1. Introduction

Continental boundaries lead to a mean oceanic stratification with both zonal and meridional structure, so by thermal wind balance local mean flows can take on any direction—this situation can be contrasted with the atmosphere in which the mean winds are primarily zonal. Midlatitude storm tracks have long been associated with baroclinic instability of that zonal mean flow (Charney 1947), and much of the theory, both linear and nonlinear, of baroclinic instability has been developed for vertically sheared zonal mean flow. Baroclinic instability of vertically sheared meridional mean flow is well understood from a linear perspective (Robinson and McWilliams 1974; Pedlosky 1987; Walker and Pedlosky 2002); the essential distinction from zonal theory is that arbitrarily weak meridional shear can lead to growing waves because the northward Coriolis gradient (β) does not prevent sign changes of the mean zonal potential vorticity gradient. In practical terms, this means that the oceanic available potential energy stored in east–west gradients of the pycnocline is more easily converted to eddy kinetic energy than that stored in north–south gradients.

The turbulence resulting from the baroclinic instability of a vertically sheared meridional current, on the other hand, has only recently been investigated. Spall (2000) and Arbic and Flierl (2004a) find that steady-state eddy energies generated by this mechanism can exceed the mean kinetic energy by more than a factor of 1000, much in excess of eddy energies obtained in simulations of zonally generated baroclinic instability (e.g., Held and Larichev 1996). Thus not only are east–west gradients of density surfaces more unstable, but the ensuing turbulence results from an extreme amplification of the transfer from mean to eddy energy. One goal of Spall (2000), and the primary goal of the present paper, is to understand the eddy generation mechanism and nonlinear equilibration in nonzonal baroclinic turbulence and to find a theory for their dependence on mean flow parameters. Given the ubiquity of meridional flow in the ocean, a closed theory for these statistics is a necessary component of any successful potential-vorticity-based ocean eddy parameterization.

Spall (2000) considers baroclinic turbulence driven by nonzonal mean flows in a series of simulations, using both a meridional channel and a wind-driven basin, and provides a theory that successfully describes the scaling of the obtained eddy energies with the parameters of the problem. The theory derived depends critically on the zonal width of the domain, which is assumed to set the length scale in a diffusive closure for the eddy potential vorticity flux (the details will be reviewed in the next section). The implication is that the width of the basin sets the zonal eddy scales that result from nonzonal baroclinic instability in the real ocean and, by extension, that “local” theories for midocean eddy statistics are inapplicable. In other words, the theory implies that the “homogeneous limit” of Haidvogel and Held (1980) does not really exist.

Arbic and Flierl (2004a), on the other hand, investigate baroclinic turbulence driven by nonzonal flows in a doubly periodic model, where no domain scale is present. They find that eddy energies do saturate but that the equilibrated energy level is very sensitive to bottom drag, and moreover increases with increasing β, just the opposite of the trend predicted by Spall’s theory. It is also opposite the trend for zonal baroclinic instability in which eddy energy decreases with increasing β (e.g., Held and Larichev 1996). The fact that a steady state is obtained in these experiments implies that, in lieu of boundaries or zonal inhomogeneities, some local mechanism must act to set the zonal eddy length scale.

The complexity of the boundary-free problem must lie in understanding the effects of the mean vorticity gradient, β. Were it not for β, the eddying flow would be isotropic and there could be no statistical dependence on the mean flow direction. The linear theory leads one to consider the effects of β on suppressing the linear instability, but in fully developed turbulence β has a second, unrelated effect on the flow: the inverse cascade of energy produced by baroclinic instability is impeded in the meridional direction by the Rossby dispersion relation, leading to the formation of zonal jets (Rhines 1975; Williams 1978; Vallis and Maltrud 1993). A slew of recent papers (Maximenko et al. 2005; Richards et al. 2006; Nadiga 2006) have revealed the presence of jets in both ocean observations and numerical models. How does this eddy anisotropy affect the energy levels in nonzonal baroclinic turbulence?

Eddy energy generation by baroclinic instability is proportional to the eddy flux of eddy potential vorticity,

where the overbar represents a horizontal spatial average (assuming homogeneous statistics), H is the depth, E is the eddy energy, U = u = U(z)i + V(z)j is the mean velocity, u′ = u′(x, y, z, t)i + υ′(x, y, z, t)j is the eddy velocity, and q′ is the eddy quasigeostrophic potential vorticity (PV). In the classic problem, V = 0 and, when the zonal shear is unstable, the generation is due to the northward eddy flux of eddy potential vorticity. When the turbulence is either isotropic or zonally elongated, υ′ is random and uncorrelated, and turns out to be well approximated by a mixing-length hypothesis and downgradient mixing of potential vorticity (Larichev and Held 1995; Held and Larichev 1996; Smith and Vallis 2002).

In the special case that U = 0, however, one can see from (1.1) that the generation will be driven by correlations between the zonal eddy flow and the eddy PV. When the eddies are zonally elongated or jetlike, u′ is coherent and there is no reason to expect a mixing length hypothesis to describe the eastward flux of eddy PV. Rather, for scales large compared to the deformation scale, the PV is primarily baroclinic, and its flux is primarily due to passive advection by barotropic eddies (Salmon 1980). In this case, one should consider the transport of a passive scalar by jet flow in a turbulent background. Particle pairs will be sheared apart, but will remain correlated over large flight distances, until randomness in the turbulent background decorrelates them. Thus the weaker the background turbulent diffusion, the larger the effective along-jet transport. This is the shear dispersion problem first described by Taylor (1953), and the quantitative result is that the flux along the jet is inversely proportional to the background diffusivity. When the background diffusion is due to turbulent mixing, the diffusivity can be related to the across-jet flux.

The specific application of shear dispersion to the flux of a tracer by β-plane jet flow is worked out in Smith (2005). It is shown in the present paper that a shear dispersion model for the flux of eddy potential vorticity is sufficient to explain the trends and high energy levels found by Spall (2000) and Arbic and Flierl (2004a) for nonzonal baroclinic turbulence. Specifically, because the cross-jet flux decreases with increasing β, the along-jet flux does just the opposite, so the energy generation increases with increasing β (so long as the eddying flow is zonally elongated, which can be prevented if the bottom drag is strong enough). The present theory falls short, however, of providing a complete closure for the problem. An intensely nonlinear feedback mechanism makes it extremely difficult to change the jet scale in the weak drag limit, even over more than a decade of values of β. The details are described in the body of the paper.

The paper is organized as follows. The background to the problem of fully developed baroclinic turbulence driven by a nonzonal mean flow is worked out in section 2. Numerical experiments designed to explore the problem and the theory to describe the results are described in section 3. Conclusions are given in section 4.

2. Background

The quasigeostrophic (QG) approximation is appropriate for small Rossby number phenomena with length scales of order of the deformation scale, and so is an appropriate model for nonequatorial mesoscale ocean eddy dynamics. Given a background flow U = U(z)i + V(z)j, assumed to be steady on eddy time scales, and set by large-scale wind and buoyancy forcing (i.e., we do not require that it be a quasigeostrophic solution), the mean streamfunction is

Quasigeostrophic perturbations (dropping primes) about this mean will satisfy the potential vorticity conservation equation

where D represents drag and dissipation terms,

is the eddy QG potential vorticity, N = N(z) is the background buoyancy frequency, f is the Coriolis parameter, and

is the mean PV gradient. Multiplying (2.1) by −ψ and integrating over space results in the energy generation equation (1.1) discussed in the introduction.

To address the issue at hand, we specialize to the case of two equal-depth layers. Denoting the upper and lower layer eddy streamfunctions as ψ₁ and ψ₂, respectively, the two-layer system can be represented in terms of baroclinic and barotropic modes with barotropic eddy streamfunction ψ = (ψ₁ + ψ₂)/2 and baroclinic eddy streamfunction τ = (ψ₁ − ψ₂)/2. The barotropic and baroclinic eddy potential vorticities are then q_ψ = ∇²ψ and q_τ = ∇²τ − λ²τ, respectively, where λ is the internal deformation wavenumber (the inverse of the internal Rossby deformation radius). Assuming without loss of generality that the background velocities are purely baroclinic, with U = (U₁ − U₂)/2 and V = (V₁− V₂)/2, and invoking a linear vorticity drag in the bottom layer, the equations of motion for this system are

where D_ψ and D_τ are small-scale dissipation terms.

As originally noted by Salmon (1980), at horizontal length scales l ≫ λ⁻¹ the equations (ignoring drag since here we are concerned with energy generation) can be approximated as forced two-dimensional barotropic flow advecting a passive scalar with a linear mean gradient

where Fτ = −J(τ, q_τ) can be thought of as a forcing term from the baroclinic dynamics and we have defined the approximate baroclinic PV as q̃_τ ≡ −λ²τ ≃ q_τ.¹

To put this approximation to use, we first relate the energy generation rates to the PV transport. Multiplying (2.3a) by −ψ and (2.3b) by −τ and integrating each over horizontal space and summing, one obtains the energy budget for the two-layer system

analogous to (1.1). Anticipating use of the passive scalar analogy, we define diffusivities in each direction

where we have used ∂/∂x = −λ²V and ∂/∂y = λ²U. Thus the generation of eddy energy due to a gradient in the x direction is λ²V² κ_x and that due to a gradient in y is λ²U² κ_y, which demonstrates that downgradient PV fluxes indeed generate eddy energy.

In the standard case with V = 0, Held and Larichev (1996) suggest an approximate closure for the eddy energy level and scale utilizing the passive scalar analogy for the baroclinic dynamics and the phenomenology of β-plane turbulence for the barotropic flow. In particular, they assume: 1) the diffusivity (and so energy generation rate) can be approximated as κ_y ∼ υ_el_e, where υ_e is the root-mean-square meridional barotropic eddy velocity and l_e is the meridional eddy scale; 2) the horizontal eddy scale is approximately the Rhines scale l_e ∼ (υ_e/β)^1/2; and 3) the upscale energy flux, eddy scale, and eddy velocity are related by Taylor’s estimate v³_e/l_e ∼ κ_yλ²U². Together these lead to the estimates υ_e ∼ λ²U²/β, l_e ∼ λU/β, and κ_y ∼ λ³U³/β².

In the special case that U = 0, Spall (2000) alters the Held and Larichev theory by assuming that, while barotropic eddies still have a typical meridional scale given by the Rhines scale, the mixing length l_e instead scales with the zonal width of the domain L_x. Spall motivates this choice by arguing that baroclinic potential vorticity can develop arbitrarily long zonal anomalies without being influenced by β. Using the definitions above, this amounts to assuming 1) κ_x ∼ υ_eL_x, 2) υ³_e/l_e ∼ κ_xλ²V², and, as before, 3) l_e ∼ (υ_e/β)^1/2. The predicted estimates are thus υ_e ∼ (λ⁴V⁴L²_x/β)^1/3, l_e ∼ (λ²V²L_x/β²)^1/3, and κ_x ∼ L_x(λ⁴V⁴L²_x/β)^1/3.

The dramatic increase of eddy energy with increasing β found by Arbic and Flierl (2004a), as well as the fact that a steady state can be obtained without boundaries (L_x → ∞), demands that we seek an alternate mechanism for the zonal eddy length scale. Returning to the passive scalar analogy, which should hold all the more at high eddy energy levels, energy generation in the U = 0 case results from the passive transport of eddy baroclinic PV by the jet-dominated barotropic eddy flow u_ψ. Following Taylor (1920), one can relate the eddy transport, or effective eddy diffusivity, to the velocity correlation. In a perfectly smooth jet, the velocity correlation between two neighboring particles (in the cross-stream direction) would grow linearly with time to infinity, so the diffusivity would diverge. When the jet is embedded in a turbulent background, as in the present problem, there is some length scale over which particles will be swept apart by the jet, but then decorrelated by the turbulent background, and this is the effective mixing length. The effective eddy diffusivity in the along-jet direction will thus increase as the background turbulent diffusivity decreases. This is the shear-dispersion problem of Taylor (1953) (see also Young et al. 1982).

Smith (2005) considered the system (2.3) with F_τ replaced by a small-scale random forcing (with total upscale energy generation g) and derives predictions for the diffusivities κ_x and κ_y in β-plane turbulence. It is shown therein that the best estimate for the background turbulent diffusivity (the cross-jet diffusivity κ_y) is the Held and Larichev estimate (see also Smith et al. 2002; Barry et al. 2002)

The along-jet effective diffusivity is found by considering a tracer χ with a zonal mean gradient stirred by β-plane turbulence, and approximating its dynamics with an advection–diffusion equation for a large-scale, periodic, smooth jet flow u = U_jet(y)i and a turbulent diffusivity κ_y, is written

where we have used the linearity of the tracer equation to set ∂χ/∂x = 1 without loss of generality. The equilibrated along-jet diffusivity κ_x = lim_t→∞ − takes on the constant value

where Û_n are the Fourier components of U_jet, and k_n = nπ/L are the wavenumbers, such that

The derived along-jet diffusivity increases with decreasing κ_y and in fact diverges in the limit κ_y → 0, in support of the expectations from the thought experiment.

The next obvious thing to do is incorporate this result into the scaling for the energy generation in the closure theory by assuming a sinusoidal jet flow. Before proceeding, we test the underlying idea of a shear-dispersion energy generation mechanism in a set of numerical experiments.

3. Numerical experiments

In this section the results of a set of numerical simulations using a standard two-layer spectral quasigeostrophic model are described. The model has equal layer depths and a maximum resolved wavenumber of k_max = 127. Nonlinear terms are calculated using the method of (Orszag 1971), with isotropic truncation at (8/9)k_max, and physical space fields are represented on a 256² grid. The enstrophy cascade is absorbed by an exponential cutoff filter that acts only on wavenumbers k > k_cut = 90 [though the effective scale at which dissipation becomes active is even smaller; see Smith and Vallis (2002), Smith et al. (2002), for model details]. The domain is 2π periodic, so that the k = 1 wave fills the domain, and the deformation wavenumber is fixed at λ = 40 for some runs and λ = 60 for others. All simulations have U = 0 and V = 0.1 (nondimensional units) so that the Eady time scale is (Vλ)⁻¹ = 0.25. The control parameters explored are the nondimensional Coriolis gradient β̃ = β/(λ²V) and the nondimensional bottom drag r̃ = r/(λV).

As discussed originally by Haidvogel and Held (1980), in the homogeneous limit, these are the only parameters of the problem. Each simulation was first spun up to steady state and, at that point, a passive tracer was added to each simulation. The passive tracer was stirred only by the barotropic velocity, and its variance was generated by a linear, meridional mean gradient. The tracer acts as an independent measure of the meridional diffusivity κ_y. Specifically, the modeled tracer is

where u_bt is the barotropic velocity and D_ϕ is the dissipation (modeled using an exponential cutoff filter similar to that for vorticity), so the diffusivity in the meridional direction is

where, as with the equation for χ, we have set ∂/∂y = 1.

Thirty simulations were completed and included in the analysis with the intention of covering a wide swath of r̃–β̃ parameter space. An even larger set would have been performed, and at higher resolution, but for the fact that in the most interesting part of parameter space (r̃ < 1), up to 2 × 10⁷ time steps were required to achieve equilibration for some simulations, and no simulation in the set took less than 10⁶ timesteps.

b. Scaling theory

To close the problem, we need independent predictions for the two diffusivities, the eddy length scale and the eddy velocity. We will consider each assumption separately and test them against the numerical results where possible.

1) Diffusivity predictions

First, we can immediately test the Held and Larichev prediction (2.4). Figure 6a plots κ̃_y against the predicted value, using the generation rate produced by the model for each simulation. The fit is not exact but demonstrates that the scaling is roughly correct, consistent with past studies. The values calculated from the numerical experiments also show that the meridional diffusivities are many orders of magnitude smaller than the zonal diffusivities, rationalizing the neglect of zonal mean shear in this research. Simulations with both flow components present have been performed, but are not included here—these simulations demonstrate that the contribution of zonal flow instabilities is negligible in the small- to medium-drag limit.

To make a closed prediction for the zonal diffusivity, we consider the simplest possible jet structure

where α is an arbitrary constant phase, Û₀ is the maximum jet speed, and k_jet is the meridional wavenumber of the jet. The along-jet diffusivity (2.6) in this case simplifies to

where U_jet = ()^1/2 = Û₀/2^1/2. We can check this estimate directly against the model results by estimating U_jet as the zonal and time average of the zonal velocity and again using the values of κ_y measured from the passive tracer, given by (3.1). The results are plotted in Fig. 6b. Particularly for larger values of κ̃_x, this estimate does a remarkably good job at describing the results.

Substituting the expression (2.4) for κ_y into (3.2), setting g = λ²V²κ_x, and solving for κ_x we have the nondimensional result

This prediction is plotted against the measured zonal diffusivity in Fig. 7. Apart from the simulation with the smallest value, this expression is better than even the raw calculation of (2.6) shown in Fig. 5. The author can find no apparent reason that expression (3.3), in which κ_y has been eliminated using the Held and Larichev prediction and the assumed structure of the jets has been simplified, should fit the data so well.

2) Energy constraint

The presence of linear drag on the bottom layer vorticity affords us a constraint on the total energy generation

where we have neglected the small-scale dissipation term. Thompson and Young (2006) derive a new conservation law for the cross term, which in steady state implies that³ . Thus to a good approximation we can write

Since all of the simulations satisfy the assumption of large scales (compared to the deformation scale), we can safely neglect the baroclinic kinetic energy term and write

The two sides of this equation are plotted against one another in Fig. 8. The fit is remarkably good.

3) Separating jets from turbulence

To apply the shear-dispersion generation prediction, we need to separate the jets from the turbulence in the barotropic mode and so write

At this point we are still in good shape, but now must make independent estimates of the two terms on the left-hand side. Smith et al. (2002) show that the turbulent energy of two-dimensional, β-plane flow in the small-drag limit is nearly independent of drag. Smith [2005, (4.3)] uses this result to make the estimate

where C ≃ 6 is a Kolmogorov constant. The numerical value of the turbulent barotropic energy for each simulation is plotted against this estimate in Fig. 9. Apparently this is not particularly accurate, perhaps because of the assumption of independence from drag (which is harder to justify in the present baroclinic case), and things only get worse from here.

One can attempt to relate the jet energy to the jet scale via the Rhines relation, calculated as in Smith et al. (2002) as the integral of the putative −5 spectrum (see, e.g., Chekhlov et al. 1996; Smith et al. 2002)

where a ≃ 0.1. Considering Fig. 10, for the system at hand this is the poorest approximation yet. Figure 10a plots the nondimensionalized version of (3.8) directly, and Fig. 10b plots 1/k_jet against the Rhines scale β/U_jet. Neither a plausible explanation of this result nor a better theory for the jets is apparent to the author.

4) Closure

Despite the imperfections of the last two relations, one might attempt to put the above relations together to close for the diffusivity, or eddy energy generation rate. However, if one puts the relations together, the resulting prediction actually has the generation decreasing with increasing β, just the opposite of what is found in the simulations. Therefore, given the accuracy of the predictions for the diffusivities and for the barotropic energy budget, the Rhines relationship cannot be correct in this particular system.

Moreover, for simulations with small drag, which have their energy near the domain scale, it seems that the unavoidable conclusion, given that in this regime the stopping scale is nearly constant with β, is that the cascade cannot be controlled by β and that some additional mechanism is necessary. Given that the available potential energy is always at the domain scale in these simulations, even though the kinetic energy is not, perhaps thermal drag would close the system. This is left for future work.

Note in passing that, if one assumes small drag and takes k_jet to be constant, the closed solution does yield a diffusivity and a jet velocity that increase with β, as observed. In lieu of a complete closure or understanding of the jet dynamics, this is however not very useful and so is not included.

4. Discussion and conclusions

Numerical simulations of the two-layer model with a mean background shear directed along the mean vorticity gradient (the pure nonzonal flow problem) demonstrate a host of behaviors that are qualitatively different from the zonal mean flow problem to a remarkable degree. The root of the distinction between the two limits is the nature of the eddy generation, and in particular its interaction with jets that arise in the flow due to β. Nonzonal mean shear leads to eddy generation driven by correlations with the zonal eddy velocity and the baroclinic potential vorticity. When jets are present, this eddy flux is best described by a shear-dispersion model. The result is primarily that the eddy energy in the nonzonal flow case is orders of magnitude larger than that in the zonal flow problem and, moreover. that the energy and the generation rate increase with increasing β, just the opposite of the behavior in the zonal flow case.

When the drag parameter r/λV is small compared to unity, a strong feedback arises in the jet scale. On the one hand, increasing β leads to faster Rossby waves, so for fixed eddy turn-around time jets should form at smaller scale. But the turbulent time scale is not fixed since as β is increased the eddy generation and hence energy increase, driving a faster turbulent time scale and a more vigorous inverse cascade. These two effects work against each other and lead to a balance at some length scale that, at least in the presently discussed simulations, is apparently invariant with β. It could be the case that an even larger separation between the domain scale and the deformation scale would lead to an observed variation of the jet-scale with β at low drag since the former turns out to be so near the domain scale in the present simulations. However, the same results are obtained when the deformation wavenumber is increased from 40 to 60 at r̃ = 0.3, so this seems unlikely.

One could interpret the lack of separation between the eddy and domain scale in the small-drag limit as supporting the theory proposed by Spall (2000). Alternatively, one could ask whether sufficiently high drag could yield a local limit in Spall’s calculations. Most of the simulations presented by Spall used a sponge layer at the western boundary to dissipate energy, but in one set a weak linear bottom drag is also included. The result is that the overall eddy energy is reduced, and the scaling theory provides a less accurate description at high eddy energies, implying some dependence on drag not accounted for by the theory. Therefore, one might conclude that drag and domain effects are at least in competition in his experiments. Using (3.2), we can compute the effective zonal eddy length scale. Setting κ_x = l_eu_e, and taking u_e = U_jet, we have that l_e = U_jetL²_jet/κ_y. Using ballpark estimates U_jet ∼5 cm s⁻¹, L_jet = 300 km, and κ_y = 1000 m² s⁻¹ we have that l_e ∼2200 km, very similar to the domain scale L_x used in Spall (2000).

The incredibly effective eddy energy generation mechanism found in the present idealized problem should lead one to question whether this effect is present in the ocean. Observed midocean eddy velocities are much stronger than mean shears, but not by two or more orders of magnitude. Either bottom drag must not be small, as hypothesized by Arbic and Flierl (2004a), or some other dissipation mechanism must act to quell eddy energy. Quadratic bottom drag is a more truthful representation of the effects of a bottom boundary layer on the interior flow and is also naturally scale-selective (Grianik et al. 2004). One could also argue that a direct sink on eddy available potential energy, such as thermal drag, should be included. Unbalanced flows have also been entirely neglected.

A full closure for the problem was not obtained, due to the difficulty in finding a suitable theory for the jet scale—the Rhines scale is a poor predictor of the jet scales that evolved in these simulations. At small drag, this is likely related to the feedback discussed above, but the Rhines scaling is also inaccurate at large drag. It may not be a worthwhile pursuit to attempt to close the present system since, as pointed out, the energies are too large to be relevant to the ocean without a very strong dissipation mechanism. Moreover, the energy generation is so large that surely the instability will not remain localized and so the homogeneous, local limit may not apply.

The aspect of the problem that is potentially most useful for future research is the unique eddy energy generation mechanism demonstrated here. Regardless of their genesis and maintenance, jets and elongated eddies lead to shear dispersion of baroclinic potential vorticity that, in the presence of even a weak cross-jet mean shear, leads to intense eddy energy growth.