Mesoscale Eddy - Internal Wave Coupling. II.Energetics and Results from PolyMode

Kurt L. Polzin ∗

∗Corresponding author address:Kurt L. Polzin, MS#21 WHOI Woods Hole MA, 02543.E-mail: kpolzin@whoi.edu

Abstract

The issue of internal wave–mesoscale eddy interactions is revisited. Previousobservational work identified the mesoscale eddy field as a possible source of in-ternal wave energy. Characterization of the coupling as a viscous process providessomewhat smaller transfer coefficients than previously obtained, withνh

∼= 50 m2

s−1 andνv + f2

N2 Kh∼= 2.5 × 10−3 m2 s−1, in contrast toνh

∼= 200 − 400 m2 s−1

andνv + f2

N2 Kh = 0 ± 10−2 m2 s−1. Current meter data from the Local DynamicsExperiment of the PolyMode field program indicate mesoscaleeddy–internal wavecoupling through horizontal interactions (i) is a significant sink of eddy energy and(ii) plays anO(1) role in the energy budget of the internal wavefield.

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1. Introductiona. Preliminaries

Winds and air-sea exchanges of heat and fresh water are ultimately responsible for the basin-scale currents, or general circulation of the oceans. In order to achieve a state where the energyand enstrophy (vorticity squared) of the ocean is not continuously increasing, some form ofdissipation is required to balance this forcing. While the above statement may seem obvious,little is known about how and where this dissipation occurs.

Early theories of the wind driven circulation [Stommel (1948), Munk (1950)] view the west-ern boundary as a region where energy and vorticity input by winds in mid-gyre could be dis-sipated. Apart from rationalizing why western boundary currents are located on the westernboundary, even the crudest perusal of observations suggests distinct discrepancies with suchmodels. One example is that an intensely energetic mesoscale eddy field is associated with theGulf Stream after its separation (e.g. Schmitz (1976) with the perception (Moore 1963) that oneof reasons for the existence of such an intense eddy field is that insufficient dissipation actuallyhappens at the boundary. Thus the question of closing an energy budget for the general circula-tion (e.g. Wunsch and Ferrari (2004)) is to remove dissipation from the mean budget and placeit in the eddy budget. Where upon one can follow the thread andask,“how does the mesoscaleeddy field dissipate?”. One among many possible mechanisms (Ferrari and Wunsch 2009) is theexchange of energy between the mesoscale eddy field and the internal wave field. The rest ofthis Introduction attempts to discuss the phenomenology ofmesoscale eddy – internal coupling,review previous results and lay groundwork for the rest of the paper.

b. Mesoscale Eddy - Internal Wave Coupling

A cornerstone of theoretical understanding for wave problems concerns zonal mean the-ory and the analysis of wave propagation in parallel shear flows. A basic constraint, typicallyreferred to as Andrews and McIntyre’s generalized Eliassen-Palm (EP) flux theorem:

d kA

dt+ ∇ · F = D + O(α3); (1)

states that in the absence of dissipationD and nonlinearity (small wave amplitudeα limit),and for steady conditions, the Eliassen-Palm fluxF is spatially nondivergent,∇ · F = 0. Interms of either linear internal wave or linear Rossby wave kinematics, the Eliassen-Palm fluxF = kCgA, with streamwise (zonal) wavenumberk, group velocityCg and wave actionA.With respect to the mean fields, the attendant nonacceleration theorem (Andrews et al. 1987)states that the mean flow remains steady if∇ · F = 0.

My opinion is that this cornerstone represents a considerable barrier to conceptual under-standing of the mesoscale eddy – internal wave coupling problem. If one understands why, thenone has a good grasp of the coupling mechanism.

1) SYMMETRY

A convenient starting place to examine eddy-wave coupling is to invoke a decomposition ofthe velocity [u = (u, v, w)], buoyancy [b = −gρ/ρo with gravitational constantg and densityρ] and pressure [π] fields into a quasigeostrophic ‘mean’ () and small amplitude internal wave(′′) perturbations on the basis of a time scale separation:φ = φ + φ′′ with φ = τ−1

∫ τ0 φ dt in

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which τ is much longer than the internal wave time scale but smaller than the eddy time scale.The double prime notation for internal wavefield variables is retained for consistency with acompanion paper (Polzin 2008b).

Employing this averaging process to the equations of motionreturns the result that the right-hand side of the mean equations represents the flux divergence of psuedomomentum. FollowingMuller (1976), the pseudomomentum flux associated with a wave-packet is:

CjgP

i =∫

dk n(k,x, t) ki Cjg (2)

with 3-D wave action spectrumn ≡ E/ω, intrinsic frequencyω = σ − k · u, group velocityCg, wave vectork = (k, l, m) having horizontal magnitudekh = (k2 + l2)1/2 and energydensityE = Ek + Ep. Superscripted indices indicate particular vector/tensor components andrepeated indices imply summation. For example, the vertical flux of zonal momentum can beidentified asu′′w′′ = C3

gP1. This formulation assumes a slowly varying plane wave solution

and manipulations involving the algebraic factors relating (u, v, w, b, and π) for linear internalwaves.

A fundamental insight is that psuedomomentum is not,in general, conserved in a wave-mean interaction problem, (Buhler and McIntyre 2005; Polzin 2008a). For steady conditions,small amplitude waves in a slowly varying background have a nondivergent action flux,

∫

dk∇·Cgn = 0. In a zonally oriented parallel shear flow, the streamwise component (k) of the hori-zontal wavevector is constant following the ray trajectory. Thus∇·CgP

x =∫

dk k ∇·Cgn = 0and the Eliassen-Palm flux theorem (1) is nothing more than anaction flux conservation state-ment.

What is not immediately obvious is that the generalized Eliassen-Palm flux theoremdoesnot apply to asymmetric (3-d) flows. If the background velocity field contains horizontal gradi-ents in both (x, y) dimensions, the streamwise component of the horizontal wavevector evolvesfollowing a ray trajectory and thus the pseudomomentum flux is, in general, divergent. A sim-ple rationalization of the difference in behavior between 2-d (symmetric) and 3-d (asymmetric)systems comes from theoretical physics: Each symmetry exhibited by a Hamiltonian systemcorresponds to a conservation principle [Nother’s theorem, e.g. Shephard (1990)]. For spatialsymmetries the conservation principle concerns momentum:axisymmetric flows preserve theflux of pseudomomentum in the symmetric coordinate.

2) ASYMMETRY, THE SHRINKING CATASTROPHE AND WAVE CAPTURE

It turns out that the filamentation of waves by a strain field provides the essential mechanismthrough which the streamwise wavenumber varies and streamwise pseudomomentum is not con-served. Buhler and McIntyre (2005) point to an analogy between internal wave propagation inhorizontally nondivergent flows and the problem of particlepair separation in incompressible 2-D turbulence. In this relative dispersion problem, particle pairs undergo exponential separationwhen the rate of strain variance exceeds relative vorticityvariance:

S2s + S2

n > ζ2 (3)

with Ss ≡ vx + uy the shear component of strain,Sn ≡ ux − vy the normal componentandζ ≡ vx − uy relative vorticity, Polzin (2008a). Equation (3) is simplythe Okubo-Weisscriterion [e.g., Provenzale (1999)]. Buhler and McIntyre(2005) argue that the problem of smallamplitude waves in a horizontally nondivergent flow field is kinematically similar to particle

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pair separation under the hypothesis of a scale separation.In this case the ray equations for theevolution of the wavevector following a ray path,

dk

dt= −∇(ω + k · u) (4)

have solutionsk ∝ e±(S2n+S2

s−ζ2)1/2t/2. Thus a dominance of rate of strain variance over relativevorticity variance leads to an exponential increase/decrease in the density of phase lines, i.e., anexponential increase/decrease in horizontalwavenumber, kh = (k2 + l2)1/2, Fig. 1. Vorticitysimply tends to rotate the horizontalwavevector(k, l) in physical space. This provides a simplepicture of pseudomomentum flux divergence associated with an internal wave packet interactingwith an eddy strain field.

Buhler and McIntyre (2005) further argue for a scenario which they term ‘wave capture’ andJones (1969) labels a ’shrinking catastrophe’. Simply put,the vertical wavenumber is slaved tothe horizontal wavenumber,dm

dt= −kuz − lvz, so that exponential growth of the horizontal

wavenumber implies exponential growth or decay of verticalwavenumber in the presence ofthermal wind shear. Those waves with growing vertical wavenumber will tend to be trapped(captured) within the extensive regions of the eddy strain field. In two dimensions the zonalwavenumber is constant and the meridional wavenumber growslinearly in time. The verticalwavenumber also experiences linear growth but is decoupledfrom the evolution of the horizon-tal component. In two dimensions one has the notion of a critical layer conditionσ − ku = ±fas the intrinsic frequency approaches the lower bound for freely propagating wavesf (f is theCoriolis parameter). Such critical layers arenot a part of the phenomenology in three dimen-sions.

The ray-tracing arguments about internal wave packets interacting with mesoscale eddiesreturn useful pictures about a mechanism, but they do not provide information about the nettransfer rates for the observations. This requires consideration of the energy balance, whichis done below within the quasigeostrophic limit. Note that quasigeostrophy invokes both asmallness of Rossby numberand that the two horizontal length scales are similar. It is thislastassumption that is crucial to scaling the horizontal divergence asO(Rossby number squared),i.e. crucial to the statement that the background field is horizontally nondivergent in the contextof ray tracing (4) and its summary (3). The length scale requirement of quasigeostrophy furtherunderscores the issue of how asymmetry in the background influences the character of the wave-mean interaction.

c. Linearized Wave Energy Balances

The internal wave energy equation is [Muller (1976)]:

( ∂∂t

+ u · ∇h)(Ek + Ep) + ∇ · π′′u′′ =

−u′′u′′ux − u′′v′′uy − u′′w′′uz − v′′u′′vx − v′′v′′vy − v′′w′′vz − N−2b′′u′′ bx − N−2b′′v′′ by

(5)

with kinetic [Ek = (u′′2 + v′′2 + w′′2)/2] and potential [Ep = (N−2b′′2)/2] energies. Temporalvariability and advection of internal wave energy by the geostrophic velocity field are balancedby wave propagation and energy exchanges between the quasigeostrophic and internal wavefields. Nonlinearity and dissipation are assumed to be higher order effects. With the exceptionof vertical buoyancy fluxes,b′′w′′, which are negligible for linear waves, the energy exchanges

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are adiabatic. Similarly, spatial gradients of the vertical velocity w do not appear aswz issmall [order Rossby number squared] in the quasigeostrophic approximation. The thermal windrelation can be invoked to cast the vertical Reynolds stressand horizontal buoyancy flux asthe rate of work by an effective vertical stress acting on thevertical gradient of horizontalmomentum:

u′′w′′uz + N−2b′′v′′ by = [u′′w′′ −f

N2b′′v′′]uz

v′′w′′vz + N−2b′′u′′ bx = [v′′w′′ +f

N2b′′u′′]vz

These two terms in the effective stress will cancel each other in the limit thatω → f , Ruddickand Joyce (1979).

The character of the horizontal terms for quasigeostrophicflows can be made more apparentby expressing the right-hand-side of (5) as:

−u′′u′′ux − v′′v′′vy = −[u′′u′′ − v′′v′′]Sn/2

−u′′v′′uy − v′′u′′vx = −u′′v′′Ss .

Within the wave capture scenario, linear wave kinematics implies a negative stress-strain cor-relation for internal waves (i.e. a positive horizontal viscosity in a flux-gradient closure) and apositive stress-strain correlation for Rossby waves (i.e.a negativehorizontal viscosity), Fig. 1.For internal waves, the negative correlation follows from the fact that the major axis of the hor-izontal velocity trace isparallel to the projection of the wavevector onto the horizontal plane.For Rossby waves, geostrophy implies that the horizontal velocity is normal to the horizontalwavevector. Upgradient transfers (a negative viscosity) have long been recognized as a propertyof many different planetary scale systems, Starr (1968).

Two further points are to be made. First, the right-hand sideof (5) represents the rateat which the pseudomomentum fluxCgP is doing work against mean gradients. It is thus adescription of how internal wave energy is altered through wave radiation (work and an energyflux divergence). Exchanges of energy through the vertical terms can be either positive ornegative, depending upon the sign of the vertical group velocity, horizontal wavenumbers andvertical shear. The standard parallel shear flow critical layer has, withuz > 0, Cz

g < 0 andk < 0 so thatσ − ku → f and the wave looses energy at the expense of the mean flow. Theenergetics of a wave capture event admits to a similar characterization in the vertical coordinate.But such events may not dominate the net energy exchange. Second, the horizontal couplingcan be written as the product of a momentum flux and the rate of strain tensor, just as one wouldin the context of isotropic turbulence [e.g. Tennekes and Lumley (1972)]. Thus in thiswaveproblem, the filamentation of a wave by a larger scale background described in the previoussubsection plays a role analogous to turbulent energy transfer by vortex stretching.

If closure of mesoscale eddy - internal wave coupling through flux gradient relations can bejustified, in which−2u′′v′′ = νh(vx + uy), −u′′w′′ = νvuz, −u′′u′′ = νhux, −v′′v′′ = νhvy,−u′′b′′ = Khbx, and−v′′b′′ = Khby, considerable simplification results. The right-hand-sideside of the internal wave energy equation becomes, after a little manipulation, a simple sourceterm:

So = νh[v2x + v2

y + u2x + u2

y] + [νv +f 2

N2Kh][u

2z + v2

z] (6)

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d. Observations

It remains to enquire whether the coupling is sufficiently large that interior damping ofthe mesoscale be considered as an important process. The first step is a characterization of themagnitude of the coupling, which has been investigated observationally using current meter datafrom moored arrays bracketing the Gulf Stream. This includes individual moorings at Site-D(39◦N, 70◦W), arrays at (28◦N, 70◦W) [MODE (the Mid-Ocean Dynamics Experiment), IWEX(the Internal Wave Experiment) and the LDE (Local Dynamics Experiment; 31◦N, 69◦30′ W)component of PolyMode], and individual moorings that were part of PolyMode arrays I (35-36◦N, 55◦W) and II (28◦N, 56◦W and 28◦N, 65◦W).

Frankignoul (1974) reports horizontally anisotropic conditions at high frequencies to beassociated with large mean currents (Site-D) and correlation betweenu′′2 − v′′2 andux − vy

(MODE-0). Using data from MODE, Frankignoul (1976) reports(1) νh = O(10−1000) m2 s−1

using direct estimates ofu′′v′′ vs(vx+uy) andu′′2−v′′2 vs(ux−vy), and (2)νv+f2

N2 Kh ≤ O(0.1m2 s−1) from correspondences between fluctuations ofE andu2

z + v2z. The later estimates of

vertical coupling are refined using data from IWEX to be| νv + f2

N2 Kh |≤ O(0.01 m2 s−1)Frankignoul and Joyce (1979), and Ruddick and Joyce (1979) interpret data from Polymode Iand II as an effective eddy viscosity ofνv + f2

N2 Kh = 0 ± 2 × 10−2 m2 s−1. Finally, Brownand Owens (1981) use direct estimates of−2u′′v′′ regressed against(vx + uy) to obtainνh =200 − 400 m2 s−1 (PolyMode LDE).

e. Forward

The problem of wave forcing of the oceanic general circulation is much richer than is to beinferred from taking the EP flux theorem (1) at face value and assuming an adiabatic limit. Akey contribution of this work is the recognition of the role that asymmetry and strain play in thephenomenology, energetics and dynamics of wave–mean interactions. Note that strain is notregarded as such until Buhler and McIntyre (2005).

The purpose of this paper is to review the previous calculations of horizontal (Section 2)and vertical (Section 3) coupling, and to place those estimates into the context of the Bryden(1982) LDE eddy energy budget (Section 4). A summary and discussion concludes the paper. Acompanion paper (Polzin 2008b) examines dynamical issues and the LDE potential enstrophybudget.

2. Horizontal CouplingBrown and Owens (1981) present a scatter plot of horizontal stressu′′v′′ versusvx + uy us-

ing data from a current meter array deployed during the LocalDynamics Experiment (LDE) ofthe Polymode program. The LDE array is likely the most appropriate data set for such a study.The array was deployed for 15 months and consisted of two crosses centered about a centralmooring located at 31◦ N, 69◦ 30′ W. The inner cross moorings had a nominal spacing of 25 kmas the array was designed to resolve mesoscale velocity gradients. These moorings were instru-mented with current meters at two levels, 600 and 825 m water depth. The outer moorings wereinstrumented with a single current meter at 600 m depth and had a larger horizontal spacing.Only the inner cross data are used here. Instrument failureslimit estimation of relative vorticityand rate of strain to involving the center, northeast, northwest and southwest mooring over thefull 15 month deployment period at the the 825 m level. For estimates involving both vertical

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and horizontal gradients, estimates are available from thecenter-northwest-northeast trianglefor a 225 day period.

The analysis presented here is similar to, though not identical to, Brown and Owens in thefollowing: First, data from the Center, Northeast, Northwest, and Southwest moorings of theLDE inner cross at 825 m are used for the internal wave estimates and mesoscale gradients areestimated as first differences using the two possible triangles. Brown and Owens included datafrom the shorter (108 day) southeast current meter record and estimated the mesoscale gradientsfrom centered differences when possible. Secondly, Brown and Owens defined the internalwave band with a filter having a half power point at0.8f . Here mesoscale velocity gradientsare defined by a low-pass filter with 1/2 power at 10 day periodsand the high frequency dataare detrended. A filter having a 2 day 1/2 power point has also been used. No appreciabledifferences were noted. Finally, Brown and Owens estimate the u′′v′′ cospectrumCuv withfour-day piece length transform intervals. A transform interval of 512 points (equivalent to 5.5inertial periods at 15-minute sampling) is used here.

The major stated difference is that Brown and Owens estimatethe horizontal stress as∫ N2f

σ2−f2

σ2+f2 Cuv(σ) dσ. Multiplication of the stress estimate by the transfer function (ω2−f 2)/(ω2+

f 2) is appropriate only for the vertical stress estimate, in which there is a cancelation betweenthe Reynolds stressu′′w′′ and the buoyancy flux,v′′b′′, Ruddick and Joyce (1979).

My regressions ofu′′v′′ versusSs return horizontal viscosity estimates ofνh∼= 50 m2

s−1, significantly smaller than the Brown and Owens’ estimate of4(±1) × 102 m2 s−1 at 825m. Consequently, regressions between

∫

(Puu − Pvv) dσ vs Sn and∫

(Puu + Pvv) dσ vs ζ areboth investigated Fig. (2), in whichPxx represents the power spectrum ofx′′. The results areconsistent with the expectation for a flux-gradient closure: the

∫

(Puu−Pvv) dσ vsSn regressionis similar to the relation betweenu′′v′′ andSs while

∫

(Puu + Pvv) dσ is uncorrelated withζ .The difference is not simply in Brown and Owens’ multiplication by a transfer function

and limitation of the domain of integration to frequencies greater than 2f . A second way ofestimating the viscosity coefficient is to averagesgn(Ss)Cuv, in whichsgn represents the signof its argument. Cumulative integration of the cospectrum [divided by the estimate of rmsstrain, Fig. (4)] provides a characterization of how each frequency contributes to the viscosityoperator. Here, integration over2f ≤ σ ≤ N returns viscosity estimates of about 18 m2 s−1.

Coherence estimates (Fig. 3) are about 0.1 forσ > 2f . This result is consistent with thecharacterization of Frankignoul (1976) that the relation between stress and horizontal strainis subtle. The Brown and Owens estimate ofνh = 400 m2 s−1 would imply O(1) values ofcoherence. My estimate based upon frequenciesσ > 2f differs from Brown and Owens (1981)by the number of hours in a day. I am tempted to posit an algebraic error.

3. The Vertical DimensionCorrelations between the vertical flux of horizontal pseudo-momentum and the mesoscale

eddy field were pursued by Ruddick and Joyce (1979) using current meter data from Polymodearrays I and II. For the relatively energetic moorings of thePolymode II array, they foundthat: (i) wavefield energy levels modulated with the strength of the eddy vertical shear and(ii) a significant correlation between wave stress and low frequency velocity. No significantvertical stress–vertical shear correlations were apparent in either the Polymode II data or theless energetic Polymode I data.

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Ruddick and Joyce (1979) interpret the observations as, “consistent with generation of short(∼ 1 km horizontal wavelength) internal waves by the mean shearnear the thermocline, result-ing in an effective viscosity ofνv + f2

N2 Kh∼= 0.01 m2 s−1.” Their interpretation depends upon

accepting the observed stresses as being significantly impacted by Doppler shifting, which inturn is consistent with the generation of internal waves from a critical layer. The less energeticmoorings would be less prone to contamination by Doppler shifting, but it was perceived that therequired stress-strain correlation could not be resolved there because of statistical uncertainty.This obstacle can be circumvented simply by increasing the number of degrees of freedom inthe data record. Thus the vertical coupling process was investigated here with the LDE arraydata. The eddy energy at the LDE sight is comparable to the less energetic PolyMode I data.

Evaluation of the LDE data returns a positive vertical shear– vertical stress correlation atlow frequency (σ < 1.2f ) and negative correlation for high frequencies (σ > 10 cph), Fig. 5.The positive stress-shear correlations at near-inertial frequencies imply the transfer of energyfrom the wavefield to the mean, which is consistent with either critical layer or wave capturescenarios. Coherence estimates areO(1) at near-inertial frequencies, but this part of the spec-trum carries little effective momentum flux. Consequently the estimates of vertical viscosityand associated energy transfer are dominated by high frequency contributions having small co-herence estimates. Integration of the cospectra returnsνv + f2

N2 Kh∼= 0.0025 m2 s−1.

There are two landmarks of possible consequence in the frequency domain. The positivestress–shear correlation occurs for frequencies in which the wave aspect ratio is equal to orsmaller than the aspect ratio of the mesoscale eddy field. Thenegative stress–shear correlationoccurs for waves that potentially encounter a buoyancy frequency turning point: the minimumbuoyancy frequency for the observed density profile is approximately 10 cpd (N = 7 × 10−4

s−1).

4. LDE Energy Budgetsa. Internal Wave - Eddy Coupling as Eddy Dissipation

As part of the LDE, moored current and temperature measurements were made for 15months in the main thermocline of the Gulf Stream Recirculation Region near 31◦N, 69◦ 30′Wto assess the energetics and dynamics of the mesoscale eddy field [Bryden (1982), Brown et al.(1986)]. Here we expand slightly on the energetics documented in the study of Bryden (1982),Figure 7.

Bryden infers that the mean density field serves as a source ofeddy kinetic energy at a rateof 3.3 × 10−9 W/kg through baroclinic instability and that the mean velocity field representsa counter gradientsink (see Fig. 1) at the rate of1.5 × 10−9 W/kg. The residual representsdissipation, propagation, advection and possibly time dependence. Brown and Owens (1981)estimate that the internal wavefield serves as a sink of eddy energy at a rate ofνh[v

2x + v2

y +u2

x + u2y] = 1.2 × 10−9 W/kg usingνh = 200 m2 s−1 and a gradient variance estimate of

6 × 10−12 s−2. Thus approximate closure of the eddy energy budget betweengeneration viabaroclinic instability, conversion of eddy kinetic to meankinetic energy and interior dissipationwas implied.

The estimates here of interior eddy dissipation are somewhat smaller:

νh[v2x + v2

y + u2x + u2

y]∼= 50 m2s−1 [6 × 10−12 s−2]

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[νv +f 2

N2Kh][u

2z + v2

z]∼= 0.0025 m2s−1 [4 × 10−8 s−2]

so that the total transfer of energy from eddies to internal waves is

Si = 4 × 10−10 W/kg .

Some other mechanism is required to close the eddy budget. Time dependence and propagation(through the divergence of pressure work) may play a role, and Bryden (1982) notes that the sumof baroclinic production and conversion of eddy kinetic to mean kinetic energy is smaller thanthe estimated uncertainty. There is, however, an obvious dissipation mechanism associated withviscous stresses in the bottom boundary layer. Standard boundary layer theory parameterizeswork done against viscous stresses in the bottom boundary layer asρCDU3 in which U is thefarfield velocity at the bottom boundary,ρ is density and typical estimates of the drag coefficientCD are2 − 3 × 10−3. The central mooring’s 5332-m current meter at a height of 23m abovebottom returns an estimate ofU3 = (u2 + v2)3/2 = (0.097 m s−1)3, which in turn implies anenergy loss to viscous stress in the bottom boundary layer inthe range of1.7 < ρCDu3 < 2.5mW/m2.

A possible interpretation is that eddy energy associated with baroclinic production radiatesdownward through the water column and is dissipated in the bottom boundary layer.If thisdissipation was distributed throughout the water column inproportion toN2, as is the case fordissipation associated with internal wave breaking in the background internal wavefield (Polzin2004), it would be equivalent to an interior dissipation of(6− 9)× 10−10 W/kg at 800 m depth.

This calculation, albeit based upon crude extrapolation, implies that the internal wave -eddycoupling mechanism is a significant part of the total dissipation (interior plus boundary) of eddyenergy. Recent comparisons of an idealized quasigeostrophic numerical model with mid-gyreobservations [Arbic and Flierl (2004)] have focussed simply on the issue of Ekman dissipation.

b. Mesoscale Eddy – Internal Wave Coupling as Internal Wave Forcing

If mesoscale eddy – internal wave coupling represents a sinkof eddy energy, then it rep-resents a source of internal wave energy. An open question iswhether it can be considered assignificant in the internal wave energy budget. The tack hereis to compare the identified sourcerate with dissipation estimates from finestructure data obtained as part of the Frontal Sir-SeaInteraction Experiment (FASINEX). Field work took place inFebruary-March of 1986 in thevicinity of an upper-ocean frontal system in the subtropical convergence zone of the northwestAtlantic (28◦ N, 69◦ W), Weller et al. (1991). Further analysis of the data appears in Polzinet al. (1996) and Polzin et al. (2003). The Mid-Ocean Dynamics Experiment (MODE) and In-ternal Wave Experiment (IWEX) studies were also located here. Sampling during FASINEXtook place over several degrees of latitude and longitude. It is assumed that the sampling israndom relative to the underlying eddy field and so is not spatially biased. The high frequencyinternal wavefield in the main thermocline at 34◦N, 70◦W [the Long Term Upper Ocean Study(LOTUS) site] is known to exhibit an annual cycle with maximum energy in the late winter inthis region, Briscoe and Weller (1984). It is not clear how this annual cycle would appear in thefinestructure data, but the FASINEX data were obtained during the more energetic part of theannual cycle.

The finestructure parameterization of Polzin et al. (1995) assigns a turbulent production rate

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of:

production = (1 + Rf) ǫof

fo

N2

N2o

E2

√

2

Rω − 1

3(Rω + 1)

4Rω(7)

with flux Richardson numberRf = 0.15, ǫo = 7×10−10 W/kg,fo the Coriolis parameter at 31.5degrees latitude,No = 3 cph, shear spectral levelE relative to a reference of7N2 (1/cpm), andshear-strain ratioRω. The factor

√

2Rω−1

represents a scaled aspect ratio under the hydrostatic

approximation. The main thermocline finestructure observations provide the estimates:N2 =0.70N2

o , E = 1.2 andRω = 6, which in turn imply a production of:

production(z = 800 m) = 4 × 10−10 W/kg

This represents a value numerically similar to the identified eddy forcing of the internal wave-field in Section 4.a.

Additional dissipation of internal wave energy will be present in the bottom boundary layer.This contribution to the energy budget can be quantified through a linearized formulation:

dissipation = 2ρCD | u | u′′2.

With (u′′2)1/2 = 0.025 m s−1 the observed near-bottom internal wave speed, the dissipationestimate becomes:

dissipation = 0.2 − 0.3 mWm−2

For comparison, dissipation in the background internal wavefield can be estimated by integrat-ing (7) with the observedN2 profile and amounts to a depth-integrated dissipation of 1 mW/m2.The estimated forcing through wave-eddy coupling is the same order of magnitude as the an-ticipated dissipation. Application of this finestructure parameterization scheme to AbsoluteVelocity Profiler (AVP) data obtained during the PolyMode LDE (Kunze and Sanford 1996)returns similar results.

With caveats about this being a crude extrapolation and withknowledge that the upper-ocean frontal regime present in this region [Weller et al. (1991); Polzin et al. (1996)] representsan ill-defined departure from the stated assumptions in the extrapolation used to define bothsources and sinks, I conclude that the mesoscale eddy-internal wave coupling is a dominantenergy source for the internal wavefield in the Gulf Stream Recirculation.

5. Summary and Discussiona. Summary

Current meter array data from the Local Dynamics Experiment(LDE) of the PolyModefield program were used in investigate the coupling of the mesoscale eddy and internal wavefields in the Southern Recirculation Gyre of the Gulf Stream.The coupling was characterizedas a viscous process. Viscosity coefficients inferred from regressions of horizontal stress vshorizontal rate of strain return an estimate ofνh = 50 m2 s−1. The regression of effectivevertical stress estimates against vertical shear returnνv + f2

N2 Kh = 2.5 × 10−3 m2 s−1. Interms of energy budgets, eddy-wave coupling represents a significant mechanism by whicheddy energy is dissipated and plays anO(1) role in the energy budget of the internal wavefield.

These results may be specific to the LDE region, which is situated at the exit of the South-ern Recirculation Gyre. Variability of the viscosity coefficients acting on the mesoscale field

10

will depend upon variability in the background wavefield. Such variability exists [Polzin et al.(2007)]. A possible implication of this work is that the variability in both the mesoscale eddyfield (Zang and Wunsch 2001) and the internal wavefield (Polzin et al. 2007) are related throughmesoscale eddy–internal wave coupling. In terms of understanding the geographic variabilityof viscosity coefficients, there is a simplicity if the energetics of the internal wave field aredominated by an interior coupling to the mesoscale eddy field, as appears to be the case in theSouthern Recirculation Gyre. This represents the dynamic balance advocated by Muller andOlbers (1975), albeit at somewhat reduced interaction rates.

This study comes with many caveats:

• The maximum observational record length for the LDE array data is 15 months, but thefailure of certain instruments reduces the usable record length to 225 days. Stable es-timates of time mean quantities typically require averaging periods of order 500 days(Schmitz 1977). The mean quantities quoted here represent record length means withassociated record length uncertainties. See Bryden (1982)and Brown et al. (1986) forfurther discussion of these uncertainties. I note, however, that the available 15 monthestimates are consistent with the 225 day record means (to within uncertainty). Any dif-ferences do not change the interpretation presented here.

• The LDE array does not fully spatially resolve the mesoscaleeddy velocity gradient vari-ances. This issue is examined in the companion manuscript where resolution of the en-strophy gradient variance is even more problematic. Again,consideration of such issuesdoes not change the interpretation presented here.

• The characterization of the coupling through a flux gradientrelation applies only to quasi-geostrophic flows in which the flow field is horizontally nondivergent toO(Rossby num-ber squared). Symmetric flow structures such as rings and jets are not coupled in the samemanner.

6. DiscussionThe discussion below tries to flesh out some of the broader implications of eddy-wave cou-

pling as they appear in the context of the PolyMode field program.

a. Coherent Vortices

One of the surprises of the Local Dynamics Experiment was theprevalence of coherent sub-mesoscale lenses, Elliot and Sanford (1986a). These lensestypically had temperature-salinityproperties that distinguished them as having relatively long (several-year) life spans. These sub-mesoscale features had horizontal scales ofL =15 km, significantly smaller than the mesoscalefield (∼= 100 km). Such life spans are not consistent with a viscous decay andνh = 50 m2s−1,which provides a temporal spin down scaleL2/νh

∼= of 50 days.However, note that the relative vorticity variance in the core of such lenses (Elliot and San-

ford 1986b) dominates the rate of strain variance, so by the criterion (3) one does not expecta strong coupling between the internal wavefield and the mesoscale eddy field. Moreover, asa symmetric feature, the assumptions leading to a flux-gradient relation are not valid and theexpectations of a short temporal spin down scale are incorrect. The viscosity operators here areappropriate for a 3-dimensional field, not two dimensional fields.

11

A viscous closure cannot be anticipated from analytic results with axisymmetric (e.g., zonal)flows. In that instance, apart from diabatic processes or critical layers, the pseudomomentumflux is nondivergent. Extrapolation of this result to non-axisymmetric background flows givesthe misleading impression that the only consequence of internal waves for geostrophically bal-anced flows is through a diabatic link.

The existence of long-lived, coherent submesoscale vortices is consistent with the proposedmodel of wave-eddy interaction.

b. Topographic Rossby Waves

Price and Rossby (1982) document that the sub-thermocline velocity field at the start ofthe LDE is dominated by highly polarized, oscillatory flow consistent with topography Rossbywave characteristics: the horizontal wavelength (340 km),intrinsic frequency (1/61 cpd) andphase propagation toward 300◦T agree with a the dispersion relation of barotropic planetarywaves modified by topography. The group velocity is approximately 0.05 m s−1 directed toward100◦T (eastward). The LDE was situated at the exit of the SouthernRecirculation Gyre and themean rate of strain is non-zero. This topographic wave is oriented so that energy is beingtransferedfrom the waveto the mean field, Fig. 1. Such events dominate the record lengthestimates of energy transfer in Bryden (1982).

A large part of the mesoscale eddy field at the LDE site arelinear waves and the energetictransfer estimates of (Bryden 1982) fit nicely into a wave-mean interaction paradigm. Thisstatement, though, comes with major caveats. Observed particle speeds (0.12 m s−1) exceed thephase speed by about a factor of two.

c. Vertical Mode Coupling

A second caveat regarding nonlinearity is that planetary wave fits to the mesoscale eddy fieldrequire a superposition of several barotropic and baroclinic plane waves to match the horizontalstructure of the observed velocity field, Hua et al. (1986). The vertical modes are dynamicallycoupled and the potential vorticity balance of the baroclinic mode has significant nonlinearcontributions, especially at the smallest resolved scales. Of two interaction events, one is de-scribed as the straining of an antecedent, large–scale baroclinic flow into a baroclinic jet bythe barotropic wave described above. The consequences of nonlinearity in terms of potentialvorticity dynamics is addressed in the companion manuscript (Polzin 2008b).

12

Acknowledgments.

Much of the intellectual content of this paper evolved out ofdiscussions with R. Ferrari.The manuscript benefitted from discussions with Brian Arbic, Rob Scott and a review providedby Peter Rhines. Salary support for this analysis was provided by Woods Hole OceanographicInstitution bridge support funds.

13

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16

List of Figures1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

17

aSs=constant

−1 −0.5 0 0.5−0.5

0

0.5

1b ζ=constant

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

u‘v‘ ∝ − Ss

c Internal Wave

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

u‘v‘ ∝ Ss

d Rossby Wave

FIG. 1. Phase lines (dashed) of waves being passively advected by two realizations of a steadymean flow having streamlines denoted by the solid contours: (a) a spatially constant shearstrainSn, and (b) a spatially constant relative vorticityζ . For internal waves, parcel velocitiesprojected onto the horizontal plane (see arrows) are elliptically polarized with the major axisparallel to the horizontal wavevector and normal to wave crests (lines of constant phase), panel(c). For Rossby waves, the horizontal velocity trace is linearly polarized (see arrows) andparallel to wave crests (lines of constant phase), panel (d). The tendency of background rateof strain to create anisotropic wavefields and wave stressesby preferentially orienting the phaselines along the extensive axis of a strain field can be inferred [(c) and (d)]. However, there is animportant distinction in the two cases. Parameterization of the stress-strain relation in terms ofa flux–gradient relation leads to a positive eddy viscosity for internal waves and a negative eddyviscosity for Rossby waves. The tendency of vorticity to notresult in wave stresses can also beinferred (b).

18

−4 −3 −2 −1 0 1 2 3 4

x 10−6

−4

−3

−2

−1

0

1

2

3

4x 10

−4

strain: Vx+U

y = S

s (s−1)

shea

r st

ress

−2C

uv (

m2 s

−2 )

64±20 m2 s−1

Polymode LDE 825 m Inner Cross f ≤ ω ≤ N

−4 −3 −2 −1 0 1 2 3 4

x 10−6

−4

−2

0

2

4x 10

−4

strain: Ux−V

y = S

n (s−1)

norm

al s

tres

s P

uu−

Pvv

(m

2 s−

2 )

37± 16 m2 s−1

−5 0 5

x 10−6

0

2

4

6

8x 10

−3

vorticity: Vx−U

y = ζ (s−1)

Puu

+P

vv (

m2 s

−2 )

FIG. 2. Upper panel: scatter plots of shear stress∫ Nf Cuv dω against the shear component

of strainSs. Middle panels: scatter plots of the normal stress∫ Nf (Puu − Pvv) dω against the

normal component of strainSn. Lower panels: scatter plots of velocity variance∫ Nf (Puu +

Pvv) dω plotted against vorticityζ . The regression lines of stress vs strain return horizontalviscosity coefficients ofνh

∼= 50 m2 s−1. Solid dots are 10 day averages. The uncertaintyestimates represent 95% confidence levels. In the lower panel, no trends of kinetic energy vs.relative vorticity are apparent. The lack of an apparent trend i this case is consistent with simplecharacterization of the coupling as a viscous process.

19

100

101

102

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

a

frequency (cpd)

−[P

uu−

Pvv

]/[sq

rt(P

uu)s

qrt(

Pvv

)]

a

100

101

102

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

b

frequency (cpd)

−2

Re[

Cuv

] / [s

qrt(

Puu

)sqr

t(P

vv)]

bb

FIG. 3. Coherence functions created by averaging (a)−sgn(Sn)[Puu − Pvv]/P1/2uu P 1/2

vv and(b) −sgn(Ss)Cuv/P

1/2uu P 1/2

vv with Cuv being the real part of theuv cross-spectrum. Estimatesare based upon 512 point transform intervals and averaging over both triangles at 825 m waterdepth.

20

100

101

102

0

10

20

30

40

50

60

70

frequency (cpd)

Vis

cosi

ty (

m2 s

−1 )

shear

normal

FIG. 4. Cumulative integrals of the spectral functions−sgn(Sn)[Puu − Pvv] (thick line) and

−2sgn(Ss)Cuv (dashed line), divided by estimates of the corresponding rate of strain| Sn and

| Ss |, to provide estimates of the horizontal viscosityνh.

21

100

101

102

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

frequency (cpd)

Coh

eren

ce

FIG. 5. Coherence function created by averaging−sgn(Uz)[Cuw − fN−2Cvb]/T (ω)P 1/2uu P 1/2

ww

and−sgn(Vz)[Cvw + fN−2Cub]/T (ω)P 1/2vv P 1/2

ww . The factorCxy represents the real part of thexy cross-spectrum. The transfer functionT (ω) = (ω2−f 2)/(ω2 +f 2) accounts for cancelationof the Reynolds stress by the buoyancy flux and renders the denominator consistent with thenumerator. The coherence estimates are based upon 1024 point transform intervals of data atboth 600 and 825 m levels. Data are from the Center, Northeastand Northwest moorings.

22

100

101

102

−1

−0.5

0

0.5

1

1.5

2

2.5

3x 10

−3

frequency (cpd)

Vis

cosi

ty (

m2 s

−1 )

FIG. 6. Cumulative integrals of the spectral function−sgn(Uz)[Cuw − fCvb/N2]/(u2

z)1/2 −

sgn(Vz)[Cvw + fCub/N2]/(v2

z)1/2, to provide estimates of the vertical viscosity(νv + f2

N2 Kh).Vertical velocity was estimated by assuming a vertical balance in the temperature equation,T ′′

t + w′′T z∼= 0 and were corrected for the roll-off associated with a centerdifference operator.

23

Mean

APE

79

Mean

MKE

3

Mesoscale Eddies

EPE + EKE

104 61

Internal Wave

IPE + IKE

6 16

3.3x10-9 W/kg BC Instability 1.5x10-9 W/kg Work

1.4-2.1 mW/m2

bbl dissipation

4x10-10 W/kg & 1 mW/m2

interior wave breaking

0.2-0.3 mW/m2

bbl dissipation

νv - 1x10-10 W/kg

other sources?

Propagation?

Propagation?

Wind Work and Buoyancy Forcing

Energy Budget

νh - 3x10-10 W/kg

Flux Div.

FIG. 7. A schematic of energy conversion rates derived from the LDE array, following Bryden(1982). Estimates in units of W/kg refer to energy conversions at the depths of 600–800 m. Esti-mates in units of W/m2 refer to boundary inputs or depth integrated means. Non-labeled energyestimates are in units of10−4m2s−2. Energy is input into the subtropical gyre by wind work andbuoyancy forcing. It is converted from the mean density field(by baroclinic instability) to eddyenergy. About 40% of this is converted into mean kinetic energy Bryden (1982). The eddy fieldis damped by dissipation in the bottom boundary layer and forcing of the internal wavefield.Bottom boundary layer dissipation may play a somewhat larger role in the eddy energy budgetthan interactions with the internal wave field. However, eddy-wave coupling provides a sourcethat is in approximate balance with the estimated dissipation, implying eddy-wave couplingplays anO(1) in the internal wave energy budget. With regards to the mean kinetic energy bud-get, Bryden (1982) finds an approximate balance between the gain associated with eddy workagainst the mean strain and a flux divergence,∂x(uu′u′ + vu′v′) + ∂y(uu′v′ + vv′v′)

24