁) domain utilized in Müller et al. (1986) and the previous sections. We emphasize that the present paper concerns itself with scale-invariant wave action spectrum (2). The scale-invariant wave action can be translated from (k₁, m₁) to (ω₁, m₁) domain with out loss of generality. This statement would not be true for the realistic Garrett–Munk spectrum or any other oceanic spectrum because of the non-scale-invariant form of the linear dispersion relation (12). Such spectra will be analyzed in subsequent publication (LPY).

Therefore, to proceed, we assume a power-law spectrum, similar to Eq. (2), but in the (ω, m) space,

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We need to transform the wave action as a function of k and m to a function of ω and m. This is done in appendix B. The relation between a, b and ã, b̃ reads as

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Thus, we need to express both the kinetic equation and the kinematic box in terms of frequency and vertical wavenumber. For this, we use the dispersion relation (12) to express k in terms of ω in the description of the kinematic box (24),

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where we have introduced the four curves in the (ω₁, m₁) domain that parameterize the kinematic box,

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The kinematic box in the (ω, m) domain is shown in Fig. 5. Note that the region (38b) and (38c) can be transferred to each other by interchanging indices 1 and 2; consequently, two disconnected ω₁ < ω regions look like mirrored and shifted copies of each other. To help in the transition from the traditional kinematic box (24) to the kinematic box in (ω, m) domain, the following limits were identified:

• ID1 is the ID limit of Eq. (25b) with indices 1 and 2 being flipped;

• ID2 is the ID limit of Eq. (27a) with indices 1 and 2 being flipped;

• ID3 is the ID limit of Eq. (25b);

• ID4 is the ID limit of Eqs. (29b) and (30b);

• PSI1 is the PSI limit of Eq. (28a);

• PSI2 is the PSI limit of Eq. (28b);

• PSI3 is the PSI limit of Eq. (26a);

• PSI4 is the PSI limit of Eq. (26b);

• ES1 is the ES limit of Eq. (25a);

• ES2 is the ES limit of Eq. (27b) with indices 1 and 2 being flipped;

• ES3 is the ES limit of Eq. (27b) with indices 1 and 2 being flipped; and

• ES4 is the ES limit of Eq. (29a).

Equation (18) transforms into

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We have used the dispersion relation k_i = m_i and defined J as the Jacobian of the transformation from (k₁, k₂) into (ω₁, ω₂) times the kk₁k₂ factor,

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In Fig. 5, there are three ID regions (or corners) with significant contribution to the collision integral in the IR limit:

1. ID1 region: In this region, m₁ is slightly bigger than m, ω₁ is slightly smaller than ω, and ω₂ and m₂ are both very small. This region can be obtained from the region (38b) above by interchanging indices 1 and 2. In this region, the wave action n₂ is much smaller than wave action n and n₁:

   

   

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2. ID2 region: In this region, ω₁ is slightly bigger than ω + f, where ω₂ = ω₁ − ω is small and m₂ = m₁ − m is negative and small. This is the region (38d). Also,

   

   

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3. ID3 region: In this region, ω₁ is small and m₁ is small and negative. This corresponds to the region (38b), where wave action obeys

   

   

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   Note that this region can be obtained from the region ID1 by flipping indices 1 and 2. Consequently, only one of the ID1 and ID3 should be taken into account, with a factor of 2 multiplying the respective contribution. This also can be seen from Fig. 5 as ID1 and ID3 regions are shifted mirror images of each other.

Making these simplifications and taking into account the areas of integration in the kinematic box, we obtain

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where the small parameter ω_s is introduced to restrict the integration to a neighborhood of the ID corners. The arbitrariness of the small parameter will not affect the end result below.

To quantify the contribution of near-inertial waves to a (ω, m) mode, we write

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Subsequently, near the region ID3 of the kinematic box, we write

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whereas near the ID2 corners of the kinematic box we write

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We then expand the resulting analytical expression (40) in powers of ϵ and f without making any assumptions on their relative sizes. These calculations, including the integration over vertical wavenumbers m₁, are presented in appendix C. The resulting expression for the kinetic equation is given by

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The integral over ϵ diverges at ϵ = 0, if f = 0 and if 6 + ã + b̃ > −1.⁴

However, if we postpone taking f = 0 limit, we see that the integral is 0 to leading order if

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or, in terms of a and b,

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This is the family of power-law steady-state solutions to the kinetic equations dominated by infrared ID interactions. These steady states are identical to the ID stationary states identified by McComas and Bretherton (1977), who derived a diffusive approximation to their collision integral in the infrared ID limit. Note that McComas and Müller (1981) interpreted b = 0 as a no-action flux in vertical wavenumber domain, whereas 9 − 2a − 3b = 0 is a constant action flux solution.

We note that one can use the eikonal approach to describe these types of interactions (Flatté et al. 1985; Broutman et al. 2004; Müller et al. 1986; Henyey et al. 1986). An advantage of the eikonal approach versus our scale-invariant analysis is that it allows us to consider not only scale-invariant interactions in separable spectrum. The possible disadvantage of the eikonal approach is that construction of a transport theory is far less rigorous. For a more detailed discussion on the differences between the resonant interaction approximation and the eikonal approach, we refer the reader to Polzin (2004).

What is presented in this section is a rigorous asymptotic derivation of Eq. (43). These ID solutions help us to interpret observational data of Fig. 1 that are currently available to us. The value added associated with a rigorous asymptotic derivation is the demonstration that the ID stationary states are meaningful only in the IR divergent part of the (a, b) domain.

7. 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.

A nonrotating scale-invariant analysis provides two subdomains in which the kinetic equation converges in either the UV or the IR limit (light gray shading), two subdomains in which the kinetic equation diverges in both UV and IR limits but with oppositely signed values (dark gray shading) and a single domain with similarly signed UV and IR divergences (black shading). In this nonrotating analysis, a stationary state is possible only for oppositely signed divergences: that is, within the dark gray shaded regions. Six of the observational points lie in IR and UV divergent subdomains, and a seventh (Site-D) is on the boundary with the IR divergent–UV convergent subdomain. Two of the observational points lie in the domain of IR and UV divergence having similar signs, which does not represent a possible solution in the nonrotating analysis. These two points lie close to the boundaries of the “forbidden” (black shaded) region, and subtracting the vortical contribution from one (NATRE¹) returns a “best” estimate of the internal-wave spectrum (NATRE²) that lies outside of the forbidden region.

All the data lie in an IR divergent regime and hence a regularization of the kinetic equation is performed by including a finite lower frequency of f. This produces a family of stationary states, the induced diffusion stationary states. These stationary states collapse much of the observed variability. The exception is the NATRE spectrum.

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 at zero, at infinity, or at both ends. This means that, in the wave turbulence kinetic equation framework, the energy transfer is dominated by the scale-separated interactions with large and/or small scales.

The only exception where the integral converges is a segment of a line, , 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 are two subdomains of power-law exponents that 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 approximately balance each other. The Pelinovsky–Raevsky spectrum is a special case of this scenario.

The 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.

This possibility was further investigated by introducing rotation and then pursuing a rigorous asymptotic expansion of the kinetic equation. In doing so, we obtain the induced diffusion stationary states that appear as white lines in Fig. 6, which had previously been determined through a diffusive approximation. Much of the observed oceanic variability lies about these stationary states in the IR divergent subdomain.

A more detailed review of available observational data used for this study appears in PL. Numerical evaluation of the complete non-scale-invariant kinetic equation of the Garrett–Munk spectrum is presented in LPY in which we also consider waves that are slightly off resonant interactions. The theory, experimental data, and results of numerical simulations in Lvov and Yokoyama (2009) all 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, such as the local value of the Coriolis parameter and specific features of the spectrum, specifically for near-inertial frequencies.