TMCnet News
Nonlinear Equilibration of Baroclinic Instability: The Growth Rate Balance Model [Journal of Physical Oceanography](Journal of Physical Oceanography Via Acquire Media NewsEdge) ABSTRACT A theoretical model is developed, which attempts to predict the lateral transport by mesoscale variability, generated and maintained by baroclinic instability of large-scale flows. The authors are particularly concerned by the role of secondary instabilities of primary baroclinically unstable modes in the saturation of their linear growth. Theory assumes that the fully developed equilibrium state is characterized by the comparable growth rates of primary and secondary instabilities. This assumption makes it possible to formulate an efficient algorithm for evaluating the equilibrium magnitude of mesoscale eddies as a function of the background parameters: vertical shear, stratification, beta effect, and bottomdrag. The proposed technique is applied to two classical models of baroclinic instability-the Phillips two-layer model and the linearly stratified Eadymodel. Theory predicts that the eddy-driven lateral mixing rapidly intensifies with increasing shear and weakens when the beta effect is increased. The eddy transport is also sensitive to the stratification pattern, decreasing as the ratio of upper/lower layer depths in the Phillips model is decreased below unity. Theory is successfully tested by a series of direct numerical simulations that span a wide parameter range relevant for typical large-scale currents in the ocean. The spontaneous emergence of large-scale patterns induced by mesoscale variability, and their role in the cross-flow eddy transport, is examined using a suite of numerical simulations. (ProQuest: ... denotes formulae omitted.) 1. Introduction One of the most fundamental, and not yet fully resolved, problems in physical oceanography concerns the dynamics of mesoscale variability and its impact on larger-scale motions. The abundance of eddies in the ocean and the significance of eddy-driven transport of buoyancy, momentum, and energy are now firmly established. Eddies have been shown to affect the structure of the thermocline (Rhines and Young 1982; Radko and Marshall 2004; Henning and Vallis 2005), the strength of the meridional overturning circulation (Nikurashin and Vallis 2011; Radko and Kamenkovich 2011; Wolfe and Cessi 2010, 2011), and biogeochemical processes in the ocean (e.g., McGillicuddy et al. 1998). Numerous studies of eddy-driven transport have focused on the Antarctic Circumpolar Current (ACC) of the Southern Ocean- an area where mesoscale variability is both intense and plays the zero-order role in large-scale dynamics (e.g., Marshall and Radko 2003). The spontaneous formation of long-lived coherent nearly zonal jets in the ocean is yet another example of large-scale phenomena induced by mesoscale variability (Hogg andOwens 1999; Maximenko et al. 2005; Kamenkovich et al. 2009; Berloffet al. 2009; 2011). Parallel developments in the meteorological context (e.g., Larichev andHeld 1995; Frisius 1998; Farrell and Ioannou 2009) underscore the universal nature of the problem and its broad geophysical significance. Many previous studies of eddy-induced mixing were largely motivated by the interest in developing adequate parameterizations ofmesoscale eddies in coarse-resolution numerical models (Gent and McWilliams 1990; Visbeck et al. 1997; Killworth 1997, 2005; Eden 2011; Marshall et al. 2012). Such parameterizations are still required for multicentury climate simulations (e.g., Stouffer et al. 2006; Meehl et al. 2007). However, it appears that continuous advancements in computing capabilities may bring eddyresolving climate simulations into the mainstream in the not-too-distant future. While the incentive to parameterize mesoscales will probably carry less weight in the upcoming era of ocean modeling, the more challenging and rewarding task of explaining eddy dynamics from first principles remains as pressing as ever. The primary source of mesoscale variability in the ocean is the baroclinic instability. While the linear theory of baroclinic instability is well developed and generally understood (Eady 1949; Phillips 1951), intriguing questions arise with regard to the saturation of its linear growth and equilibrium levels of the resulting eddy activity. The equilibration problem is complicated and fundamentally nonlinear. Although the general analytical description of the finite-amplitude baroclinic instability is still lacking, several cases have already been successfully treated (e.g., Benilov 1993, 1995a,b). Weakly nonlinear models (Pedlosky 1970, 1971, 1981; with numerous extensions) are particularly illuminating in terms of explaining equilibration mechanisms. Perturbation expansions that are focused on the slightly supercritical regime reduce the governing equations to much simpler asymptotic models, which allow direct physical interpretation of the processes at play. For instance, the weakly nonlinear theory reveals the tendency of eddyinduced fluxes to reduce the mean vertical shear, which has a stabilizing effect on the baroclinic system. The price paid for the analytical tractability of weakly nonlinear models is their limited accuracy beyond the region of marginal instability. Therefore, several studies attempted to describe key properties of large-amplitude baroclinic instability by focusing on simple scaling laws, usually derived based on energy arguments, and supporting numerical simulations (e.g., Larichev and Held 1995; Held and Larichev 1996; Frisius 1998; Lapeyre and Held 2003; Thompson and Young 2006, 2007). An important effect analyzed in these studies involves the interaction between barotropic and baroclinic modes, the former (latter) characterized by the upscale (downscale) cascade of energy. The barotropic cascade is ultimately arrested by the beta effect and/or bottom drag, and the equilibration mechanisms associated with the nonlinear baroclinic/barotropic interactions can, in certain cases, control the eddy-induced transport in fully developed instability. A distinct class of phenomenological models seeks to bring additional insight into the dynamics of eddy-induced mixing by representing the turbulent ocean by a regular array of eddies. The specific spatial pattern of individual vortices is prescribed a priori, and then an attempt is made to evaluate the cumulative transport characteristics of this array and its impact on larger scales of motion (Spall and Chapman 1998; Manfroi and Young 1999, 2002; Radko 2011a,b). Techniques of multiscale analysis [reviewedmost recently by Mei and Vernescu (2010)] are particularly suitable for the analytical treatment of largescale eddy-driven effects (Gama et al. 1994; Novikov and Papanicolaou 2001). Models of this nature offer deterministic and physically transparent views of the dynamics at play. However, the practical value of this approach is limited by the sensitivity of model predictions-even of such fundamental characteristics as the direction of eddyinduced momentum transfer-to the assumed pattern of individual vortices. The present study does not question the significance of equilibration mechanisms identified in earlier models, but rather argues for adding a new candidate to the list. The mechanism considered herein differs from earlier propositions in its focus on the local mesoscale dynamics of equilibration. Importantly, we address the problem in the context of an effectively unbounded ocean model. This configuration can be dynamically dissimilar to channel models commonly used in studies of baroclinic instabilities. For instance, the linearly fastest growing instabilities of vertically sheared zonal flows in the laterally unbounded ocean take the form of meridionally uniform modes, as indicated in Fig. 1a. These modes represent exact, exponentially growing solutions of the fully nonlinear governing equations. The meridionally uniform modes have no effect on the mean properties of the flow and therefore the mechanisms of equilibration associated with the nonlinear modification of a basic state (Malkus and Veronis 1958) are not engaged. The equilibration of such systems occurs through the generation of secondary instabilities, which act to disrupt primary modes and thereby arrest their linear growth (see the schematic in Fig. 1). The present study examines their dynamics in an attempt to develop a predictive mechanistic model for the equilibrium eddy-driven transport. Our theory assumes that the equilibrium state is controlled by the competition between linear mechanisms involved in the growth of meridional modes and the disruptive action of their secondary instabilities. This proposition is concisely phrased in terms of the growth rate balance: ... (1) where l1 is the typical growth rate of primary baroclinic modes, l2 is the growth rate of their secondary instabilities, and C is a dimensionless order one quantity that can be calibrated using numerical simulations. The primary growth rate l1 is determined by the background parameters (the vertical shear, beta effect, bottom drag, and stratification pattern). The secondary instability l2 is also affected by these quantities, but, additionally, it depends very strongly on the amplitude of primary modes. Thus, for any given set of background parameters and value of C, the growth rate balance (1) implicitly determines the equilibrium amplitude of baroclinic instability. The physical reasoning behind (1) is straightforward. As baroclinic instability modes, emerging from random perturbations, grow in time, they develop secondary instabilities. The growth rate of secondary instabilities monotonically increases with the amplitude of primary modes. At first, the growth of secondary instabilities is too slow to inflict any significant damage to primary modes-the evolution of small-amplitude perturbations is adequately captured by the linear theory. However, at some point the growth rate of secondary instabilities significantly exceeds the primary growth rates. As a result, the secondary instabilities gain in magnitude, rapidly reaching the level of primary modes, and suppress their growth. At this stage, the system reaches statistical equilibrium. In this study, condition (1) is used as the basis for an algorithm evaluating the equilibrium amplitude of mesoscale eddies generated by baroclinic instability. The growth of primary modes (l1) is computed from linear instability theory (Eady 1949; Phillips 1951). For any given amplitude of primary modes, the growth rate of secondary instabilities (l2) can be determined using the method based on the Floquet theory. The amplitude of primary modes is iteratively adjusted until l1 and l2 satisfy the growth rate balance (1). Themodel predictions can be conveniently expressed in terms of the meridional potential vorticity (PV) fluxes or rms of perturbation velocity. The growth rate balance (GRB) theory is applied to two baroclinically unstable systems: the two-layer quasigeostrophic Phillips model and the continuously stratified Eady model. In both cases, we find that the growth rate balance theory (1) with C'324 adequately describes the equilibrium magnitude of eddies and its dependencies on the background parameters. The utility of the GRB theory has already been demonstrated, albeit in a very different context. The idea originated in the study (Radko and Smith 2012) of nonlinear equilibration of double-diffusive instability- a small-scale mixing process driven by the difference in the molecular diffusivities of heat and salt (Stern 1960). For that problem, completely unrelated to baroclinic instability, the model based on (1) adequately represented the magnitude and dependencies of vertical mixing induced by salt fingers for C'2:7 and C'4:3 in two and three dimensions, respectively. These encouraging results, first for double diffusion and now for baroclinic instability, point toward a very general nature of the proposed equilibration mechanism and the potential applicability of our method to a wide class of instability problems. It should be emphasized that our theory, in its present form, makes no attempt to describe the dynamics of large-scale eddy-driven patterns (LEDPs hereafter) that emerge spontaneously in certain parameter regimes and can substantially affect the net eddy-induced transport. To properly interpret model predictions, it becomes important to delineate the parameter regions where eddy activity is largely contained within the mesoscale range from the LEDP-controlled regimes. This task is accomplished using a suite of numerical simulations covering a wide range of governing parameters. This paper is organized as follows. In section 2, we present preliminary direct numerical simulations for the two-layer quasigeostrophic model (Phillips configuration), focusing our inquiry on the evolution of primary baroclinic instabilities. The theoretical model of equilibration, motivated by these experiments, is presented in section 3 and tested numerically in section 4. In section 5, we explore the conditions for spontaneous emergence of LEDPs and discuss the impact of such events on the predictive capabilities of theGRBmodel. To demonstrate that the proposed equilibration theory is sufficiently general and that our specific predictions are not configuration dependent, the GRB theory is also applied (section 6) to the continuously stratified Eady model. We summarize and draw conclusions in section 7. 2. Preliminary calculations Our starting point for the analysis of baroclinic instability of a zonal flow is the two-layer quasigeostrophic model (e.g., Pedlosky 1987): ... (2) where (C1, C2) are the streamfunctions, (Q1, Q2) are the potential vorticities, and (H1, H2) are the depths of the upper and lower layer, respectively; g0 5(Dr/r)g is the reduced gravity, n is the lateral viscosity, and g is the bottom drag coefficient. The flow field (C1, C2) is separated into the basic state (C1, C2), representing the laterally homogeneous zonal current, and the perturbation (c1, c2). Without loss of generality, we consider a basic state in which the lower layer is motionless (C2 50) and express the governing Equations (2) in terms of (c1, c2). To reduce the number of governing parameters, the system is nondimensionalized using Rd 1, jUj, and Rd 1/jUj as the scales of length, velocity, and time, respectively; U is the basic velocity of the upper layer, and ... is the radius of deformation of the upper layer. The resulting nondimensional system takes the form ... (3) where bnd 5bR2 d 1/jUj, nnd 5n/(Rd 1jUj), gnd 5gRd 1/jUj, and r5H1/H2 are the key nondimensional parameters, s5U/jUj is the sign of the background velocity, and (q1, q2) are the perturbation PV fields. To gain a preliminary understanding of the mechanics of equilibration, the system (3) was solved numerically. We assume doubly periodic boundary conditions for (c1, c2) in x and y and integrate the governing equations using a dealiased pseudospectral method, analogous to the model employed in Radko and Stern (1999, 2000). In the following calculation, we used (bnd, r, gnd, nnd, s)5(0:25, 1/ 3, 0:5, 0:005, 1). In dimensional units, these parameters can describe, for instance, an eastward basic current with U 50:05ms21, b52310211 m21 s21, H1 51km, H2 53 km, g51026 s21, n56m2 s21, and Rd 1 525km-scales that are representative of typical midocean flows. The computational domain (Lx, Ly)5(150, 75), which is equivalent to 3750km31875km, was resolved by a uniform mesh with Nx 3Ny 515363768 elements, and the model was initialized from rest by a small-amplitude, random, computer-generated (c1, c2) distribution. The active statistically steady eddying motion driven by baroclinic instability was established after a few growth rate periods. Figure 2a shows a typical instantaneous (t 5 250) potential vorticity field (q1) in the initial stage of linear growth. As expected from linear stability theory, the most rapidly growing perturbations take the form of meridional harmonics. Figure 2b (t 5 305) illustrates the second evolutionary stage-development of secondary instabilities, which act to distort the primary modes, adversely affecting their growth. The secondary instabilities operate on the same spatial scales as primary modes, but are characterized by strong meridional variability. These structures have been implicated, for instance, in the spontaneous formation of zonal jets (Berloffet al. 2009). Finally, by t 5 450 (Fig. 2c) the system enters the quasi-equilibrium stage, characterized by irregular transient patterns. The transition to the statistically steady regime is illustrated in Fig. 3, which shows the evolution of meridional eddy-driven PV fluxes: ... (4) where (y1, y2)5(?c1/?x, ?c2/?x) are the meridional velocities in the upper and lower layers, respectively; the overbar denotes the spatial average over the entire domain. The initial stage of the simulation (t , 250) is characterized by the exponential growth of fluxes, which is then followed by their equilibration and transition to the statistically steady regime (t.350). In the interest of identifying the essential mechanism of equilibration, it is important to emphasize that the point of transition from the exponential growth of fluxes to the quasi-steady regime (t;300) coincides with the emergence of secondary instabilities (Fig. 2b). Analogous evolutionary patterns have been observed in numerous other simulations of this type. The equilibration dynamics realized in these experiments support the GRB hypothesis (1), which is next examined in greater detail. 3. Theoretical model The numerical simulations in section 2 suggest that secondary instabilities of meridional modes play an essential role in arresting the linear growth of primary perturbations. Therefore, the growth rate balance theory holds the promise of predicting the equilibrium transport from first principles. However, the development of (1) into a quantitative and testable model of baroclinic instability requires us to examine the stability of the new basic state represented by the primary instability modes: ... (5) where (A1, A2) are the upper- and lower-layer perturbation amplitudes, and u is the phase shift. The secondary stability problem can be formulated in different ways. Even the choice of the wavenumber k in (5) is not immediately obvious. For instance, this wavenumber can be evaluated at the marginally unstable point (k0) corresponding to the zero growth rate. This restriction would define a well-posed stability problem, characterized by a regular basic steady state. Themain reason to question this formulation is that simulations (e.g., Fig. 2) reveal very clearly that the horizontal wavenumber of emerging primary modes is much better described by kmax, the wavenumber corresponding to the maximum growth rate than by k0. Use of k5kmax in (5), however, results in a time-dependent basic state. Therefore, the question arises whether the conventional methods of stability analysis can be applied directly to such systems. This problem is not uncommon and not insurmountable. Notable examples of analyzing the stability of timedependent flows come from studies based on the Kolmogorov model-the sinusoidal parallel flow in viscous fluid (Sivashinsky 1985; Manfroi and Young 1999, 2002; Balmforth and Young 2002, 2005). Since the unforced viscous parallel flow would inevitably decay, the Kolmogorov model circumvents this problem by introducing artificial forcing in the momentum equation that maintains the steady state. Variations on the same principle, often referred to as the quasisteady state approximation or the ''frozen flow'' method, have been successfully applied to numerous stability problems (e.g., Lick 1965; Robinson 1976; Kimura and Smyth 2011; Radko and Smith 2012). In the context of the Phillips model of baroclinic instability, the frozen flow approximation has been used and numerically validated by Berloffet al. (2009). The same principle is used in the present study. To analyze secondary instabilities using the frozen flow model, we separate the total flow field into the dominant basic state, consisting of primary meridional modes, and a small perturbation, ... (6) and then linearize the governing equations (3) with respect to (~c1, ~c2): ... (7) The governing equations are then rewritten in the coordinate system associated with the propagating primary modes. Therefore, the time dependence associated with the propagation of primary instability is not a part of the frozen flow approximation. The only aspect of primary time dependence that leads to errors in linear analysis is the amplitude growth. To examine the stability properties of the linear system (7), we use the Floquet technique in which the perturbation is sought in the following form: ... (8) where l is the growth rate, m is the meridional wavenumber, and f0 is the Floquet coefficient, which controls the fundamental wavelength in x. Substituting (8) into the linear system(7) and collecting the individual Fourier components allows us to express the governing equations in the matrix form: ... (9) where ... (10) and A is the square matrix whose elements are functions of (k, m, s, f0, bnd, nnd, gnd, A1, A2, u, N). The eigenvalues of A correspond to the growth rates of the linear system (7). For each set of governing parameters, we determine the eigenvalue with the maximum real part, which represents the fastest growing mode. This eigenvalue is then maximized with respect to m and f0. The symmetry and periodicity properties of our system are such that the Floquet coefficient f0 needs to be varied only within the range 0#f0 #0:5. Various Fouriertruncated models of this type have been commonly used to describe instabilities of the Rossby waves (e.g., Gill 1974; Connaughton et al. 2010). The linear theory of primary baroclinic instability (Phillips 1951) predicts the fastest growing wavelength k5kmax, the ratio of coefficients A1 and A2, and the phase shiftu between the upper/lower harmonics in the normal mode (5). Therefore, the primary mode is fully determined by a single measure of its magnitude-for instance, by the amplitude of the upper-layer streamfunction A1. This, in turn, determines the solution of the eigenvalue problem (9) for given N. This solution for l2 rapidly converges with increasing resolution (N/'), as indicated in Table 1, and therefore the fastest growing secondary instability for large N is effectively determined by six parameters: ... (11) The next step is to solve the growth rate balance (1). For each set of governing parameters, we evaluate the primary growth rate l1 using linear theory for primary instabilities. Suppose now that the value of C in (1) is known. Then, by varying the amplitude of the normal mode A1, we can readjust (11) until the growth rate balance is satisfied. An iterative procedure for solving the growth rate balance (1) was coded in Maple, and it typically requires 10-11 iterations for the model to converge, within a relative error of 1025 to the sought after solution for A1. Knowledge of A1, in turn, makes it possible to evaluate any desired measure of the equilibrium magnitude of primary modes. For instance, we can conveniently evaluate such key transport characteristic of eddying flows as the PV fluxes (4). The parameterization of PV fluxes is one of the avenues for representing mesoscale variability in the ocean and a subject of ongoing research (e.g., Marshall et al. 2012). The PV fluxes are plotted as a function of sbnd 5bR2 d 1/U in Figs. 4a and 4b and as a function of r in Figs. 4c and 4d for various values of C. Fluxes (Fq 1,2) rapidly increase with increasing C and with increasing r. The dependence of Fq 1,2 on sbnd takes a more complicated nonmonotonic form as fluxes change sign at the point of transition from the eastward (sbnd . 0) to the westward (sbnd , 0) flow. 4. Calibration and testing of the equilibration theory An obvious limitation of the GRB model in section 3 is related to its inability to determine the constant C internally-theory assumes that C is comparable to unity but does not provide an exact value. To evaluate C, we turn to numerical simulations. The selection of specific experiments was guided by the following considerations. The explicit lateral dissipation nnd has a relatively minor effect on the flow structure and eddy transport characteristics. Therefore, in all the following experiments we use constant values nnd 50:005, which is sufficient to maintain numerical stability of the code-in dimensional units, these values correspond to n5nc ;10m2 s21. The role of the bottom drag coefficient gnd is more subtle. The bottom drag in the two-layer model is meant to represent the interaction of the lower layer with bottom topography, and its values are poorly constrained. Some rough guidance with regard to g is provided by the bottom drag model of wind-driven gyres (Stommel 1948; Veronis 1966a,b) that offer a reasonable representation of largescale circulation for g;1026 s21. Another consideration involves the spontaneous generation of large-scale eddydriven patterns that occurs for sufficiently low values of bottom drag. Since themagnitude of LEDPs in the ocean is limited, it is sensible to restrict our analysis to the regime in which variability is dominated by the mesoscale component. In our nondimensional units, g;1026 s21 translates to gnd ;0:5 and is the value used in the following experiments. However, the effect of spontaneous generation of LEDPs is interesting in its own right and warrants some experimentation with gnd (section 5). The two key nondimensional parameters that govern the mesoscale dynamics of baroclinic instability and profoundly affect its transport characteristics are bnd and r. Deducing the pattern of variation of mixing characteristics with bnd and r-and theoretically rationalizing the observed dependencies-is our primary goal. Figure 5 presents a series of simulations in which bnd and the direction of the background flow (s) were varied, with other governing parameters kept constant (r51/ 3, gnd 50:5, and nnd 50:005). Here, the numerically inferred PV fluxes (Fq 1,2) are plotted as a function of sbnd along with the prediction based on the growth rate balance theory (section 3) for C 5 3.5 and C 5 4.0. In each experiment, the PV fluxes were averaged in time over the period following equilibration (see Fig. 3) and, overall, numerically inferred fluxes closely follow the theoretical pattern. Both simulations and theory predict a complicated nonmonotonic dependence, with larger fluxes obtained for a westward-flowing basic current (sbnd , 0). Inclusion of the beta effect has a generally stabilizing effect on the system, and the meridional PV transport becomes particularly strong for small bnd. The dependence of the meridional transport on bnd in Fig. 5 can be rationalized by inspecting typical patterns of primary and secondary instabilities (Fig. 6). The largest amplitude of primary instabilities in Fig. 6 is realized for bnd 5 0. In this case, the secondary instability takes the form of closed recirculating patterns localized entirely in the northward or southward flow regions of the primary modes (Fig. 6a). The perturbation of this type has a relatively limited effect on the primary modes and, therefore, is inefficient in drawing energy from them. In order for secondary modes in Fig. 6a to attain substantial growth rates, the amplitude of primarymodes should be large. In contrast, the secondary instabilities for the eastward basic flow (Fig. 6b) are represented by irregular, predominantly zonal, interconnected patterns. Structures of this type are fully capable of disrupting the primary modes, which would supply sufficient energy for the growth of secondary instabilities. In this case, large secondary growth rates, sufficient to satisfy the GRB balance, can be attained for relatively modest primary amplitudes. The situation for the westward flow (Fig. 6c) is somewhere in between. The secondary instability in this case is more localized and recirculating than in Fig. 6b, but more distributed and therefore potentially more disruptive than that in Fig. 6a. These features are consistent with the pattern of meridional transport in Fig. 5, which is weakest for the eastward flow (finite positive sbnd), strongest for small sbnd, and intermediate for finite negative sbnd. The agreement between the GRB model and numerics, suggested by the analysis in Fig. 5, is also confirmed by the diagnostics in Fig. 7, which examine the effects of variation in the layer thickness ratio (r). Theory and simulations predict mutually consistent, rapidly increasing patterns of Fq 1,2(r). The ability of the theoretical model, containing only one adjustable parameter, to capture details of the numerical simulations to such an extent is impressive. It lends credence to the proposed GRB assumption (1) as the most relevant constraint on the eddy-driven transport in baroclinically unstable flows. It is also encouraging that the values of C suggested by the numerical calibration, while finite, still significantly exceed unity. This implies that in the quasi-equilibrium regime, the linear growth rate of secondary instabilities significantly exceeds that of the primary instabilities, which provides a posteriori justification of the frozen flow approximation used in the theoretical model (section 3). It should be emphasized at this point that the GRB model is based on the interaction of primary and secondary baroclinically unstable modes. Therefore, its agreement with simulations of fully developed eddying flows in Figs. 5 and 7 suggests that the rate of meridional eddy-induced transport is largely controlled by low-order mesoscale dynamics. This suggestion is also supported by the spectral analysis presented in the appendix. Other modes, excited by nonlinear multistage interactions, could be important for other aspects of the problem (e.g., the energetics), but their role in meridional mixing is surprisingly limited. Many practical applications, including environmental and climatological problems, also demand some knowledge of the lateral diffusivity of dynamically passive tracers (KC 1,2). This quantity can be estimated from the PV flux by assuming that KC 1,2 is close to the eddy diffusivity of PV (Kq 1,2), where ... (12) To test this assumption, the governing system (3) was supplemented by the evolutionary equations for passive tracers c1(c2) located in the upper (lower) layers. The tracer concentrations were separated into uniform unbounded background gradients (c1, c2), which do not vary in time, and the perturbation fields (c01, c02). Without loss of generality, the background gradients were set to unity (?c1/?y5?c2/?y51), and the resulting tracer equations took the following form: ... (13) The system (13) was initiated by a random smallamplitude distribution of (c01, c02) and integrated, in parallel with the dynamic equations (3), until statistical equilibration of tracer transport. To compare the equilibrium diffusivities of PV and the passive tracer c, we performed a series of simulations with various background parameters. Figure 8 presents the upperlayer tracer diffusivity, diagnosed from these simulations, along with the corresponding PV diffusivities as functions of sbnd (Fig. 8a) and r (Fig. 8b). In most simulations, Kq1-diffusivity of a dynamically active field-is within 5% of the passive tracer diffusivity KC1. In the second layer, the PV and tracer diffusivity patterns (not shown) become noticeably different. This is attributed to the action of the bottom drag, which directly affects the PV distribution but not the passive tracer c2. 5. Mesoscales and LEDPs: Who is in charge? Our theoretical model of equilibration (section 3) is focused on mesoscale dynamics and makes no attempt to incorporate the effects of LEDPs that occasionally emerge in simulations. Experiments in section 4 indicate that the GRB theory adequately captures the key properties of eddy-driven transport as long as it is dominated by the mesoscale component. Thus, for completeness, it behooves us to delineate conditions for mesoscale-dominated and LEDP-dominated regimes and thereby specify the parameter range for the applicability of our theoretical model. The contribution of LEDPs to the net meridional transport varies greatly with background parameters and can be surprisingly large. For instance, Fig. 9 presents three sets of analogous numerical experiments performed for (bnd, gnd)5(0:25, 0:5), (bnd, gnd)5 (0:25, 0:005), and (bnd, gnd)5(0:125, 0:005) in Figs. 9a, 9b, and 9c, respectively. The total flow field, represented by the upper-layer perturbation PV, is shown in the upper panels. The central panels show the corresponding LEDP components, defined here as structures consisting of Fourier harmonics with wavelengths that are larger than Lcr 520Rd, where ... is the first baroclinic radius of deformation in the two-layer ocean model. The remaining mesoscale components are shown in the bottom panels of Fig. 9. Visual inspection of the flow patterns in Fig. 9 reveals major structural differences. The PV field in Fig. 9a is clearly dominated by mesoscale variability. The simulation in Fig. 9b is characterized by comparable contributions from mesoscales and LEDPs. The third experiment (Fig. 9c) is controlled by LEDPs. The mechanics of LEDP generation (cf. Fig. 9c) is a challenging problem, interesting in its own right and requiring an in-depth investigation (e.g., Kamenkovich et al. 2014, manuscript submitted to J. Fluid Mech.). At this point, we only present a preliminary numerical exploration of the parameter space, which is aimed at identification of the regimes that can be adequately described by the GRB theory (section 3). Figure 10 presents a summary of 40 numerical simulations with varying sbnd and gnd; all other parameters are kept the same as in our baseline calculation (Fig. 2). The relative importance of mesoscales and LEDPs is quantified by introducing the following discriminator variable: ... (14) where FMS 1 represents the time-mean upper-layer PV flux driven by mesoscale components (L,Lcr), and FLEDP 1 is the LEDP-driven flux (L.Lcr). We consider only the meridional fluxes, and no attempt is made to isolate the divergent and rotational components. Positive (negative) values of R indicate that the net eddy-induced transport is controlled by the LEDP (mesoscale) variability. As expected, R varies strongly with both sbnd and gnd. An increase in bottom drag primarily affects LEDPs, and therefore their transport contribution rapidly decreases with increasing gnd. The beta effect is detrimental for both mesoscales and LEDPs, although the latter, owing to their large spatial scales, are more affected by- and respond more readily-to the increase in bnd. It is interesting to note that parameters realized for typical oceanographic conditions (bnd ;0:2, gnd ;0:0520:5) are located in the intermediate regions, where R is comparable to, but less than, unity. This can be interpreted as a sign that both components are substantial, and even amodest variation in the background parameters can easily shiftthe balance in favor of LEDPs or mesoscale eddies. 6. Continuous stratification effects: The Eady problem Calculations discussed in section 5 demonstrated that the GRB theory adequately represents the equilibration mechanism-and predicts the mesoscale transport-for the two-layer model of baroclinic instability. To ensure that our conclusions are sufficiently general and not model dependent, we now turn to a more complicated and substantially different ocean model characterized by continuous stratification. a. Numerics The mechanics of equilibration in the fully stratified ocean is first illustrated qualitatively, by inspection of the simulation in Fig. 11. The computation was made using the Massachusetts Institute of Technology general circulation model (MITgcm) (Marshall et al. 1997a,b) in a rectangular domain. The model integrates the incompressible Navier-Stokes equations of motion, and in the present implementation we used the hydrostatic version of the model. Rigid boundary conditions (y 5 0) were implemented at the meridional walls (y50, Ly), and periodic boundary conditions were assumed in the zonal direction (x50, Lx). The contribution of salinity to density was not taken into account, and therefore the buoyancy distribution is controlled by the temperature field. Fluid was set in motion by relaxing buoyancy at the meridional walls to the prescribed linear distributions [bS(z), bN(z)], which, by virtue of thermal wind balance, engages the mean vertical shear: ... (15) For the calculation in Fig. 11, we used the uniform target cross-flow buoyancy variation Db5gaDT 50:01ms22, where g59:8ms22 is gravity and a5231024 K21 is the thermal expansion coefficient. The simulations were carried out on an f plane with Coriolis parameter f 51024 s21. The computational domain (Lx, Ly, H) 5 (1024 km, 1024 km, 1075 m) was discretized on a uniform mesh with 5123512344 elements. For these parameters, the mean shear (15) is ?u/?z'1024 s21. The mean bottom velocity is effectively set to zero by imposing quadratic bottom drag. The resulting mean velocity profile is ... (16) where the largest mean velocity (U) is attained at the surface (z50). The vertical stratification imposed on the meridional boundaries corresponds to the buoyancy frequency ..., and the radius of deformation is Ld 5f21NH 530 km. The model was initialized from rest with a buoyancy pattern that is linear in y and z and uniformin x (Fig. 11a). After a few rotation periods, a steady zonal flow develops that closely conforms to (16). The next stage (e.g., Fig. 11b) is characterized by baroclinic destabilization (t ; 3-4 months). The perturbation takes the form of regular, elongated, meridionally oriented modes. Upon reaching finite amplitude, these primary modes also become unstable. These secondary instabilities manifest themselves through the wavelike meridional variability, starting to distort primary instabilities at t ; 5 months (Fig. 11c). Finally, the flow transitions to the statistically steady regime (Fig. 11d) characterized by intense mesoscale variability in the form of disorganized, irregular patterns (t . 6 months). Overall, the observed evolution of the stratified system bears strong resemblance to baroclinic instabilities observed, at the corresponding stages, in two-layer experiments (e.g., Fig. 2). Importantly, the emergence of finite-amplitude secondary instabilities (Fig. 11c) coincides in time with the equilibration of the baroclinic instability. The latter proposition is supported by Fig. 12, which presents the globally averaged meridional temperature flux ... (17) where angular brackets denote the volume mean. The link between equilibration and the development of secondary instabilities revealed in Figs. 11 and 12 suggests that the statistically steady characteristics of the continuously stratified, baroclinically unstable flow can also be described by a model based on the GRB hypothesis (1). The development of such a model is described next. b. Theory To simplify the following analysis, we adopt the continuously stratified quasigeostrophicmodel (e.g., Pedlosky 1987). Following Eady (1949), the total streamfunction is separated into the background component ... (18) and the perturbation c. The result is nondimensionalized using the radius of deformation Ld 5f21NH as the horizontal length scale, U as the scale of velocity, H as the depth scale, and Ld/U as the time scale, which yields ... (19) whereas the top (z50) and bottom (z521) boundary conditions become ... (20) The stability problem [(19), (20)] will be solved under the assumption that the flow is effectively unbounded laterally, which differs from the original formulation (Eady 1949) and, for that matter, from the foregoing simulations (Figs. 11-12). However, we anticipate that the distinction between bounded and unbounded dynamics will be relatively minor as long as the rigid walls in the bounded model are greatly separated from each other (Ly Ld). The procedure for development of the GRB model for the continuously stratified model is analogous to its two-layer counterpart (section 3), and therefore it is presented in abbreviated form. The linear stability analysis of (19), (20) indicates that the largest growth rates are attained by the meridionally oriented modes: ... (21) where ... (22) Without any loss of generality, we assume that B1 is real, and therefore it can be used as a convenient measure of the magnitude of primary modes. We also note that the (nondimensional) growth rate of primary instability is now fully determined: l1 5Re(l)50:31. The analysis of the secondary instability is more involved. First, the governing equations are rewritten in the coordinate system associated with the propagating primary modes (which move with the phase speed cbc 50:5 in the x direction). Next, we invoke the frozen flow approximation and separate c into the primary mode ~c and a small perturbation c0. Since the stability analysis is performed in the moving coordinate system, we are assured that the only component of primary time dependence that leads to errors in linear analysis is the amplitude growth. The governing equations, (19) and (20), are linearized with respect to ~c, and the perturbation is expressed in terms of the (x, y) Floquet series: ... (23) The Fourier coefficients are also discretized vertically on a uniform grid: ... (24) Substituting (23) and (24) into the linearized system, in which z derivatives are represented by their finitedifference counterparts, and collecting the Fourier components at individual levels allows us to reduce the eigenvalue problemto the equivalent matrix formulation: ... (25) where the vector j consists of c(n) m (n52N, . . . , N; m50, 1, . . . , M) and A is the square matrix whose elements are determined by the amplitude of the primary mode as measured, for instance, by B1. For any given value of B1, the growth rate of secondary instability l2 is evaluated by computing real parts of all eigenvalues of A and selecting the largest one. The foregoing procedure makes it possible to solve the growth rate balance (1) iteratively. For each value of C, the amplitude of primary mode B1 is varied until the secondary growth rate l2 matches the target growth rateCl1. This iterative algorithmwas coded inMaple and requires only 8-10 iterations to converge, within a relative error of 1025, to the sought after solution forB1. This, in turn, fully determines the primary mode at its equilibrium level, making it possible to quantify the statistically steady magnitude of eddy transfer. c. Calibration Since the value of C is unknown a priori, we now proceed-as for the layered model in sections 3 and 4- with the calibration of C based on simulations. A complication arising in the GRB analysis of the Eady model is related to the choice of the relevant quantity on which we should base the comparison of theory and numerics. For the two-layer case, the calibration was based on the meridional PV flux. However, for the Eady model this choice has to be ruled out since the basic state (18) represents a rather special case with zero lateral PV gradients. If the eddy diffusivity of PV is finite, then the lateral PV fluxes should vanish as well. Therefore, in the following analysiswe express theGRBmodel predictions in terms of the horizontally and time-averaged rms perturbation velocity-another convenient and widely used quantity that is readily available from simulations. In Fig. 13, we plot the predicted vertical patterns of nondimensional rms velocity yrms for various values of C. These patterns are symmetric in z, with maximum values of yrms attained near the surface and near the bottom and minima at middepth. The equilibrium velocity monotonically increases with C. To test the predictive capabilities of the theoretical model and to identify the appropriate value of C, we have performed a series of numerical simulations, analogous to that in Fig. 11, in which we varied the surface speed U and the background stratification N. This was accomplished by changing the target horizontal and vertical variation in temperature (DyT and DzT). The results are summarized in Fig. 14. In all cases, the theory with C53 predicts fairly accurately both the magnitude and the pattern of rms velocity, particularly in the upper part of the water column. In the lower part, there is a noticeable deviation between the predicted and simulated patterns, which is attributed to the effects of the bottom drag. The bottom drag was incorporated in the numerical model (primarily to suppress the spontaneous generation of LEDPs), but not in the Eady model and its GRB-based analysis. It should also be noted that the numerical model contains several other ingredients that are absent in the theory. For instance, the latter is based on the quasigeostrophic approximation, whereas the former uses more general Navier-Stokes equations. The rigid meridional boundaries are present in simulations but absent in the theory. In view of these differences, the consistency of the predicted and simulated equilibrium patterns is impressive. It is interpreted as evidence of a very robust character of the theoretical model and of its cornerstone-the growth rate balance argument. 7. Discussion This paper presents an attempt to develop a simple mechanistic model for the meridional eddy-induced transport in large-scale baroclinically unstable zonal flows. The crux of the proposed approach is the growth rate balance (GRB) hypothesis. We assume-and subsequently verify numerically-that the statistical equilibrium is reached when the growth rate of primary baroclinically unstable modes (l1) becomes comparable to that of their secondary instabilities (l2). This assumption is based on a fairly intuitive argument. In the unbounded ocean, primary baroclinic instabilities take the form of meridionally uniform modes that represent an exact solution of our governing (quasigeostrophic) equations. When the amplitude of these primary modes is small, their evolution follows the linear scenario: primary instabilities grow exponentially, unaffected by their relatively weak secondary instabilities. This regime persists for as long as l1 $l2. However, as the amplitude of primary modes increases, so does the growth rate of their secondary instabilities. When l2 significantly exceeds l1, secondary instabilities start to gain in magnitude, rapidly reach the level of primary modes and disrupt them. At this point, the system transitions from the linear regime, marked by the exponential growth of perturbations, to the quasi-equilibrium state with statistically steady transport characteristics. Thus, one is led to a conclusion that the approximate balance between the growth rates of primary and secondary instabilities is inevitable for the statistical equilibrium and should be generally satisfied for a wide range of governing parameters. The typical growth rates of primary instabilities (l1) are estimated from the well-known linear instability theory (Eady 1949; Phillips 1951) and the growth rate of secondary instabilities (l2) can be evaluated using the Floquet-based method. Since l2 is controlled by the amplitude of the primary modes, whereas l1 is fully determined by the basic state, the growth rate balance l1 ;l2 implicitly determines the equilibrium amplitude of primary modes. Our ability to determine the point of equilibration, in turn, makes it possible to evaluate various integral characteristics of the fully developed baroclinic instability, including the meridional PV fluxes, eddy diffusivity, and rms velocity. It should be understood that the growth rate of the secondary instability (l2) can only be estimated in the order of magnitude sense and not determined exactly. One of the obvious limitations of our approach is the frozen flow approximation, which assumes that the instability of slowly evolving patterns can be assessed using conventional techniques developed for regular steady states. Therefore, we can only claim that the growth rates l1 and l2, as evaluated by our theory, should be comparable but not necessarily equal. This uncertainty forced us to consider a weak form of the growth rate balance in (1). The nondimensional orderone coefficient C in (1) cannot be determined internally, and therefore it was calibrated using direct numerical simulations. In this study, the GRB model has been applied to two classical baroclinically unstable systems-the two-layer configuration (Phillips 1951) and linearly stratified f-plane model (Eady 1949). In both systems, which differ in several key aspects, theory was compared with corresponding numerical simulations and was shown to adequately represent the intensity of eddy-induced mixing and its dependencies. These comparisons were performed for the parameter regimes in which temporal and spatial variability is dominated by mesoscale eddies. The GRB theory makes no attempt to represent the effects of large-scale eddy-induced patterns (LEDPs) that spontaneously emerge in some simulations. For instance, the model is not meant to accurately predict the eddy-induced transport in configurations dominated by strong zonal jets. However, the consistency of the GRB model with simulations of relatively homogeneous flows is encouraging. It instills confidence in our ability to capture the essential dynamics of mesoscale equilibration.We might also speculate that this study can open new opportunities for purely analytical explorations of baroclinic instability. In its present form, our algorithm still contains numerical elements, such as the Floquet-based calculation of the secondary growth rates. However, it is our belief that investigations focused on (1) have a higher potential for analytical tractability than those based on more complicated original systems. Our optimism is partially based on the pronounced low-order dynamics of the growth rate balance model. The calculations made with highly truncated (N 5 2) Fourier series in the Floquet calculation are close to their fully resolved counterparts (Table 1). Thus, it is likely that key equilibration effects could be captured by even more explicit low-dimensional models. Finally, we note that the current investigation represents the second example of successful application of the GRB theory. The first problem treated in this manner (Radko and Smith 2012) addressed the equilibrium dynamics of double-diffusive instability-see the discussion in Schmitt (2012). Interestingly, the equilibrium transport in both double-diffusive and baroclinic instability studies, which are as dissimilar as two instability problems could be, can be adequately represented by the growth rate balance with C;324. This may be related to the most general physical basis for (1), independent of the specifics of a particular stability problem. It is therefore likely that the GRB theory will find applications even beyond these two areas. Hydrodynamic instabilities are numerous and diverse; they present some of the most challenging and intriguing theoretical problems in fluidmechanics. The fundamental question arising for each type of instability is what controls the equilibrium amplitude of a fully developed state. Such problems are usually solved on a case-by-case basis. However, the GRBbased modeling may prove to be as profitable for other instabilities as it is for baroclinic instability and for double diffusion. Acknowledgments. The authors thank Pavel Berloff, Igor Kamenkovich, and the anonymous reviewers for helpful comments. Support of the National Science Foundation (Grant OCE 1155866) is gratefully acknowledged. This research was performed while J. Flanagan held a National Research Council award at the Naval Postgraduate School. REFERENCES Balmforth, N. J., and Y.-N. Young, 2002: Stratified Kolmogorov flow. J. Fluid Mech., 450, 131-167, doi:10.1017/ S0022111002006371. -, and -, 2005: Stratified Kolmogorov flow. Part 2. J. Fluid Mech., 528, 23-42, doi:10.1017/S002211200400271X. Benilov, E. S., 1993: Baroclinic instability of large-amplitude geostrophic flows. J. Fluid Mech., 251, 501-514, doi:10.1017/ S0022112093003490. -, 1995a: Stability of large-amplitude geostrophic flows localized in a thin layer. J. Fluid Mech., 288, 157-174, doi:10.1017/ S0022112095001108. -, 1995b: Baroclinic instability of quasi-geostrophic flows localized in a thin layer. J. Fluid Mech., 288, 175-200, doi:10.1017/ S002211209500111X. Berloff, P., I. Kamenkovich, and J. Pedlosky, 2009:Amechanism of formation of multiple zonal jets in the oceans. J. Fluid Mech., 628, 395-425, doi:10.1017/S0022112009006375. -, S. Karabasov, T. Farrar, and I. Kamenkovich, 2011: On latency of multiple zonal jets in the oceans. J. Fluid Mech., 686, 534-567, doi:10.1017/jfm.2011.345. Charney, J., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, 1087- 1095, doi:10.1175/1520-0469(1971)028,1087:GT.2.0.CO;2. Connaughton, C. P., B. T. Nadiga, S. V. Nazarenko, and B. E. Quinn, 2010: Modulational instability of Rossby and driftwaves and generation of zonal jets. J. Fluid Mech., 654, 207- 231, doi:10.1017/S0022112010000510. Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1, 33-52, doi:10.1111/j.2153-3490.1949.tb01265.x. Eden, C., 2011: A closure for mesoscale eddy fluxes based on linear instability theory. OceanModell., 39, 362-369, doi:10.1016/ j.ocemod.2011.05.009. Farrell, B. F., and P. J. Ioannou, 2009: A theory of baroclinic turbulence. J. Atmos. Sci., 66, 2444-2454, doi:10.1175/ 2009JAS2989.1. Frisius, T., 1998: A mechanism for the barotropic equilibration of baroclinic waves. J. Atmos. Sci., 55, 2918-2936, doi:10.1175/ 1520-0469(1998)055,2918:AMFTBE.2.0.CO;2. Gama, S., M. Vergassola, and U. Frisch, 1994: Negative eddy viscosity in isotropically forced two-dimensional flow: Linear and nonlinear dynamics. J. Fluid Mech., 260, 95-126, doi:10.1017/ S0022112094003459. Gent, P., and J. McWilliams, 1990: Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20, 150-155, doi:10.1175/ 1520-0485(1990)020,0150:IMIOCM.2.0.CO;2. Gill, A. E., 1974: The stability on planetary waves on an infinite beta-plane. Geophys. Fluid Dyn., 6, 29-47, doi:10.1080/ 03091927409365786. Held, I.M., and V. D. Larichev, 1996: Scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta plane. J.Atmos. Sci., 53, 946-952, doi:10.1175/1520-0469(1996)053,0946: ASTFHH.2.0.CO;2. Henning, C. C., and G. Vallis, 2005: The effects of mesoscale eddies on the stratification and transport of an ocean with a circumpolar channel. J. Phys. Oceanogr., 35, 880-896, doi:10.1175/ JPO2727.1. Hogg, N., and B. Owens, 1999: Direct measurement of the deep circulation within the Brazil basin. Deep-Sea Res., 46, 335-353, doi:10.1016/S0967-0645(98)00097-6. Kamenkovich, I., P. Berloff, and J. Pedlosky, 2009: Anisotropic material transport by eddies and eddy-driven currents in a model of the North Atlantic. J. Phys. Oceanogr., 39, 3162- 3175, doi:10.1175/2009JPO4239.1. Killworth, P. D., 1997: On the parameterization of eddy transfer Part I. Theory. J. Mar. Res., 55, 1171-1197, doi:10.1357/ 0022240973224102. -, 2005: Parameterization of eddy effects on mixed layers and tracer transport: A linearized eddy perspective. J. Phys. Oceanogr., 35, 1717-1725, doi:10.1175/JPO2781.1. Kimura, S., and W. D. Smyth, 2011: Secondary instability of salt sheets. J. Mar. Res., 69, 57-77, doi:10.1357/002224011798147624. Lapeyre, G., and I. M. Held, 2003: Diffusivity, kinetic energy dissipation, and closure theories for the poleward eddy heat flux. J. Atmos. Sci., 60, 2907-2916, doi:10.1175/1520-0469(2003)060,2907: DKEDAC.2.0.CO;2. Larichev, V., and I. Held, 1995: Eddy amplitudes and fluxes in a homogeneous model of fully developed baroclinic instability. J. Phys. Oceanogr., 25, 2285-2297, doi:10.1175/ 1520-0485(1995)025,2285:EAAFIA.2.0.CO;2. Lick, W., 1965: The instability of a fluid layer with timedependent heating. J. Fluid Mech., 21, 565-576, doi:10.1017/ S0022112065000332. Malkus, W. V. R., and G. Veronis, 1958: Finite amplitude cellular convection. J. Fluid Mech., 4, 225-261, doi:10.1017/ S0022112058000410. Manfroi, A., and W. Young, 1999: Slow evolution of zonal jets on the beta plane. J. Atmos. Sci., 56, 784-800, doi:10.1175/ 1520-0469(1999)056,0784:SEOZJO.2.0.CO;2. -, and -, 2002: Stability of beta-plane Kolmogorov flow. Physica D, 162, 208-232, doi:10.1016/S0167-2789(01)00384-0. Marshall, D. P., J. R. Maddison, and P. S. Berloff, 2012: A framework for parameterizing eddy potential vorticity fluxes. J. Phys. Oceanogr., 42, 539-557, doi:10.1175/ JPO-D-11-048.1. Marshall, J., and T. Radko, 2003: Residual-mean solutions for the Antarctic Circumpolar Current and its associated overturning circulation. J. Phys. Oceanogr., 33, 2341-2354, doi:10.1175/ 1520-0485(2003)033,2341:RSFTAC.2.0.CO;2. -, A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997a: A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res., 102, 5753-5766, doi:10.1029/96JC02775. -, C. Hill, L. Perelman, and A. Adcroft, 1997b: Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. Res., 102, 5733-5752, doi:10.1029/96JC02776. Maximenko, N., B. Bang, and H. Sasaki, 2005: Observational evidence of alternating zonal jets in the world ocean. Geophys. Res. Lett., 32, L12607, doi:10.1029/2005GL022728. McGillicuddy, D. J., Jr., and Coauthors, 1998: Influence of mesoscale eddies on new production in the Sargasso Sea. Nature, 394, 263-266, doi:10.1038/28367. Meehl, G. A., and Coauthors, 2007: Global climate projections. Climate Change 2007: The Physical Science Basis, S. Solomon et al., Eds., Cambridge University Press, 747-845. Mei, C. C., and M. Vernescu, 2010: Homogenization Methods for Multiscale Mechanics. World Scientific Publishing, 330 pp. Molemaker, M. J., J. C. McWilliams, and X. Capet, 2010: Balanced and unbalanced routes to dissipation in an equilibrated Eady flow. J. Fluid Mech., 654, 35-63, doi:10.1017/ S0022112009993272. Nikurashin, M., and G. Vallis, 2011: A theory of deep stratification and overturning circulation in the oceans. J. Phys. Oceanogr., 41, 485-502, doi:10.1175/2010JPO4529.1. Novikov, A., and G. Papanicolaou, 2001: Eddy viscosity of cellular flows. J. Fluid Mech., 446, 173-198. Pedlosky, J., 1970: Finite-amplitude baroclinic waves. J. Atmos. Sci., 27, 15-30, doi:10.1175/1520-0469(1970)027,0015: FABW.2.0.CO;2. -, 1971: Finite-amplitude baroclinic waves with small dissipation. J. Atmos. Sci., 28, 587-597, doi:10.1175/ 1520-0469(1971)028,0587:FABWWS.2.0.CO;2. -, 1981: Nonlinear dynamics of baroclinic wave ensembles. J. Fluid Mech., 102, 169-209, doi:10.1017/S0022112081002590. -, 1987: Geophysical Fluid Dynamics. Springer, 710 pp. Phillips, N. A., 1951: A simple three-dimensional model for the study of large scale extra tropical flow pattern. J. Meteor., 8, 381-394, doi:10.1175/1520-0469(1951)008,0381: ASTDMF.2.0.CO;2. Radko, T., 2011a: On the generation of large-scale structures in a homogeneous eddy field. J. Fluid Mech., 668, 76-99, doi:10.1017/S0022112010004568. -,2011b: Eddy viscosity and diffusivity in the modon-sea model. J. Mar. Res., 69, 723-752, doi:10.1357/002224011799849426. -, and M. E. Stern, 1999: On the propagation of oceanic mesoscale vortices. J. Fluid Mech., 380, 39-57, doi:10.1017/ S0022112098003371. -, and -, 2000: Self-propagating eddies on the stratified f plane. J. Phys. Oceanogr., 30, 3134-3144, doi:10.1175/ 1520-0485(2000)030,3134:SPEOTS.2.0.CO;2. -, and J. Marshall, 2004: Eddy-induced diapycnal fluxes and their role in the maintenance of the thermocline. J. Phys. Oceanogr., 34, 372-383, doi:10.1175/1520-0485(2004)034,0372: EDFATR.2.0.CO;2. -, and I. Kamenkovich, 2011: Semi-adiabatic model of the deep stratification and meridional overturning. J. Phys. Oceanogr., 41, 757-780, doi:10.1175/2010JPO4538.1. -, and D. P. Smith, 2012: Equilibrium transport in doublediffusive convection. J. Fluid Mech., 692, 5-27, doi:10.1017/ jfm.2011.343. Rhines, P. B., and W. R. Young, 1982: A theory of the wind-driven circulation. Part I: Mid-ocean gyres. J. Mar. Res., 40 (Suppl.), 559-596. Robinson, J. L., 1976: Theoretical analysis of convective instability of a growing horizontal thermal boundary layer. Phys. Fluids, 19, 778-791, doi:10.1063/1.861570. Schmitt, R. W., 2012: Finger puzzles. J. Fluid Mech., 692, 1-4, doi:10.1017/jfm.2011.468. Sivashinsky, G., 1985:Weak turbulence in periodic flows. PhysicaD, 17, 243-255, doi:10.1016/0167-2789(85)90009-0. Spall, M. A., and D. C. Chapman, 1998: On the efficiency of baroclinic eddy heat transport across narrow fronts. J. Phys. Oceanogr., 28, 2275-2287, doi:10.1175/1520-0485(1998)028,2275: OTEOBE.2.0.CO;2. Stern, M. E., 1960: The ''salt-fountain'' and thermohaline convection. Tellus, 12, 172-175, doi:10.1111/j.2153-3490.1960.tb01295.x. Stommel, H., 1948: The westward intensification of winddriven ocean currents. Trans. Amer. Geophys. Union, 29, 202-206. Stouffer, R. J., and Coauthors, 2006: Investigating the causes of the response of the thermohaline circulation to past and future climate changes. J. Climate, 19, 1365-1387, doi:10.1175/ JCLI3689.1. Thompson, A. F., and W. R. Young, 2006: Scaling baroclinic eddy fluxes: Vortices and energy balance. J. Phys. Oceanogr., 36, 720-738, doi:10.1175/JPO2874.1. -, and -, 2007: Two-layer baroclinic eddy heat fluxes: Zonal flows and energy balance. J. Atmos. Sci., 64, 3214-3231, doi:10.1175/JAS4000.1. Veronis, G., 1966a: Wind-driven ocean circulation-Part 1. Linear theory and perturbation analysis. Deep-Sea Res. Oceanogr. Abstr., 13, 17-29, doi:10.1016/0011-7471(66)90003-9. -, 1966b: Wind-driven ocean circulation-Part 2. Numerical solutions of the non-linear problem. Deep-Sea Res. Oceanogr. Abstr., 13, 31-55, doi:10.1016/0011-7471(66)90004-0. Visbeck, M., J. Marshall, T. Haine, and M. Spall, 1997: Specification of eddy transfer coefficients in coarse-resolution ocean circulation models. J. Phys. Oceanogr., 27, 381-402, doi:10.1175/ 1520-0485(1997)027,0381:SOETCI.2.0.CO;2. Wolfe, C. L., and P. Cessi, 2010: What sets the strength of the middepth stratification and overturning circulation in eddying ocean model? J. Phys. Oceanogr., 40, 1520-1538, doi:10.1175/ 2010JPO4393.1. -, and -, 2011: The adiabatic pole-to-pole overturning circulation. J. Phys. Oceanogr., 41, 1795-1810, doi:10.1175/ 2011JPO4570.1. T. RADKO, D. PEIXOTO DE CARVALHO, AND J. FLANAGAN Department of Oceanography, Naval Postgraduate School, Monterey, California (Manuscript received 16 November 2013, in final form 14 April 2014) Corresponding author address: T. Radko, Department of Oceanography, Naval Postgraduate School, 883 Dyer Road, Bldg. 232, Room 344, Monterey, CA 93943. E-mail: [email protected] APPENDIX Spectral Composition of the Equilibrium Eddy Field An attempt to represent the dynamics of fully developed baroclinic instability in terms of a low-order GRB model (section 3) raises the question of the role and significance of higher-order modes-modes generated through nonlinear multistage and multiscale interactions in the turbulent ocean. Some insight is provided by the following spectral analysis of the numerical simulations for the two-layer Phillips model. Spectral diagnostics are applied to the spatially averaged energy density E and the meridional upper-layer PV flux Fq1, given by ... (A1) Both quantities are expressed in terms of their spectral density distribution: ... (A2) where, according to Parseval's identity, ... (A3) The carets in (A3) denote the Fourier transformed fields, and the asterisk denotes the complex conjugate. Figure A1 presents the patterns of ekl and fkl diagnosed from the simulation in Fig. 2 and time averaged over the quasi-equilibrium period (t.400). Both energy and flux patterns reveal strong mesoscale signals. However, the distribution of fkl is significantly more localized. For instance, we estimated the fraction of spectral power contained in the spectral window jlj, 2lmax, 0:5kmax ,jkj,2kmax in which kmax is the zonal wavenumber of the fastest growing primary instability and lmax is the meridional wavenumber of the fastest growing secondary instability. These modes are directly represented by our equilibration theory, and therefore such calculation quantifies the ability of the GRB model to describe the fully developed eddying field. For the simulation in Fig. A1, we find that this spectral window contains 58.5% of the total energy and 96.7% of the total PV flux. These numbers suggest that the meridional eddy transport is controlled by modes located in spectral space near primary and secondary instabilities, and therefore it can be accurately described by the GRB theory. Of course, the non-GRB modes, excited by nonlinear high-order interactions, could be important for other aspects of the problem, including, for instance, the energetics of baroclinic instability. However, the primary objective of this study is the development of a predictive model formeridional eddy transport, and the GRB theory appears to be adequate in this regard. Our next series of calculations addresses the importance of the energy and enstrophy cascades for the lateral eddy mixing in quasi-equilibrated baroclinically unstable flows. The spectral dynamics of geostrophic turbulence (Charney 1971) is characterized by the upscale energy and downscale enstrophy cascades. In simulations performed in support of the GRB theory (section 4), the upscale cascade was intentionally inhibited by assuming sufficiently large bottom drag. The downscale cascade, on the other hand, is arrested at rather moderate scales by explicit lateral friction. Since the lateral viscosity n due to unresolved scales is poorly constrained, it is of interest to examine the sensitivity of our solutions to n. The immediate consequence of decreasing n is the reduction of the dissipation scale and the associated broadening of the inertial range. The question arises whether these effects can impact the intensity of the meridional transfer and the applicability of the GRB model. This concern was addressed by performing a series of small-domain simulations (Lx 5Ly 520) in which viscosity was systematically decreased from the baseline value of nnd 5531023 (experiment 1) to nnd 5531024 (experiment 2) and then to nnd 5531025 (experiment 3). To resolve the dissipation scale we have employed a much finer grid with Dx5Dy56:531023. Other parameters were kept the same as for the experiment in Fig. 2: (sbnd, r, gnd)5(0:25, 1/ 3, 0:5). In each case, the simulations were initiated by small-amplitude random noise and simulations were extended into the equilibrated statistically steady regimes. In Fig. A2, the results of these simulations are expressed in terms of zonal spectra of energy and fluxes ek 5 Ð ekl dl and fk 5 Ð fkl dl. As the lateral friction decreases, the inertial range broadens and the energy spectrum approaches the power law ek } k23, as expected for quasi-two-dimensional turbulence. The reduction in nnd has an even lesser effect on the flux spectrum, which is largely limited to the mesoscale. The rapid ( fk } k27) dropofffor shorter wavelengths, which is much steeper than either the energy or potential enstrophy spectra, should be attributed to decorrelation between PV and velocity perturbations at small scales. Note the change in sign of fk (at k;6) from negative at mesoscale to positive at submesoscale, which reflects the tendency for weak upgradient transport by small-scale perturbations. Overall though, the presence of fully developed submesoscale variability in the highresolution experiments described in this section has a rather minor effect on the integral characteristics of the system. For instance, the two order of magnitude variation in nnd only results in ;22% change in energy and ;10% change in meridional PV transport. It should be realized that the quasigeostrophic approximation used herein becomes highly suspect for exceedingly small spatial scales. The small-scale variability is geostrophically unbalanced, which allows alternative pathways to the dissipation of energy and very dissimilar spectral interactions (e.g., Molemaker et al. 2010). However, the foregoing experiments are highly significant. They demonstrate that widening of the spectrum of geostrophically balanced variability has a very limited impact on the overall intensity of lateral eddy transfer. Therefore, it is likely that the mixing dynamics can be captured by low-dimensional models, exemplified here by the GRB theory. (c) 2014 American Meteorological Society |