a. Background

The observed decadal variability in the midlatitude ocean and atmosphere (e.g., Wallace 2000; Deser and Blackmon 1993; Kushnir 1994) can be explained in terms of either intrinsic variabilities of the ocean and atmosphere or as a coupled ocean–atmosphere phenomenon. Coupled ocean–atmosphere general circulation models (GCMs) do not yet discriminate between these alternatives: some models suggest that the coupling is important (e.g., Latif and Barnett 1994) while the others do not (e.g., Pierce et al. 2001). It is thus sensible to analyze idealized coupled and uncoupled models in order to get insight into potential mechanisms of the observed low-frequency variability. Recent advances in understanding coupled dynamics have been made with midlatitude, quasigeostrophic models that resolve mesoscale turbulence in both the oceanic and atmospheric components (Hogg et al. 2003; Kravtsov et al. 2006, 2007; Hogg et al. 2005, 2006). Kravtsov et al. (2006, 2007) have shown that atmospheric transitions between the pair of statistically preferred states and the nonlinear adjustment of the ocean (Dewar 2003) to these states can be the essential mechanisms of decadal coupled variability. An important point is that the oceanic adjustment is controlled by the mesoscale eddies. If the eddies are parameterized in terms of eddy viscosity, coupled variability is suppressed. This is presumably because the physical process, by which the eddies maintain sea surface temperature anomalies feeding back on the atmosphere, is not fundamentally a viscous one.

Although the role of oceanic mesoscale eddies in shaping the large-scale ocean circulation has been known for a long time (e.g., Holland 1978), accurate mathematical models (i.e., parameterizations) of the eddy effects are not yet generally available for use in non-eddy-resolving GCMs. The most common eddy parameterization, which in different forms is present in virtually all GCMs, is eddy viscosity (Taylor 1921). However, when the large-scale currents are enhanced rather than smoothed by the eddies, such parameterizations can be incorrect. Such situations are common in geophysical flows (Starr 1968). Progress in simulating “negative viscosity” has been made in terms of random forcing added to an otherwise non-eddy-resolving ocean model (Berloff 2005b). This approach falls into a broad class of fluid dynamic models in which the small-scale nonlinear interactions are replaced by stochastic processes (e.g., Herring and Kraichnan 1971). Other eddy models are based on representation of the eddies by ensembles of particles advected by and interacting with the large-scale flow (Laval et al. 2003), or on the assumption that the eddies are passively advected by the large-scale flow (Holm and Nadiga 2003).

b. Statement of the problem

The main goals of the present paper are to diagnose and understand the oceanic mesoscale eddy effects in an idealized ocean–atmosphere coupled model (Kravtsov et al. 2006, 2007) and to develop a stochastic parameterization of these eddy effects for use in a coupled model with a non-eddy-resolving ocean component. The following specific steps are made. First, the history of the eddy forcing is diagnosed from the eddy-resolving solution and then analyzed statistically and dynamically. The mechanism of the eddy/mean flow interaction is uncovered, and the effects of the eddies on the large-scale flow are separated from the effect of the wind. Statistics of the eddy forcing are treated as nonstationary, because the coupled solution is characterized by significant decadal variability.

Next, in a non-eddy-resolving ocean component of the coupled model, eddy forcing is simulated as a stochastic, space–time correlated process with spatially inhomogeneous variance. It is found that imposed stationarity of the random-forcing variance suppresses the low-frequency variability but has small impact on the climatology. On the other hand, the low-frequency variability is recovered if the parameterization allows for nonstationarity of the eddy-forcing variance. This is achieved by relating the variance to the large-scale flow, and amounts to a simple parameter closure. The random forcing is ultimately related to the large-scale flow, thus yielding a closed stochastic model of the eddy effects.

The problem is posed so that statistics (i.e., parameters) of the random-forcing parameterization come from the reference eddy-resolving solution. Here, the goals pursued are to identify the appropriate mathematical model for and the relevant physics behind future parameterizations. A related, if distinct, result of this paper is the illustration of a systematic strategy for developing eddy parameterization that can be employed in a hierarchy of eddy-resolving models, from idealized to comprehensive. Our ultimate result is close to an internally closed parameterization in that the implementation of the parameterization is related in a self-consistent way to a measure of the ocean state. On the other hand, we stop short of developing a fully closed parameterization because details of the parameterization are diagnosed from an analysis of an eddy-resolving run. It is our hope ultimately to complete the closure, but in these early days of experimentation with stochastic closures, the demonstration of their utility represents a significant contribution.

c. Coupled model

The coupled model consists of a closed rectangular 5120 km × 5600 km ocean basin and an overlying zonally periodic 20 480 km × 6400 km atmospheric channel on the beta plane (Kravtsov et al. 2006). Both oceanic and atmospheric components are quasigeostrophic, with two isopycnal layers (H_1a = 3000 m and H_2a = 7000 m) in the atmosphere and three isopycnal layers (H₁ = 300, H₂ = 700, and H₃ = 3000 m) in the ocean. The interior ocean and the atmosphere are coupled through a 50-m-deep oceanic mixed layer, in which sea surface temperature (SST) is advected by both geostrophic and Ekman currents. The SST and the atmosphere are connected by surface heat exchange. The atmospheric model’s parameters are as in experiment 2 of Kravtsov et al. (2007, their Table 1).

The ocean model’s potential vorticities Q_i (index i = denotes the layers) are given by

where Ψ_i are the corresponding isopycnal-layer velocity streamfunctions, and β = 2 × l0⁻¹¹ m⁻¹ s⁻¹ is the planetary vorticity gradient. (The most common symbols are cited in Table 1.) The stratification parameters are

where g′₁ = 0.042 m s⁻² and g′₂ = 0.022 m s⁻² are reduced gravities, associated with the density jumps across the upper and lower internal interfaces between the isopycnal layers, and f₀ = 10⁻⁴ s⁻¹ is the Coriolis parameter. Given (4), the first and second internal Rossby radii are equal to 46 and 26 km, respectively. Evolution of the potential vorticity anomalies is governed by

where J(A, B) ≡ A_xB_y − A_yB_x is the Jacobian operator, H_bot = 30 m is the thickness of the bottom boundary layer, and ν = 200 m² s⁻¹ is the horizontal viscosity coefficient. The forcing consists of a time-dependent Ekman pumping W_E and heat exchange W_D between the ocean interior and mixed layer (the latter is a crude parameterization of the meridional overturning circulation). Both of these quantities are diagnostically related to the states of the atmosphere and mixed layer. Equations (5)–(7), subject to partial-slip conditions on the lateral boundaries and mass conservation (McWilliams 1977), are discretized on a 513 × 561, 10-km-resolution grid and integrated using leapfrog time stepping with a 20-min interval.

a. Climatology and low-frequency variability

The eddy-resolving coupled solution, calculated over 400 yr, is referred to as the full reference solution. Its time-mean structure is shown in Fig. 1. The ocean is dominated by subtropical and subpolar gyres of unequal strengths. The gyres are separated by the eastward-jet extension of the pair of merging western boundary currents. The axis of the eastward jet coincides with a sharp SST front. With depth, the gyres weaken and retreat to the western boundary. There is also a third, southern gyre, but it is relatively weak. The atmospheric component of the model is dominated by an eastward jet with a reasonable storm track and meridional temperature profile. In the ocean, the time-mean Ekman pumping, W_E, is strongly dependent on longitude as well as latitude (Fig. 2). The time-mean thermodynamic forcing, W_D, is set to zero through iterative corrections (Kravtsov et al. 2006).

The low-frequency variability of the reference solution around its time-mean state is analyzed with empirical orthogonal function (EOF) decomposition (Preisendorfer 1988). It is found that the atmospheric variability is dominated by zonally shifting winds (Fig. 3), and the corresponding leading EOF accounts for about half of the variances of W_E and W_D. The three leading oceanic EOFs contain about ⅔ of the total oceanic variance, with the first EOF accounting for nearly half (Fig. 4). The first EOF describes irregular but roughly decadal transitions of the ocean between a more preferred low-latitude state and a more transient high-latitude state. These states correspond to positive and negative values of the first principal component (Fig. 4d). The most dramatic transitions to the low-latitude state are associated with shifts of the eastward-jet axis by a few hundred kilometers to the south. As shown by Kravtsov (2007), the transition time scale is set by the nonlinear adjustment of the ocean. The principal components of the second and third EOFs are time-lag correlated; thus they describe a westward propagating signal with a 4-yr period. This signal is identified in Kravtsov et al. (2006, 2007 to be a first baroclinic basin mode amplified by coupling with the atmosphere (Goodman and Marshall 1999).

b. Diagnostics of the eddies

The dynamical eddy analysis (Berloff 2005a), generalized to allow for time-dependent external forcing of the ocean, is now applied. The reference ocean solution is projected on a uniform 129 × 141, 40-km-resolution grid by subsampling all three layer streamfunctions in both space and time, the latter at an interval of 5 days. The goal is to decompose the projected reference solution, denoted by ψ_i and q_i, as

where overbar and asterisk denote large-scale and eddy components of the flow, respectively. Here, potential vorticity components are found from the streamfunction according to (5)–(7). Continuous time series of both ψ_i and the forcing, W = (W_E, W_D), are obtained by cubic interpolation in time. The history of W is then used to force a non-eddy-resolving ocean model, which has the same form as (5)–(7), with the addition of an eddy-forcing term as discussed below. The non-eddy-resolving model is discretized on the coarse 40-km grid; it has a horizontal viscosity¹ ν = 4000 m² s⁻¹, and it is integrated using an 80-min time step. The additional eddy-forcing term is calculated interactively, on each time step of the non-eddy-resolving model, in terms of the difference between the non-eddy-resolving and the projected reference solutions. In each isopycnal layer, the eddy forcing is found as

where J( ) and ∇ are coarse-grid operators, u_i is the isopycnal-layer velocity vector, and F_i is the effective eddy potential vorticity flux. Both the eddies and the eddy forcing are found simultaneously and interactively by integrating the eddy-corrected non-eddy-resolving model. The large-scale component of the reference flow, , is defined as the non-eddy-resolving solution, and the eddy component, ψ*_i, is automatically found as the difference between ψ_i and . Thus, the mesoscale eddies are viewed as flow fluctuations projected on the coarse grid and dynamically accounted for by the eddy forcing. Here, the non-eddy-resolving model is initialized with = ψ_i(0), and integrated for 400 yr. The fields of , ψ′_i, and f_i are saved every 5 days.

The eddy-corrected non-eddy-resolving model captures main features of the reference solution very well (Fig. 5). When the atmospheric jet is predominantly in the low-latitude regime, the eddy forcing helps to spin up a narrow and intense oceanic eastward jet—this state is referred to as the low-latitude state of the ocean. The atmospheric high-latitude regime corresponds to the weakened oceanic eastward jet in the high-latitude state. Both states are characterized by the presence of many coherent eddies, which are analyzed in the next section.

a. Role of the eddies in climatology

The gross effect of the eddies on the ocean climatology is estimated by comparing the reference solution and the solution of the non-eddy-resolving model without the eddy forcing (Fig. 6). The latter solution has a significantly weaker eastward jet, a stronger cyclonic circulation in the northern part of the subpolar gyre, and a weaker cyclonic circulation near the southern boundary. The absence of the eastward jet results in suppression of the coupled variability (Kravtsov et al. 2006, 2007). This indicates the fundamental importance of the eddies and motivates a more detailed analysis of the eddy/large-scale flow interaction.

The eddy forcing, f_i(t, x, y), diagnosed in section 2, is decomposed into time-mean, 〈f_i〉(x, y), and fluctuating, f ′_i(t, x, y), components. The dynamical roles of these components can be recovered by analysis of the non-eddy-resolving ocean-only solutions forced by these components. The ocean solution forced by W(t, x, y) and no f_i (Fig. 6b) has a climatology similar to that of the coupled non-eddy-resolving solution without eddy-forcing correction. The benchmark solution, which is driven by the time-interpolated histories of both W and f_i (Fig. 7a), is the closest approximation to the eddy-corrected non-eddy-resolving ocean solution obtained without the interpolation. Removing either 〈f_i〉 or f ′_i results in a substantially different flow pattern (Figs. 7b,c). When W is imposed, effects of the eddy forcing are masked by the wind-driven gyres, but they become more evident when W is turned off (Figs. 7d–f). The strongest eddy effect is due to f ′_i, which is surprising because traditional analysis of the eddy effects focuses on 〈f_i〉. The f ′_i effect is due to the nonlinear rectification of the flow (e.g., Haidvogel and Rhines 1983; Feliks et al. 2004; Berloff 2005b, c) by a spatially inhomogeneous eddy forcing discussed in section 3b. An important property of f ′_i is its variance (Fig. 8):

Here, it has large values in the upper ocean, and in the western boundary currents with their eastward-jet extension.

The main upper-ocean contribution of the 〈f〉 effect (i.e., the time-mean eddy-forcing effect) is spinning down the gyres and weakening the western boundary currents, but it is also responsible for meandering of the eastward jet (Figs. 7b,e). The former effect, which is particularly strong in the subpolar gyre, is consistent with the baroclinic instability of the westward return currents of the gyres and the western boundary currents. In terms of spinning down the gyre, the f-driven response is weaker than the sum of the 〈f〉 and f ′ responses, but because of the nonlinearity one should not expect any linear summation here. This nonlinear upper-ocean effect is consistent with the idea that fluctuations driven by f ′ tend to homogenize potential vorticity anomalies induced by 〈f〉, and thus weaken the flow response.

Analysis of the eddy forcing is extended by decomposing f_i into its relative vorticity R and buoyancy (i.e., isopycnal thickness) B components:

where buoyancy is

and

The time-mean forcings, 〈R_i〉 and 〈B_i〉, induce qualitatively different flows (Fig. 9). The 〈R〉 forcing is mainly responsible for driving standing eddies that cause meandering of the eastward jet, and its effect on the gyres is small. On the contrary, the main effect of the 〈B〉 forcing is weakening of the upper-ocean gyres, whereas its role in shaping the eastward jet is modest and opposite to that of the 〈R〉 forcing. The 〈B〉-induced pattern is consistent with the baroclinic instability of the westward flows of the gyres, and corresponds to the gyre spin down. Overall, in the eastward-jet region, the buoyancy-forcing effect is weaker than in Berloff (2005a), and the difference can be attributed to the differences in boundary conditions [no slip in Berloff (2005a) versus partial slip here] and stratification. The upper- and second-layer eddy-forcing variances are dominated by the R- and B-term contributions, respectively. (The second- and third-layer eddy effects are found to be qualitatively similar; therefore discussion of the deep ocean is focused on the second layer only.) This is so because the upper ocean is dominated by relatively short scale instabilities of the sharp potential vorticity front associated with the eastward jet, whereas the deep ocean is driven mostly by fluctuations in layer thicknesses, which are characterized by larger scales.

The above conclusions about the eddy-forcing components and their dynamical effects are supported by calculating the potential Π and kinetic K energy conversion terms in the conventional way (e.g., Berloff and McWilliams 1999):

Here, the angle brackets denote ensemble averages and the primes denote fluctuations around these averages. The ensembles on which the averages formed are collected by conditional sampling of the model results as described below. The conversion terms are the corresponding energy exchange rates between the mean (i.e., ensemble average) state and the fluctuations, which are due to the work done either by isopycnal form stresses (Π) or by horizontal Reynolds stresses (K). Barotropic instability (i.e., extracting fluctuation energy from the mean horizontal shear) is associated with a positive K, and baroclinic instability (i.e., extracting fluctuation energy from the mean vertical shear) corresponds to a positive Π. Conditional sampling is performed by defining low- and high-latitude states as occurring when the normalized principal component (PC) of EOF1 exceeds +1 and −1 respectively. Through ensemble averaging,² we find the energy conversions in the low- and high-latitude states (Fig. 10), and the former conversion pattern is qualitatively similar to the one calculated with respect to the global time average of the flow. In both states, the basin-averaged values of Π and K, that is, Π_av and K_av, are positive, thus indicating that the fluctuations feed on the mean state. In the low-latitude state, Π_av ≈ K_av, which indicates that the baroclinic and barotropic instability mechanisms are equally efficient in generating the eddies. However, the former dominates in the return westward flows. In the high-latitude state, Π_av increases by 80% and K_av increases by 20%. Thus, the high-latitude state generates eddies more efficiently and mostly through the baroclinic instability mechanism. Negative values in the spatial distributions of Π reveal that in the eastward-jet region the fluctuation energy returns to the mean flow, which is consistent with the mean flow rectification driven by the eddy forcing. The negative values of Π are more negative and more confined to the eastward jet in the low- rather than high-latitude state. The spatial distributions of K are more confined to the boundaries, and they exhibit no broad negative regions, thus indicating that fluctuations transfer kinetic energy to the mean.

The importance of the vertical structure of the eddy forcing is addressed by calculating the non-eddy-resolving solutions driven by either f ′₁ or f ′₂. Without the upper-layer eddy forcing, the eastward-jet enhancement does not emerge; hence, this enhancement is mainly an upper-ocean phenomenon. Without the middle-layer eddy forcing, the eastward jet is noticeably exaggerated in the upper layer. This is so because the eddy buoyancy flux fluctuations tend to oppose each other in the first and second layers; hence there is some mutual cancellation between their large-scale responses. Thus, given the coarse and idealized stratification, the vertical structure of the eddy forcing is only moderately important.

The middle-layer flows driven by the eddy-forcing components 〈f〉, 〈f₂〉, f ′₁, and f ′₂ are shown in Fig. 11. The 〈f〉-induced flow, which is noticeable in the northwestern corner of the basin and around the eastward jet, and the f ′₁-induced flow are both due to the baroclinic instabilities of the eddy-induced upper-layer circulation patterns. The time-mean gyres of the 〈f₂〉-induced flow are the dominant components of the eddy-driven deep-ocean circulation. They are associated with the baroclinic instability that spins down the upper-ocean gyres but spins up the deep ocean. The middle-layer fluctuations of the eddy forcing are relatively small; therefore the rectified flow consists of several zonally elongated recirculation cells (Fig. 11c) rather than a recirculation dipole enhancing the eastward jet (Berloff 2005c). To summarize, the roles of all the eddy inputs are comparable in structuring the middle-layer mean flow.

b. Dynamical mechanism of the eddy–mean flow interaction

Fluctuating eddy forcing with zero time mean is capable of driving rectified (i.e., nonzero time mean) large-scale flows, as shown originally in laboratory experiments (Whitehead 1975). Two regimes of rectification are relevant to the double gyres, and both of them result in formation of zonal currents (Rhines 1975, 1994). Consider the potential vorticity dynamics forced by localized zero-mean external forcing, Φ(t, x), with positive amplitude A:

Integrating (16) over a short time and multiplying by the flow velocity yields

At the latitudes outside of the localized forcing, taking into account that υ = δy/δt and averaging over an ensemble of realizations (denoted by angular brackets) reveals

Assuming irreversible³ Lagrangian dispersion implies the asymptotic diffusive limit, in which 〈y²〉 ∼ t. In this limit, the rhs of (18) is negative; hence, the ensemble-average meridional flux of q, which is equivalent to the Eulerian flux of q, is negative. This flux results in the westward acceleration of the background ensemble-average flow, as can be seen from the zonally averaged vorticity equation:

The resulting flow regime is due to the meridional mixing and partial homogenization of the absolute potential vorticity.

The second regime occurs when the forcing is sufficiently strong and υ〈∫Φ dt〉 > 0, which is a typical phenomenon for the β-plane dynamics (16) in the near-Sverdrup regime. An argument for the positive correlation is the necessity to have a countercurrent that closes the westward current and enforces the conservation of mass. Thus, in the second regime the meridional q flux is positive; hence, it drives the eastward background current. An alternative explanation (Thompson 1971) is that eastward jet is driven by convergence of the Reynolds stress associated with Rossby waves radiating away from the forced latitudes.

In the double-gyre ocean, both regimes are present simultaneously because the eddy forcing is strong around the eastward-jet axis but weak elsewhere. Therefore, the rectified flow response consists of the cyclonic/anticyclonic recirculations to the north/south of the eastward jet. The underlying eddy/mean flow interaction is characterized by the following ingredients. First, as suggested by the energy conversion arguments (section 3a), the eddies are generated by instabilities in the western boundary currents and the upstream part of their eastward-jet extension. Then, the eddies are advected downstream by the background flow. Along the eastward jet, and particularly near its eastern end,⁴ the eddies backscatter into the enhancement of the large-scale eastward jet and its adjacent recirculation zones through the rectification mechanism described above.

It is useful to compare the double-gyre eastward-jet dynamics with the classical situation of a zonal, baroclinically unstable eastward jet (e.g., Vallis 2006). In the latter case, the time-averaged equation for the enstrophy, E′ = q′²/2, can be written as

where D′ represents dissipation, and angular brackets and prime denote conventional definitions of the time mean and fluctuations around it. In the stationary case, the tendency term disappears. Zonal averaging of (20) removes the x derivatives, and an additional integration in the meridional direction yields

In the above integral balance, the first term is positive; therefore, the second term must be negative. This implies that eddy PV flux is downgradient in the integral sense, which is also true in a closed basin. On the other hand, the downgradient property of the eddy flux is not necessarily true locally, because advection of the enstrophy enters the balance.

c. Role of the eddies in the low-frequency variability

Given that nonlinear adjustment of the oceanic eastward jet is an essential part of the coupled variability (Kravtsov 2007), and that the jet is driven mainly by the eddy-forcing fluctuations, we have made a low-frequency analysis of the eddy-forcing variance in order to relate that variance to the evolving large-scale flow. With that in mind, the total eddy forcing is decomposed into low-passed and residual high-passed fluctuations, respectively,

as defined by a 1000-day running average. The high-passed fluctuations are characterized by their finite-time variance, σ( f_i)^HP(t, x, y), calculated over the same time interval. The spatial structure and temporal variability of σ( f_i)^HP are described in terms of the EOFs and their principal components (Fig. 12). Most of the low-frequency variability is captured by the first EOF. The corresponding principal component is strongly correlated (correlation coefficient is C ≈ 0.8) with the first oceanic EOF principal component of the reference solution (section 2a). The spatial pattern of the leading EOF of the eddy-forcing variance describes the dramatic reduction of the variance associated with ocean transitions to the high-latitude state. This reduction occurs because the low-latitude state is characterized by the least energetic currents, due to the change of the Ekman pumping pattern. The second EOF corresponds to the coherent north–south shifts of the variance, and its principal component is weakly but noticeably (C ≈ 0.3) correlated with the second and third oceanic EOF principal components.

To characterize the reference solution and simply relate the eddy-forcing variance and the large-scale flow, the following indices are defined and calculated: I^A_BT, I^O_BC, I^O_BT, and I^f_L1. Here, the superscript indicates whether the index describes the ocean (O), the atmosphere (A), or the ocean eddy forcing ( f ), and the subscripts refer to the barotropic-mode streamfunction (BT), oceanic baroclinic streamfunction, ψ_BC = ψ₁ − ψ₂ (BC), or the top oceanic isopycnal layer (L1). Index I^A_BT is nondimensional latitude of the maximum of zonally averaged, barotropic zonal velocity, and it is found by cubic interpolation. Index I^f_L1 is found similarly, but for the nondimensional latitude of the maximum amplitude of the eddy-forcing fluctuations. The indices I^O_BC and I^O_BT are found by averaging the corresponding streamfunctions over the middle ⅔ of the basin (i.e., without the basin margins near the zonal boundaries), and then by removing the means from the resulting time series. All indices are normalized by their variances. The time-lag correlation function between any two indices I₁ and I₂ is defined as

where σ₁ and σ₂ are the corresponding variances.

The low-frequency variability of the ocean appears most clearly in I^O_BC, whereas I^O_BT is highly correlated with the atmosphere [C(I^O_BT, I^A_BT) = 0.9] and, therefore, noisy. The difference between these indices is set by the difference between the fast barotropic and slow baroclinic ocean adjustments to variable atmospheric forcing. Index I^O_BC lags I^A_BT by a few years, with high correlation,⁵ and it is also strongly correlated with I^f_L1 at zero lag. The latter correlation is high because I^O_BC reflects variation of the potential energy storage that feeds the eddies. In summary, the above observations make the basis for relating the eddy-forcing variance to I^O_BC—this is the basis for the parameter closure developed in section 4.