1. Introduction
The ubiquitous mesoscale eddies in the ocean affect the transport and structure of tracers: salt, potential temperature, potential density, and potential vorticity (PV). The turbulent character of these eddies makes for few tractable observational, analytical, and numerical approaches. Formal mathematical treatments of eddy fluxes have revealed that careful manipulation of the definition of an eddy may provide advantages in capturing the eddy–mean flow interaction. The Transformed Eulerian Mean (TEM: e.g., Andrews et al. 1987; Vallis 2006) and Generalized Lagrangian Mean (GLM; Andrews and McIntyre 1978) frameworks are two examples.
Here the temporal residual mean formulation (TRM; McDougall and McIntosh 2001) is used to formulate a model consistent with the mean density structure of the ocean while retaining some important effects of time-dependent eddies. In an isopycnal-coordinate model, the TRM velocity is simply the thickness-weighted mean velocity, and the layer mass and tracer transports are natural. The TRM approach has been used successfully to study eddy–mean flow interactions in the atmosphere (Andrews et al. 1987). Here we show that similar advantages arise in the study of gyre circulations in enclosed domains. In particular we use the TRM formalism to include eddy feedbacks in models akin to Parsons (1969) and later extensions by other investigators.
Unlike much of the recent literature, this paper does not provide detailed maps of eddy diffusivities or viscosities: instead the TRM Parsons model compellingly illustrates the roles of eddies in a gyre circulation. In the TRM formulation, eddy effects appear as a force in the momentum equations—a fact extensively exploited here and in many atmospheric eddy–mean flow interaction studies. It is shown that eddies transfer momentum downward in the ocean interior and buoyancy poleward in the ocean mixed layer. The TRM Sverdrup relation is simply modified in the presence of eddies. The next section introduces the Parsons model and subsequent sections discuss impacts of eddies on the resulting ocean circulation.
2. Eddy–mean flow interaction in layered models
Equations (5)–(6) are identical to the steady form of (1)–(2), except the velocity in (5)–(6) is the thickness-weighted mean velocity3 and the eddy terms collected in
a. A broad eddy closure
Here the general form (12) will be carried throughout the analysis, but the solutions plotted use κ = 2000 m2 s−1 and 𝗔 as the identity matrix.
3. A noninertial, adiabatic, reduced gravity solution
So far, the mean flow remains arbitrarily inertial and nonlinear ( f ∼
The interfacial drag ϵi is tiny compared to ϵe, the “eddy form drag.”9 Lacking ϵe, Huang (1987a) uses an inflated value of ϵi ≈ 10−3 and suggests that this crudely parameterizes baroclinic instability. Here, a more careful treatment of eddies is attempted. While ϵi ≪ ϵe allows neglecting ϵi altogether, ϵi will be retained for ready comparison to earlier studies.
First, we will seek a solution to the equivalent to the Huang and Flierl (1987) model, which has a fixed volume of water in the upper layer and a motionless deeper layer. Toward the end of the section, generalizations of this model will be made.
a. Interior solution
b. Western boundary current
The Parsons (1969) model is exceptional among analytic solutions in that it possesses a western boundary current that can be attached to the coast (as is the Gulf Stream) or detached from the coast (as is the North Atlantic Drift). The solution for (16)–(17) is briefly described here; the reader is referred to Parsons (1969), Huang and Flierl (1987), and Nurser and Williams (1990) for a detailed presentation.
Figure 1 compares the uniform solution (33) to the results of Holland and Rhines (1980). Note that the changes to layer depth are consistent with the numerical solution, and likewise the potential vorticity perturbations against the western boundary are similar in both layers. The solution reproduces the simulation results without the introduction of an unrealistically large interfacial drag. The boundary layer structure results from lateral eddy PV flux.
The analytic solution misses a number of features. Obviously, the recirculation gyres are missing, largely because of the neglect of advection of mean flow momentum by the mean flow (these gyres form even in steady solutions; e.g., Cessi and Ierley 1995). Furthermore, Holland and Rhines (1980) note that the eddy kinetic energy and effective eddy diffusivity are much larger in the recirculation gyres than elsewhere, implying that κ1 should be larger there. While (27) can be used with spatially varying eddy statistics, the numerical implementation of this effect is beyond the scope of this work. PV homogenization (Rhines and Young 1982a) below the recirculation gyre area is also less effective. These missing features are expected given the assumed scale separation between the boundary layer width (31) and the deformation radius. This separation is not large for the parameters used here.
The effects of f on widening boundary layer width are minor in this simulation. The basin is small—the change in f is only two-fifths of f0—so the error in approximating ϵe precludes a close comparison.
The eddy PV flux added here is crucial, however, in obtaining reasonable solutions. The PV flux allows reasonable boundary current width and velocity under typical oceanic parameters. The Huang and Flierl (1987) steady solution with the same parameters (except ϵe = 0) has a boundary current 400 times thinner and velocities 400 times faster.
4. Eddy implications
Now that a solution with realistic eddy effects is complete, the remainder of this paper illustrates some of the implications.
a. An important condition on eddy fluxes
b. A weakly moving lower layer
Thus, the amount of vertical shear balances downward eddy form drag and bottom friction to determine the equilibrated flow. Recent equilibrated quasigeostrophic turbulence simulations agree (Arbic and Flierl 2004; Thompson and Young 2006).
c. Diabatic eddies in the mixed layer
Pedlosky (1987) adds a diabatic transport in a model similar to the Parsons (1969) model by capping it with an Ekman flow within an independent surface mixed layer. The Ekman flow provides pumping to the geostrophic interior and is allowed to change its density as it moves, resembling that of the Luyten et al. (1983) model. The resulting model does not have a separated western boundary current. By contrast, the adiabatic model above combines the Ekman and geostrophic flows in the upper layer to cause separation in (34)–(36).
Nurser and Williams (1990) explore the thermodynamic implications of allowing the Ekman layer water to change density classes within the mixed layer (Uc in Fig. 2). Similarly, Veronis (1978) was successful in perturbing the Veronis (1973) adiabatic solution by adding flow to the western boundary currents with localized diabatic sinking near the poles (Sn in Fig. 2) and localized diabatic upwelling near the equator (Ss in Fig. 2). These authors impose heating or cooling in particular locations and infer the layer transport based on the loss or gain of fluid.
What is the physical basis of this mixed layer flow? Nurser and Williams (1990) do not specify, but it is directed across the mean density gradient at the outcrop so as to flatten the isopycnal slope, closes the residual circulation transport budget (44)–(45), and responds promptly to balance surface cooling events. These are characteristic behaviors of near-surface eddy fluxes, as shown by Ferrari and Plumb (2003) and Plumb and Ferrari (2005). These fluxes are cast as an eddy parameterization scheme by Ferrari et al. (2008).
d. The depth-integrated equations
Rhines and Holland (1979) and Radko and Marshall (2004) note that in numerical simulations the eddy fluxes leak away from the boundary into the interior and affect the vorticity balance there. They analyze an eddy-induced vortex stretching in addition to the Ekman wind forcing for the upper-layer vorticity balance. This section shows that TRM naturally includes this effect.
Similarly, the depth-integrated momentum equations here resemble the steady Stommel (1948) model, as eddy momentum transfers from layer to layer cancel out, leaving only wind stress and bottom drag (the Antarctic Circumpolar Current has a similar depth-integrated balance as shown by Johnson and Bryden 1989). Unlike the barotropic Stommel model, vertical shear is permitted at density interfaces with a magnitude consistent with the Margules relation and controlled by the eddy form drag as in (42). Only in an eddy-resolving or eddy-parameterizing model can the shear be quantified.
e. Boundary conditions and global constraints
The treatment of eddies here is more realistic than in previous models. However, it should be noted that inertial terms are neglected by assuming that eddy and mean scales exceed the deformation radius. Relaxing this constraint defies eddy parameterization, because the neglected terms (e.g., ∇
The eddy-resolving models of Berloff (2005) and Fox-Kemper (2004) reveal a layered structure of eddy fluxes near the boundary, which one expects to be true with O(1) depth variations as well. The seaward boundary layer is the one parameterized here, where potential vorticity fluxes of thickness tend to flatten the isopycnals. At a deformation radius from the boundary, thickness PV fluxes are exchanged for relative vorticity PV fluxes. Finally, in a viscous sublayer, relative vorticity PV fluxes are removed from the basin frictionally.
Interestingly, the outcrop line does not behave as a rigid boundary in this regard: the eddy flux into the separated boundary current
5. Conclusions and summary
A reconciliation of the residual, or thickness-weighted, mean and the traditional steady wind-driven gyre model initially studied by Parsons (1969) reveals that most of the traditional solution methods proceed with little modification. The interpretation of the solutions, however, is unfamiliar as the thickness-weighted mean velocity (not the Eulerian mean velocity) plays the role of the steady velocity. The thickness-weighted mean velocity appears because its transport is nondivergent even in an eddying flow, unlike the Eulerian mean. It is shown that an “eddy form drag” results from a realistic parameterization of along-isopycnal eddy fluxes of potential vorticity and plays the role of the unrealistically large interfacial friction typically found in steady models.
This model sheds light on the behavior of eddies in the gyre circulation. An important constraint—that eddies only rearrange and do not create momentum—is shown to be crucial to recover the traditional Sverdrup (1947) relation upon integration over the total ocean depth. However, eddies do redistribute momentum in the vertical, as shown by (49). To maintain a Sverdrup-like upper-layer vorticity equation under the effects of eddies, the thickness-weighted mean velocity may be used.
Inertia is a crucial aspect of gyre flows missing from this simple model, which was chosen primarily to illustrate, not simulate. Relatedly, the eddy parameterization here allows eddy PV fluxes through the boundary, while resolved eddies must transfer these fluxes to frictional or diffusive ones as in the second case studied by Fox-Kemper and Pedlosky (2004). Perhaps an extension including the eddy advection of eddy relative vorticity—at least for weak eddies—is possible using a similar form of eddy parameterization to that used here [see (10) and Plumb 1990]. Even so, the mean advection of mean relative vorticity usually requires numerical simulations (e.g., Cessi and Ierley 1995; Speich et al. 1995) or analytic methods beyond the treatment here (Charney 1955; Huang 1990).
Acknowledgments
This paper is a contribution in honor of Joe Pedlosky, who teaches us all how to learn from simple models. The authors received a beautiful introduction to residual mean formulations from R. A. Plumb. One anonymous reviewer offered many excellent suggestions. BFK was partially supported by a NOAA Climate and Global Change postdoctoral fellowship, NSF DMS-0855010, and NSF OCE-0825614. BFK and RF were partially supported by NSF OCE-0612143.
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Reynolds averaging is assumed, so
This is the Eulerian mean velocity following a fixed location in density coordinates, not z coordinates (see McDougall and McIntosh 2001 for discussion).
Note that the velocity contained in
Rotational flux components result in an antisymmetric contribution, but can be absorbed into the bolus velocity by a judicious choice of gauge for small amplitude eddies (Plumb 1990).
Killworth notes that an ensemble average of 𝗔k over all angles would result in 𝗔k being one-half the identity tensor.
However, the numerical results of Nadiga (2008) indicate that finding θ may be a very challenging task.
Lengths are scaled by basin dimension L, times by 1/(βL), heights by mean upper-layer depth H1, velocities by |u1| = g′H1/L2β, and wind stress by W.
Typical ocean values (Holland and Rhines 1980): L = 1000 km, H1 = 1000 m, H = 5000 m, f0 = 9.3 · 10−5 s−1, β = 2 × 10−11 s−1 m−1, ρ0 = 1000 kg m−3, W = 0.1 N m−2, g′ = 0.02 m s−1, and basin dimensions are 0 < x < L, −L < y < L, τ = W cos(πy/L), Cb = H2r, and r = 10−7 s−1. Additional parameters used here are κ1 = 2000 m2 s−1, Cd ≈ ν/H1, and ν = 10−4 m2 s−1.
At the risk of being “obscurantist” (Warren et al. 1996), we refer to this effect as an “eddy form drag.” This term is becoming traditional (Vallis 2006), and stems from the resemblance of the correlation of eddy velocity and layer thickness to the correlation of bottom velocity with bottom topography of true form drag. However, it is not a true “form drag,” and does not even appear in the Eulerian mean momentum equation. Only in the thickness-weighted mean equations does it appear as a force. In this context, it will be shown to resemble an interfacial drag.
The thickness-weighted mean has been used to approximate the interfacial drag term rather than the Eulerian mean velocity. This approximation is acceptable on the along-current velocities; both the Eulerian and thickness-weighted along-current mean velocities are geostrophic to first order.
This width is close to the deformation radius (40 km) and depends on the assumed value of κ1, which questions the validity of the scalings here. However, the parameters are chosen here to reproduce Holland and Rhines’ (1980) values, not a clean scale separation.
This is assuming that relative vorticity fluxes are neglected, consistent with eddies being larger than the deformation radius.
Bottom drag magnitude is again approximated with
A term −∇ ·
This is aside from relative vorticity contributions.
κ ≡ κ1𝗔n,n.