documentclass[12pt]article par usepackagegraphicx usepackageamsmath usepackagebm usepackagenatbib usepackagefullpage newedcommandrewrite bfrewrite here newedcommandfinish bfend of rewrite par topskip 2cm par newedcommandtodo[1]vspace5 mmpar noindent marginpartextscToDo frameboxbeginminipage[c]0.9 textwidth tt #1 endminipagevspace5 mmpar par titleWave-wave interactions in stratified fluids:
A comparison of approaches. authorYuri V Lvov , Kurt Polzin$^2$ and Naoto Yokoyama$^3$
small Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy NY 12180
small $^2$ Woods Hole Oceanographic Institution, MS#21, Woods Hole, MA 02543
small $^3$ Department of Mechanical Engineering, Doshisha University, Kyotanabe, Kyoto 610-0394 JAPAN par begindocument par maketitle par beginabstract Various approaches have been developed over the last four decades to characterize the magnitude of nonlinear interactions between triads of internal waves in stratified oceanic flows. The present manuscript compares some of these approaches and their predictions for internal wave nonlinearity parameter and Boltzman rate. We demonstrate that, for em resonant triads in the limit of long internal waves in hydrostatic balance and in the absence of rotation, these various approaches predict em equivalent rates of energy transfer between waves. However, with the inclusion of background rotation and off-resonant interactions, these approaches lead to qualitatively different predictions. In particular, a noncanonical approach in Lagrangian coordinates leads to higher levels of nonlinearity at high frequencies and large wavenumbers than a canonical approach in isopycnal coordinates. par endabstract par sectionIntroduction par Wave-wave interactions in stratified oceanic flows have been a subject of intensive research in the last four decades. Of particular importance is the existence of a ``universal'' internal-wave spectrum, the Garrett and Munk 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. Due to 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 with wavenumber $bm{p}$ interact with wavevectors $bm{p}_1$ and $bm{p}_2$ , where $bm{p} = bm{p}_1 + bm{p}_2$ ? What are the oceanographic consequences of this interaction? par Various approaches have been developed to characterize the magnitude of such interactions citepH66,K66,K68,McC75,MO75,O74,O76,PR77,MB77,PMW80,Voronovich,Milder,Zeitlin,LT,LT2 (see Table refTABLEOFELEMENTS for a summary of the major distinctions.) par All these approaches represent various attempts to derive a closed equation representing the slow time evolution of the wave field's wave action spectrum. Such an equation is called a em kinetic equation citepZLF. par begintable[htbp] labelTABLEOFELEMENTS captionA list of various kinetic equations. Results from citeO76, MB77, PMW80 are reviewed in citeM86, who state that citeO76, citeMB77 and an unspecified Eulerian representation are consistent on the resonant manifold. citePMW80 utilizes Langevin techniques to assess nonlinear transports. citeM86 characterizes those Langevin results as being mutually consistent with the direct evaluations of kinetic equations presented in citeO76, MB77. citeK68 states (without detail) that citeK66 and citeH66 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, citePMW80 use an exponential profile and the others assume a constant stratification rate. The kinetic equations marked by $^{dag}$ are investigated in SrefResonantInteractions, while kinetic equations marked by $^{ddag}$ are investigated further in SrefOffResonant. labeldefault begincenter begintabularcccccc hline source & coordinate & vertical & rotation & hydro- & special
& system & structure & & static &
hline citetH66 & Lagrangian & discrete & no & no &
citetK66, K68 & Eulerian & discrete & no & no & non-Hamiltonian
citetMO75 $^{dag ddag}$ & Lagrangian & cont. & yes & no &
citetMcC75, McComas & Lagrangian & cont. & yes & yes &
citetPR77 & Lagrangian & cont. & no & no & Clebsh
citetVoronovich$^{dag}$ & Eulerian & cont. & no & no & Clebsh
citetPMW80 & Lagrangian & discrete & yes & no & Langevin
citetMilder & Isopycnal & n/a & no & no &
citetZeitlin$^{dag}$ & Eulerian & cont. & no & no & non-Hamiltonian
citetLT$^{dag}$ & Isopycnal & cont. & no & yes & canonical
citetLT2$^{ddag}$ & Isopycnal & cont. & yes & yes & canonical
hline endtabular endcenter endtable par In this manuscript we concentrate on four of these different versions of the internal-wave kinetic equation: beginitemize item a noncanonical description using Lagrangian coordinates citepO74,O76,MO75, item a canonical Hamiltonian description using Clebsh variables in Eulerian coordinates citepVoronovich, item a dynamical derivation of a kinetic equation without use of Hamiltonian formalisms in Eulerian coordinates citepZeitlin, item a canonical Hamiltonian description in isopycnal coordinates citepLT,LT2. enditemize par Our intent is to compare these approaches, and in particular, compare the predictions for the wavenumber-dependent characteristic nonlinear time scale of the Garrett and Munk wave action spectrum. To achieve this goal, we give necessary background in Section refBackground, briefly review approaches of Table refTABLEOFELEMENTS in Section (refVariousApproaches), and then we demonstrate in Section refResonantInteractions that, under assumption of hydrostatic balance and under the assumption of em resonant wave-wave interactions, the interaction matrices associated with the listed approaches are em equivalent. While one, with sufficient experience, might regard this as an intuitive statement, it is far from trivial. We will then demonstrate in Section refOffResonant that, if the assumption of resonant wave-wave interactions is relaxed, both quantitatively and qualitatively different transfer rates are predicted. In particular, we show that the Boltzman rate ${cal epsilon}_{bm{p}}$ , defined below in (refNonlinearTime) is, in fact, representation dependent for near-resonant interactions. par We have em not, at this time, achieved a detailed mathematical understanding of how these differences arise and consequently do not digress into a detailed discussion of why, for example, the radius of convergence of two consecutive series expansions in one coordinate system differs so dramatically from a single series expansion in a different coordinate system. We conclude in Section refConclusion. par sectionBackgroundlabelBackground 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 weak turbulence theory citepZLF 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 resonant redistribution of the spectral energy density among space- and time-scales, and is described by a kinetic equation. par Typical assumptions needed for the derivation of kinetic equations are: beginitemize item Weak nonlinearity, item Gaussian statistics of the interacting wave field in wavenumber space and item Resonant wave-wave interactions. enditemize par The derivation of the kinetic equation for general nonlinear systems is well studied and understood, and thus will not be repeated here. Three wave kinetic equations take the form citepZLF,NoisyNazarenko,LLNZ: begineqnarray fracd n_bmpdt = 4pi int |V_bmp_1,bmp_2^bmp|^2 f_p12 delta_bmp - bmp_1-bmp_2 delta(omega_bmp -omega_bmp_1-omega_bmp_2) d bmp_12 nonumber
-4piint |V_bmp_2,bmp^bmp_1|^2 f_12p delta_bmp_1 - bmp_2-bmp delta(omega_bmp_1 -omega_bmp_2-omega_bmp) d bmp_12 nonumber
-4piint |V_bmp,bmp_1^bmp_2|^2 f_2p1 delta_bmp_2 - bmp-bmp_1 delta(omega_bmp_2 -omega_bmp-omega_bmp_1) d bmp_12 ,nonumber
rm with   f_p12 = n_bmp_1n_bmp_2 - n_bmp(n_bmp_1+n_bmp_2) . labelKineticEquation endeqnarray Here $n_{bm{p}} = n(bm{p})$ is a three-dimensional wave action spectrum (spectral energy density divided by frequency) and the interacting wavevectors $bm{p}$ , $bm{p}_1$ and $bm{p}_2$ are given by

$$bm{p} = (bm{k}, m),$$

i.e. $bm{k}$ is the horizontal part of $bm{p}$ and $m$ is its vertical component. We assume the wavevectors are signed variables and wave frequencies $omega_{bm{p}}$ are restricted to be positive. The magnitude of wave-wave interactions $V_{bm{p},bm{p}_1}^{bm{p}_2}$ is a matrix representation of the coupling between triad members. It serves as a multiplier in the nonlinear convolution term in what is now commonly called the Zakharov equation - equation in the Fourier space for the waves field variable. This is also an expression that multiplies the cubic convolution term in the three-wave Hamiltonian. par For internal waves in the ocean such kinetic equation was derived by the approaches in Table refTABLEOFELEMENTS. The development of a kinetic equation is facilitated by transforming to canonical coordinates in a Hamiltonian framework, for which one can demonstrate that the symmetries and hence conservation principles of the original equation set in the spatial/temporal domains have been preserved in the spectral domain citep[e.g.][]ZLF. Finding canonical coordinates, however, can be highly nontrivial. Transformations to canonical coordinates have been found using Clebsch variables in Eulerian coordinates citepVoronovich and in isopycnal coordinates citepLT, LT2. Kinetic equations in Lagrangian coordinates start by averaging the Lagrangian of the stratified fluid (averaging of the variational principle) and then transform the Lagrangian to the Hamiltonian. The Lagrangian coordinate kinetic equations considered here are noncanonical. We also note that it is possible to obtain a kinetic equation directly from the dynamical equations of motion, without the use of the Hamiltonian structure. Such an approach was executed by citetZeitlin. The conservation properties of non-canonical and non-Hamiltonian representations are not guaranteed unless explicitly demonstrated. The issue of conservation properties is greatly compounded for non-canonical and non-Hamiltonian representations off the resonant manifold. par A typical restriction is to exclude interactions with potential vorticity carrying members of the fluid dynamical system, for which we refer the reader to citeLelong and citeZeitlin. Even in these extended analyses a plane wave formulation is assumed that eliminates the potential vorticity associated with a slowly varying wave-packet structure citepBM05, P08. par Note that the kinetic equation allows us to numerically estimate the life time of any given spectrum. In particular, we can define a wavenumber dependent nonlinear time scale proportional to the inverse Boltzman rate: beginequation tau^mathrmNL_bmp = fracn_bmpdot n_bmp . labelNonlinearTime endequation This time scale characterizes the net rate at which the spectrum changes and can be directly calculated from the kinetic equation. par One can also define the characteristic linear time scale,

$$tau^{mathrm{L}}_{bm{p}}=2pi/omega_{bm{p}}.$$

The non-dimensional ratio of these time scales can characterize the level of nonlinearity in the nonlinear system: beginequation cal epsilon_bmp = frac tau_bmp^mathrmL tau_bmp^mathrmNL = frac 2pi dot n_bmp n_bmp omega_bmp labelNonlinearRatio endequation We refer to (refNonlinearRatio) as the nonlinearity parameter. par The nonlinear parameter serves as a low order consistency check for the various kinetic equation derivations. An $O(1)$ value of ${cal epsilon}_{bm{p}}$ implies that the derivation of the kinetic equation is internally inconsistent. The Boltzman rate represents the net rate of transfer for wavenumber $bm{p}$ and is an appropriate measure of nonlinearity for smooth, isotropic and homogeneous spectra. The individual rates of transfer into and out of $bm{p}$ maybe significantly larger for spectral spikes citepM86 and potentially for smooth, homogeneous but anisotropic spectra. Estimates of the Boltzman rate and ${cal epsilon}_{bm{p}}$ require integration of Eq. (refKineticEquation). In this manuscript such integration is performed numerically. par In this paper we concentrate on four approaches, namely citetMO75,Voronovich,Zeitlin,LT,LT2. We show that on the resonant manifold they produce em equivalent results. par Resonant interaction approximation is self-consistent for small level of nonlinearities. However, as the nonlinearity parameter increase, near-resonant interactions start to play a role. par For realistic estimates the effects of rotation must be included, and this restricts our investigations to two approaches that allow inclusion of background rotations. Therefore, we concentrate in more details on the citeLT2 and citeMO75 representations. par We show that for the near-resonant interactions, these two approaches returns qualitatively different predictions for transfer rates. This is the main physical result of the present paper. par There is a multitude of reasons for possible differences. First and foremost, we view the distinction between Lagrangian, isopycnal and Eulerian coordinates as the most dynamically significant difference. The use of a Lagrangian coordinate system requires an expansion in powers of small fluid parcel displacements in addition to an assumption of weak nonlinearity, whereas formulations in isopycnal or Eulerian coordinates require only an assumption of weak nonlinearity. An issue with extant Lagrangian coordinate representations is that the small amplitude assumption represents an unconstrained approximation whose domain of validity em vis-a-vis the weak interaction approximation is not well defined, citepM86. A subsidiary issue is that the use of a Lagrangian coordinate system places the nonlinearity in the incompressibility constraint, and a single plane wave is not an exact solution of the equations of motion, citepS85. Similarly, a single plane wave also does not constitute a solution to the isopycnal equations of motion. In Eulerian coordinates the nonlinearity is advective and a single plane wave is an exact solution of the equations of motion. On the other hand, 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, citepSandP91. Thus differences in the approaches may represent physical effects rather than technical issues such as the proper implementation of a potential vorticity conservation statement citepZeitlin. par We emphasize that our intent is to estimate transport rates for various approaches within a common framework and to compare those results. Our goal is a qualitative physical explanation of the possible reasons for the similarities and differences rather than a quantitative analytical explanation of how those differences arise. par sectionVarious Approaches labelVariousApproaches par In this section we list the approaches that we use. We do so for completion and to transfer everything to a uniform notation. Our attention is restricted to the hydrostatic balance case, for which beginequation |bmk| ll |m| . labelhydrostatic endequation A minor detail is that the linear frequency has different algebraic representations in isopycnal and Cartesian coordinates. The Cartesian vertical wavenumber, $k_z$ , and the density wavenumber, $m$ , are related as $m = - g/(rho_0 N^2) k_z$ where $g$ is gravity, $rho$ is density with reference value $rho_0$ , $N$ is the buoyancy (Brunt-Väisälä) frequency and $f$ is the Coriolis frequency. In isopycnal coordinates the dispersion relation is given by, begineqnarray omega(bm p) = sqrtf^2 + fracg^2rho_0^2 N^2 frac|bmk|^2m^2. labeleq:dispersionISO endeqnarray In Cartesian coordinates, begineqnarray omega(bmp) = sqrtf^2 + N^2 frac|bmk|^2k_z^2 . labeleq:dispersionCAR endeqnarray In the limit of $f=0$ these dispersion relations assume the form beginequation omega_bmp propto frac |bmk ||m| propto frac |bmk ||k_z| labeldispersion endequation par subsectionKenyon and Hasselmann The first kinetic equations for wave-wave interactions in a continuously stratified ocean appear in citetK66, citetH66 and citetK68. citetK68 states (without detail) that citetK66 and citetH66 give numerically similar results. We have found that citetK66 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 citetK66. We have not rederived this non-Hamiltonian representation and thus exclude it from this study. par subsection Müller and Olbers par Matrix elements derived in citetO74 are given by $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}^{mathrm{MO}}\vert^2 = T^{+} / (4pi)$ and $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}^{mathrm{MO}}\vert^2 = T^{-} / (4pi)$ . We extracted $T^{pm}$ from the Appendix of citetMO75. In our notation, in the hydrostatic balance approximation, their matrix elements are given by par beginalign |V^bmp_bmp_1, bmp_2^mathrmMO|^2 =frac(N_0^2-f^2)^232 rho_0 omega omega_1 omega_2 left| frac|bmk| |bmk_1| |bmk_2|omega omega_1 omega_2 |bmp||bmp_1||bmp_2| right. nonumber
left( - fracleft(-m_1 fracbmk_1 cdot bmk_2 - i f bmk_2 cdot bmk_1^perp/omega_1k_1^2 + m_2right) left(-m_2 fracbmk_1 cdot bmk_2 - i f bmk_1 cdot bmk_2^perp/omega_2k_2^2 + m_1right)m right. nonumber
- fracleft(-m_2 fracbmk_2 cdot bmk + i f bmk_2 cdot bmk^perp/omega_2k_2^2 + mright) left(-m fracbmk_2 cdot bmk - i f bmk cdot bmk_2^perp/omegak^2 + m_2right)m_1 nonumber
left.left. - fracleft(-m fracbmk cdot bmk_1 - i f bmk cdot bmk_1^perp/omegak^2 + m_1right) left(-m_1 fracbmk cdot bmk_1 + i f bmk_1 cdot bmk^perp/omega_1k_1^2 + mright)m_2 right) right|^2 . labelVMO endalign par Taking a $f=0$ limit we get: par begineqnarray |V^bmp_bmp_1, bmp_2^mathrmMO|^2 propto frac|bmk||bmk_1||bmk_2||m m_1 m_2| left( - frac1m left(-fracm_2 bmk_1 cdot bmk_2|bmk_2|^2 + m_1 right) left(-fracm_1 bmk_2 cdot bmk_1|bmk_1|^2 + m_2 right) right. nonumber
left. + frac1m_1 left(fracm_2 bmk cdot bmk_2|bmk_2|^2 - m right) left(-fracm bmk_2 cdot bmk|bmk|^2 + m_2 right) + frac1m_2 left(-fracm bmk_1 cdot bmk|bmk|^2 + m_1 right) left(fracm_1 bmk cdot bmk_1|bmk_1|^2 - m right) right)^2 endeqnarray par subsectionPelinovsky and Raevsky An important paper on internal waves is citetPR77. Clebsh variables are used to obtain the interaction matrix elements for both constant stratification rates, $N=mathrm{const.}$ , and arbitrary buoyancy profiles, $N=N(z)$ . Not much details are given, but there are some similarities in appearance with citetVoronovich. The most significant result is the identification of a scale invariant (non-rotating, hydrostatic) stationary state. It is stated in the paper that their matrix elements are equivalent to those derived in their citation [11], which is citetB75. Because citetB75 and citePR77 are in Russian and not generally available, we refrain from including them in this comparison. par subsection Voronovich Voronovich used Clebsh-like variables to derive the Hamiltonian for incompressible stratified flows in the ocean. It is probably the first canonical Hamiltonian structure derived for such kind of flows. A detailed explanation of Voronovich's method appears in section 7.1 of the textbook citetMir It is a straightforward task to write down the kinetic equation associated with this Hamiltonian structure. par We formulate the matrix elements for Voronovich's Hamiltonian using his formula (A.1). This formula is derived for general boundary conditions. To compare with other matrix elements of this paper, we assume a constant stratification profile and Fourier basis as the vertical structure function $phi(z)$ . That allows us to solve for the matrix elements defined via Eq. (11) and above it in his paper. Then the convolutions of the basis functions give delta-functions in vertical wavenumbers. Vornovich's equation (A.1) transforms into: begineqnarray |V^bmp_bmp_1, bmp_2^mathrmV|^2 propto frac|bmk||bmk_1||bmk_2||m m_1 m_2| left( - m left( frac1|bmk| |m| left(fracbmk cdot bmk_1 |m_1||bmk_1| + fracbmk cdot bmk_2 |m_2||bmk_2| right) + fracomega_1 + omega_2 - omegaomega right) right. nonumber
left. + m_1 left( frac1|bmk_1| |m_1| left(fracbmk cdot bmk_1 |m||bmk| + fracbmk_1 cdot bmk_2 |m_2||bmk_2| right) - fracomega_1 + omega_2 - omegaomega_1 right) right. nonumber
left. + m_2 left( frac1|bmk_2| |m_2| left(fracbmk cdot bmk_2 |m||bmk| + fracbmk_2 cdot bmk_1 |m_1||bmk_1| right) - fracomega_1 + omega_2 - omegaomega_2 right) right)^2 . nonumber
labeleq:Voronovich endeqnarray par Note that Eq. (refeq:Voronovich) shares structural similarities with the interaction matrix elements in em isopycnal coordinates, Eq. (refHamiltonian) below. par subsectionMilder An alternative Hamiltonian description was developed in citetMilder, 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. Since those calculations were not done in citetMilder, we do not pursue the comparison further. par subsectionCaillol and Zeitlin A non-Hamiltonian kinetic equation for internal waves was derived in citetZeitlin, Eq. (61). To make it appear equivalent to more traditional form of kinetic equation, as in citetZLF, we make a change of variables $bm{l}to -bm{l}$ in the second line, and $bm{k}to -bm{k}$ in the third line of (61) of citetZeitlin. If we further assume that all spectra are symmetric, $n(-bm{p}) = n(bm{p})$ , then the kinetic equation assumes traditional form, as in Eq. (refKineticEquation), see citetMO75,ZLF,LT,LT2. par The matrix elements according to citetZeitlin are shown as $X_{k,l,p}$ and $Y_{k,l,p}^{pm}$ in Eqs. (62) and (63), where $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}^{mathrm{CZ}}\vert^2 = X_{bm{p}_1,bm{p}_2,bm{p}}$ and $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}^{mathrm{CZ}}\vert^2 = Y_{bm{p}_1,-bm{p}_2,bm{p}}^{+}$ . In our notation it reads begineqnarray |V^bmp_bmp_1, bmp_2^mathrmCZ|^2 propto (|bmk| mathrmsgn(m) + |bmk_1| mathrmsgn(m_1) + |bmk_2| mathrmsgn(m_2))^2 frac(m^2 - m_1 m_2)^2|m| |m_1| |m_2| |bmk||bmk_1||bmk_2| nonumber
timesleft( frac|bmk|^2 - |bmk_1| mathrmsgn(m_1) |bmk_2| mathrmsgn(m_2)m^2 - m_1 m_2 m - frac|bmk_1|^2m_1 - frac|bmk_2|^2m_2 right)^2 nonumber
.labeleq:VCZ endeqnarray par subsectionIsopycnal Hamiltonian par Finally, in citetLT2 the following wave-wave interaction matrix element was derived based on a canonical Hamiltonian formulation in isopycnal coordinates: par beginalign |V^0_1,2 ^mathrmH |^2 = fracN^232 g left( left( frack bmk_1 cdot bmk_2k_1 k_2 sqrtfracomega_1 omega_2omega + frack_1 bmk_2 cdot bmkk_2 k sqrtfracomega_2 omegaomega_1 + frack_2 bmk cdot bmk_1k k_1 sqrtfracomega omega_1omega_2 right. right. nonumber
left. left. + fracf^2sqrtomega omega_1 omega_2 frack_1^2 bmk_2 cdot bmk - k_2^2 bmk cdot bmk_1 - k^2 bmk_1 cdot bmk_2k k_1 k_2 right)^2 right. nonumber
left. + left( f fracbmk_1 cdot bmk_2^perpk k_1 k_2 left(sqrtfracomegaomega_1 omega_2 (k_1^2 - k_2^2) - sqrtfracomega_1omega_2 omega (k_2^2-k^2) - sqrtfracomega_2omega omega_1 (k^2-k_1^2)right) right)^2 right)  .nonumber
labelLTV endalign citetLT is a rotationless limit of citetLT2. Taking the $fto 0$ limit, the citeLT2 reduces to citeLT, and (refLTV) reduces to begineqnarray |V^bmp_bmp_1, bmp_2^mathrmH|^2 propto frac1|bmk||bmk_1||bmk_2| left( |bmk| bmk_1 cdot bmk_2 sqrtleft|fracmm_1 m_2right| + |bmk_1| bmk_2 cdot bmk sqrtleft|fracm_1m_2 mright| + |bmk_2| bmk cdot bmk_1 sqrtleft|fracm_2m m_1right| right)^2 .nonumber
labelHamiltonian endeqnarray par Observe that in this form, these equations share structural similarities with Eq. (refeq:Voronovich). par In this section we gave brief review of the various approaches that were developed for describing wave-wave interactions of internal waves in the ocean. While this review is necessarily brief, this is the first time all these papers are cited together by a single manuscript. par sectionResonant wave-wave interactions labelResonantInteractions par How one can compare the function of two vectors $bm{p}_1$ and $bm{p}_2$ , and their sum or difference? First one realizes that out of 6 components of $bm{p}_1$ and $bm{p}_2$ , only relative angles between wavevectors enter into the equation for matrix elements. That is because the matrix elements depend on the inner products of wavevectors. The overall horizontal orientation of the wavevectors does not matter: relative angles can be determined from a triangle inequality and the magnitudes of the horizontal wavevectors $bm{k}$ , $bm{k}_1$ and $bm{k}_2$ . Thus the only needed components are $\vert bm{k}\vert$ , $\vert bm{k}_1\vert$ , $\vert bm{k}_2\vert$ , $m$ 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. That is to say that it is sufficient to choose an arbitrary scalar value for $\vert bm{k}\vert$ , and $m$ , since only $\vert bm{k}_1\vert/\vert bm{k}\vert$ , $\vert bm{k}_2\vert/\vert bm{k}\vert$ and $m_1/m$ enter the expressions for matrix elements. We make the particular (arbitrary) choice that $\vert bm{k}\vert=m=1$ for the purpose of numerical evaluation, and thus the only independent variables to consider are $\vert bm{k}_1\vert$ , $\vert bm{k}_2\vert$ 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. par subsectionReduction to the Resonant Manifold When confined to the traditional form of the kinetic equation, wave-wave interactions (scattering) are constrained to the resonant manifolds defined by begineqnarray a)   begincases bmp = bmp_1 + bmp_2
omega = omega_1 + omega_2 endcases b)   begincases bmp_1 = bmp_2 + bmp
omega_1 = omega_2 + omega endcases c)   begincases bmp_2 = bmp + bmp_1
omega_2 = omega + omega_1 endcases. labelRESONANCES endeqnarray To compare matrix elements on the resonant manifold we are going to use the above resonant conditions and the internal-wave dispersion relation (refdispersion). To determine vertical components $m_1$ and $m_2$ of the interacting wavevectors, 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. par Resonances of type (refRESONANCESa) give beginsubequations allowdisplaybreaks beginalign & begincases m_1 = fracm2 |bmk| left(|bmk| + |bmk_1| + |bmk_2| + sqrt(|bmk| + |bmk_1| + |bmk_2|)^2 - 4 |bmk| |bmk_1|right)
m_2 = m - m_1. endcases , labeleq:sol1
& begincases m_1 = fracm2|bmk| left(|bmk| - |bmk_1| - |bmk_2| - sqrt(|bmk| - |bmk_1| - |bmk_2|)^2 + 4 |bmk| |bmk_1|right)
m_2 = m - m_1. endcases , labeleq:sol2 endalign endsubequations Note that because of the symmetry, (refeq:sol1) translates to (refeq:sol2) if wavenumbers $1$ and $2$ are exchanged. par Resonances of type (refRESONANCESb) give beginsubequations allowdisplaybreaks beginalign & begincases m_2 = - fracm2 |bmk| left(|bmk| - |bmk_1| - |bmk_2| + sqrt(|bmk| - |bmk_1| - |bmk_2|)^2 + 4 |bmk| |bmk_2|right)
m_1 = m + m_2. endcases , labeleq:sol3
& begincases m_2 = - fracm2|bmk| left(|bmk| + |bmk_1| - |bmk_2| + sqrt(|bmk| + |bmk_1| - |bmk_2|)^2 + 4 |bmk| |bmk_2|right)
m_1 = m + m_2. endcases , labeleq:sol4 endalign endsubequations par Resonances of type (refRESONANCESc) give beginsubequations allowdisplaybreaks beginalign & begincases m_1 = - fracm2|bmk| left(|bmk| - |bmk_1| - |bmk_2| + sqrt(|bmk| - |bmk_1| - |bmk_2|)^2 + 4 |bmk| |bmk_1|right)
m_2 = m + m_1. endcases , labeleq:sol5
& begincases m_1 = - fracm2|bmk| left(|bmk| - |bmk_1| + |bmk_2| + sqrt(|bmk| - |bmk_1| + |bmk_2|)^2 + 4 |bmk| |bmk_1|right)
m_2 = m + m_1. endcases . labeleq:sol6 endalign endsubequations Because of the symmetries of the problem, (refeq:sol3) is equivalent to (refeq:sol5), and (refeq:sol4) is equivalent to (refeq:sol6) if wavenumbers $1$ and $2$ are exchanged. par subsectionComparison of matrix elements par As explained above, we assume $f=0$ and hydrostatic balance. Such a choice makes the matrix elements to be scale-invariant functions that depend only upon $\vert bm{k}_1\vert$ and $\vert bm{k}_2\vert$ . As a consequence of the triangle inequality we need to consider matrix elements only within a ``kinematic box'' defined by

$$\vert\vert bm{k}_1\vert - \vert bm{k}_2\vert\vert < \vert bm{k}\vert < \vert bm{k}_1\vert + \vert bm{k}_2\vert.$$

The matrix elements will have different values depending on the dimensions so that isopycnal and Eulerian approaches will give different values (refeq:dispersionISO)-(refeq:dispersionCAR). To address this issue in the simplest possible way, we multiply each matrix element by a dimensional number chosen so that all matrix elements are equivalent for some specific wavevector. In particular, we choose the scaling constant so that $\vert V(\vert bm{k}_1\vert=1,\vert bm{k}_2\vert=1)\vert^2=1$ . This allows a transparent comparison without worrying about dimensional differences between various formulations. par subsubsectionResonances of the ``sum'' type (refRESONANCESa) par Figure refFIGRESONANTa presents the values of the matrix element $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}_{mathrm{(ref{eq:sol2})}}\vert^2$ on the resonant sub-manifold given explicitly by (refeq:sol2). All approaches give equivalent results. This is confirmed by plotting the relative ratio between these approaches, and it is given by numerical noise (not shown). The solution (refeq:sol1) gives the same matrix elements but with $\vert bm{k}_1\vert$ and $\vert bm{k}_2\vert$ exchanged owing to their symmetries. par subsubsectionResonances of the ``difference'' type (refRESONANCESb) and (refRESONANCESc) par We then turn our attention to resonances of ``difference'' type (refRESONANCESb) for which (refRESONANCESc) could be obtained by symmetrical exchange of the indices. All the matrix elements $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol3})}}\vert^2$ on the resonant sub-manifold (refeq:sol3), are shown in Fig. refFIGRESONANTb. All the matrix elements are equivalent. The relative differences between different approaches are given by numerical noise (not shown). Finally, $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol4})}}\vert^2$ on the resonant sub-manifold (refeq:sol4) are shown in Fig. refFIGRESONANTc. Again, all the matrix elements are equivalent. par The solutions (refeq:sol5) and (refeq:sol6) give the same matrix elements but with $\vert bm{k}_1\vert$ and $\vert bm{k}_2\vert$ exchanged as the solutions (refeq:sol3) and (refeq:sol4) owing to their symmetries. par subsubsectionSpecial triads Three simple interaction mechanisms are identified by citetMB77 in the limit of an extreme scale separation. In this subsection we look in closer detail at these special limiting triads to confirm that all matrix elements are indeed asymptotically consistent. The limiting cases are: par beginitemize item 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 is called elastic scattering (ES). The solution (refeq:sol1) in the limit $\vert bm{k}_1\vert to 0$ corresponds to this type of special triad. item 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 (ID). The solution (refeq:sol2) in the limit that $\vert bm{k}_1\vert to 0$ corresponds to this type of special triad. item 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 (refeq:sol3) in the limit that $\vert bm{k}_1\vert to 0$ corresponds to this type of triad. enditemize par To study the detailed behavior of the matrix elements in the special triad cases, we choose to present the matrix elements along a straight line defined by

$$(\vert bm{k}_1\vert, \vert bm{k}_2\vert) = (epsilon, epsilon/3+1) \vert bm{k}\vert.$$

This line originates from the corner of the kinematic box in Figs. refFIGRESONANTa-refFIGRESONANTc at $(\vert bm{k}_1\vert, \vert bm{k}_2\vert) = (0,\vert bm{k}\vert)$ and has a slope of 1/3. The slope of this line is arbitrary. We could have taken $epsilon/4$ or $epsilon/2$ . The matrix elements here are shown as functions of $epsilon$ in Fig. refFigureThree. We see that all four approaches are again em equivalent on the resonant manifold for the case of special triads. par In this section we demonstrated that all four approaches we considered produce it 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. par sectionSmearing of the resonance manifold and near-resonant InteractionslabelOffResonant par subsectionNonlinear frequency renormalization as a result of nonlinear wave-wave interactions Above we have compared the values of matrix element on the em resonant manifold. The resonant interaction approximation is a mathematical simplification which reduces the complexity of the problem. In this subsection we examine transfers including near resonant interactions. Our interest in near-resonant interactions has significant physical motivations. For example a major unresolved issue is the importance of Doppler shifting citepPolzin2004a. Of particular interest here is the variable effects of Doppler shifting in different coordinate systems. The resonant interaction approximation assumes, perforce, an expansion in terms of a non-advected wavefield, with dispersion relation given by Eq. (refeq:dispersionISO) or Eq. (refeq:dispersionCAR). In the limit of extreme time scale separation between high frequency waves and a low frequency background, one is tempted to replace the non-advected frequency by its Doppler shifted intrinsic frequency counterpart, $omega rightarrow omega - bm{k} cdot overline{bm{u}}$ , in which $omega$ and $bm{k}$ are the frequency and wavevector of the high frequency wave and $overline{bm{u}}$ is the velocity field of the low frequency wavefield. This is the genesis of the eikonal approach citepM86 to internal wave-wave interactions. Then the resonant approximation is self-consistent for small values of nonlinearities. Indeed, change in the wave amplitude will be small, and the Doppler shift cancels from the frequency delta function. Yet, as nonlinearity increases, the near-resonant interactions become more and more pronounced, consequently the issue of Doppler shifting more and more important. Furthermore, near-resonant interactions play a major role in numerical simulations on a discrete grids citepLvovNazarenkoPokorni, for time evolution of discrete systems  citepGersh2007, in acoustic turbulence citepLLNZ, surface gravity waves citepJansenXXX,yuen_lake, and internal waves citep2006JFM...568..273V,Shrira. par To take into account the effects of near-resonant interactions self-consistently, the energy conserving delta-functions in Eq. (refKineticEquation), $delta({omega_{bm{p}} -omega_{{bm{p}_1}} - omega_{{bm{p}_2}} })$ , need to be ``broadened''. The physical motivation for this broadening is the following: when the resonant kinetic equation is derived, it is assumed that the amplitude of each plane wave is constant in time, or, in other words, that the lifetime of single plane wave is infinite. Resulting kinetic equation, nevertheless, predicts that the amplitude of the wave do change. Consequently the wave lifetime is finite. For small level of nonlinearity this distinction is not significant, and resonant kinetic equation constitutes a self-consistent description. For larger values of nonliterary this is no longer the case, and the wave lifetime is finite and amplitude changes need to be taken into account. Consequently interactions may not be strictly resonant. This statement also follows from the Fourier uncertainty principle. In other words, the waves with varying amplitude can not be represented by a single Fourier component. This effect is larger for stronger level of nonlinearity parameter. par Derivation of the kinetic equation with a broadened delta function is given in details in citepLLNZ, and is not going to be repeated here. The result is that begineqnarray fracd n_bmpdt = 4 int |V_bmp_1,bmp_2^bmp|^2 f_p12 delta_bmp - bmp_1-bmp_2 cal L(omega_bmp -omega_bmp_1-omega_bmp_2) d bmp_12 nonumber
-4int |V_bmp_2,bmp^bmp_1|^2 f_12p delta_bmp_1 - bmp_2-bmp cal L (omega_bmp_1 -omega_bmp_2-omega_bmp) d bmp_12 nonumber
-4int |V_bmp,bmp_1^bmp_2|^2 f_2p1 delta_bmp_2 - bmp-bmp_1 cal L(omega_bmp_2 -omega_bmp-omega_bmp_1) d bmp_12 ,nonumber
labelKineticEquationBroadened endeqnarray Here ${cal L}$ is defined as beginequation calL(Deltaomega) = fracGamma_k12(Deltaomega)^2 + Gamma_k12^2, labelscriptyL endequation where $Gamma_{k12}$ is the total broadening of each particular resonance, and is given below. If the nonlinear frequency renormalization tends to zero, i.e. $Gamma_{k12} to 0$ , ${cal L}$ reduces to the delta function:

$$limlimits_{Gamma_{k12}to 0} {cal{L}}(Deltaomega) =pi delta(Deltaomega).$$

Consequently, in the limit resonant interactions (i.e. no broadening) (refKineticEquationBroadened) reduces to Eq. (refKineticEquation) . We have shown in citeLLNZ that the broadening in Eq. (refscriptyL) is given by beginequation Gamma_k12=gamma_bmp+gamma_bmp_1+gamma_bmp_2. labelGammak12 endequation It means that the total resonance broadening is the sum of individual frequency broadening, and can be thus seen as the ``triad interaction'' frequency. par The single frequency renormalization can be calculated em self-consistently from begineqnarray gamma_bmp = 4 int |V_bmp_1,bmp_2^bmp|^2 (n_bmp_1+ n_bmp_2) delta_bmp - bmp_1-bmp_2 cal L(omega_bmp -omega_bmp_1-omega_bmp_2) d bmp_12 nonumber
-4int |V_bmp_2,bmp^bmp_1|^2 (n_bmp_2 - n_bmp_1) delta_bmp_1 - bmp_2-bmp cal L (omega_bmp_1 -omega_bmp_2-omega_bmp) d bmp_12 nonumber
-4int |V_bmp,bmp_1^bmp_2|^2 (n_bmp_1- n_bmp_2) delta_bmp_2 - bmp-bmp_1 cal L(omega_bmp_2 -omega_bmp-omega_bmp_1) d bmp_12 . nonumber
labelGamma endeqnarray The interpretation of this formula is the following: nonlinear wave-wave interactions lead to the change of wave amplitude, which in turn makes the lifetime of the waves to be finite. This, in turn, makes the interactions to be near-resonant. par A self-consistent estimate of $gamma_{bm{p}}$ requires the iterative solution of (refKineticEquationBroadened) and (refGamma) over the entire field: the width of the resonance (refGamma) depends on the lifetime of an individual wave [from (refKineticEquationBroadened)], which in turn depends on the width of the resonance (refGammak12). This numerically intensive computation is beyond the scope of this manuscript. Instead, we make the uncontrolled approximation that: beginequation gamma_bmp = delta omega_bmp. labelGammaFraction endequation par We note that this choice is made for illustration purposes only, we certainly do not claim that it represents a self consistent choice. Below, we will take $delta$ to be $10^{-2}$ and $10^{-3}$ . These values are rather small, therefore we remain in the closest proximity to the resonant interactions. par subsectionCharacteristic nonlinear time scale of the Garrett and Munk Spectrum par Estimates of near-resonant transfers are obtained by assuming horizontal isotropy integrating (refKineticEquationBroadened) over horizontal azimuth: begineqnarray fracpartial n_bmppartial t = 4pi int frack_1 k_2S_p12 |V_bmp_1,bmp_2^bmp|^2 f_p12 delta_bmp - bmp_1-bmp_2 cal L(omega_bmp -omega_bmp_1-omega_bmp_2) dk_12 dm_1 nonumber
-4pi int frack_1 k_2S_12p |V_bmp_2,bmp^bmp_1|^2 f_12p delta_bmp_1 - bmp_2-bmp cal L (omega_bmp_1 -omega_bmp_2-omega_bmp) dk_12 dm_1 nonumber
-4pi int frack_1 k_2S_2p1 |V_bmp,bmp_1^bmp_2|^2 f_2p1 delta_bmp_2 - bmp-bmp_1 cal L(omega_bmp_2 -omega_bmp-omega_bmp_1) dk_12 dm_1 , labelIntKineticEquationBroadened endeqnarray where $S_{p12}$ is the area of the triangle $bm{k} = bm{k}_1 + bm{k}_2$ . We numerically integrated (refIntKineticEquationBroadened) for $bm{p}$ 's which have frequencies from $f$ to $N$ [specifically (33/32, 17/16, 9/8, 5/4, 3/2, 2, 4, ...)$f$ ] and vertical wavenumbers from $4pi/(2b)$ to $200pi/(2b)$ ([2,4,6, ... 98] $pi/b$ ) . The limits of integration are restricted by horizontal wavenumbers from $2pi/10^5$ to $2pi/10$ meters$^{-1}$ , vertical wavenumbers from $2pi/(2b)$ to $2pi/10$ meters$^{-1}$ , 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 $5m$ , where $m$ is the vertical wavenumber. par Below we calculate the nonlinear time scale (refNonlinearTime) and nonlinearity parameter (refNonlinearRatio). To calculate this parameter, we need to choose a form of spectral energy density of internal waves. We we utilize the Garrett and Munk spectrum as an agreed-upon representation of the internal waves: beginequation E(omega,m) = frac4 fpi^2 m_ast E_0 frac11+(fracmm_ast)^2 frac1omega sqrtomega^2-f^2 . labelGM endequation Here the reference wavenumber is given by beginequation m_ast = pi j_ast / b,labelJstarendequation in which the variable $j$ represents the mode number of an ocean with an exponential buoyancy frequency profile having a scale height of $b$ . par We choose the following set of parameters: beginitemize item $b$ = 1300 m in the GM model item The total energy is set as: beginequation E_0 = 30 times 10^-4 rm  m^2 rm  s^-2 . nonumber endequation item Inertial frequency is given by $f=10^{-4}$ rad/sec, and buoyancy frequency is given by $N_0=5 times 10^{-3}$ rad/sec. item The reference density is taken to be $rho_0=10^{3}$ kg/m$^{3}$ . enditemize par We then calculate the nonlinearity parameter (refNonlinearRatio) and the nonlinear time scale (refNonlinearTime). To do so we substitute the Garrett and Munk spectrum (refGM) into the kinetic equation with broadening (refKineticEquationBroadened). For matrix elements we use citeMO75, Eq. (refVMO), and citeLT2, Eq. (refLTV). We also use the dispersion relation of internal waves, (refeq:dispersionISO) for the isopycnal Hamiltonian, and (refeq:dispersionCAR) for Lagrangian coordinates. We use two values of $delta$ in (refGammaFraction): $delta=10^{-2}$ and $delta=10^{-3}$ . We therefore make four calculations: beginitemize item Run I citeLT2 with $delta=10^{-3}$ item Run II citeMO75 with $delta=10^{-3}$ item Run III citeLT2 with $delta=10^{-2}$ item Run IV citeMO75 with $delta=10^{-2}$ enditemize Results appear in Figs. refNonlinearityParameter and refNonlinearTimeFigure. par For Run 1 the nonlinearity parameter is uniformly small, smaller than $10^{-1}$ . Such value of the nonlinearity parameter indicates that the kinetic equation is a self-consistent approach for the Garrett and Munk Spectrum. Increasing values of the nonlinearity parameter are noted with increasing vertical wavenumbers. This is consistent with intuition that we have about such systems. The nonlinear time scale is of the order of one hundred wave periods at low vertical wavenumber and of order ten wave periods at high vertical wavenumber. We also define a ``zero curve'' - It is the locus of wavenumber-frequency where the nonlinearity parameter and time-derivative of waveaction is exactly zero. The zero curve clearly delineates a pattern of energy gain for frequencies $f < omega < 2f$ , energy loss for frequencies $2f < omega < 5 f$ and energy gain for frequencies $5f < omega < N$ . This seems to be a characteristic pattern that appears in our calculations. Note that the zero curves are nearly independent of vertical wavenumber. par The citeMO75, matrix element (refVMO), Run II ( $delta=10^{-3}$ ) results are qualitatively similar to Run I. Factor of 2-3 faster decorrelation times and levels of nonlinearity are noted in the high-frequency and high-wavenumber part of the spectrum. par Therefore we conclude that when near-resonant interactions are included, the transfer rates are representation dependent. Furthermore, Lagrangian approaches predict higher level of nonlinearity. par To investigate in more details results of near-resonant interactions, we perform numerical calculations for $delta=10^{-2}$ . Results for the canonical Hamiltonian formulation in isopycnal coordinates Run III are nearly identical to those with $delta=10^{-3}$ . Results for the Lagrangian coordinate representation are both em quantitatively and em qualitatively different. The Lagrangian coordinate formulation (refVMO) now predicts $O(1)$ nonlinearity for high frequencies, while the isopycnal coordinate formulation still returns nonlinearity parameter and much slower decorrelation time estimates. The zero curves for the Lagrangian coordinate representation are no longer simple functions of frequency at this higher level of nonlinearity. The zero curves in the isopycnal coordinate system are relatively independent of $delta$ . par To investigate the differences between approaches and the sensitivity of our results to the value of $delta$ , in more details, we plot in Fig. refDifferences the differences of the nonlinearity parameter for these runs. In particular, we calculate the differences between Run I and Run II, Run I and Run III, and finally between Run II and Run IV. Differences associated with increased resonance broadening are minimal, $10^{-3}$ or smaller, for the isopycnal Hamiltonian. As the nonlinearity parameter estimates are representation dependent, differences between isopycnal coordinate and Lagrangian coordinate representations are much larger and increase with increasing $delta$ . par We have found that transports for the canonical Hamiltonian representation are not too sensitive to near-resonant interactions. We have also found in Section refResonantInteractions that all approaches are equivalent on the resonant manifold. We therefore conclude that all approaches will converge to Hamiltonian one as delta decreases. We have not undertaken such calculations as such small values of $delta$ would require significant modifications to our numerical algorithm. par Note that the Fig. refNonlinearTimeFigure, especially Runs I and II, bear a strong resemblance to Fig. 4 of citeO76. These figures contain two positive and one negative lobe with similar boundaries separating these regions, consistent with the characteristic pattern mentioned above. citeO76 does not make the hydrostatic approximation, used the GM75 model as the basis of his evaluations and is constrained to the resonance manifold. We have made the hydrostatic approximation, base our evaluations on the GM76 model and have included resonance broadening. Similarities are also apparent with Fig. 12 of citepMB77 and Fig.s 10 and 11 of McComas and Müller (1981). In those resonant evaluations using the GM76 model, the hydrostatic approximation was invoked and interactions with frequencies greater than $N/3$ were excluded. The major difference is that the zero line separating the positive and negative lobe at high frequencies has moved to $10f$ . par sectionDiscussion labelConclusion par In this paper we have review different approaches for wave-wave interactions that have been presented in the literature in the last three decades. Namely, we have concentrate on the approaches of citetMO75,Voronovich,Zeitlin,LT,LT2. In the absence of background rotation, we demonstrate that these four approaches produce em equivalent results on the resonant manifold. par 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. par This result is less surprising for the canonical Hamiltonian approaches citepVoronovich,LT. A canonical Hamiltonian representation is the gold standard of wave turbulence. It guarantees that the symmetries and hence conservation principles of the original equation set in the spatial/temporal domains have been preserved in the spectral domain,citep[e.g.][]ZLF. Thus, if Voronovich's Clebsh variable representation in Eulerian coordinates and the Lvov and Tabak isopycnal Hamiltonian describe the same physical system, then there is a canonical transformation that connects these two Hamiltonians. It is well known that such a canonical transformation reduces to the identity transformation on the resonant manifold. To prove this statement one constructs a near-identical canonical transformation, which is applicable for weakly nonlinear systems (See Appendix A3 in citeZLF). The Hamiltonian on the resonant manifold is invariant under a canonical near-identity transformation. par That is why Voronovich's matrix elements (refeq:Voronovich) look identical to the interaction matrix element in em isopycnal coordinates (refHamiltonian) em on the resonant manifold. par We can argue that, while the other two matrix elements (citeZeitlin and citeMO75) are not in a canonical Hamiltonian formulation, they nevertheless do describe the same physical system. Consequently, they also can be approximated by a certain Hamiltonian structure, at least for small nonlinearities. This is explicitly the case for the noncanonical Hamiltonian of citepMO75. It appears to be implicitly true of the citetZeitlin non-Hamiltonian kinetic equation. Therefore equivalence of the scattering matrix element on the resonant manifold is an intuitive, yet not trivial statement. par On the other hand, it is also intuitive that there will be coordinate representation dependent differences. 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 citepSandP91. Further differences are anticipated in a fully Lagrangian coordinate system citepPinkel08. 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. In particular, differences may represent the effects of resonance broadening associated with Doppler shifting. par We also demonstrate that the isopycnal and Lagrangian coordinate system approaches predict qualitatively different results with the inclusion of the near-resonant interactions and background rotation. The canonical Hamiltonian isopycnal formalism is insensitive to off-resonant interactions: Broadening the resonance width by an order of magnitude does not create significant differences in the nonlinearity parameter. The noncanonical Lagrangian coordinate representation is, in contrast, quite sensitive to these changes. par As explained above, the Hamiltonian on the resonant manifold is invariant under near-identity canonical transformations. The kinetic equation describes the spectral transfers associated with the cubic terms of the Hamiltonian and conserves the energy associated with quadratic terms of the Hamiltonian. The kinetic equation should therefore return representation independent results on the resonant manifold. This statement is no longer true for near-resonant interactions. par Indeed, since the structure of the Hamiltonian may be altered off the resonant manifold by a near-identity canonical transformations, one em should anticipate representation dependent differences in spectral energy transfer when near-resonant interactions are included. Such differences become more and more significant as nonlinearity increases and cubic parts of the Hamiltonian become increasingly large. par We would like to suggest that the differences between citetMO75 and citetLT2 off the resonant manifold represent physical effects. However, an issue with extant Lagrangian coordinate representations is that they require a small amplitude assumption that represents an unconstrained approximation whose domain of validity em vis-a-vis the weak interaction approximation is not well defined, citepM86. On the basis of estimates of how horizontal Doppler shifting contributes to isopycnal spectra, we would anticipate that the Lagrangian coordinate stationary state would have typically steeper spectral slopes in the frequency domain than frequency spectra in isopycnal coordinates. The results presented here indicate that resonance broadening will quickly whiten the high frequency Lagrangian coordinate spectrum, in direct contradiction to our intuition regarding physical effects. par In this paper we have shown that while on the resonant manifold (i.e. for weakly nonlinear interactions) all approaches we considered do agree, inclusion of the near-resonant interactions (for stronger nonlinearities) should be done with care. Results with near resonant interactions are representation dependent. This observations warrants further study. par vspace*baselineskip par We thank V. E. Zakharov for presenting us with a book citepMir and for encouragement. We also thank E. N. Pelinovsky for providing us with citetPR77. This research is supported by NSF CMG grants 0417724 and 0417466. Y. L. was also supported by NSF CAREER DMS 0134955. We are grateful to YITP in Kyoto University for permitting use of their facility. par newpage par beginfigure[tp] begincenter includegraphicsfig/MO.0_psfrag.epsincludegraphicsfig/V.0_psfrag.eps includegraphicsfig/Z.0_psfrag.epsincludegraphicsfig/H.0_psfrag.eps captionMatrix elements $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}_{mathrm{(ref{eq:sol2})}}\vert^2$ given by the solution (refeq:sol2). upper left: $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}_{mathrm{(ref{eq:sol2})}}^{mathrm{MO}}\vert^2$ according to citetMO75, upper right: $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}_{mathrm{(ref{eq:sol2})}}^{mathrm{V}}\vert^2$ according to citetVoronovich, bottom left: $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}_{mathrm{(ref{eq:sol2})}}^{mathrm{CZ}}\vert^2$ according to citetZeitlin, bottom right: $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}_{mathrm{(ref{eq:sol2})}}^{mathrm{H}}\vert^2$ according to citetLT. labelFIGRESONANTa endcenter endfigure par beginfigure[tp] begincenter includegraphicsfig/MO.1_psfrag.epsincludegraphicsfig/V.1_psfrag.eps includegraphicsfig/Z.1_psfrag.epsincludegraphicsfig/H.1_psfrag.eps captionMatrix elements $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol3})}}\vert^2$ given by the solution (refeq:sol3). upper left: $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol3})}}^{mathrm{MO}}\vert^2$ according to citetMO75, upper right: $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol3})}}^{mathrm{V}}\vert^2$ according to citetVoronovich, bottom left: $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol3})}}^{mathrm{CZ}}\vert^2$ according to citetZeitlin, bottom right: $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol3})}}^{mathrm{H}}\vert^2$ according to citetLT. labelFIGRESONANTb endcenter endfigure par beginfigure[tp] begincenter includegraphicsfig/MO.6_psfrag.epsincludegraphicsfig/V.6_psfrag.eps includegraphicsfig/Z.6_psfrag.epsincludegraphicsfig/H.6_psfrag.eps captionMatrix elements $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol4})}}\vert^2$ given by the solution (refeq:sol4). upper left: $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol4})}}^{mathrm{MO}}\vert^2$ according to citetMO75, upper right: $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol4})}}^{mathrm{V}}\vert^2$ according to citetVoronovich, bottom left: $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol4})}}^{mathrm{CZ}}\vert^2$ according to citetZeitlin, bottom right: $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{(ref{eq:sol4})}}^{mathrm{H}}\vert^2$ according to citetLT. labelFIGRESONANTc endcenter endfigure par beginfigure[tp] begincenter includegraphicsfig/ES.0_psfrag.eps includegraphicsfig/ID.0_psfrag.eps includegraphicsfig/PSI.1_psfrag.eps captionupper: Matrix elements $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}_{mathrm{ES}}\vert^2$ given by the solution (refeq:sol1). middle: Matrix elements $\vert{V^{bm{p}}_{bm{p}_1, bm{p}_2}}_{mathrm{ID}}\vert^2$ given by the solution (refeq:sol2). bottom: Matrix elements $\vert{V^{bm{p}_1}_{bm{p}_2, bm{p}}}_{mathrm{PSI}}\vert^2$ given by the solution (refeq:sol3), which gives PSI as $\vert bm{k}_1\vert to 0$ ( $epsilon to 0$ ). The matrix elements here are shown as functions of $epsilon$ such that $(\vert bm{k}_1\vert, \vert bm{k}_2\vert) = (epsilon, epsilon/3+1) \vert bm{k}\vert$ . All four versions of the Matrix elements are plotted here: the appearance of a single line in each figure panel testifies to the similarity of the elements on the resonant manifold. labelFigureThree endcenter endfigure par beginfigure[tp] begincenter includegraphics[scale=0.5]fig/NewEpsilon.eps captionNonlinearity parameter (refNonlinearRatio) for the Garrett and Munk spectrum (refGM) calculated via (refKineticEquationBroadened). The upper figures represent the value of nonlinearity parameter calculated using citeLT2, equation (refLTV) with $delta=10^{-3}$ , Run I (upper left) and $delta=10^{-2}$ Run III (upper right). The bottom two pictures represent the value of nonlinearity parameter calculated via citeMO75, (refVMO) with $delta=10^{-3}$ Run II (bottom left) and $delta=10^{-2}$ Run IV (bottom right). labelNonlinearityParameter endcenter endfigure par beginfigure[tp] begincenter includegraphics[scale=0.5]fig/NewTau.eps captionNonlinear time (refNonlinearTime) for the Garrett and Munk spectrum (refGM) calculated via (refKineticEquationBroadened). The upper figures represent the value of nonlinearity parameter calculated using citeLT2, equation (refLTV) with $delta=10^{-3}$ , Run I (upper left) and $delta=10^{-2}$ Run III (upper right). The bottom two pictures represent the value of nonlinearity parameter calculated via citeMO75, (refVMO) with $delta=10^{-3}$ Run II (bottom left) and $delta=10^{-2}$ Run IV (bottom right). On this bottom right figure white region to the left of the $0.1$ contour corresponds to extremely fast time scales, faster then $0.1$ of a day. On these figures, $omega$ in cpd, $m$ in cycle/m, and nonlinear time $tau^{mathrm{NL}}$ is measured in days. labelNonlinearTimeFigure endcenter endfigure par beginfigure[tp] begincenter includegraphics[scale=.5]fig/NewDifference.eps captionDifferences between nonlinearity parameter (refNonlinearRatio) calculated via citeMO75 and citeLT2 with $delta=10^{-3}$ , i.e. between Run I and Run II, (a), calculated with citeLT2 with $delta=10^{-2}$ and $delta=10^{-3}$ , i.e. the difference Run I and Run III (b), and finally between citeMO75 with $delta=10^{-2}$ and $delta=10^{-3}$ , i.e. between Run II and Run IV (c). labelDifferences endcenter endfigure par clearpage par bibliographystyleametsoc par beginthebibliography100 par bibitem[Brehovsky(1975)]B75 Brehovski. 1975: On interactions of internal and surface waves in the ocean. em Oceanology, bf 15 (in Russian). This is citation [11] of citetPR77. par bibitem[Bühler and McIntyre, 2005]BM05 Bühler, O. and M. E. 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