The Internal Wave Action Model IWAM

Peter Muller and Andrei Natarov

Department of Oceanography, University of Hawaii, Honolulu, HI 96822

Abstract. This article describes the overall modeling strategy for the InternalWave Action Model (IWAM), which is a hybrid dynamical model that predictsthe internal wave field in a basin or region for given forcing and environmentalfields. Large-scale internal tides and near-inertial internal waves will bemodeled by integrating appropriate hydrodynamic equations. Intermediateand small-scale internal waves will be modeled by integrating the radiativebalance equation (RBE). The RBE describes changes of the action densityspectrum along wave group trajectories as a result of forcing, interaction,and dissipation processes; RBE makes the random phase, geometric optics(WKB), and weak interaction approximations. The action density spectrumis a reduced description of the internal wave field; it is a statistical second-order moment from which all other second-order moments can be calculated.IWAM will combine different dynamical processes into a coherent framework.Methodologically, it is similar to the WAM model developed by the surfacewave modeling community. One important task in constructing IWAM isspecifying the source terms in the radiation balance equation. The specificationof the dissipation source function is discussed in some detail. It involves the“derivation” of a functional form with free parameters and solutions of theRBE for the reflection off a straight slope, where propagation and dissipationare assumed to balance. Comparisons with observations and numerical processstudies then allow the calibration of the free parameters. The calibration andvalidation of the RBE are the first tasks in constructing IWAM. The nexttasks include the integration of the radiative balance equations and embeddingit into a circulation model. The major issues and problems are discussed.

Introduction

Sufficient progress in internal wave theory and ob-servation has been made to attempt the construction ofa dynamical internal wave model. Here we describeone such modeling effort: the hybrid Internal WaveAction Model (IWAM). Large-scale internal tides andnear-inertial internal waves will be modeled by integrat-ing appropriate hydrodynamic equations in an oceanbasin or a region. Intermediate- and small-scale inter-nal waves will be modeled by integrating the radiativebalance equation (RBE). Together, these componentswill predict the internal wave field in a basin or a regionfor given forcing and environmental fields. Specifically,IWAM will

• provide understanding of the internal wave field asa balance of generation, transfer, and dissipationprocesses,

• predict changes in the internal wave field in re-sponse to changes in forcing and environmentalfields, and

• determine internal wave induced stresses, fluxes,mixing, and dispersion.

Overall, IWAM is expected to focus research on inter-nal wave dynamics, similar to the way that the intro-duction of the Garrett and Munk model spectrum fo-cused research on internal wave kinematics. Eventually,IWAM is envisioned to be run in conjunction with cir-culation, turbulence, tracer, population, and acousticmodels where it will provide the internal-wave-inducedfluxes, transports, dispersion, mixing, noise, and inter-nal wave environment.

In this article we first reiterate some of the reasonsthat motivate the construction of a dynamical internalwave model. Chief among these reasons are the link

95

96 MULLER AND NATAROV

between internal waves and diapycnal mixing, the pos-sible role of internal waves (IW) in cascading energyfrom large to small scales, and important processes suchas acoustic propagation and biological production thatare affected by IW. We then briefly discuss some rele-vant aspects of the kinematics of IW. One must distin-guish between near-inertial IW, internal (or baroclinic)tides, the IW continuum, and nonlinear waves such assolitons and bores. These different kinds of IW willbe modeled differently. Our current dynamical under-standing of open ocean IW is reviewed next. Internalwaves are generated as large-scale near-inertial IW bythe atmosphere and as large-scale internal tides by thebarotropic tide at topography. These large-scale wavespropagate in physical space while nonlinear wave-waveinteractions and other scattering processes cascade theirenergy through wavenumber space to small scales wherethey break and dissipate. The generation and propa-gation of the large-scale near-inertial IW and internaltides will be modeled by integrating appropriate hy-drodynamic equations on a spatial grid. The cascadethrough the internal-wave continuum will be modeledby integrating the radiative balance equation (RBE).The RBE is a reduced description. It foregoes deter-ministic information and describes the evolution of theenergy (or action) density spectrum, a statistical sec-ond order moment of the IW field from which all othersecond order moments can be calculated. The RBEmakes the random phase, geometric optics (WKB), andweak interaction approximations. Central to the RBEare the source terms representing the various dynam-ical processes. These source terms can be “derived”from underlying hydrodynamical equations under moreor less restrictive assumptions. Some of these deriva-tions contain free parameters that must be calibratedby observations or results from numerical process stud-ies. As an explicit example we describe in some detailthe construction of the dissipation source function. Weconclude with a discussion of the major issues and prob-lems facing the IWAM modeling effort.

Motivation

There are many reasons to study and understandoceanic internal waves. Here we point out the link be-tween IW and diapycnal mixing, the possible participa-tion of IW in the overall energy cascade from the largegeneration scales to the small dissipation scales, and theeffect of IW on acoustic propagation.

Diapycnal mixing. The link between diapycnal mix-ing and IW is by now well established (e.g., Mullerand Briscoe, 2000). Diapycnal mixing is an integralfactor in the meridional overturning circulation of theocean. The slow upwelling in the ocean interior across

density surfaces requires counteracting diapycnal dif-fusion. Diapycnal diffusion in a stably stratified fluidrequires mechanical energy, and the availability of thisenergy might well be the controlling factor in settingthe strength of the overturning circulation as, e.g., dis-cussed by Munk and Wunsch (1998). It has been foundthat

• Diapycnal mixing in the ocean interior is drivenby intermittent patches of small-scale turbulence.

• The turbulent patches have a vertical extent ofup to a few meters and are caused by breakinginternal waves.

• Internal waves break by either shear or convectiveinstabilities that are caused by chance superposi-tions or encounters with critical layers.

Diapycnal mixing can also be caused by other pro-cesses such as double diffusion. The link between IWand diapycnal mixing has come out of wave-wave inter-action theories (McComas and Muller, 1981; Henyey etal., 1986) and has led to parameterizations of the dia-pycnal mixing coefficient or turbulent dissipation ratein terms of the internal wave shear and strain. Differ-ent schemes have been suggested (Gregg, 1989; Henyey,1991; Wijesekera et al., 1993; Polzin et al., 1995; Sunand Kunze, 1999; Gregg et al., 2003) and are currentlybeing used to estimate the diapycnal mixing coefficientor turbulent dissipation from ADCP or CTD measure-ments; an example is given in Figure 1. Thus, by un-derstanding and predicting internal waves we may un-derstand and predict diapycnal mixing.

Routes to dissipation. The general circulation of theocean is forced by surface fluxes of momentum, heat,and fresh water at large, basin-wide horizontal scalesof the order of 1000 km and at long time-scales of theorder of 1 year and longer. The circulation has compa-rable large and long scales, as well as intermediate onesassociated with equatorial zonal and lateral boundarycurrents and various mesoscale instabilities. In steadystate, the large scale input of variance into the veloc-ity, temperature, and salinity fields of the circulationmust be dissipated at small scales by molecular fric-tion, heat conduction, and salt diffusion. For molecularfriction the dissipation scale is the Kolmogorov scale, ofthe order 10−2 m. We presently do not know by whichprocesses the variances cascade across the eight ordersof magnitude separating the generation and dissipationscales.

For the kinetic energy - the variance of the velocityfield - the first steps in the down-scale cascade are theprocesses of barotropic and baroclinic intability. These

IWAM 97

-3

-2

-1

0

1 cm /s 2log(K)

Munk

GM

N Atlantic eastern polar gyre (A24)

GM

6000

5000

4000

2000

1000

z (m)

50N 60N 60N 50N40W 40W 30W 20W 20W20W

2 12 25 35 45 55 78 92 102 121 141 152 162 176

-3

-2

-1

0

1

Munk

GMM2

1000 2000 3000 4000 5000 6000 6000

5000

4000

2000

1000

r (km)

z (m)

Rey

kjan

es R

idg

e

Rey

kjan

es R

idg

e

Figure 1. Eddy diffusivity K inferred from internalwave strain using the Gregg et al. (2003) parameteri-zation. The data are from one of the WOCE/ACCEA24 cruises in the eastern polar gyre of the N. At-lantic with multiple crossings of the Mid-Atlantic andReykjanes Ridges. Strain is inferred from CTD pro-file segments, either from buoyancy frequency squaredor potential temperature gradient.The upper panel as-sumes a shear/strain ratio of 3 (corresponding to theGM model value) while the lower panel uses a semidiur-nal shear/strain ratio under the assumption that inter-nal tides may be important. The latter increases withlatitude so that high-latitude diffusivities in the lowerpanel are higher than those at low latitude. For bothpanels, white values correspond to expectations for atypical (GM) internal wave field, blue values are lower,red values are higher. (Courtesy of E. Kunze)

generate mesoscale eddies with a spatial scale of the or-der of the internal Rossby radius of deformation, whichis about 50 km on average. These instabilities thusprovide a forward (downscale) cascade of energy frombasin-wide generation scale to the internal deformationradius. But any further down-scale cascade is inhib-ited by the fact that mesoscale eddies, like the generalcirculation, are very anisotropic flows, constrained byrotation and stratification to be nearly horizontal (i.e.,to be nearly two-dimensional in this sense). These mo-tions approximately satisfy hydrostatic and geostrophicmomentum balances, and their evolution is governed bythe potential vorticity equation. The key point in thepresent context is that the advective dynamics of suchbalanced motions - called geostrophic turbulence - ex-hibits a forward or down-scale cascade of enstrophy (i.e.,variance of potential vorticity) to its dissipation at smallscales but an inverse (up-scale) turbulent cascade of en-ergy toward larger scales, hence away from dissipation

by molecular viscosity at small scales (Charney, 1971);this behavior is analogous to 2D turbulence, i.e., theturbulence in a two-dimensional fluid. Thus balanceddynamics do not provide an efficient root to energy dis-sipation in the ocean interior.

At smaller scales, stratified shear flows are known tobecome unstable, due to Kelvin-Helmholtz instability,once the Richardson number falls below 1/4. This in-stability is observed to set in at vertical scales of about10 m. This instability starts a downscale energy cas-cade - called stratified turbulence - where the overturn-ing eddies work against buoyancy forces, mix the wa-ter column, and increase the potential energy at thesame time as they are being dissipated by molecularfriction. Once the down-scale cascade reaches the Os-midov or buoyancy scale, which is of the order of 1 m,the stratification becomes unimportant and the turbu-lence becomes a three-dimensional, isotropic turbulencethat efficiently cascades energy down-scale through aKolmogorov inertial range to the dissipation scales.

As depicted in Figure 2, there is thus a gap of roughlythree orders of magnitude between the anisotropic, bal-anced motions at large scales and the isotropic, small-scale motions that complete the forward energy cascadeto dissipation. This gap needs to be bridged for an equi-librium kinetic energy balance in the ocean. Figure 2depicts three possible routes:

• the internal gravity wave route (II)

• the instability route (III), and

• the boundary route (IV)

The internal wave route assumes that pre-existing in-ternal gravity waves interact with large- and mesoscalemotions and catalyze a cascade of their energy to smallscales. This and the other routes are more fully dis-cussed in Muller et al. (2003). None of these routesto dissipation, or any other route, have strong theo-retical or observational confirmation yet. But it is es-sential that we determine and understand the routesto dissipation. Simply inserting horizontal and verti-cal eddy viscosity coefficients in models of the oceanicgeneral circulation and tuning their values to some de-sired outcome is hazardous and diminishes the fidelityof the model. Any circulation model that aspires topredictive capabilities must base its parametrizationson a dynamical understanding of the parametrized mo-tions. Internal waves are part of these parameterizedmotions—perhaps a crucial part.

Acoustic propagation. The sound speed c is the mostfundamental quantity that determines acoustic propa-

98 MULLER AND NATAROV

10-2

100

102

104

106

Lh (m

)

Climate forcingGeneral

circulation

I

Meso-scaleeddies

StormsTidesCurrents over topography

Inertia-Gravitywaves

Stratified turbulence

Kolmogorovcascade

GAP?II

?III

?IV

102

100

10-2

10-4

10-6

- kK

- ko

- kKH

- kaa

- Rd

-1

kh (m

-1)

Figure 2. Possible energy pathways of the general cir-culation. Energy must be transferred across the gapbetween large-scale balanced flows and small-scale im-balanced flows that exhibit a forward cascade towardsviscous dissipation. I. Baroclinic/barotropic instability.II. Interaction with inertia-gravity waves. III. Unbal-anced instabilities. IV. Dissipation in boundary layers.(From Muller et al., 2003)

gation. It is defined by the thermodynamic relation

c−2 =∂ρ(p, θ, s)

∂p(1)

where ρ(p, θ, s) is the equation of state, i.e., the den-sity of sea water as a function of pressure p, potentialtemperature θ, and salinity s. The sound speed in-creases with pressure, temperature and salinity. Theeffect of salinity is relatively small. In the ocean, pres-sure, potential temperature, and salinity are functionsof position x and time t. So is the sound speed

c(x, t) = c (p(x, t), θ(x, t), s(x, t)) (2)

The sound speed at a specific horizontal position anda specific time as a function of the vertical or depthcoordinate z is called a sound speed profile c(z). Thegradient of the sound speed profile is given by

dc

dz=

(∂c

∂θ

)

p,s

dθ

dz+

(∂c

∂s

)

p,θ

ds

dz+

(∂c

∂p

)

θ,s

dp

dz(3)

The first two terms constitute the potential gradient(dc

dz

)

p

=(∂c

∂θ

)

p,s

dθ

dz+

(∂c

∂s

)

p,θ

ds

dz(4)

The third term is called the adiabatic gradient. Internalgravity waves - and often other oceanic motions - causesurfaces of constant sound speed to move up and downand not to be level in space. An internal wave of verticaldisplacement ζ thus causes a change in the sound speedgiven to first order by

δc = −(dc

dz

)

p

ζ (5)

The sound speed variation δc enters the acousticwave equation and affects acoustic propagation. Theforward or direct problem is to calculate the acousticfield for given internal wave displacements. The inverseproblem consists of inferences about ζ from acousticmeasurements. Internal wave velocities u also affectacoustic propagation, through a Doppler shift

δc = k · u (6)

The effect is small and generally ignored, except in re-ciprocal transmissions where the difference in transmis-sion times is related to the water velocity along theacoustic ray path.

Kinematics

Free linear gravity waves have frequencies betweenthe Coriolis frequency f and the buoyancy or Brunt-Vaisala frequency N . In the ocean their vertical scalesrange from about 1 km (first baroclinic mode) to 10m, and horizontal scales range from tens of kilome-ters to tens of meters. Since internal waves are disper-sive, there can also exist nonlinear waves of permanentform, where nonlinear steepening is balanced by disper-sion. There exist different kinds of such nonlinear wave-forms, including bores and solitons. In the ocean suchnonlinear waves are formed when internal tides shoaland steepen as they propagate onto the shelf. Figure3 shows a frequency spectrum of the horizontal kineticenergy. The spectrum shows a peak near the inertialfrequency f , another peak at the semidiurnal tidal fre-quency M2 and a broad continuum with a −2 slope.The continuum is often well described by the modelspectrum of Garrett and Munk (1972). The GM spec-trum, as it is now called, describes a wavefield that ishorizontally isotropic (waves coming in from all hori-zontal directions equally) and vertically symmetric (asmany waves propagate upward as downward). Oftenupward and downward propagating waves are assumedto be phase-locked and to produce vertically standing“normal modes.” The most surprising and significantaspect of the GM spectrum is its universality. To ob-serve significant (say larger than a factor of three) de-viations from the GM spectrum one has to go to very

IWAM 99

Figure 3. Frequency spectrum of the horizontal kineticenergy at 140 m depth at a site in the northeast Pacificduring the Ocean Storms Experiment, 1987-88. Internalwaves exist between the Coriolis frequency f and thebuoyancy frequency N . M2 is the semi-diurnal tidalfrequency. The dashed line labeled GM is the Garrettand Munk model prediction (Courtesy of M. Levine).

special places such as the Arctic Ocean or submarinecanyons. The dashed line in Figure 3 represents theGM spectrum and shows how well the observed con-tinuum is represented by it. The GM spectrum doesnot account for the variance in the internal (or baro-clinic) tidal peaks, which contain a considerable buthighly variable fraction of the variance. The GM spec-trum also does not properly represent many of the ob-served features of the inertial peak. It neither reflectsits large temporal variability, nor its strong dependenceon depth, nor its vertical asymmetry (most near-inertialIW are observed to propagate downward). The model-ing strategy of IWAM will take these differences in thekinematic structure into account. It will treat sepa-rately

• the large scale near-inertial waves and large scaleinternal tides,

• the internal wave continuum, and

• nonlinear soliton- or bore-like internal waves.

Figure 4. The conventional dynamic balance of theoceanic internal wave field in physical (x) and wavenum-ber (k) space. The wind and tides generate large-scalewaves of near inertial and tidal frequencies. Theselarge-scale waves propagate away from their sources inphysical space and cascade towards small-scale waves inwavenumber space. The cascade is caused by wave-waveinteractions and scattering at fronts, mesoscale eddies,topography and other scatterers. The small-scale wavesbreak and cause turbulence and mixing.

Dynamics

The conventional dynamic balance is depicted in Fig-ure 4. It has only two energy sources for internal waves:

• changes in the atmospheric windstress cause iner-tial oscillations in the oceanic surface mixed layer;part of their energy propagates into the ocean in-terior as large-scale, near-inertial internal waves,

• barotropic (or surface) tidal currents flowing acrosstopography in a stratified ocean generate inter-nal waves of tidal frequency, the baroclinic tides.Their large-scale components propagate into theocean interior.

There is good observational evidence for both thesegeneration processes. Near-inertial internal waves havebeen observed to propagate away from the surface un-derneath storm and cyclone tracks and quantitativeestimates of the energy input are becoming available(D’Asaro, 1985; Alford, 2001 and 2003; Watanabe andHibiya, 2002). Baroclinic tidal beams have been ob-served to emanate from certain topographic featuresand can often be traced for hundreds of kilometers (Rayand Mitchum, 1997; Dushaw et al., 1995) and quantita-tive estimates of the energy input are again becomingavailable (Sjoberg and Stigebrandt, 1992; Morozov, 1995;Egbert and Ray, 2000 and 2001; St.Laurent and Garrett,2002).

As these large-scale waves propagate away from theirgeneration region, they are assumed to interact non-linearly with themselves and other internal waves and

100 MULLER AND NATAROV

to scatter at mesoscale eddies, fronts, and topogra-phy. These nonlinear interaction and scattering pro-cesses transfer energy out of the large-scale waves intoever smaller scale waves that eventually break and causeturbulence and mixing. In a breaking event, the waveenergy is partly converted to potential energy, since themixing of a stably stratified fluid increases its potentialenergy.

Overall modeling strategy

Internal waves have horizontal scales down to tens ofmeters, vertical scales down to meters, and time scalesdown to tens of minutes. To adequately resolve IWone thus would have to integrate some appropriate hy-drodynamic equation on a grid with horizontal spacing∆x,∆y = O(10 m) and vertical spacing ∆z = O(1 m)with a time step ∆t = O(1 min). This is possible ina box of the order of 10 km × 10 km × 1 km but isclearly beyond present and foreseeable future computercapabilities for regions or basins. Various “box” studieshave been carried out to study specific processes.

Large-scale IW can be and have been resolved basin-wide and regionally. Niwa and Hibiya (2001) studiedthe generation and propagation of near-inertial IW inthe North Pacific in a model with horizontal grid spac-ing of 1/6 (18.5 km), 28 vertical levels, and 16-mintime steps. The generation and propagation of inter-nal tides has been modeled by Xing and Davies (1998),Holloway (2001), Merrifield et al. (2001), and Merri-field and Holloway (2002), among others. Johnson et al.(2003) studied the scattering of the internal tide at bot-tom topography in a region southwest of the HawaiianIsland chain.

To include into these basin-wide and regional mod-els intermediate and small-scale IW not only requiresexorbitant computer resources but also details in theforcing and environmental fields (e.g., topography) thatare generally not available. However, this point canbe turned to one’s advantage. All the detailed unre-solved influences tend to randomize the phases of thewaves. Because of all these unaccountable influencesone can employ the random phase approximation. Itassumes that the phases of the waves are random andthat the only predictable quantity is the amplitude orenergy of the wave, which is proportional to the am-plitude squared. Instead of describing the IW field byits velocity vector, vertical displacement, and pressure,one thus describes the internal wave field by its energydensity spectrum E(k) which gives the distribution ofwave energy among different wavenumbers. All othersecond-order moments of the IW field can be inferredfrom the energy density spectrum E(k) by using IWkinematic relations.

A second property of the IW field also ought to beexploited to reduce the computational load: its near lin-earity. IW are by definition a nearly linear phenomenon.When the nonlinearities become large the field turnsinto stratified turbulence. The zeroth order physics ofnearly linear internal wave is energy propagation alongwave group trajectories. This zeroth order physics doesnot need to be calculated, wasting computer resources,but should be presupposed. Any modeling effort shouldassume (rather than calculate) that IW propagate alongtheir trajectories. This assertion leads to the radiationbalance equation discussed next.

Radiative balance equation

The radiative balance equation makes three basic ap-proximations:

• the random phase approximation,

• the WKB or geometric optics approximation, and

• the weak interaction approximation.

The random phase approximation assumes that thevarious dynamical processes affecting the wave field willdistort the wave phases in an irregular way such that itis neither possible nor useful to predict wave phases.Instead, the wave field is described by its energy oraction density spectrum.

The WKB approximation assumes that the wave-lengths of the waves are small compared to the scalesof environment. The action density spectrum as a func-tion of wavenumber then varies slowly with position andtime on the scales of the environment.

The weak interaction approximation assumes that in-ternal waves are basically a linear phenomenon. Thewaves propagate along their group trajectories, beingonly slowly modified by dynamical processes. The dy-namical evolution of the action density spectrum alongwave group trajectories is caused by generation, trans-fer, and dissipation processes.

How good are these approximations? The randomphase approximation has worked well for the kinematicdescription of oceanic internal waves, except for someinternal tide phenomena which are driven by and phase-locked to surface tide.

The weak interaction approximation holds well formost generation and transfer processes. It can evenbe expected to hold for processes such as wave break-ing that are strongly nonlinear but localized in space.These processes are ”weak-in-the-mean” in that theyonly cause moderate changes in the spectrum.

The WKB approximation is not a real problem butmore a matter of definition. The environmental fields

IWAM 101

are divided into a mean component with scales largerthan the wavelengths in which waves propagate in WKBfashion and a fluctuating component with scales compa-rable to wavelengths at which the waves are scattered.

Because the radiation balance equation does notcarry any wave phase information and assumes (as op-posed to calculates) wave propagation it can eventuallybe integrated globally or regionally with a horizontalresolution of ∆x = O(100 km). This is different fromdirect numerical simulation (DNS) or large eddy sim-ulation (LES) of internal waves in which phase infor-mation is retained and wave propagation is calculated.These studies have grid sizes of order ∆x =1 cm (DNS)or of order ∆x =1 m (LES) if they correctly interprettheir diffusion coefficients as molecular coefficients oras representing the transfer across Kolmogorov’s iner-tial range. There are of course studies that purportto simulate internal waves at a coarser resolution byintroducing an eddy diffusion coefficient. These stud-ies, however, assume that the high Reynolds numberoceanic flow field can be simulated by a low Reynoldsnumber model, an unsupported assumption so far.

Formally, the radiative balance equation takes theform (Muller and Olbers, 1975)

(∂t + x · ∂/∂x + k · ∂/∂k

)A(k,x, t)

= Sgen + Strans + Sdiss (7)

where A(k,x, t) is the action density spectrum, k is thewavenumber vector, x is the position vector, and t thetime. The frequency of the waves can be obtained fromthe dispersion relation

ω = Ω(k,x, t) (8)

which depends on the local values of the mean environ-mental fields which in turn depend in a slowly varyingmanner on position x and time t. The wave group tra-jectories are defined by the group velocity

v = x = ∂Ω/∂k (9)

and by the rate of refraction

r = k = −∂Ω/∂x (10)

The left-hand side of the radiation balance equationdescribes changes of the action density spectrum alongwave trajectories. These changes are given by the dy-namical source terms on the right hand side where Sgen

describes the generation of internal wave action, Strans

the transfer of action across the spectrum, and Sdiss

the dissipation of wave action. Without any dynami-cal sources or sinks the action density spectrum does

not change along wave group trajectories. The radia-tion balance equation has to be augmented by boundaryconditions.

Generally, the radiation balance equation becomesless accurate as the scales of the waves increase. Theexact scale where the statistical description and theradiation balance equation need to be replaced by adeterministic description and hydrodynamic equationshas not yet been determined. The radiation balanceequation is clearly not applicable to tidally generatedsolitary waves propagating away from their generationsites.

Source terms

An important task is the specification of the sourceterms in the radiation balance equation and its bound-ary conditions. Some of these source terms can be de-rived from the basic hydrodynamic equations by mak-ing appropriate assumptions. Some examples are con-sidered in this section. For highly nonlinear processessuch as wave breaking, such a derivation is not possible.Instead one only arrives at functional forms with freeparameters. These parameters must then be calibratedby comparing solutions of the RBE with observationsor process simulations. This approach is demonstratedfor the dissipation source function in the next section.

Reflection off a straight slope. At rigid boundariesthe hydrodynamic boundary condition for inviscid flowsis u · n = 0 where u is the velocity vector and n is thenormal vector of the boundary. The normal velocitycomponent must be zero. For internal waves this con-dition gets transformed into reflection laws. One partof these laws relates the wavenumber and frequency ofthe reflected wave to those of the incident wave; the fre-quency and tangential wavenumber do not change uponreflection

ωr = ωi (11)kr

t = kit (12)

The normal component of the reflected wavenumbercan then be inferred from the “dispersion” relation

krn = kn (kr

t , ωr) (13)

Explicit formulas can be found in Eriksen (1982).Since the frequency does not change, the angle of thewave rays with respect to the vertical axis remainsthe same, rather than with respect to the normal vec-tor. When this angle and the bottom slope add up toπ/2 then the reflection becomes critical. The reflectedwavenumber becomes infinite. The second part of thereflection laws determines the amplitude of the reflected

102 MULLER AND NATAROV

wave. Under the assumptions of the radiation balanceequation it becomes

n · (Fi + Fr) = 0 (14)

whereFi,r = v (ki,r) E (ki,r) (15)

is the energy flux of the incident and reflected wave.The normal energy flux must be zero at the boundary.Equation (14) provides the boundary conditions for theradiative balance equation.

Nonlinear wave-wave interactions. The hydrody-namic equations contain advective nonlinearities of theform u · ∇ψ where ψ is any of the prognostic variables.Under various weak interaction assumptions these non-linearities lead to resonant wave-wave interactions. Thegeneral theory was developed by Hasselmann and Za-kharov and first applied to internal waves by Olbers(1976) and McComas and Bretherton (1977). Subse-quent investigations of nonlinear interactions among in-ternal waves are reviewed in Muller et al. (1986). Newanalyses are pursued by Lvov and Tabak (2003). Forinternal waves, wave-wave interactions take the form oftriad interactions. Two waves k1 and k2 interact togenerate a third wave k3. The source term in the RBEequation takes the form

Snl (k) =∫ ∫

d3k′d3k′′ T (k,k′,k′′)A (k′)A (k′′)

(16)where T (k,k′,k′′) is a transfer function that can be cal-culated from the underlying hydrodynamic equations.The source term describes the time rate of change ofthe action density spectrum at wavenumber k. Waveaction is gained at k due to the interaction of waves k′

and k′′ that generate waves at k. Wave action is lostat k due to the interaction of waves k and k′ that gen-erate waves at k′′. The double integral sums over allthese possible interactions. The transfer function con-tains two delta functions because the triad interactionssatisfy the resonant conditions

ω = ω′ + ω′′ (17)k = k′ + k′′ (18)

The six-dimensional integral thus reduces to an integralover two-dimensional resonant surfaces.

The structure of the source terms for various otherprocesses has been derived in Muller and Olbers (1975).Attempts have been made to evaluate some of thesesource terms for quasi-realistic conditions to assess theimportance of the underlying process for the overall dy-namics of IW. Such studies include the scattering of IWat bottom topography (Muller and Xu, 1992).

In a series of papers, Polzin (2003a,b) represents thenonlinear source function as a flux through wavenumberspace. The functional form of the flux is based on a com-bination of theoretical and dimensional arguments withfree parameters constrained by observations. The fluxis toward high wavenumbers and assumed to be “dissi-pated.” This form is then utilized to calculate solutionsof the radiation balance equation for the IW field aboverough topography in the Brazil Basin, where IW gener-ated by currents across topography propagate upwardand flux their energy across the spectrum to dissipation.

Dissipation source function

Dissipation of internal wave energy is assumed to oc-cur by wave breaking. Internal waves break by eithershear or convective instabilities that are caused by ei-ther chance superposition or encounter of critical lay-ers. Breaking internal waves not only generate turbu-lence but they mix the water column. Part of the IWenergy is thus converted into potential energy. The en-ergy in the turbulence is partly dissipated into heat andpartly emitted as small-scale IW waves as the turbulentpatch collapses and spreads laterally. Wave breakingis a highly nonlinear, localized, and sporadic processwhich has defied so far any rigorous derivation of theassociated source function. However, its localized andsporadic nature in space and time makes it a processthat is ”weak-in-the-mean,” meaning that although theprocess itself is strongly nonlinear its effect on the over-all spectrum is weak. This fact, together with the re-quirement that the source function must be negative,implies that the dissipation source function must havethe form

Sdiss (k) = −γA (k) (19)

boundary where the dissipation coefficient γ depends onwavenumber k and integral properties of the wave field,such as the total energy, shear, or inverse Richardsonnumber Ri−1 (Hasselmann, 1974, Komen et al., 1994).Thus

γ = γ(k, Ri−1

)(20)

and 19 is only quasi-linear. To arrive at a more specificform Natarov and Muller (2003) proceeded as follows.First they assumed a separable functional form for thedissipation coefficient

γ(k, Ri−1

)= c (k) f

(Ri−1

)(21)

where the function c (k) describes the wavenumber dis-tribution of dissipation and the function f

(Ri−1

)de-

scribes the intensity of dissipation, increasing with in-creasing inverse Richardson number. This functionalform contains as a limit the Garrett and Gilbert (1988)scenario where all energy fluxed past a certain critical

IWAM 103

wavenumber is instantaneously dissipated. The func-tions c (k) and f

(Ri−1

)need to be calibrated by ob-

servations or process simulations. To do so Natarovand Muller considered reflection of an incoming GMspectrum at a straight slope. Because of critical reflec-tion, the reflected spectrum has (infinitely) high shearand inverse Richardson number with associated vigor-ous wave breaking. The dominant balance in RBE forthe reflected spectrum can thus be assumed to be be-tween propagation and dissipation. Hence

vn (k)∂

∂xnA (k, xn) = −c (k) f

(Ri−1

)A (k, xn) (22)

where xn is the coordinate normal to the slope and vn

the group velocity in the normal direction. This equa-tion is subject to the condition

A (k, xn = 0) = Ar (k) (23)

where Ar (k) is the reflected spectrum obtained fromthe prescribed incoming GM spectrum. A solution canbe obtained if the contribution to the inverse Richard-son number from the incoming spectrum is neglected.This is indeed a good approximation since the incomingGM spectrum carries only little shear. The solution isfacilitated by the fact that the function f

(Ri−1

)can

be separated from the problem by introducing a scaleddistance ξ = ξ(xn) such that

dξ

dxn= f

(Ri−1

)(24)

The RBE then reduces to the linear form

vn (k)∂

∂ξA (k, ξ) = −c (k)A (k, ξ) (25)

which is solved by

A (k, ξ) = A (k, ξ = 0) exp− c(k)vn(k)

ξ

(26)

If c(k) is known one can calculate from this solution

Ri−1 (ξ) =∫d3k w(k)A (k, ξ) (27)

where w(k) is a known weighting function. Comparingthis solution to observations (or results from processstudies)

Ri−1(ξ) = Ri−1obs(xn) (28)

then yields ξ = ξ(xn) and hence f(Ri−1

). Conversely,

if ξ = ξ(xn) is known one can solve equation (25) forc(k) and substitute observations (or results of processstudies). The result is

c(k) = −vn(k)c(k)

logAobs (k, ξ)

Aobs (k, ξ = 0)(29)

The right hand side of this equation needs to be in-dependent of xn. This provides a test of the validityof the assumptions that went into the derivation. Ofcourse, the solutions for f

(Ri−1

)and c(k) have to be

found simultaneously, requiring some iterative solutiontechnique. Also, the complete action density spectrumas a function of wavenumber vector and distance xn isnot observed, nor easily calculable in process simula-tions. Generally, only moments like the energy density,inverse Richardson number and dissipation rate are ob-served. One thus assumes functional forms for c(k) andf

(Ri−1

)with a certain number of free parameters, cal-

culates the observed fields (the forward problem) andthen determines the free parameters by minimizing the“distance” between these calculated fields and the ob-served fields (the inverse problem). Details are forth-coming in Natarov and Muller (A dissipation functionfor IWAM, submitted, 2003).

Initial tasks and outlook

The construction of IWAM requires a variety oftasks. The major ones are these:

• the determination, evaluation, calibration andvalidation of the source terms of the radiative bal-ance equation,

• the integration of the radiative balance equation,and

• the embedding of the radiation balance equationinto a high resolution circulation model.

The first step of the first task is the determination ofthe source terms. Most of the source terms in the RBEcan be and have been “derived” from the hydrodynamicequations under more or less restrictive assumptions.Some of these “derivations” contain free parametersthat need to be calibrated. The source terms often havea fairly complicated form. The wave-wave interactionsource term is an integral over a two-dimensional reso-nance surface. Efficient algorithms need to be designedor approximations and simplifications be implemented.Such approximations and simplifications are also neces-sary when the source term requires information aboutenvironmental fields that is not readily available. Scat-tering at bottom topography depends on the bottomspectrum which is not everywhere available at the rele-vant wavenumbers.

The second step is the calibration and validation ofthe source terms by comparison with observations or nu-merical process studies. This should be done in circum-stances where only a limited number of processes dom-inate the RBE and where transparent solutions of the

104 MULLER AND NATAROV

RBE can be obtained. This step is envisioned to be avery productive and fruitful basic research phase, sinceit will allow us to explicitly test dynamical balances un-derlying various sets of observations. Examples includetests of whether the decay of internal tides or near-inertial IW is due to dissipation, nonlinear wave-waveinteraction, bottom scattering, or any other process, orthe identification of the processes by which a spectrumreflected off topography adjusts. Comparisons of solu-tions of the RBE with results from numerical processstudies will allow us to determine the range of valid-ity of the RBE approach and its calibration. Overall,this phase will encode the knowledge that we have fromobservational and numerical process studies into the ra-diation balance equation.

The second task is to put all processes togetherand integrate the radiation balance. While the WAMproject for surface waves is a precedent for this task itshould be emphasized that the internal wave problemconstitutes a propagation problem in a six-dimensionalphase space (three space coordinates and three wavenum-ber components) whereas the surface-wave problem isa four-dimensional problem. Straightforward extensionof the 4D to the 6D problem exceeds current computercapacities and ways need to be explored that reducethe computational load. The third task is to embedthe RBE model in a circulation model. Again, there isprecedence from WAM.

All these tasks are challenging, theoretically, obser-vationally, and computationally, but not unsurmount-able. The result of the efforts will be a model thatdetermines the state of the internal wave field from thegoverning physics.

Acknowledgments. Work on the development ofIWAM has been supported by the Office of Naval Re-search.

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This preprint was prepared with AGU’s LATEX macros v4,

with the extension package ‘AGU++ ’ by P. W. Daly, version 1.6b

from 1999/08/19.