How Rossby Waves Break. Results from POLYMODE and the End of the Enstrophy Cascade

KURT L. POLZIN Woods Hole Oceanographic Institution

Submitted, Journal of Physical Oceanography, May 31, 2005

Corresponding author address: Kurt L. Polzin, MS#21 WHOI Woods Hole MA, 02543 E-mail: [email protected]

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ABSTRACT

The issue of internal wave–mesoscale eddy interactions in Physical Oceanogra- phy is revisited. Previous theoretical work identified the mesoscale eddy field as a possible source of internal wave energy and pseudo-momentum. Adiabatic pseudo- momentum flux divergences, in turn, serve as a sink of eddy potential vorticity. The exchange of internal wave momentum with potential vorticity in the eddy field is an intrinsic component of linear wave propagation in geostrophically balanced mean fields. Previous estimates of internal wave - eddy coupling using data from the POLY- MODE field program are assessed. Characterization of the coupling as a diffusive process provides somewhat smaller transfer coefficients than previously obtained, ∼ 2 −1 ∼ −3 2 −1 ∼ with νh = 50 m s and νv = 2.5 × 10 m s , in contrast to νh = 200 − 400 2 −1 2 −1 m s and νv =∼ 0.01 m s . Current meter data from the Local Dynamics Experiment of the Polymode field program indicate (i) this internal wave – mesoscale eddy coupling through horizontal interactions plays an O(1) role in the energy budget of the internal wavefield and is a significant sink of eddy energy, and (ii) eddy-internal wave interactions play an O(1) role in the eddy potential enstrophy budget. MAY 2005 3

1. Introduction a. 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 energy and enstrophy (vorticity squared) of the ocean is not continuously increasing, some form of energy dissipation 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 western boundary as a region where energy and vorticity input by winds in mid-gyre could be dissipated. Those theories predict Gulf Stream transports that are approximately equal to the interior Sverdrup transport [about 30 Sv, Schmitz et al. (1992)] and that are much smaller than observed Gulf Stream transports after the Stream separates from the coast [about 150 Sv, Johns et al. (1995)]. Subsequent theories of the wind driven circulation have attempted to address the role of nonlinearity and baroclinicity in increasing Gulf Stream transports above that given by the Sverdrup relation. Hogg (1983) proposed the existence of two relatively barotropic recirculation gyres on either side of the Stream that combine to increase the total transport. It is now generally ac- cepted that the recirculation gyres result from the absorption of Planetary and Topographic Rossby Waves generated by the meandering of a baroclinicly unstable Gulf Stream. The absorption process is uncertain. In the atmospheric context, radiative damping [Stone (1972)] is typically invoked to provide damping of planetary scale motions. Much of the atmosphere is sufficeintly close to radiative equilibrium, in which heating associated with incoming solar radiation is balanced by cooling associated with black body radiation, that thermal perturbations associated with eddy motions are damped. Such a process does not operate within the ocean and we are left searching for other mechanisms by which planetary waves could be damped. If one assumes the ocean interior is adiabatic, the energy sinks would neccesarily be located at the boundaries. The oceanic eddy field could be closed by some combination of bottom Ekman layers, eddy-mixed layer interactions, and an interior enstrophy cascade. Despite the intellectual predjudice that views the oceanic interior as adiabatic, or perhaps more precisely because, it seems prudent to enquire whether interior processes could provide a damping of both enstrophy and energy. b. Eddy-Wave Coupling as an Eddy Dissipation Mechanism A convenient starting place to examine eddy-wave coupling is to invoke a decomposition of the velocity [u = (u,v,w)], buoyancy [b = −gρ/ρo with gravitational constant g and density ρ] and pressure [π] fields into a quasi-geostrophic ‘mean’ ( ) and small amplitude internal wave (′′) perturbations on the basis of a time scale separation: φ = φ+φ′′ with −1 τ φ = τ 0 φ dt in which τ is much longer than the internal wave time scale but smaller R 4 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

than the eddy time scale. Employing this averaging process to the equations of motion returns the result that the right-hand-side of the mean momentum and buoyancy equations represents the flux divergence of psuedo-momentum. Following Müller (1976), the pseudo-momentum flux associated with a wave-packet as:

CjP i = d3k n(k, x, t) ki Cj (1) g Z g

with 3-D wave action spectrum n ≡ E/ω, intrinsic frequency ω = σ −k·u, group velocity 2 2 1/2 Cg, wave vector k = (k,l,m) having horizontal magnitude kh = (k + l ) and energy density E = Ek + Ep. Superscripted indices indicate particular vector/tensor components and repeated indices imply summation. This formulation assumes a slowly varying plane wave solution and manipulations involving the algebraic factors relating (u,v,w,b, and π) for linear internal waves. Psuedo-momentum is not, by itself, conserved. For small amplitude waves in a slowly varying background, it is the action flux which is nondivergent, ∇ · Cgn = 0. If the background velocity field contains horizontal gradients, the horizontal wavenumber of an internal wave packet evolves following a ray trajectory and thus the pseudo-momentum flux is, in general, divergent. The most familiar conservation statement is for a zonal back- z x ground flow, in which case Cg P is nondivergent, Eliassen and Palm (1961); Jones (1967). The zonal problem is, in this respect, trivial. Bühler and McIntyre (2004) point to an analogy between internal wave propagation and the problem of particle pair separation in 2-D turbulence. In this relative dispersion problem, particle pairs undergo exponential separation when the rate of strain exceeds relative vorticity: 2 2 2 Ss + Sn >ζ (2)

with Ss ≡ vx + uy the shear component of strain, Sn ≡ ux − vy the normal component and ζ ≡ vx − uy relative vorticity. Equation (2) is simply the Okubo-Weiss criterion [e.g., Provenzale (1999)]. Bühler and McIntyre (2004) argue that the problem of small amplitude waves in a larger scale flow field is kinematically similar to particle pair separation under the hypothesis of action conservation. In this case the ray equations lead to exponential increase/decrease in the density of phase lines, i.e., an exponential increase/decrease in 2 2 1/2 horizontal wavenumber, kh =(k + l ) , Fig. 1. Vorticity simply tends to rotate the horizontal wavevector in physical space. This provides a simple picture of pseudo-momentum flux divergence associated with an internal wave packet interacting with an eddy strain field. Bühler and McIntyre (2004) further argue for a scenario which they term ‘wave capture’. Simply put, exponential growth of the horizontal wave number implies exponential growth or decay of vertical wave number in the presence of geostrophic shear. Those waves with growing vertical wavenumber will tend to be trapped (captured) within the extensive FIG.1 regions of the eddy strain field. MAY 2005 5

The ray-tracing arguments about internal wave packets interacting with mesoscale eddies return useful pictures, but they do not provide information about the net transfer rates for the observations. This requires consideration of energy and potential vorticity balances.

1) INTERNAL WAVE ENERGY The internal wave energy equation is [M¨uller (1976)]:

∂ ( + u · ∇ )(E + E )+ ∇· π′′u′′ = ∂t h k p ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ −2 ′′ ′′ −2 ′′ ′′ −u u ux − u v uy − u w uz − v u vx − v v vy − v w vz − N b u bx − N b v by (3)

′′2 ′′2 ′′2 −2 ′′2 with kinetic [Ek =(u + v + w )/2] and potential [Ep =(N b )/2] energies. Tem- poral variability and advection of internal wave energy by the geostrophic velocity field are balanced by wave propagation and energy exchanges between the quasi-geostrophic and internal wave fields. Nonlinearity and dissipation are assumed to be higher order ef- fects. With the exception of vertical buoyancy fluxes, (b′′w′′), which are negligable for linear waves, the energy exchanges are adiabatic. Similarly, wz does not appear as it is small [order Rossby number squared] in the quasigeostrophic approximation. The thermal wind relation can be invoked to cast vertical Reynold’s and horizontal buoyancy flux as the rate of work by an effective vertical stress acting on the vertical grandient of horizontal momentum: f u′′w′′u + N −2b′′v′′ b = [u′′w′′ − b′′v′′]u z y N 2 z f v′′w′′v + N −2b′′u′′ b = [v′′w′′ + b′′u′′]v z x N 2 z The character of the coupling mechanism can be made more apparent by expressing the right-hand-side of (3) as:

i j ij i 3 i3 rhs (1.3) −→ k Cg ns |i,j=1,2 +k Cg ns |i,j=1,2

The factor 1 sij = [∂j ui + ∂i uj] 2 x x represents the rate of strain tensor. Two points are to be made. First, the energy exchange represents the rate at which the pseudo-momentum flux CgP, is doing work against mean gradients. Equation (3) is thus a description of how internal wave energy is altered through radiative processes (work and an energy flux divergence). Second, the horizontal coupling can be written as the product of a momentum flux and the rate of strain tensor, just as one would in the context of isotropic turbulence [e.g. Tennekes and Lumley (1972)], so that in this wave problem there is an analogy to turbulent energy transfer by vortex stretching. 6 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

2) POTENTIAL VORTICITY The (quasi-geostrophic) potential vorticity equation is [Müller (1976)]: f 2 (∂ + u∂ + v∂ )(∂2Φ+ ∂2Φ+ ∂ [ ∂ Φ] + βy)= t x y x y z N 2 z f ∂ [∂ v′′u′′ + ∂ v′′v′′ + ∂ (v′′w′′ + b′′u′′)] x x y z N 2 f −∂ [∂ u′′u′′ + ∂ u′′v′′ + ∂ (u′′w′′ − b′′v′′)] + Q (4) y x y z N 2 in which Q represents modification of the mean buoyancy profile through diabatic processes, Φ is the geostrophic streamfunction (Φ = π/f with Coriolis parameter f) and pressure π is defined in the absence of the internal wavefield. The factor ∇h is the 2-D horizontal gradient operator. For parcels that do not make excursions into the mixed layer, Q = −ρo∂z(Kρ∂zzρ/∂zρo). Assuming phase conservation (or equivalently a scale separation and local plane wave solution) Müller (1976) demonstrates that the source terms in the potential vorticity equation can be cast as the flux of pseudo-momentum: f ∂ [∂ v′′u′′ + ∂ v′′v′′ + ∂ (v′′w′′ + b′′u′′)] x x y z N 2 f −∂ [∂ u′′u′′ + ∂ u′′v′′ + ∂ (u′′w′′ − b′′v′′)] = y x y z N 2 3 i j −∇h ×∇· d k n(k, x, t) k C (5) Z g The right-hand side of (5) directly states that a spatially localized internal wave packet carries with it a potential vorticity perturbation. This vorticity perturbation is reversible in the sense that the mean state is unchanged after the passage of a linear packet, Bühler and McIntyre (2004). Müller (1976) provides the insight that, if the action of nonlinearity is to relax the wavefield back to an isotropic state, it is possible for the associated vorticity perturbation to become permanent: nonlinearity enables a net transfer of energy and vorticity between mesoscale eddies and internal waves.

c. Enstrophy The eddy enstrophy budget is obtained by first decomposing the time mean fields above into mean and fluctuating components φ = φ + φ′ with φ given by either a spatial scale −1 λ separation, φ = λ 0 φ dx, or an additional time average. Multiplying (4) by the quasi- ′ geostrophic perturbationR potential vorticity q and averaging returns:

1 d q′2 + u′q′ · ∇ q = −q′ ∇ ×∇· P i Cj (6) 2 dt h h g d in which dt represents the material derivative following the geostrophic flow and ∇hq is the background potential vorticity gradient. MAY 2005 7 d. Coupling Characterized as Viscosity Operators If the characterization of eddy-wave coupling as viscosity operators can be justified, 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-side side of the internal wave energy equation becomes, after a little manipulation, a simple source term: f 2 S = ν [v2 + v2 + u2 + u2] + [ν + K ][u2 + v2] (7) o h x y x y v N 2 h z z The right-hand-side of the enstrophy equation, using the thermal wind relation and similarly integrating by parts, can be rewritten as:

2 2 1 d 1 f f 1 2 2 q′2 + u′q′ · ∇ q = − ν [(ζ2 + ζ2)+ o ζ2] − [ν + K ][ζ2 + (b + b )] . (8) 2 dt h 2 h x y z v N 2 h z xz yz bz bz The characterization of eddy-internal wave coupling as a viscous process implies spe- ciﬁc correlations (M¨uller 1976) between:

′′ ′′ u v and vx + uy ≡ Ss ′′ ′′ ′′ ′′ u u − v v and ux − vy ≡ Sn ′′ ′′ ′′ ′′ (u w , v w ) and (uz, vz) ′′ ′′ ′′ ′′ (u b , v b ) and (bx, by). (9)

A zero correlation is implied between

′′ ′′ ′′ ′′ u u + v v and vx − uy ≡ ζ . e. Observations It remains to enquire whether the coupling is sufﬁciently large that interior damping of the mesoscale be considered as an important process. The ﬁrst step is a characterization of the magnitude of the coupling, which has been investigated observationally using current meter data from moored arrays bracketing the Gulf Stream. This includes individual moorings at Site-D (39N, 70W), arrays at (28N, 70W) [MODE(the Mid-Ocean Dynamics Experiment), IWEX(the Internal Wave Experiment) and the LDE(Local Dynamics Exper- iment; 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 be ′′2 ′′2 associated with large mean currents (Site-D) and correlation between u − v and ux −vy (MODE-0). Using data from MODE, Frankignoul (1976) reports (1) νh = O(10 − 1000) 2 −1 ′′ ′′ ′′2 ′′2 m s using direct estimates of u v vs (vx + uy) and u − v vs (ux − vy), and (2) 8 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

2 −1 2 2 νv ≤ O(0.1 m s ) from correspondances between ﬂuctuations of E and uz + vz. The later estimates of vertical coupling are reﬁned using data from IWEX to be νv ≤ O(0.01 m2 s−1) Frankignoul and Joyce (1979), and Ruddick and Joyce (1979) interpret data from 2 −1 Polymode I and II as an effective eddy viscosity of νv = 0.01 m s . Finally, Brown ′′ ′′ and Owens (1981) use direct estimates of −2u v regressed against (vx + uy) to obtain 2 −1 νh = 200 − 400 m s (Polymode LDE).

f. Forward In this observational work there are both inconsistancies of method and uncertainties in interpretation. 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) and the potential vorticity ﬂux estimates of Brown et al. (1986) (Section 5). A summary concludes the paper.

2. Horizontal Coupling Brown and Owens (1981) present scatter plots of the horizontal stress u′′v′′ versus shear strain Ss using data from a current meter array deployed during the Local Dynamics Exper- iment (LDE) of the 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 central mooring located at 31 N, 69 30 W. The inner cross moorings had a nominal spacing of 25 km as the array was designed to resolve mesoscale velocity gradients. These moorings were instrumented with current meters at two levels, 600 and 825 m water depth. The outer moorings were instrumented with a single current meter at 600 m depth and had a larger nominal spacing. Only the inner cross data are used here. Instrument failures limit estimation of vorticity and strain to involing the center, northeast, northwest and southwest mooring over the full 15 month deployment period at the the 825 m level. For estimates involving both vertical and horizontal gradients, estimates are available from the center-northwest-northeast traingle for a 225 day period. The analysis presented here is similar to, though not identical to, Brown and Owens in the following: First, data from the Center, Northeast, Northwest, and Southwest moorings of the LDE inner cross at 825 m are used for the internal wave estimates and mesoscale gradients are estimated as first differences using the two possible triangles. Brown and Owens included data from the shorter (108 day) southeast current meter record and estimated the mesoscale gradients from centered differences when possible. Secondly, Brown and Owens defined the internal wave band with a filter having a half power point at 0.8f. Here the mesoscale is defined by a low-pass filter with 1/2 power at 10 day periods and the internal wave data are detrended. A filter with 2 day 1/2 power point has also been used. No appreciable differences were noted. Finally, Brown and Owens’ estimate the ′′ ′′ u v cospectrum Cuv with four-day piece length transform intervals. A transform interval of 512 points (equivalent to 5.5 inertial periods at 15-minute sampling) is used here. N ω2−f 2 The major stated difference is that Brown and Owens estimate the stress as 2f ω2+f 2 Cuv(ω) dω. R MAY 2005 9

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 between the Reynold’s stress u′′w′′ and the buoyancy flux, v′′b′′, Ruddick and Joyce (1979). ′′ ′′ The resulting regressions between u v and Ss return horizontal viscosity estimates of ∼ 2 −1 νh = 50 m s , significantly smaller than the Brown and Owens’ estimate of 4(±1) × 2 2 −1 10 m s at 825 m. Consequently, regressions between (Puu − Pvv) dω vs Sn and (Puu + Pvv) dω vs ζ are both investigated Fig. (2), in whichR Pxx represents the power ′′ spectrumR of x . The results are consistant with the expectation for a diffusive closure: the ′′ ′′ (Puu − Pvv) dω vs Sn regression is similar to the relation between u v and Ss while R (Puu + Pvv) dω is uncorrelated with ζ. R 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 of estimating the viscosity coefficient is to average sgn(Ss)Cuv, in which sgn represents the sign of its argument. Cumulative integration of the cospectrum (divided by the estimate of rms strain, Fig. (4) provides a characterization of how each frequency contributes to the viscosity operator. Here, integration over 2f ≤ ω ≤ N returns viscosity estimates of about 15 m2 s−1, more than an order magnitude smaller than Brown and Owens’ estimate. Coherence estimates (Fig. 3) are about 0.1. This result is consistent with the characterization of Frankignoul (1976) that the relation between stress and horizontal strain is 2 −1 subtle. The Brown and Owens estimate of νh = 400 m s would imply O(1) values of coherence. FIG.2

FIG.3 3. The Vertical Dimension FIG.4 Correlations 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 Poly- mode arrays I and II. For the relatively energetic moorings moorings of the Polymode II array, they found that: (i) wave field energy levels modulated with the strength of the eddy vertical shear and (ii) a significant correlation between wave stress and low frequency velocity. No significant vertical stress–vertical shear correlations were apparent in either the Polymode II data or the less energetic Polymode I data. Ruddick and Joyce (1979) interpret the observations as, “consistent with generation of short (∼ 1 km horizontal wavelength) internal waves by the mean shear near the thermo- f 2 ∼ 2 −1 cline, resulting in an effective viscosity of νv + N 2 Kh = 0.01 m s .” Their interpretation depends upon accepting the observed stresses as being significantly impacted by Doppler shifting, which in turn is consistent with the generation of internal waves from a critical layer. The less energetic moorings would be less prone to contamination by Doppler shifting, but it was perceived that the required 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 in the data record. Thus the vertical coupling process was investigated here with the LDE array data. Evaluation of the LDE data returns a positive shear-stress correlation at low frequency 10 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

(ω < 2f) and negative correlation for high frequencies (ω > 10 cph), Fig. 5. Coherence estimates are O(1) at near-inertial frequencies, but this part of the spectrum carries little effective momentum flux. Consequently the diffusivity estimates are dominated by high frequency contributions having small coherence estimates. Integration of the cospectra f 2 ∼ 2 −1 returns νv + N 2 Kh = 0.0025 m s . Wave energy is transfered from the eddies to the high frequency wavefield. A small and possibly insignificant transfer of near-inertial internal wave energy to the eddy field is implied by the positive shear-stress correlation at low FIG.5 frequency.

FIG.6 4. The LDE Energy Budget

a. Internal Wave - Eddy Coupling as Eddy Dissipation As part of the LDE, moored current and temperature measurements were made for 15 months in the main thermocline of the Gulf Stream Recirculation Region near 31◦N, 69◦ 30′W to 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 of eddy kinetic energy at a rate of 3.3 × 10−9 W/kg and that the mean velocity field represents a counter gradient sink at the rate of 1.5 × 10−9 W/kg. The residual represents dissipation, propagation and possibly time dependence. Brown and Owens (1981) estimate that the internal wavefield 2 2 2 2 −9 serves as a sink of eddy energy at a rate of νh[vx + vy + ux + uy]=1.2 × 10 W/kg using 2 −1 −12 −2 νh = 200 m s and a gradient variance estimate of 6 × 10 s . Thus an approximate closure of the eddy energy budget between generation via baroclinic instability, conversion of eddy kinetic to mean kinetic energy and interior dissipation was implied. The estimates here of interior eddy dissipation are somewhat smaller: 2 2 2 2 ∼ 2 −1 −12 −2 νh[vx + vy + ux + uy] = 50 m s [6 × 10 s ]

2 f − − − [ν + K ][u2 + v2] ∼= 0.0025 m2s 1 [4 × 10 8 s 2] v N 2 h z z so that the total transfer of energy from eddies to internal waves is

−10 So =4 × 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 sum of baroclinic production and conversion of eddy kinetic to mean kinetic energy is smaller than the estimated uncertainty. There is, however, an obvious dissipation mechanism associated with viscous stresses in the bottom boundary layer. Standard boundary layer theory parameterizes work done against viscous stresses in the bottom boundary layer 3 as ρCDu in which u is the farﬁeld velocity at the bottom boundary, ρ is density and typical MAY 2005 11

−3 estimates of the drag coefficient CD are 2 − 3 × 10 . The rms subinertial speed recorded at the central mooring’s 5332-m current meter at a height of 23 m above bottom was 0.092 ms−1, which in turn implies an energy loss to viscous stress in the bottom boundary layer 3 2 in the range of 1.4 < ρCDu < 2.1 m W/m . A possible interpretation is that eddy energy associated with baroclinic production ra- diates downward through the water column and is dissipated in the bottom boundary layer. If this dissipation was distributed throughout the water column in proportion to N 2, as is the case for dissipation associated with internal wave breaking in the background internal wavefield Polzin (2004), 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 -eddy coupling mechanism is a significant part of the total dissipation (interior plus boundary) of eddy energy. Recent comparisons of an idealized quasi-geostrophic numerical model with mid-gyre observations [Arbic and Flierl (2004)] have focussed simply on the issue of Ekman dissipation. b. Internal Wave - Eddy Coupling as Internal Wave Forcing If internal wave-eddy coupling represents a sink of eddy energy, then it represents a source of internal wave energy. An open question is whether it can be considered as significant in the internal wave energy budget. The tack here is to compare the identified source rate with dissipation estimates from finestructure data obtained as part of the Frontal Sir- Sea Interaction Experiment (FASINEX). Field work took place in February-March of 1986 in the vicinity of an upper-ocean frontal system in the subtropical convergence zone of the northwest Atlantic (28 N, 69 W), Weller et al. (1991). Further analysis of the data appears in Polzin et al. (1996) and Polzin et al. (2003). Sampling took place over several degrees of latitude and longitude. It is assumed that the sampling is random relative to the underlying eddy field and so is not spatially biased. The high frequency internal wavefield in the main thermocline is known to exhibit an annual cycle with maximum energy in the late winter in this region, Briscoe and Weller (1984). It is not clear how this annual cycle would appear in the finestructure data, but the FASINEX data were obtained during the more energetic part of the annual cycle. The finestructure parameterization of Polzin et al. (1995) assigns a turbulent production rate of: 2 f N 2 2 3(Rω + 1) P = (1+ Rf ) ǫo 2 E (10) fo No rRω − 1 4Rω −10 with flux Richardson number Rf =0.15, ǫo =7 × 10 W/kg, fo the Coriolis parameter 2 at 31.5 degrees latitude, No = 3 cph, shear spectral level E relative to a reference of 7N (1/cpm), and shear-strain ratio . The factor 2 represents a scaled aspect ratio under Rω Rω−1 the hydrostatic approximation. The main thermoclineq finestructure observations provide 2 2 the estimates: N = 0.70 No , E =1.2 and Rω =6, which in turn imply a production of: P =4 × 10−10 W/kg 12 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

This represents a value similar to the identified eddy forcing of the internal wavefield. 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:

′2 dissipation =2ρCDuu .

With u′ = 0.025 m s−1 the observed near-bottom internal wave speed, the dissipation estimate becomes: − dissipation =0.2 − 0.3 mWm 2

For comparison, dissipation in the background internal wavefield can be estimated by integrating (10) with the observed N 2 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 anticipated dissipation. With caveats about this being a crude extrapolation and with knowledge that the upper- ocean frontal regime present in this region [Weller et al. (1991); Polzin et al. (1996)] represents an ill-defined departure from the stated assumptions in the extrapolation used to define both sources and sinks, I conclude that the mesoscale eddy-internal wave coupling FIG.7 is a dominant energy source for the internal wavefield in the Gulf Stream Recirculation.

5. The LDE Enstrophy Budget

We are now in a position to ask how the generation of internal wave pseudomomentum by eddy–wave coupling can be cast as a nonconservative term in the eddy enstrophy budget (6) and thereby be linked to potential vorticity dynamics. Brown et al. (1986) use the LDE ′ array data to estimate the eddy thickness (η = f0(ρ /ρz)z) and relative vorticity (ζ) ﬂuxes:

| u′η′ |=1.65 ± 0.77 × 10−7 m s−2 | u′ζ′ |=2.5 ± 2.1 × 10−8 m s−2. (11)

Maps of planetary vorticity in McDowell et al. (1982) are not inconsistant with the use of ∼ −11 −1 −1 qy = β =2 × 10 m s at the level of the current meter data (825 m), so that

′ ′ ∼ −18 −2 −2 u q · ∇hq = 3 × 10 m s .

If a steady state is to be assumed, these ﬂuxes are balanced by nonconservative mechanisms acting on the eddy ﬁeld. These nonconservative terms will be evaluated assuming MAY 2005 13 that the internal wave - eddy coupling can be cast as a viscous process:

1 f 2 f 2 1 r.h.s. of (6) = − ν [(ζ2 + ζ2)+ o ζ2] − [ν + o K ][ζ2 + (b2 + b2 )] 2 h x y z v N 2 h z xz yz bz bz 2 2 −20 −2 −2 (ζx + ζy )=3.2 × 10 m s 2 fo 2 −20 −2 −2 ζz =1.5 × 10 m s bz 2 −17 −2 −2 ζz =5.3 × 10 m s 1 ′2 ′2 −16 −2 −2 (bxz + byz)=1.2 × 10 m s bz

2 2 The horizontal vorticity gradient variance (ζx + ζy ) was estimated as twice the squared difference of the two possible vorticity estimates at 825 m, divided by the separation between the northeast-center-northwest and northwest-center-southwest triangle centers. 2 The vertical vorticity gradient variance (ζz ) and the horizontal thickness gradient variance 2 2 (bxz + byz) were estimated from the one possible (northeast-center-northwest) triangle. Fol- lowing the previous discussion, the interior viscosity coefficients are taken to be νh = 50 2 −1 −3 2 −1 m s and νv =2.5 × 10 m s . The grand total is: r.h.s. of (6)= 1.6 × 10−18 s−3 which is of appropriate order of magnitude to balance the observed down gradient flux of potential vorticity. Internal waves are thus inferred to play a significant role in the momentum/enstrophy/vorticity balances of the Gulf Stream Recirculation.

6. Summary and Discussion a. Summary Current meter array data from the Local Dynamics Experiment (LDE) of the Poly- MODE field program were used in investigate the coupling of the mesoscale eddy and internal wave fields in the Southern Recirculation Gyre of the Gulf Stream. The coupling was characterized as a diffusive process. Viscosity coefficients inferred from regressions 2 −1 of horizontal stress vs horizontal strain return an estimate of νh = 50 m s . The regression of estimates of effective vertical stress against vertical shear return an estimate −3 2 −1 of νv = 2.5 × 10 m s . In terms of energy budgets, the eddy-wave coupling likely represents a significant mechanism by which eddy energy is dissipated. The eddy-wave coupling likely plays an O(1) role in the energy budget of the internal wavefield. In terms of momentum budgets, the transfer of eddy vorticity to internal wave pseudomomentum likely plays an O(1) role in the eddy potential enstrophy budget. This study comes with many caveats: 14 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

• The estimated viscosity coefficients could varying significantly in response to variability in the amplitude and spectral characteristics of the internal wavefield. In the scenario considered by Müller (1976), the net transfers of energy and momentum between eddies and waves is accomplished by nonlinearity relaxing wavefield perturbations back to an isotropic state. The strength of the nonlinearity will varying in response to variability in the background wavefield. Characterization of such variability in response to spatial/temporal variations in sources and sinks is an open question. • The diffusive characterization need not completely characterize the eddy-wave coupling process. • The characterization of enstrophy dissipation (9) necessarily omitted flux divergence terms. The issue of eddy-wave coupling is sufficiently important that a more heavily instrumented LDE should be repeated in a more strongly forced region. Denser and redundant sampling would help avoid the limitations of the PolyMODE LDE associated with instrument failures. In particular, it would permit a direct evaluation of the potential vorticity dissipation terms and assessment of the uncertainty associated with a diffusive characterization. Stronger forcing of the mean field implies stronger dissipation, which may in turn imply larger correlations between potential vorticity perturbations and perturbations of the wave field. The discussion below tries to flesh out some of the broader implications of eddy-wave coupling.

b. Discussion

1) THE END OF THE ENSTROPHY CASCADE Potential vorticity is conserved in the absence of diabatic processes and friction. In the oceanographic context, virtually every frictional process has been regarded as being turbulent (diabatic) in nature and sufficiently weak within the oceanic interior that potential vorticity is conserved and significantly modified only at the boundaries. Diabatic mixing occurs in the ocean when small-scale shears become strong enough to overturn the stable stratification. Balanced eddies are not efficient at generating small-scale shear (enstrophy conservation arrests the flux of energy to small scales). At small scales the −3 energy spectrum generated by mesoscale eddies scales with a power law of E(kh) ∝ kh . Thus the shear variance is independent of scale and typically so small that small-scale overturns (e.g., Kelvin-Helmhotz billows) do not develop. The picture is one of potential vorticity anomalies in the enstrophy cascade being filamented, collapsing, and eventually removed by ambient diabatic processes. If, on the other hand, eddies are coupled to the internal wavefield, then the enstrophy cascade is short-circuited. The exchange of eddy vorticity for wave pseudomomentum MAY 2005 15 accomplishes enstrophy dissipation. The removal of energy into the internal wavefield implies a diabatic dissipation mechanism (ultimately through internal wavebreaking) even though the direct eddy–wave interaction is adiabatic. The terminology referring to the right-hand-side of the potential vorticity equation as ‘frictional’ is misleading. In view of the Reynold’s decomposition that produces (4), friction is any process giving rise to a momentum flux divergence, adiabatic or otherwise.

2) THE ROLE OF NONLINEARITY Small amplitude waves propagating in a larger scale geostrophic flow field obey an action conservation priciple. Since the horizontal wavevector evolves following a wave- packet in non-axisymmetric background flows, it follows almost trivially from (4) and (5) that a momentum flux divergence will induce a potential vorticity perturbation, and that this can be accomplished adiabatically. Bühler and McIntyre (2004)’s momentum conservation statement is similar in content. In that work, changes in pseudo-momentum associated with the momentum flux divergence are compensated by an equivalent hydrodynamic impulse associated with a remotely located vortex pair. This vorticity perturbation is reversible in the sense that the mean state is unchanged after the passage of a linear packet, Bühler and McIntyre (2004). Müller (1976) provides the insight that, if the action of nonlinearity is to relax the wavefield back to an isotropic state, it is possible for the associated vorticity perturbation to become permanent: nonlinearity enables a net transfer of energy and vorticity between mesoscale eddies and internal N 2 waves. Theoretical predictions for the wave-eddy coupling coefficients νh and (νv + f 2 Kh) are presented in Müller (1976). Ruddick and Joyce (1979) and Brown and Owens (1981) find that observations differ from the theoretical predictions by 1-2 orders of magnitude. Further consideration of the theoretical problem is called for.

3) RESIDUAL CIRCULATIONS AND THE EDDY PROBLEM The content of Andrew and McIntyre’s generalized Eliassen and Palm flux theorem is that a residual meridional circulation in an erstwhile zonal mean background requires an Eliassen-Palm flux divergence, which in the context of linear planetary wave theory is directly proportional to the rate of enstrophy dissipation, Andrews et al. (1987). The work here regarding eddy-internal wave coupling thus bears directly upon the the mean circulation in that problem. Idealized eddy resolving models are used to explore eddy dynamics and the conse- quences eddies have for the mean circulation [e.g. Wilson and Williams (2004) See Kuo et al. (2005) for further discussion about potential vorticity homogenization.]. A tentative generalization appears to be that eddy fluxes typically are along, rather than across, mean potential vorticity contours, and that the mean tends to a state of potential vorticity homogenization, in which the down gradient potential vorticity flux contribution to the enstrophy budget is small. Wilson and Williams (2004) attribute this behavior to the eddy advection of enstrophy. The model used in Wilson and Williams (2004) employs a biharmonic mix- 16 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

ing of momentum. This functional representation permits the creation of high enstrophy at the deformation scale and its diffusion at the smallest scales: large enstrophy gradients are permitted and eddy advection of these gradients balance the down gradient potential vorticity ﬂuxes. Such model behavior may not be realistic. The ocean may be in a much more highly damped state and less sensitive to eddy dynamics. Given a diffusive representation to the eddy-wave coupling mechanism, a de- 4 formation scale (Ld =5 × 10 m) Reynolds number is: ′ ∼ u LD/νh = 25

′ −1 2 −1 with rms velocity u =0.05 ms and νh = 50 m s . In Wilson and Williams (2004), the 9 4 −1 biharmonic operator with coefﬁcient Ah = 2.5 × 10 m s implies a deformation scale Reynolds number of: ′ 3 ∼ u LD/Ah = 2500 I suspect that the truncation of the enstrophy cascade has substantial implications for the behavior of numerical models.

4) RADIATION BALANCE Further insight into the role of dissipation in the enstrophy equation (6) could be gained by considering its formulation in terms of a radiation balance equation, e.g. ?, in which a mixed spectral/spatial-temporal representation is invoked. The role of eddy advection u′ · ∇q′2 in Wilson and Williams (2004) is that it has a time mean that can balance the downgradient flux. In the spectral context, the role of eddy advection has an alternate interpretation as nonlinearity, in which triple correlations u′q′2 act as a source function in the spectral domain transporting enstrophy to dissipation scales. At small scales, dissipation balances a divergence of the spectral transport. At larger scales, a convergence of spectral transport is associated with a source term, the equivalent of u′q′ ·q. The enstrophy transport is set by the spectral amplitude at this larger eddy scale. This larger eddy scale is not externally determined. Eddies will also undergo an inverse cascade of energy, and thus the outer scale of the enstrophy cascade is free to evolve. It is possible that this outer scale depends upon the dissipation function. In an idealized model of two-layer quasi-geostrophic turbulence with bottom friction, Arbic and Flierl (2004) find that, under weak damping, model eddies evolve toward larger horizontal scales, and consequently support larger fluxes of potential vorticity. The budgets are not independent of the viscosity formulation. MAY 2005 17

Acknowledgment. Much of the intellectual content of this paper evolved out of discussions with R. Ferrari. 18 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

REFERENCES Andrews, D. G., J. R. Holton and C. B. Leovy, 1987: Middle Atmosphere Dynamics, Aca- demic Press, Orlando, pp. 1–489. Arbic, B. K. and G. R. Flierl, 2004: Baroclinically unstable geostrophic turbulence in the limits of strong and weak bottom Ekman friction: Application to mid-ocean eddies. J. Phys. Oceanogr., 34, 2257–2273. Bell, T. H., 1975: Topographically-generated internal waves in the open ocean. J. Geophys. Res., 80, 320–327. Briscoe, M. G. and R. A. Weller, 1984. Preliminary results from the long-term upper-ocean study (LOTUS). Dyn. Atmos. Oceans, 8, 243–265. Brown, E. D. and W. B. Owens, 1981: Observations of the horizontal interactions between the internal wave field and the mesoscale flow. J. Phys. Oceanogr., 11, 1474–1480. Brown, E. D., W. B. Owens and H. L. Bryden, 1986: Eddy-potential vorticity fluxes in the Gulf Stream Recirculation. J. Phys. Oceanogr., 16, 523–1531. Bryden, H. L., 1982: Sources of eddy energy in the Gulf Stream Recirculation Region. J. Mar. Res., 40, 1047–1068. Bühler, O. and M. E. McIntyre, 2003: Remote recoil: a new wave-mean interaction effect. J. Fluid. Mech., 492, 207–230. Bühler, O. and M. E. McIntyre, 2004: Wave capture and wave-vortex duality. submitted to J. Fluid. Mech.. Eliassen, A. and E. Palm, 1961: On the transfer of energy in stationary mountain waves. Geophys. Publ., Oslo 22, 1–23. Frankignoul, C., 1974: Observed anisotropy of spectral characteristics of internal waves induced by low-frequency currents. J. Phys. Oceanogr., 4, 625–634. Frankignoul, C., 1976: Observed interation between oceanic internal waves and mesoscale eddies. Deep-Sea Res. I, 23, 805-820. Frankignoul, C., and T. M. Joyce, 1979: On the internal wave variabiliy during the Internal Wave Experiment (IWEX). J. Geophys. Res., 84. 769–776. Hogg, N. G., 1983: A note on the deep circulation of the western North Atlantic: Its nature and causes, Deep-Sea Res. I, 30, 945–961. Johns, W. E., T. J. Shay, J. M. Bane, and D. R. Watts, 1995: Gulf Stream structure, transport, and recirculation near 68◦W, J. Geophys. Res., 100 817–838. Jones, W. L., 1967: Propagation of internal gravity waves in fluids with shear flow and rotation. J. Fluid. Mech., 30, 439–448. Kuo, A., R. A. Plumb and J. Marshall, 2005: Transformed eulerian mean theory. Part II: Potential vorticity homogenization and the equilibrium of a wind and buoyancy-driven zonal flow. J. Phys. Oceanogr., 35, 175–187. McDowell, S., P. Rhines and T. Keffer, 1982: North Atlantic potential vorticity and its relation to the general circulation. J. Phys. Oceanogr., 12, 1417–1436. Müller, P., 1976: On the diffusion of momentum and mass by internal gravity waves. J. Fluid. Mech., 77, 789–823. Munk, W., 1950: On the wind-driven circulation. J. Meteor., 7, 79-93. MAY 2005 19

Provenzale, A., 1999: Transport by coherent barotropic vortices, Ann. Rev. Fluid Mech., 31, 55–93. Polzin, K. L., 2004: A heuristic description of internal wave dynamics. J. Phys. Oceanogr., 34(1), 214–230. Polzin, K. L., J.M. Tooleand R. W. Schmitt, 1995: Finescale parameterizations of turbulent dissipation, J. Phys. Ocean., 25, 306–328. Polzin, K. L., N. S. Oakey, J. M. Toole and R. W. Schmitt, 1996: Fine- and microstructure characteristics across the northwest Atlantic Subtropical Front. J. Geophys. Res., 101, 14,111–14,121. Polzin, K. L., E. Kunze, J. M. Toole and R. W. Schmitt, 2003: The partition of finescale energy into internal waves and subinertial motions. J. Phys. Oceanogr., 33, 234-248. Ruddick, B. R. and T. M. Joyce, 1979: Observations of interaction between the internal wavefield and low-frequency flows in the North Atlantic, J. Phys. Oceanogr., 9, 498– 517. Schmitz, W. J., J. D. Thompson, and J. R. Luyten, 1992: The Sverdrup circulation for the Atlantic along 24◦, J. Geophys. Res., 97, 7251–7256. Stommel, H., 1948: The westward intensification of wind-driven ocean currents. Trans. American. Geophys. Union, 29, 202-206. Stone, P. H., 1972: A simplified radiative-dynamical model for the satic stability of rotating atmospheres. J. Atmos. Sci., 29, 405–418. Tennekes, H. and J. L. Lumley, 1972. A First Course in Turbulence, The MIT Press, Cam- bridge, pp. 300. Weller, R. A., D. L. Rudnick, C. C. Eriksen, K. L. Polzin, N. S. Oakey, J. M. Toole, R. W. Schmitt and R. T. Pollard, 1991: Forced ocean response during the Frontal Air-Sea Interaction Experiment. J. Geophys. Res., 96, 8611-8693. Wilson, C.and R. G. Williams, 2004: Why are eddy fluxes of potential vorticity so difficult to parameterize?. J. Phys. Oceanogr., 34, 142-155. 20 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

Figure Captions

FIG. 1. Phase lines (dashed) of waves being passively advected by three realizations of a steady mean flow having streamlines denoted by the solid contours: (a) a spatially constant deformation strain. (b) a spatially constant shear strain. (c) a spatially constant vorticity. Parcel velocities induced by internal waves are normal to wave crests (lines of constant phase). The tendency of mesoscale strain to create anisotropic wavefields and internal wave stresses by preferentially orienting the phase lines along the extensive axis of a strain field can be inferred [(a) and (b)]. Similarly, the tendency of vorticity to not result in wave stresses can also be inferred (c).

N IG F . 2. Upper panel: scatter plots of shear stress f Cuv dω against the shear component R N of strain Ss. Middle panels: scatter plots of the normal stress f (Puu −Pvv) dω against the R N normal component of strain Sn. Lower panels: scatter plots of velocity variance f (Puu + Pvv) dω plotted against vorticity ζ. The regression lines of stress vs strain returnR horizontal ∼ 2 −1 viscosity coefﬁcients of νh = 50 m s . Solid dots are 10 day averages. The uncertainty estimates represent 95% conﬁdence levels. In the lower panel, no trends of kinetic energy vs. vorticity are apparent. The lack of an apparent trend in this case is consistent with simple characterization of the coupling as a diffusive process.

1/2 1/2 FIG. 3. Coherence functions created by averaging (a) −sgn(Sn)[Puu − Pvv]/Puu Pvv 1/2 1/2 and (b) −sgn(Ss)Cuv/Puu Pvv with Cuv being the real part of the uv cross-spectrum. Estimates are based upon 512 point transform intervals and averaging over both triangles at 825 m water depth.

FIG. 4. Cumulative integrals of the spectral functions −sgn(Sn)[Puu −Pvv ] (thick line) and 2 1/2 2 1/2 −2sgn(Ss)Cuv (dashed line), divided by estimates of the rms strain (Sn) and (Ss ) , to provide estimates of the horizontal viscosity νh.

FIG. 5. Coherence function created by averaging −sgn(Uz)[Cuw − −2 1/2 1/2 −2 1/2 1/2 fN Cvb]/T (ω)Puu Pww and −sgn(Vz)[Cvw + fN Cub]/T (ω)Pvv Pww . The factor Cxy represents the real part of the xy cross-spectrum. The transfer function T (ω)=(ω2 − f 2)/(ω2 + f 2) accounts for cancelation of the Reynold’s stress by the buoyancy ﬂux and renders the denominator consistant with the numerator. The coherence estimates are based upon 1024 point transform intervals of data at both 600 and 825 m levels. Data are from the Center, Northeast and Northwest moorings. MAY 2005 21

1/2 2 2 FIG. 6. Cumulative integrals of the spectral function −sgn(Uz)[Cuw − fCvb/N ]/uz − 1/2 2 2 f 2 sgn(Vz)[Cvw +fCub/N ]/vz , to provide estimates of the vertical viscosity (νv + N 2 Kh). Vertical velocity was estimated by assuming a vertical balance in the temperature equation, ′′ ′′ ∼ Tt + w T z = 0 and were corrected for the roll-off associated with a center difference operator.

FIG. 7. A schematic of energy conversion rates derived from the LDE array, following Bry- den (1982). Estimates in units of W/kg refer to energy conversions at the depths of 600–800 m. Estimates in units of W/m2 refer to boundary inputs or depth integrated means. Non- labeled energy estimates are in units of 10−4m2s−2. Energy is input into the subtropical gyre by wind work and buoyancy forcing. It is converted from the mean density field (by baroclinic instability) to eddy energy. About 40% of this is converted into mean kinetic energy Bryden (1982). The eddy field is 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 budget than interactions with the internal wave field. However, eddy-wave coupling provides a source that is in approximate balance with the estimated dissipation, implying eddy-wave coupling plays an O(1) in the internal wave energy budget. With regards to the mean kinetic energy budget, Bryden (1982) finds an approximate balance between the gain associated with eddy work against the mean strain ′ ′ ′ ′ ′ ′ ′ ′ and a flux divergence, ∂x(uu u + vu v )+ ∂y(uu v + vv v ) 22 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

Figures

S = constant S = constant a n b s c ζ = constant 1 1 1

0.5 0.5

0 0

0.5 0.5 1 0.5 0 0.5 1 0.5 0 0.5 1 0 1 ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ u u - v v ∝ Sn u v ∝Ss u u +v v ∝ ζ

FIG. 1. Phase lines (dashed) of waves being passively advected by three realizations of a steady mean flow having streamlines denoted by the solid contours: (a) a spatially constant deformation strain. (b) a spatially constant shear strain. (c) a spatially constant vorticity. Parcel velocities induced by internal waves are normal to wave crests (lines of constant phase). The tendency of mesoscale strain to create anisotropic wavefields and internal wave stresses by preferentially orienting the phase lines along the extensive axis of a strain field can be inferred [(a) and (b)]. Similarly, the tendency of vorticity to not result in wave stresses can also be inferred (c). MAY 2005 23

−4 ≤ ω ≤ x 10 Polymode LDE 825 m Inner Cross f N

) 4 −2

s 3 2 2 (m

uv 1

−1

−2 2 −1 −3 64±20 m s

shear stress −2C −4 −4 −3 −2 −1 0 1−1 2 3 4 strain: V +U = S (s ) −6 x y s x 10

−4 ) x 10

−2 4 s 2

(m 2 vv −P

uu 0

−2 37± 16 m2 s−1 −4 normal stress P −4 −3 −2 −1 0 1−1 2 3 4 strain: U −V = S (s ) −6 x y n x 10 −3 x 10 8 )

−2 6 s 2

(m 4 vv +P

uu 2 P

0 −5 0 −1 5 vorticity: V −U = ζ (s ) −6 x y x 10

N FIG. 2. Upper panel: scatter plots of shear stress f Cuv dω against the shear component of strain R N Ss. Middle panels: scatter plots of the normal stress f (Puu − Pvv) dω against the normal compo- R N nent of strain Sn. Lower panels: scatter plots of velocity variance f (Puu +Pvv) dω plotted against ∼ vorticity ζ. The regression lines of stress vs strain return horizontalR viscosity coefﬁcients of νh = 50 m2 s−1. Solid dots are 10 day averages. The uncertainty estimates represent 95% conﬁdence levels. In the lower panel, no trends of kinetic energy vs. vorticity are apparent. The lack of an apparent trend in this case is consistent with simple characterization of the coupling as a diffusive process. 24 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

0.25 0.25 a b 0.2 0.2

0.15 0.15 ] ] 1/2 vv 1/2 vv P P 1/2 uu

1/2 uu 0.1 0.1 ]/[ P ] / [ P vv 0.05 uv 0.05 −P uu −[P 0 −2 Re[C 0

−0.05 −0.05

−0.1 −0.1 0 1 2 0 1 2 10 10 10 10 10 10 frequency (cpd) frequency (cpd)

1/2 1/2 FIG. 3. Coherence functions created by averaging (a) −sgn(Sn)[Puu − Pvv]/Puu Pvv and (b) 1/2 1/2 −sgn(Ss)Cuv/Puu Pvv with Cuv being the real part of the uv cross-spectrum. Estimates are based upon 512 point transform intervals and averaging over both triangles at 825 m water depth. MAY 2005 25

) 50 −1 s 2 40

30 Diffusivity (m

0 0 1 2 10 10 10 frequency (cpd)

FIG. 4. Cumulative integrals of the spectral functions −sgn(Sn)[Puu − Pvv] (thick line) and 2 1/2 2 1/2 −2sgn(Ss)Cuv (dashed line), divided by estimates of the rms strain (Sn) and (Ss ) , to provide estimates of the horizontal viscosity νh. 26 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

0.1

0.05

−0.05 Coherence

−0.1

−0.15

−0.2 0 1 2 10 10 10 frequency (cpd)

−2 1/2 1/2 FIG. 5. Coherence function created by averaging −sgn(Uz)[Cuw − fN Cvb]/T (ω)Puu Pww and −2 1/2 1/2 −sgn(Vz)[Cvw + fN Cub]/T (ω)Pvv Pww . The factor Cxy represents the real part of the xy cross-spectrum. The transfer function T (ω) = (ω2 − f 2)/(ω2 + f 2) accounts for cancelation of the Reynold’s stress by the buoyancy ﬂux and renders the denominator consistant with the numerator. The coherence estimates are based upon 1024 point transform intervals of data at both 600 and 825 m levels. Data are from the Center, Northeast and Northwest moorings. MAY 2005 27

−3 x 10 3

2.5

−1 1.5 s 2

0.5 Diffusivity m

−0.5

−1 0 1 2 10 10 10 frequency (cpd)

1/2 2 2 FIG. 6. Cumulative integrals of the spectral function −sgn(Uz)[Cuw − fCvb/N ]/uz − 1/2 2 2 f 2 sgn(Vz)[Cvw +fCub/N ]/vz , to provide estimates of the vertical viscosity (νv + N 2 Kh). Verti- ′′ ′′ ∼ cal velocity was estimated by assuming a vertical balance in the temperature equation, Tt +w T z = 0 and were corrected for the roll-off associated with a center difference operator. 28 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME

Energy Budget

Wind Work and Buoyancy Forcing

Flux Div.

Mean Mean APE MKE 79 3

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

Mesoscale Eddies EPE + EKE 104 61 1.4-2.1 mW/m2 bbl dissipation ν - 1x10-10 W/kg v Propagation? -10 νh - 3x10 W/kg

Internal Wave 4x10-10 W/kg & 1 mW/m2 IPE + IKE interior wave breaking 6 16 2 other sources? 0.2-0.3 mW/m bbl dissipation

Propagation?

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. Es- timates in units of W/m2 refer to boundary inputs or depth integrated means. Non-labeled energy estimates are in units of 10−4m2s−2. Energy is input into the subtropical gyre by wind work and buoyancy forcing. It is converted from the mean density field (by baroclinic instability) to eddy energy. About 40% of this is converted into mean kinetic energy Bryden (1982). The eddy field is damped by dissipation in the bottom boundary layer and forcing of the internal wavefield. Bot- tom boundary layer dissipation may play a somewhat larger role in the eddy energy budget than interactions with the internal wave field. However, eddy-wave coupling provides a source that is in approximate balance with the estimated dissipation, implying eddy-wave coupling plays an O(1) in the internal wave energy budget. With regards to the mean kinetic energy budget, Bryden (1982) finds an approximate balance between the gain associated with eddy work against the mean strain ′ ′ ′ ′ ′ ′ ′ ′ and a flux divergence, ∂x(uu u + vu v )+ ∂y(uu v + vv v )