## 1. Introduction

Wave–wave interactions in stratified oceanic flows have been a fascinating subject of research in the last four decades. Of particular importance is the existence of a “universal” internal wave spectrum, the Garrett and Munk (GM; (Garrett and Munk 1972, (1975, (1979) spectrum. It is generally perceived that the existence of a universal spectrum is, at least in part and perhaps even primarily, the result of nonlinear interactions of waves with different wavenumbers. Because of the quadratic nonlinearity of the underlying primitive equations and the fact that the linear internal wave dispersion relation can satisfy a three-wave resonance condition, waves interact in triads. Therefore, the question arises, how strongly do waves within a given triad interact? What are the oceanographic consequences of this interaction?

Wave–wave interactions can be rigorously characterized by deriving a closed equation representing the slow time evolution of the wave field’s wave action spectrum. Such an equation is called a kinetic equation (Zakharov et al. 1992), and significant efforts in this regard are listed in Table 1.

A list of various kinetic equations. Results from Olbers (1976), McComas and Bretherton (1977), and Pomphrey et al. (1980) are reviewed in Müller et al. (1986), who state that Olbers (1976), McComas and Bretherton (1977), and an unspecified Eulerian representation are consistent on the resonant manifold. Pomphrey et al. (1980) utilizes Langevin techniques to assess nonlinear transports. Müller et al. (1986) characterizes those Langevin results as being mutually consistent with the direct evaluations of kinetic equations presented in Olbers (1976) and McComas and Bretherton (1977). Kenyon (1968) states (without detail) that Kenyon (1966) and Hasselmann (1966) give numerically similar results. A formulation in terms of discrete modes will typically permit an arbitrary buoyancy profile, but obtaining results requires specification of the profile. Of the discrete formulations, Pomphrey et al. (1980) use an exponential profile and the others assume a constant stratification rate.

**p**,

**p**

_{1}, and

**p**

_{2}that satisfy

*ω*is given by a linear dispersion relation relating wave frequency

*ω*with wavenumber

**p**.

The reduction of all possible interactions between three wave vectors to a resonant manifold is a significant simplification. Even further simplification can be achieved by taking into account that, of all interactions on the resonant manifold, the most important are those that involve extreme scale separations (McComas and Bretherton 1977) between interaction wave vectors. It is shown in McComas (1977) that the high-frequency portion of the Garrett and Munk internal wave spectrum is stationary with respect to one class of such interactions, called induced diffusion (ID). Furthermore, a comprehensive inertial-range theory with constant downscale transfer of energy was obtained by patching these mechanisms together in a solution that closely mimics the empirical universal spectrum (GM) (McComas and Müller 1981a). It was therefore concluded that that Garrett and Munk spectrum constitutes an approximate stationary state of the kinetic equation.

In this paper, we revisit the question of relation between Garrett and Munk spectrum and the resonant kinetic equation. At the heart of this paper (section 6a) are numerical evaluations of the Lvov and Tabak (2004) internal wave kinetic equation demonstrating changes in spectral amplitude at a rate greater than an inverse wave period at high vertical wavenumber for the Garrett and Munk spectrum. This rapid temporal evolution implies that the GM spectrum is not a stationary state and is contrary to the characterization of the GM spectrum as an inertial subrange. This result gave us cause to review published work concerning wave–wave interactions and compare results. The product of this work is presented in sections 3 and 4. In particular, we concentrate on four different versions of the internal wave kinetic equation:

a noncanonical description using Lagrangian coordinates (Olbers 1974, 1976; Müller and Olbers 1975),

a canonical Hamiltonian description in Eulerian coordinates (Voronovich 1979),

a dynamical derivation of a kinetic equation without use of Hamiltonian formalisms in Eulerian coordinates (Caillol and Zeitlin 2000), and

a canonical Hamiltonian description in isopycnal coordinates (Lvov and Tabak 2001, 2004).

Having clarified this, we proceed to the following observation: Not only do our numerical evaluations imply that the GM spectrum is not a stationary state, the rapid evolution rates correspond to a strongly nonlinear system. Consequently, the self-consistency of the kinetic equation, which is built on an assumption of weak nonlinearity, is at risk. Moreover, reduction of all resonant wave–wave interactions exclusively to extreme scale separations is also not self-consistent.

However, we are not willing to give up on the kinetic equation. Our second paradox is that, in a companion paper (Lvov et al. 2010), we show how a comprehensive theory built on a scale-invariant resonant kinetic equation helps to interpret the observed variability of the background oceanic internal wave field. The observed variability, in turn, is largely consistent with the induced diffusion mechanism being a stationary state.

We conclude and list open questions in section 8. Our numerical scheme for evaluating near-resonant interactions is discussed in section 5. An appendix contains the interaction matrices used in this study.

## 2. Background

A kinetic equation is a closed equation for the time evolution of the wave action spectrum in a system of weakly interacting waves. It is usually derived as a central result of wave turbulence theory. The concepts of wave turbulence theory provide a fairly general framework for studying the statistical steady states in a large class of weakly interacting and weakly nonlinear many-body or many-wave systems. In its essence, classical wave turbulence theory (Zakharov et al. 1992) is a perturbation expansion in the amplitude of the nonlinearity, yielding, at the leading order, linear waves, with amplitudes slowly modulated at higher orders by resonant nonlinear interactions. This modulation leads to a redistribution of the spectral energy density among space and time scales.

Although the route to deriving the spectral evolution equation from wave amplitude is fairly standardized (section 2b), there are substantive differences in obtaining expressions for the evolution equations of wave amplitude *a*. Section 2a describes how it is done in isopycnal coordinates in Lvov and Tabak (2001, 2004) and in the appendix for all other methods discussed in the present paper.

### a. Hamiltonian structures and field variables in isopycnal coordinates

**u**is then represented as (Lelong and Riley 1992, 2000)

^{⊥}= (−∂/∂

*y*, ∂/∂

*x*), and a normalized differential layer thickness is introduced,

*q*is function of

*ρ*only, independent of

*x*and

*y*,

_{0}(

*ρ*) = −

*g*/

*N*(

*ρ*)

^{2}is a reference stratification profile with background buoyancy frequency,

*N*= [−

*g*/(

*ρ*∂

*z*/∂

*ρ*|

_{bg})]

^{1/2}, independent of

*x*and

*y*. The variable

*ψ*can then be eliminated by assuming that potential vorticity is constant on an isopycnal so that

*f*+ Δ

*ψ*=

*q*

_{0}Π and one obtains two equations in Π and

*ϕ*,

^{−1}is the inverse Laplacian and

*ρ*′ represents a variable of integration rather than perturbation. Serendipitously, the variable Π is the canonical conjugate of

*ϕ*,

*a*

**through the transformation**

_{p}*ω*satisfies the linear dispersion relation

Equation (10) is Hamilton’s equation and (11) is the standard form of the Hamiltonian of a system dominated by three-wave interactions (Zakharov et al. 1992). Calculations of interaction coefficients *U* and *V* are a tedious but straightforward task, completed in Lvov and Tabak (2001, 2004). The result of this calculation is also presented in the appendix in Eq. (A21).

We emphasize that (10) is, with simply a Fourier decomposition and assumption of uniform potential vorticity on an isopycnal, precisely equivalent to the fully nonlinear equations of motion in isopycnal coordinates (2). All other formulations of an internal wave kinetic equation considered here depend upon a linearization prior to the derivation of the kinetic equation via an assumption of weak nonlinearity.

The difficulty is that, in order to utilize Hamilton’s equation (10), the Hamiltonian (7) must first be constructed as a function of the generalized coordinates and momenta (Π and *ϕ* here). It is not always possible to do so directly, in which case one must set up the associated Lagrangian (

### b. Wave turbulence

*n*

**=**

_{p}*n*(

**p**) is a three-dimensional wave action spectrum (spectral energy density divided by frequency) and the interacting wave vectors

**p**,

**p**

_{1}, and

**p**

_{2}are given by

**k**is the horizontal part of

**p**and

*m*is its vertical component). Furthermore, 〈…〉 indicates the averaging over the statistical ensemble of many realizations of the internal waves.

*n*

**we multiply the amplitude equation (10) with the Hamiltonian given by (11) by**

_{p}*a*

**. We then subtract the two equations and average 〈…〉 the result. We get**

_{p}*n*

**/∂**

_{p}*t*therefore requires computing

^{1}

^{2}), and we use

*ω*

**are restricted to be positive. The magnitude of wave–wave interactions**

_{p}We reiterate that typical assumptions needed for the derivation of kinetic equations are

weak nonlinearity;

Gaussian statistics of the interacting wave field in wavenumber space; and

resonant wave–wave interactions.

### c. The Boltzmann rate

The normalized Boltzmann rate serves as a low-order consistency check for the various kinetic equation derivations. An *O*(1) value of *ϵ*** _{p}** implies that the derivation of the kinetic equation is internally inconsistent. The Boltzmann rate represents the net rate of transfer for wavenumber

**p**. The individual rates of transfer into and out of

**p**(called Langevin rates) are typically greater than the Boltzmann rate (Müller et al. 1986; Pomphrey et al. 1980). This is particularly true in the induced diffusion regime (defined below in section 3) in which the rates of transfer into and out of

**p**are one to three orders of magnitude larger than their residual and the Boltzmann rates we calculate are not appropriate for either spectral spikes or potentially for smooth, homogeneous but anisotropic spectra (Müller et al. 1986). Estimates of the individual rates of transfer into and out of

**p**can be addressed through Langevin methods (Pomphrey et al. 1980). We focus here simply on the Boltzmann rate to demonstrate inconsistencies with the assumption of a slow time evolution. Estimates of the Boltzmann rate and

*ϵ*

**require integration of (22). In this manuscript, such integration is performed numerically.**

_{p}## 3. Resonant wave–wave interactions: Nonrotational limit

How can one compare the function of two vectors **p**_{1} and **p**_{2} and their sum or difference? First one realizes that, out of six components of **p**_{1} and **p**_{2}, only relative angles between wave vectors enter into the equation for matrix elements. That is because the matrix elements depend on the inner and outer products of wave vectors. The overall horizontal orientation of the wave vectors does not matter: relative angles can be determined from a triangle inequality and the magnitudes of the horizontal wave vectors **k**, **k**_{1}, and **k**_{2}. Thus, the only needed components are |**k**|, |**k**_{1}|, |**k**_{2}|, *m*, and *m*_{1} (*m*_{2} is computed from *m* and *m*_{1}). Further note that, in the *f* = 0 and hydrostatic limit, all matrix elements become scale-invariant functions. It is therefore sufficient to choose an arbitrary scalar value for |**k**| and *m*, because only |**k**_{1}|/|**k**|, |**k**_{2}|/|**k**|, and *m*_{1}/*m* enter the expressions for matrix elements. We make the particular (arbitrary) choice that |**k**| = *m* = 1 for the purpose of numerical evaluation,^{3} and thus the only independent variables to consider are |**k**_{1}|, |**k**_{2}|, and *m*_{1}. Finally, *m*_{1} is determined from the resonance conditions, as explained in the next subsection below. As a result, we are left with a matrix element as a function of only two parameters, *k*_{1} and *k*_{2}. This allows us to easily compare the values of matrix elements on the resonant manifold by plotting the values as a function of the two parameters.

### a. Reduction to the resonant manifold

*m*

_{1}and

*m*

_{2}of the interacting wave vectors, one has to solve the resulting quadratic equations. Without restricting generality, we choose

*m*> 0. There are two solutions for

*m*

_{1}and

*m*

_{2}given below for each of the three resonance types described above.

### b. Comparison of matrix elements

*f*= 0 and hydrostatic balance. Such a choice makes the matrix element scale-invariant functions that depend only upon |

**k**

_{1}| and |

**k**

_{2}|. As a consequence of the triangle inequality, we need to consider matrix elements only within a “kinematic box” defined by

*V*(|

**k**

_{1}| = 1, |

**k**

_{2}| = 1)|

^{2}= 1. This allows a transparent comparison without worrying about dimensional differences between various formulations.

#### 1) Resonances of the sum type [(25a)]

Figure 1 presents the values of the matrix element **k**_{1}| and |**k**_{2}| exchanged because of their symmetries.

#### 2) Resonances of the difference type [(25b) and (25c)]

We then turn our attention to resonances of difference type (25b) for which (25c) could be obtained by symmetrical exchange of the indices. All the matrix elements **k**_{1}| and |**k**_{2}| exchanged as the solutions (27a) and (27b) because of their symmetries.

#### 3) Special triads

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 and nearly the same wavenumber magnitude. This type of scattering is called elastic scattering. The solution (26a) in the limit |

**k**_{1}| → 0 corresponds to this type of special triad.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 is called induced diffusion. The solution (26b) in the limit that |

**k**_{1}| → 0 corresponds to this type of special triad.The decay of a low-wavenumber wave into two high vertical wavenumber waves of approximately one-half the frequency. This is called parametric subharmonic instability (PSI). The solution (27a) in the limit that |

**k**_{1}| → 0 corresponds to this type of triad.

**k**

_{1}|, |

**k**

_{2}|) in such a way that they span a special triad case. We choose one such particular parameterization; that is, we choose

**k**

_{1}|, |

**k**

_{2}|) = (0, |

**k**|) and has a slope of ⅓. The slope of this line is arbitrary. We could have taken

*ϵ*/4 or

*ϵ*/2. The matrix elements here are shown as functions of

*ϵ*in Fig. 4. We see that all four approaches are again equivalent on the resonant manifold for the case of special triads.

In this section, we demonstrated that all four approaches we considered produce equivalent results on the resonant manifold in the absence of background rotation. This statement is not trivial, given the different assumptions and coordinate systems that have been used for the various kinetic equation derivations.

## 4. Resonant wave–wave interactions: In the presence of background rotations

In the presence of background rotation, the matrix elements lose their scale invariance because of the introduction of an additional time scale (1/*f*) in the system. Consequently, the comparison of matrix elements is performed as a function of four independent parameters.

*ω*,

*ω*

_{1},

*m*, and

*m*

_{1},

*ω*

_{2}, and

*m*

_{2}can be calculated by requiring that they satisfy the resonant conditions

*ω*=

*ω*

_{1}+

*ω*

_{2}and

*m*=

*m*

_{1}+

*m*

_{2}. We then can check whether the corresponding horizontal wavenumber magnitudes

*k*, given by

^{12}points on the resonant manifold. After being multiplied by an appropriate dimensional number to convert between Eulerian and isopycnal coordinate systems, the two matrix elements coincide up to machine precision.

One might, with sufficient experience, regard this as an intuitive statement. It is, however, far from trivial given the different assumptions and coordinate representations. In particular, we note that derivations of the wave amplitude evolution equation in Lagrangian coordinates (Olbers 1976; McComas 1975; Meiss et al. 1979) do not explicitly contain a potential vorticity conservation statement corresponding to assumption (4) in the isopycnal coordinate (Lvov and Tabak 2004) derivation. We have inferred that the Lagrangian coordinate derivation conserves potential vorticity as that system is projected upon the linear modes of the system having zero perturbation potential vorticity.

## 5. Resonance broadening and numerical methods

### a. Nonlinear frequency renormalization as a result of nonlinear wave–wave interactions

The resonant interaction approximation is a self-consistent mathematical simplification, which reduces the complexity of the problem for weakly nonlinear systems. As nonlinearity increases, near-resonant interactions become more and more pronounced and need to be addressed. Moreover, near-resonant interactions play a major role in numerical simulations on a discrete grid (Lvov et al. 2006), for time evolution of discrete systems (Gershgorin et al. 2007), in acoustic turbulence (Lvov et al. 1997), surface gravity waves (Janssen 2003; Yuen and Lake 1982), and internal waves (Voronovich et al. 2006; Annenkov and Shrira 2006).

_{k}_{12}is the total broadening of each particular resonance and is given below in (32) and (33).

The difference between kinetic equation (22) and the generalized kinetic equation (30) is that the energy-conserving delta functions in (22),

_{k}_{12}→ 0),

*n*

**on the right-hand side. Those terms can be loosely interpreted as a nonlinear wave damping acting on the given wavenumber,**

_{p}*γ*

**in (32) corresponds to the renormalization or dressing of bare dumping by the nonlinear dumping that appears as a result of wave–wave interactions. This methodology is well studied in the context of diagrammatic technique (Lvov et al. 1997). Consequently, the**

_{p}A rigorous derivation of the kinetic Eq. (30) with a broadened delta function (31)–(33) is given in detail for a generic three-wave Hamiltonian system in Lvov et al. (1997). The derivation is based upon the Wyld diagrammatic technique for nonequilibrium wave systems and utilizes the Dyson–Wyld line resummation. This resummation permits an analytical resummation of the infinite series of reducible diagrams for Greens functions and double correlators. We emphasize however that the approach is perturbative in nature and that there are neglected parts of the infinite diagrammatic series.

*γ*

**requires an iterative solution of (30) and (32) over the entire field: The width of the resonance (32) depends on the lifetime of an individual wave [from (30)], which in turn depends on the width of the resonance (33). This numerically intensive computation is beyond the scope of this manuscript. Instead, we make the uncontrolled approximation that**

_{p}*C*is the dimensionless constant that defines how strongly a particular frequency gets broadened by nonlinear wave–wave interactions.

We note the choice (34) is made for illustration purposes only, we certainly do not claim it to be self consistent. Below, we will take *C* to be 10^{−3}, 10^{−2}, and 10^{−1}. These values are rather small; therefore, we remain in the closest proximity to the resonant interactions. To show the effect of strong resonant manifold smearing, we also investigate the case with *C* = 0.5.

We note in passing that the near-resonant interactions of the waves were also considered in Janssen (2003). There, instead of our *πx*)/*x*. We have shown in Kramer et al. (2003) that the resulting kinetic equation does not retain positive definite values of wave action. To get around that difficulty, self-consistent formulas for broadening should be used. Here we discuss such formulas, which are based upon a rigorous diagrammatic resummation.

### b. Numerical methods

*S*

_{p12}is the area of the triangle

**k**=

**k**

_{1}+

**k**

_{2}. We numerically integrated (35) for

**p**that have frequencies from

*f*to

*N*and vertical wavenumbers from 2

*π*/(2

*b*) to 260

*π*/(2

*b*). The limits of integration are restricted by horizontal wavenumbers from 2

*π*/10

^{5}to 2

*π*/5 m

^{−1}, vertical wavenumbers from 2

*π*/(2

*b*) to 2

*π*/5 m

^{−1}, and frequencies from

*f*to

*N*. The integrals over

*k*

_{1}and

*k*

_{2}are obtained in the kinematic box in

*k*

_{1}–

*k*

_{2}space. The grids in the

*k*

_{1}–

*k*

_{2}domain have 2

^{17}points that are distributed heavily around the corner of the kinematic box. The integral over

*m*

_{1}is obtained with 2

^{13}grid points, which are also distributed heavily for the small vertical wavenumbers whose absolute values are less than 5

*m*, where

*m*is the vertical wavenumber.

*j*represents the vertical mode number of an ocean with an exponential buoyancy frequency profile having a thermocline scale height of

*b*= 1300 m.

We choose the following set of parameters:

*b*= 1300 m in the GM model;- the total energy is set asinertial frequency is given by
*f*= 10^{−4}rad s^{−1}, and buoyancy frequency is given by*N*_{0}= 5 × 10^{−3}rad s^{−1}; the reference density is taken to be

*ρ*_{0}= 10^{3}kg m^{−3}; anda rolloff wavenumber

*m*_{*}=*N*/*N*π_{o}*j*_{*}/*b*equivalent to mode 3,*j*_{*}= 3.

*C*in (34):

*C*= 10

^{−3},

*C*= 10

^{−2},

*C*= 10

^{−1}, and

*C*= 0.5.

Our simulations do show some sensitivity to the spectral boundaries and show significant sensitivity for the choice of *γ*** _{p}**, especially for relatively large values of

*γ*

**. Sorting out these sensitivities and finding a self-consistent value of**

_{p}*γ*

**is the subject of current research.**

_{p}## 6. Time scales

### a. Resonant interactions

Here, we present evaluations of the kinetic equation (35) with a broadened delta function (31) and (34). These estimates differ from evaluations presented in Olbers (1976), McComas (1977), McComas and Müller (1981a), and Pomphrey et al. (1980) in that the numerical algorithm includes a finite breadth to the resonance surface, whereas previous evaluations have been exactly resonant. Results discussed in this section are as close to resonant as we can make (*C* = 1 × 10^{−3}).

Results are presented in Fig. 5 for different values of *C*. We see that for small vertical wavenumbers the normalized Boltzmann rate is of the order of a tenth of the wave period. This can be argued to be relatively within the domain of weak nonlinearity. However, for increased wavenumbers the level of nonlinearity increases and reaches the level of wave period (red or dark blue). There is also a white region indicating values smaller than minus one.

We also define a “zero curve”: It is the locus of wavenumber–frequency where the normalized Boltzmann rate and time derivative of wave action is exactly zero. The zero curve clearly delineates a pattern of energy gain for frequencies *f* < *ω* < 2*f*, energy loss for frequencies 2*f* < *ω* < 5*f*, and energy gain for frequencies 5*f* < *ω* < *N*_{0}. We interpret the relatively sharp boundary between energy gain and energy loss across *ω* = 2*f* as being related to the parametric subharmonic instability and the transition from energy loss to energy gain at *ω* = 5*f* as a transition from energy loss associated with the parametric subharmonic instability to energy gain associated with the elastic scattering mechanism. See section 7 for further details about this high-frequency interpretation.

The *O*(1) normalized Boltzmann rates at high vertical wavenumber are surprising given the substantial literature that regards the GM spectrum as a stationary state. We do not believe this to be an artifact of the numerical scheme for the following reasons: First, numerical evaluations of the integrand conserve energy to within numerical precision as the resonance surface is approached, consistent with energy conservation property associated with the frequency delta function. Second, the time scales converge as the resonant width is reduced, as demonstrated by the minimal difference in time scales using *C* = 1 × 10^{−3} and 1 × 10^{−2}. Third, our results are consistent with approximate analytic expressions (e.g., McComas and Müller 1981b) for the Boltzmann rate. Finally, in view of the differences in the representation of the wave field, numerical codes, and display of results, we interpret our resonant (*C* = 0.001) results as being consistent with numerical evaluations of the resonant kinetic equations presented in Olbers (1976), McComas (1977), McComas and Müller (1981a), and Pomphrey et al. (1980).

As a quantitative example, consider estimates of the time rate of change of low-mode energy appearing in Table 1 of Pomphrey et al. (1980), which is repeated as row 3 of our Table 2^{4}. We find agreement to better than a factor of 2. To explain the remaining differences, one has to examine the details: Pomphrey et al. (1980) use a Coriolis frequency corresponding to 30° latitude; neglect internal waves having horizontal wavelengths greater than 100 km (same as here); and exclude frequencies *ω* > *N _{o}*/3, with

*N*= 3 cph. We include frequencies

_{o}*f*<

*ω*<

*N*with Coriolis frequency corresponding to 45° latitude. Of possible significance is that Pomphrey et al. (1980) use a vertical mode decomposition with exponential stratification with scale height

_{o}*b*= 1200 m (we use constant

*N*and assume an ocean depth equivalent to

*b*= 1300 m). Table 2 presents estimates of the energy transfer rate by taking the depth-integrated transfer rates of Pomphrey et al. (1980), assuming

*N*= 3 cph. Although this accounts for the nominal buoyancy scaling of the energy transport rate, it does not account for variations in the distribution of

*N*via

Numerical evaluations of

### b. Near-resonant interactions

Substantial motivation for this work is the question of whether the GM76 spectrum (Cairns and Williams 1976) represents a stationary state. We have seen that numerical evaluations of a resonant kinetic equation return *O*(1) normalized Boltzmann rates and hence we are led to conclude that GM76 is not a stationary state with respect to resonant interactions. The next natural question to ask is whether the GM76 could be a stationary solution of the kinetic equation with the self-consistent broadening function *γ*** _{p}**.

Our investigation of this question is currently limited by the absence of an iterative solution to (30) and (32) and consequent choice to parameterize the resonance broadening in terms of (34). As we go from nearly resonant evaluations (10^{−3} and 10^{−2}) to incorporating significant broadening (10^{−1} and 0.5), we find a significant decreases in the normalized Boltzmann rate. The largest decreases are associated with an expanded region of energy loss associated the parametric subharmonic instability, in which minimum normalized Boltzmann rates change from −3.38 to −0.45 at (*ω*, *mb*/2*π*) = (2.5*f*, 150). Large decreases here are not surprising given the sharp boundary between regions of loss and gain in the resonant calculations. Smaller changes are noted within the induced diffusion regime. Maximum normalized Boltzmann rates change from 2.6 to 1.5 at (*ω*, *mb*/2*π*) = (8*f*, 260). Broadening of the resonances to exceed the boundaries of the spectral domain could be making a contribution to such changes.

We regard our calculations here as a preliminary step to answering the question of whether the GM76 spectrum represents a stationary state with respect to nonlinear interactions within wave turbulence methodology. Complementary studies could include comparison with analyses of numerical solutions of the equations of motion.

## 7. Discussion

### a. Resonant interactions

*ω*

^{2}/

*f*

^{2}difference between the fast and slow induced diffusion time scales. It does not imply small values of the slow induced diffusion time scale, which are equivalent to the normalized Boltzmann rates. Third, the large normalized Boltzmann rates determined by our numerical procedure are associated with the elastic scattering mechanism rather than induced diffusion. Normalized Boltzmann rates for the induced diffusion and elastic scattering mechanisms are

*m*

_{*}represents the low-wavenumber rolloff of the vertical wavenumber spectrum (vertical mode-3 equivalent here);

*m*is the high-wavenumber cutoff, nominally at 10-m wavelengths; and the GM76 spectrum has been assumed. The normalized Boltzmann rates for ES and ID are virtually identical at high wavenumber. They differ only in how their respective triads connect to the

_{c}*ω*=

*f*boundary. Induced diffusion connects along a curve whose resonance condition is approximately that the high-frequency group velocity match the near-inertial vertical phase speed,

*ω*/

*m*=

*f*/

*m*. Elastic scattering connects along a simpler

_{ni}*m*= 2

*m*. Evaluations of the kinetic equation reveal nearly vertical contours throughout the vertical wavenumber domain, consistent with ES, rather than sloped along contours of

_{ni}*ω*∝

*m*emanating from

*m*=

*m*

_{*}as expected with the ID mechanism.

**p**

_{1}, assuming that the action density of the near-inertial field is much larger than the high-frequency fields, and taking the limit (

*k*,

*l*,

*m*) = (

*k*

_{2},

*l*

_{2}, −

*m*

_{2}) ≡

**p**

^{−}. Thus,

*n*

_{p}_{−}=

*n*

**. However, this is not the complete story. A more precise characterization of the resonance surface takes into account the frequency resonance requiring**

_{p}*ω*−

*ω*

_{2}=

*ω*

_{1}≅

*f*and requires

*O*(

*ω*/

*f*) differences in

*m*and −

*m*

_{2}if

*k*=

*k*

_{2}and

*O*(

*ω*/

*f*) differences in

*k*and

*k*

_{2}if

*m*= −

*m*

_{2}. For an isotropic field,

*δ*

**p**=

**p**

_{2}−

**p**.

### b. Near-resonant interactions

The idea of trying to self-consistently find the smearing of the delta functions is not new. For internal waves, it appears in DeWitt and Wright (1982), Carnevale and Frederiksen (1983), and DeWitt and Wright (1984).

**p**,

*ω*) is independent of

*ω*and show that this assumption is not self-consistent. Lvov et al. (1997) present a more sophisticated approach to a self-consistent approximation to the operator Σ(

**p**,

*ω*). In particular, DeWitt and Wright (1982) suggest

DeWitt and Wright (1984) evaluate the self-consistency of the resonant interaction approximation and find that, for high frequency and high wavenumbers, the resonant interaction representation is not self-consistent. A possible critique of these papers is that they use resonant matrix elements given by Müller and Olbers (1975) without appreciating that those elements can only be used strictly on the resonant manifold.

Carnevale and Frederiksen (1983) present similar expressions for two-dimensional stratified internal waves. There the kinetic equation is (7.4) with the triple correlation time given by Θ (our

The main advantage of our approach over Carnevale and Frederiksen (1983) is that we use systematic Hamiltonian structures that are equivalent to the primitive equations of motion rather than a simplified two-dimensional model.

### c. Direct numerical simulations of the dynamical equations of motion

Direct numerical simulations of the dynamical equations of motion are not limited by the dynamical assumptions inherent in the weakly nonlinear resonant or near-resonant representations. They are however subject to other computational restrictions and do significantly depend upon details of forcing.

D’Asaro (1997) present spindown simulations based upon the GM76 spectrum with varying amplitude. The domain considered there consists of a rectangular box 80 km × 10 km × 1 km on a side with resolved wavelengths of 1 km in the horizontal and 50 m in the vertical. Note that this domain does not include regions in Fig. 5 exhibiting large values of the normalized Boltzmann rate. Interactions in the resolved domain may be dominated by PSI transfers as discussed in McComas and Müller (1981a).

Forced nonrotating simulations are presented in Furue (2003). The computational domain is a box of horizontal size 100 m × 100 m × 128 m height. The forcing is isotropic in wavenumber and peaks at a horizontal wavelength of 25 m. The forcing is controlled so that amplitudes are consistent with GM76 and the resulting dissipation is a significant fraction of that associated with GM76.

It is an interesting task for the future research to relate these numerical simulations with evaluations of the kinetic equations we are performing here. The first steps in this direction were performed in Lvov and Yokoyama (2009).

## 8. Conclusions

Our fundamental result is that the GM spectrum is not stationary with respect to the resonant interaction approximation. This result is contrary to the point of view expressed in McComas and Müller (1981a) and Müller et al. (1986) and gave us cause to review published results concerning resonant internal wave interactions. We also arrived at the point where we can say that the resonant kinetic equation does not constitute a self-consistent approach. We then included near-resonant interactions and found significant reductions in the temporal evolution of the GM spectrum.

This is the first step in building a self-consistent theory of the interactions of internal waves. The main point of this paper is that we reopen the challenge of how to calculate from first principles the spectral energy density of internal waves.

We compared the interaction matrices for three different Hamiltonian formulations and one non-Hamiltonian formulation in the resonant limit. Two of the Hamiltonian formulations are canonical and one (Lvov and Tabak 2004) avoids a linearization of the Hamiltonian prior to assuming an expansion in terms of weak nonlinearity. Formulations in Eulerian, isopycnal, and Lagrangian coordinate systems were considered. All four representations lead to equivalent results on the resonant manifold in the absence of background rotation. The two representations that include background rotation, a canonical Hamiltonian formulation in isopycnal coordinates and a noncanonical Hamiltonian formulation in Lagrangian coordinates, also lead to equivalent results on the resonant manifold. This statement is not trivial given the different assumptions and coordinate systems that have been used for the derivation of the various kinetic equations. It points to an internal consistency on the resonant manifold that we still do not completely understand and appreciate.

We rationalize the consistent results as being associated with potential vorticity conservation. In the isopycnal coordinate canonical Hamilton formulation, potential vorticity conservation is explicit. In the Lagrangian coordinate noncanonical Hamiltonian, potential vorticity conservation results from a projection onto the linear modes of the system. The two nonrotating formulations prohibit relative vorticity variations by casting the velocity as the gradient of a scalar streamfunction.

We infer that the nonstationary results for the GM spectrum are related to a higher-order approximation of the elastic scattering mechanism than considered in McComas and Bretherton (1977) and McComas and Müller (1981b). Our numerical results indicate evolution rates of an inverse wave period at high vertical wavenumber, signifying a system that is not weakly nonlinear. To understand whether such nonweak conditions could give rise to competing effects that render the system stationary, we considered resonance broadening. We used a kinetic equation with broadened frequency delta function derived for a generalized three-wave Hamiltonian system in (Lvov et al. 1997). The derivation is based upon the Wyld diagrammatic technique for nonequilibrium wave systems and utilizes the Dyson–Wyld line resummation. This broadened kinetic equation is perceived to be more sophisticated than the two-dimensional direct interaction approximation representation pursued in Carnevale and Frederiksen (1983) and the self-consistent calculations of DeWitt and Wright (1984), which utilized the resonant interaction matrix of Olbers (1976). We find a tendency of resonance broadening to lead to more stationary conditions. However, our results are limited by an uncontrolled approximation concerning the width of the resonance surface.

Reductions in the temporal evolution of the internal wave spectrum at high vertical wavenumber were greatest for those frequencies associated with the PSI mechanism: that is, *f* < *ω* < 5*f*. Smaller reductions were noted at high frequencies.

A common theme in the development of a kinetic equation is a perturbation expansion permitting the wave interactions and evolution of the spectrum on a slow time scale (e.g., section 2b). An assumption of Gaussian statistics at zeroth order permits a solution of the first-order triple correlations in terms of the zeroth-order quadruple correlations. Assessing the adequacy of this assumption for the zeroth-order high-frequency wave field is a challenge for future efforts. Such departures from Gaussianity could have implications for the stationarity at high frequencies.

Nontrivial aspects of our work are that we utilize the canonical Hamiltonian representation of Lvov and Tabak (2004), which results in a kinetic equation without first linearizing to obtain interaction coefficients defined only on the resonance surface and the sophisticated broadened closure scheme of Lvov et al. (1997). Inclusion of interactions between internal waves and modes of motion associated with zero eigenfrequency (i.e., the vortical motion field) is a challenge for future efforts.

We found no coordinate-dependent (i.e., Eulerian, isopycnal, or Lagrangian) differences between interaction matrices on the resonant surface. We regard it as intuitive that there will be coordinate-dependent differences off the resonant surface. It is a robust observational fact that Eulerian frequency spectra at high vertical wavenumber are contaminated by vertical Doppler shifting: near-inertial frequency energy is Doppler shifted to higher frequency at approximately the same vertical wavelength. Use of an isopycnal coordinate system considerably reduces this artifact (Sherman and Pinkel 1991). Further differences are anticipated in a fully Lagrangian coordinate system (Pinkel 2008). Thus, differences in the approaches may represent physical effects and what is a stationary state in one coordinate system may not be a stationary state in another. Obtaining canonical coordinates in an Eulerian coordinate system with rotation and in the Lagrangian coordinate system are challenges for future efforts. In conclusion, the purpose of this paper is to show that the first principle explanation of internal wave spectrum in general and Garrett and Munk in particular are still yet to come.

## Acknowledgments

YL is supported by NSF DMS Grant 0807871 and ONR Award N00014-09-1-0515. We are grateful to YITP in Kyoto University for permitting use of their facility.

## APPENDIX

### Historical Review of Other Matrix Elements

*k*and the density wavenumber

_{z}*m*are related as

*m*= −

*g*/(

*ρ*

_{0}

*N*

^{2})

*k*, where

_{z}*g*is gravity,

*ρ*is density with reference value

*ρ*

_{0},

*N*is the buoyancy (Brunt–Väisälä) frequency, and

*f*is the Coriolis frequency. In isopycnal coordinates, the dispersion relation is given by

*f*= 0, these dispersion relations assume the form

#### a. Hamiltonian formalism in Clebsch variables in Voronovich (1979)

**z**defining the vertical direction. The Hamiltonian of the system is

*ρ*

_{0}(

*z*) is the equilibrium density profile;

*ρ*is the wave perturbation; and Π is a potential energy density function,

*η*(

*ξ*) being the inverse of

*ρ*(

_{o}*z*). The intent is to use

*ρ*and Lagrange multiplier

*λ*as the canonically conjugated Hamiltonian pair,

*z*−

*η*(

*ρ*+

_{o}*ρ*) being the vertical displacement of a fluid parcel and the second equation representing continuity. The issue is to express the velocity

**v**as a function of

*λ*and

*ρ*, and to this end one introduces yet another function Φ with the harmonious feature

*ξ*in (A7), and to eliminate this explicit dependence a Taylor series in density perturbation

*ρ*relative to

*ρ*

_{0}is used to express the potential energy in terms of

*ρ*and

*λ*. The resulting Hamiltonian

*z*.

The only approximations that have been made to obtain (A13) are the Boussinesq approximation in the nonrotating limit, the specification that the velocity be represented as (A12), and a Taylor series expansion. The Taylor series expansion is used to express the Hamiltonian in terms of canonically conjugated variables *ρ* and *λ*. Truncation of this Taylor series is the essence of the slowly varying approximation that the vertical scale of the internal wave is smaller than the vertical scale of the background stratification, which requires, for consistency’s sake, the hydrostatic approximation.

The procedure of introducing additional functionals (Φ) and constraints [(A11)] originates in Clebsch (1859). See Seliger and Witham (1968) for a discussion of Clebsch variables and also section 7.1 of Miropolsky (1981). Finally, the evolution equation for wave amplitude *a*** _{p}** is produced by expressing the cubic terms in the Hamiltonian with solutions to the linear problem represented by the quadratic components of the Hamiltonian. This is an explicit linearization of the problem prior to the formulation of the kinetic equation.

*ϕ*(

*z*). That allows us to solve for the matrix elements defined via (11) and above it in his paper. Then the convolutions of the basis functions give delta functions in vertical wavenumbers. Vornovich’s (A.1) transforms into

#### b. Olbers, McComas, and Meiss

**r**is the initial position of a fluid parcel at

**x**: these are Lagrangian coordinates. In the context of Hamiltonian mechanics, the associated Lagrangian density is

*x*=

_{j}*x*(

_{j}**r**,

*t*) is the instantaneous position of the parcel of fluid, which was initially at

**r**;

*J*= ∂

**x**/∂

**r**is the Jacobian, which ensures the fluid is incompressible.

*ρ*is

*ξ*/∂

_{i}*x*).

_{j}This Lagrangian is then projected onto a single wave amplitude variable *a* using the linear internal wave consistency relations^{A1} based upon plane wave solutions [e.g., Müller 1976, (2.26)], and a perturbation expansion in wave amplitude is proposed. This process has two consequences: the use of internal wave consistency relations places a condition of zero perturbation potential vorticity upon the result, and the expansion places a small-amplitude approximation upon the result with ill-defined domain of validity relative to the (later) assertion of weak interactions.

*a*

_{0}is the zeroth-order wave amplitude. After a series of approximations, this equation is cast into a field variable equation similar to (10). We emphasize that to get there small displacement of parcel of fluid was used, together with the built in assumption of resonant interactions between internal wave modes. The Lvov and Tabak (2001, 2004) approach is free from such limitations.

*T*

^{±}from the appendix of Müller and Olbers (1975). In our notation, in the hydrostatic balance approximation, their matrix elements are given by

*f*= 0 limit reduces the problem to a scale-invariant problem. We get the following simplified expression:

#### c. Caillol and Zeitlin

A non-Hamiltonian kinetic equation for internal waves was derived in Caillol and Zeitlin (2000), their (61), directly from the dynamical equations of motion, without the use of the Hamiltonian structure. Caillol and Zeitlin (2000) invoke the Craya–Herring decomposition for nonrotating flows, which enforces a condition of zero perturbation vorticity on the result.

To make it appear equivalent to more traditional form of kinetic equation, as in Zakharov et al. (1992), we make a change of variables, **l** → −**l** in the second line and **k** → −**k** in the third line of (61) of Caillol and Zeitlin (2000). If we further assume that all spectra are symmetric, *n*(−**p**) = *n*(**p**), then the kinetic equation assumes traditional form, as in Eq. (22) (see Müller and Olbers 1975; Zakharov et al. 1992; Lvov and Tabak 2001, 2004).

*X*

_{k}_{,l,p}and

#### d. Kenyon and Hasselmann

The first kinetic equations for wave–wave interactions in a continuously stratified ocean appear in Kenyon (1966), Hasselmann (1966), and Kenyon (1968). Kenyon (1968) states (without detail) that Kenyon (1966) and Hasselmann (1966) give numerically similar results. We have found that Kenyon (1966) differs from the four approaches examined below on one of the resonant manifolds but have not pursued the question further. It is possible this difference results from a typographical error in Kenyon (1966). We have not rederived this non-Hamiltonian representation and thus exclude it from this study.

#### e. Pelinovsky and Raevsky

An important paper on internal waves is Pelinovsky and Raevsky (1977). Clebsch variables are used to obtain the interaction matrix elements for both constant stratification rates, *N* = constant, and arbitrary buoyancy profiles, *N* = *N*(*z*), in a Lagrangian coordinate representation. Not many details are given, but there are some similarities in appearance with the Eulerian coordinate representation of Voronovich (1979). The most significant result is the identification of a scale-invariant (nonrotating and hydrostatic) stationary state, which we refer to as the Pelinovsky–Raevsky in the companion paper (Lvov et al. 2010). It is stated in Pelinovsky and Raevsky (1977) that their matrix elements are equivalent to those derived in their citation [11], which is Brehovski (1975). Because Brehovski (1975) and Pelinovsky and Raevsky (1977) are in Russian and not generally available, we refrain from including them in this comparison.

#### f. Milder

An alternative Hamiltonian description was developed in Milder (1982), in isopycnal coordinates without assuming a hydrostatic balance. The resulting Hamiltonian is an iterative expansion in powers of a small parameter, similar to the case of surface gravity waves. In principle, that approach may also be used to calculate wave–wave interaction amplitudes. Because those calculations were not done in Milder (1982), we do not pursue the comparison further.

#### g. Isopycnal Hamiltonian

*f*→ 0 limit, Lvov and Tabak (2004) reduces to Lvov and Tabak (2001) and (A21) reduces to

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^{2}

We will revisit this assumption in section 5a.

^{3}

To derive the interaction matrix elements in the hydrostatic balance, we assumed that *k* ≪ *m*. Once derivation is completed, values of *k* and *m* appear only as products, so it is consistent to make the choice |**k**| = *m* = 1. This choice is made only in the present section.

^{4}

A potential interpretation is that this net energy flow out of the nonequilibrium part of the spectrum represents the energy requirements to maintain the spectrum.

^{A1}

Wave amplitude *a* is defined so that *a*a* is proportional to wave energy.