1. Introduction
a. Preliminaries
Winds and air–sea exchanges of heat and freshwater are ultimately responsible for the basin-scale currents or general circulation of the oceans. To achieve a state where the energy and enstrophy (vorticity squared) of the ocean is not continuously increasing, some form of dissipation is required to balance this forcing. Although the previous 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 midgyre could be dissipated. Apart from rationalizing why western boundary currents are located on the western boundary, even the crudest perusal of observations suggests distinct discrepancies with such models. One example is that an intensely energetic mesoscale eddy field is associated with the Gulf Stream after its separation (e.g., Schmitz 1976) with the perception (Moore 1963) that one reason for the existence of such an intense eddy field is insufficient dissipation at the boundary. Thus, the question of closing an energy budget for the general circulation (e.g., Wunsch and Ferrari 2004) is to remove dissipation from the mean budget and place it in the eddy budget. One can follow the thread and ask, “how does the mesoscale eddy field dissipate?” Among many possible mechanisms (Ferrari and Wunsch 2009) is the exchange of energy between the mesoscale eddy field and the internal wave field. The rest of this introduction attempts to discuss the phenomenology of mesoscale eddy–internal coupling, review previous results, and lay groundwork for the rest of the paper.
b. Mesoscale eddy–internal wave coupling
This cornerstone summarizes a vast literature about waves in zonal or symmetric mean flows. In this restricted paradigm, the EP flux (equivalently, the pseudomomentum flux or wave stress) assumes a primary role, as it should because of the attendant nonacceleration theorems. However, there is something much more general than pseudomomentum flux conservation in symmetric flows, and this is action conservation, which is independent of the spatial structure of the background flow field. The intent of this work is to demonstrate that, if one drops the assumption of mean state spatial symmetry inherent in the EP flux theorem, mesoscale eddies and internal waves are coupled through a linear wave stress–eddy rate of strain relation, which leads to the representation of internal wave effects as an eddy viscosity.
1) Symmetry
A convenient starting place to examine eddy–wave coupling in three dimensions (3D) is to invoke a decomposition of the velocity [u = (u, υ, w)], buoyancy (b = −gρ/ρo with gravitational constant g and density ρ), and pressure (π) fields into a quasigeostrophic mean 
It is from this vantage point that one can appreciate that the generalized Eliassen–Palm flux theorem does not apply to asymmetric (3D) flows. If the background velocity field contains horizontal gradients in both (x, y) dimensions, the streamwise component of the horizontal wave vector evolves following a ray trajectory and thus the pseudomomentum flux is generally divergent. A simple rationalization of the difference in behavior between 2D (symmetric) and 3D (asymmetric) systems comes from theoretical physics: each symmetry exhibited by a Hamiltonian system corresponds to a conservation principle (Nöther’s theorem; e.g., Shepherd 1990). For spatial symmetries, the conservation principle concerns momentum: axisymmetric flows preserve the flux of pseudomomentum in the symmetric coordinate.
2) Asymmetry, the shrinking catastrophe, and wave capture
Bühler and McIntyre (2005) further argue for a scenario, which they term “wave capture” and Jones (1969) labels a “shrinking catastrophe.” Simply put, the vertical wavenumber is slaved to the horizontal wavenumber, dm/dt = −k
The ray-tracing arguments about internal wave packets interacting with mesoscale eddies return useful pictures about a mechanism, but they do not provide information about the net transfer rates for the observations. This requires consideration of the energy balance, which is done later within the quasigeostrophic limit. Note that quasigeostrophy invokes both a smallness of Rossby number and that the two horizontal length scales are similar. It is this last assumption that is crucial to scaling the horizontal divergence as O(Rossby number squared): that is, crucial to the statement that the background field is horizontally nondivergent in the context of ray tracing (4) and its summary in (5) and (6). The length scale requirement of quasigeostrophy further underscores the issue of how asymmetry in the background influences the character of the wave–mean interaction.
c. Linearized wave energy balances
Two further points are to be made. First, the right-hand side of (7) represents the rate at which the pseudomomentum flux (2) is doing work against mean gradients. It is thus a description of how internal wave energy is altered through wave radiation (work and an energy flux divergence). Exchanges of energy through the vertical terms can be either positive or negative, depending on the sign of the vertical group velocity, horizontal wavenumbers, and vertical shear. The standard parallel shear flow critical layer has, with
d. Observations
It remains to inquire whether the coupling is sufficiently large that interior damping of the mesoscale is considered 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 (39°N, 70°W), arrays at (28°N, 70°W) [the MidOcean Dynamics Experiment (MODE), the Internal Wave Experiment (IWEX), and the Local Dynamics Experiment (LDE; 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 associated with large mean currents (site D) and correlation between 
e. Forward
The problem of wave forcing of the oceanic general circulation is much richer than is to be inferred from taking the EP flux theorem (1) at face value and assuming an adiabatic limit. A key contribution of this work is the recognition of the role that asymmetry and strain play in the phenomenology, energetics, and dynamics of wave–mean interactions. Note that strain is not regarded as such until Bühler 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. A companion paper (K. L. Polzin 2009, unpublished manuscript) examines dynamical issues and the LDE potential enstrophy budget.
2. Horizontal coupling
Brown and Owens (1981) present a scatterplot of horizontal stress
The analysis presented here is similar, 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. Second, Brown and Owens defined the internal wave band with a filter having a half power point at 0.8f. Here, mesoscale velocity gradients are defined by a low-pass filter with ½ power at 10-day periods and the high-frequency data are detrended. A filter having a 2-day ½ power point has also been used. No appreciable differences were noted. Finally, Brown and Owens estimate the
The major stated difference is that Brown and Owens estimate the horizontal stress as 
My regression of
The difference is not simply in Brown and Owens’s 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)Cu,υ, in which sgn represents the sign of its argument. Cumulative integration of the cospectrum (divided by the estimate of rms rate of strain; Fig. 3) provides a characterization of how each frequency contributes to the viscosity operator. Here, integration over 2f ≤ σ ≤ N returns horizontal viscosity estimates of about 18 m2 s−1.
Coherence estimates (Fig. 4) are about 0.05 for σ > 2f. This result is consistent with the characterization of Frankignoul (1976) that the relation between stress and horizontal rate of strain is subtle. The Brown and Owens estimate of νh = 400 m2 s−1 would imply O(1) values of coherence. My estimate based on frequencies σ > 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 dimension
Correlations between the vertical flux of horizontal pseudomomentum and the mesoscale eddy field were pursued by Ruddick and Joyce (1979) using current meter data from PolyMode arrays I and II. For the relatively energetic 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) there was 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 thermocline, resulting in an effective viscosity of νυ + ( f 2/N 2)Kh ≅ 0.01 m2 s−1.” Their interpretation depends on 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 center, northeast, and northwest moorings of the LDE array. This provides some 995 days of data with stress–shear relations in two horizontal components from two vertically separated instruments, so that each cospectral estimate is associated with approximately 750 degrees of freedom. The eddy energy at the LDE site is comparable to the less energetic PolyMode I data and the consequent contamination of the stress estimates by mooring motion is small (Ruddick and Joyce 1979).
Evaluation of the LDE data returns a positive vertical shear–vertical stress correlation at low 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 energy from the wave field to the mean, which is consistent with either critical layer or wave capture scenarios. Coherence estimates are O(1) at near-inertial frequencies, but this part of the spectrum carries little effective momentum flux. Consequently, the estimates of vertical viscosity and associated energy transfer are dominated by high-frequency contributions having small coherence estimates. Integration of the cospectra returns νυ + ( f 2/N 2)Kh = 2.5 ± 0.3 × 10−3 m2 s−1 (Fig. 6). The error estimate provided here represents the compounding of the nominal 750 degrees of freedom per coherence estimate (Fig. 5) over a bandwidth of 10–40 cpd containing 320 coherence estimates.
There are two landmarks of possible consequence in the frequency domain. The positive stress–shear correlation occurs for frequencies in which the wave aspect ratio is equal to or smaller than the aspect ratio of the mesoscale eddy field. The negative stress–shear correlation occurs for waves that potentially encounter a buoyancy frequency turning point: the minimum buoyancy frequency for the observed density profile is approximately 10 cpd (N = 7 × 10−4 s−1).
4. LDE energy budgets
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; Fig. 7).
Bryden infers a conversion of available potential energy to eddy kinetic energy at a rate of 3.3 ± 1.7 × 10−9 W kg−1 through baroclinic instability and that the mean velocity field represents a countergradient “sink” (see Fig. 1) at the rate of 1.5 ± 0.9 × 10−9 W kg−1. The residual represents dissipation, propagation, advection, and possibly time dependence. Brown and Owens (1981) estimate that the internal wave field serves as a sink of eddy energy at a rate of νh(
A possible interpretation is that eddy energy associated with baroclinic production radiates 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 wave field (Polzin 2004), it would be equivalent to an interior dissipation of 6–9 × 10−10 W kg−1 at 800-m depth.
This calculation, albeit based on 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 demonstrations that eddy statistics in idealized quasigeostrophic turbulence models are strongly impacted by damping (Arbic and Flierl 2004; Arbic and Scott 2008) have focused on the issue of bottom dissipation.
b. Mesoscale eddy–internal wave coupling as internal wave forcing
If mesoscale eddy–internal wave 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 fine structure data obtained as part of the Frontal Air-Sea Interaction Experiment (FASINEX). Field work took place in February–March 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, 2003). The MODE and IWEX studies were also located here. Sampling during FASINEX 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 wave field 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 in this region (Briscoe and Weller 1984). It is not clear how this annual cycle would appear in the fine structure data, but the FASINEX data were obtained during the more energetic part of the annual cycle.
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 is a dominant energy source for the internal wave field in the Gulf Stream recirculation.
5. Summary
Current meter array data from the LDE of the PolyMode 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 viscous process. Viscosity coefficients inferred from regressions of horizontal stress versus horizontal rate of strain return estimates of νh = 64 ± 20 m2 s−1 and νh = 37 ± 16 m2 s−1. The regression of effective vertical stress estimates against vertical shear returns νυ + ( f 2/N 2)Kh = 2.5 ± 0.3 × 10−3 m2 s−1. In terms of energy budgets, eddy–wave coupling represents a significant mechanism by which eddy energy is dissipated and plays an O(1) role in the energy budget of the internal wave field.
These results may be specific to the LDE region, which is situated at the exit of the southern recirculation gyre. Variability of the viscosity coefficients acting on the mesoscale field will depend on variability in the background wave field. Such variability exists (Polzin and Lvov 2009, manuscript submitted to Rev. Geophys.). A possible implication of this work is that the variability in both the mesoscale eddy field (Zang and Wunsch 2001) and the internal wave field (Polzin and Lvov 2009, manuscript submitted to Rev. Geophys.) are related through mesoscale eddy–internal wave coupling. In terms of understanding the geographic variability of viscosity coefficients, there is a simplicity if the energetics of the internal wave field are dominated by an interior coupling to the mesoscale eddy field, as appears to be the case in the southern recirculation gyre. This represents the dynamic balance advocated by Müller and Olbers (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 the failure of certain instruments reduces the usable record length to 225 days. Stable estimates of time-mean quantities typically require averaging periods on the order of 500 days (Schmitz 1977). The mean quantities quoted here represent record length means with associated record length uncertainties. See Bryden (1982) and Brown et al. (1986) for further discussion of these uncertainties. I note, however, that the available 15-month estimates are consistent with the 225-day record means (to within uncertainty). Any differences do not change the interpretation presented here.
The LDE array does not fully spatially resolve the mesoscale eddy velocity gradient variances. This issue is examined in the companion manuscript where resolution of the enstrophy gradient variance is even more problematic. Again, consideration of such issues does not change the interpretation presented here.
The characterization of the coupling through a flux–gradient relation applies only to quasigeostrophic flows in which the flow field is horizontally nondivergent to O(Rossby number squared). Symmetric flow structures such as rings and jets are not coupled in the same manner.
The interpretive paradigm being pursued here and in related work (Polzin 2008; K. L. Polzin 2009, unpublished manuscript) is one of wave propagation in a horizontally nondivergent mesoscale eddy velocity field in which exchanges of energy and momentum are rendered permanent through wave damping or nonlinearity. Wave propagation is assumed to be consistent with ray tracing and action conservation, which in turn assumes a scale separation between wave and background and that the wave amplitude is small relative to the background. These assumptions are harmonious with (i) an enstrophy cascade regime of quasigeostrophic turbulence in which energy spectra roll off as kh−3 and (ii) observed acceleration spectra (Rupolo et al. 1996) from the main thermocline of the western North Atlantic, which suggest a slightly steeper kh−3.3 spectral slope for the eddy energy spectrum. Potential vorticity containing motions on horizontal scales of 1–10 km are quite difficult to measure, however, and such acceleration spectra do not preclude the presence of vortical modes associated with generation via topographic torques (Kunze and Sanford 1993) or internal wave breaking (Polzin et al. 2003; Polzin and Ferrari 2004). Part of the difficulty is that internal waves will tend to dominate the velocity field at horizontal wavelengths smaller than 100 km, which conflicts with the small-amplitude approximation. Finally, the focus here has been on interpreting observations from the main thermocline. Altimeter data exhibit much shallower spectral slopes in energetic western boundary current regimes, closely following a kh−5/3 spectrum that has been interpreted as an energy cascade associated with surface quasigeostrophic dynamics (LeTraon et al. 2008). Thus, the upper ocean may not fit nicely into the interpretive paradigm being pursued here. Alternative interpretive paradigms are presented in Ford et al. (2000; Lighthill radiation), Klein et al. (2004; near-inertial oscillation dynamics), and Riley and Lelong (2000; nonlinear wave–vortex interactions/rotating stratified turbulence).
6. Discussion
The discussion here tries to flesh out some of the broader implications of eddy–wave coupling as they appear in the context of the PolyMode field program.
a. Coherent vortices
One of the surprises of the Local Dynamics Experiment was the prevalence of coherent submesoscale lenses (Elliot and Sanford 1986a). These lenses typically had temperature–salinity properties that distinguished them as having relatively long (several years) life spans. These submesoscale features had horizontal scales of L = 15 km, significantly smaller than the mesoscale field (≅100 km). Such life spans are not consistent with a viscous decay and νh = 50 m2 s−1, which provides a temporal spin-down scale L2/νh ≅ of 50 days.
However, note that the relative vorticity variance in the core of such lenses (Elliot and Sanford 1986b) dominates the rate of strain variance, so by the criterion (5) one does not expect a strong coupling between the internal wave field and the mesoscale eddy field. Moreover, as a symmetric feature, the assumptions leading to a flux–gradient relation are not valid and the expectations of a short temporal spin-down scale are incorrect. The viscosity operators here are appropriate for a three-dimensional field (Fig. 1a), not two-dimensional fields (Fig. 1b).
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 pseudomomentum flux is nondivergent. Extrapolation of this result to nonaxisymmetric background flows gives the misleading impression that the only consequence of internal waves for geostrophically balanced flows is through a diabatic link. The existence of long-lived, coherent submesoscale vortices is consistent with the proposed model of wave–eddy interaction.
b. Topographic Rossby waves
Price and Rossby (1982) document that the subthermocline velocity field at the start of the LDE is dominated by highly polarized oscillatory flow that is consistent with topography Rossby wave characteristics: the horizontal wavelength (340 km), intrinsic frequency (1/61 cpd), and phase propagation toward 300°T agree with a the dispersion relation of barotropic planetary waves modified by topography. The group velocity is approximately 0.05 m s−1 directed toward 100°T (eastward). The LDE was situated at the exit of the southern recirculation gyre, and the mean rate of strain is nonzero. This topographic wave is oriented so that energy is being transferred from the wave to the mean field (Fig. 1). Such events dominate the record length estimates of energy transfer in Bryden (1982).
A large part of the mesoscale eddy field at the LDE site is linear waves, and the energetic transfer estimates of Bryden (1982) fit nicely into a wave–mean interaction paradigm described in section 1c. However, this statement comes with major caveats. Observed particle speeds (0.12 m s−1) exceed the phase speed by about a factor of 2.
c. Vertical mode coupling
A second caveat regarding nonlinearity is that planetary wave fits to the mesoscale eddy field require a superposition of several barotropic and baroclinic plane waves to match the horizontal structure of the observed velocity field (Hua et al. 1986). The vertical modes are dynamically coupled and the potential vorticity balance of the baroclinic mode has significant nonlinear contributions, especially at the smallest resolved scales. Of two interaction events, one is described as the straining of an antecedent, large-scale baroclinic flow into a baroclinic jet by the barotropic wave described earlier. Although the interpretation offered in Hua et al. (1986) is based on nonlinearity, the role of internal waves in contributing to the baroclinic jet through a frontogenetic process is an alternative hypothesis that has yet to be explored.
Acknowledgments
Much of the intellectual content of this paper evolved out of discussions with R. Ferrari. The manuscript benefited from discussions with Brian Arbic and Rob Scott and a review provided by Peter Rhines. Salary support for this analysis was provided by Woods Hole Oceanographic Institution bridge support funds.
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Phase lines (dashed) of waves being passively advected by two realizations of a steady mean flow having streamlines denoted by the solid contours: (a) a spatially constant shear strain Ss and (b) a spatially constant relative vorticity ζ. (c) For internal waves, parcel velocities projected onto the horizontal plane (see arrows) are elliptically polarized with the major axis parallel to the horizontal wave vector and normal to wave crests (lines of constant phase). (d) For Rossby waves, the horizontal velocity trace is linearly polarized (see arrows) and parallel to wave crests (lines of constant phase). (c),(d) The tendency of background rate of strain to create anisotropic wave fields and wave stresses by preferentially orienting the phase lines along the extensive axis of a strain field can be inferred. However, there is an important distinction in the two cases. Parameterization of the stress–strain relation in terms of a flux-gradient relation leads to a positive eddy viscosity for internal waves and a negative eddy viscosity for Rossby waves. (b) The tendency of vorticity to not result in wave stresses can also be inferred.
Citation: Journal of Physical Oceanography 40, 4; 10.1175/2009JPO4039.1
(top) Scatterplots of shear stress 
Citation: Journal of Physical Oceanography 40, 4; 10.1175/2009JPO4039.1
Cumulative integrals of the spectral functions −sgn(Sn)[Pu,u − Pυ,υ] (thick line) and −2sgn(Ss)Cu,υ (dashed line), divided by estimates of the corresponding rms rate of strain 
Citation: Journal of Physical Oceanography 40, 4; 10.1175/2009JPO4039.1
Coherence functions created by averaging (a) 
Citation: Journal of Physical Oceanography 40, 4; 10.1175/2009JPO4039.1
Coherence function created by averaging 
Citation: Journal of Physical Oceanography 40, 4; 10.1175/2009JPO4039.1
Cumulative integrals of the spectral function −sgn(Uz)[(Cu,w − fCυ,b)/N 2](
Citation: Journal of Physical Oceanography 40, 4; 10.1175/2009JPO4039.1
A schematic of energy conversion rates derived from the LDE array, following Bryden (1982). Estimates in units of W kg−1 refer to energy conversions at the depths of 600–800 m. Estimates in units of W m−2 refer to boundary inputs or depth-integrated means. Unlabeled energy estimates are in units of 10−4 m2 s−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 wave field. 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 rate of strain and a flux divergence, 
Citation: Journal of Physical Oceanography 40, 4; 10.1175/2009JPO4039.1
The LDE array does not fully resolve the gradient variance, but the unresolved variance is arguably acted on by a scale-dependent viscosity coefficient with consequently smaller transfer rates. These scale-dependent issues are taken up in K. L. Polzin (2009, unpublished manuscript) but do not affect the conclusions presented here regarding the energy budget.
