Abstract

The modulation of large-scale eddying flows by gentle variation in topography is examined using a combination of direct numerical simulations and theoretical arguments. The basic state is represented by a laterally uniform zonal current that is restricted to the upper layer of a baroclinically unstable quasigeostrophic two-layer system. Therefore, the observed topographically induced generation of large-scale patterns is attributed entirely to the action of mesoscale eddies. The parameter regime investigated in this study is not conducive to the spontaneous formation of stationary zonal jets. The interaction between the large-scale current, eddies, and topography is described using an asymptotic multiscale model. The ability of the model to explicitly represent the interaction between distinct flow components makes it possible to unambiguously interpret the essential dynamics of the topographic/eddy-induced modulation. The multiscale solutions obtained reflect the balance between the modification of the meridional fluxes of potential vorticity (PV) due to the variation in topography and the corresponding modification of PV fluxes due to the induced large-scale circulation. The predictions of the asymptotic theory are successfully tested by comparing to the ones obtained by direct numerical simulations.

1. Introduction

Despite the general consensus of the geophysical community regarding the significance of eddy-induced transfer of buoyancy and momentum, several key properties of mesoscale variability defy simple dynamic explanation and analytical treatments (e.g., Müller et al. 2005). The ability of eddies to transfer momentum both up and down large-scale gradients (e.g., Shepherd 1987, 1988; Gama et al. 1994; Berloff 2005) makes it difficult to represent the eddy fluxes in terms of explicit closure models (e.g., Marshall et al. 2012). The lack of accurate and reliable eddy parameterizations that can capture these effects, in turn, presents a substantial obstacle for predicting and explaining feedbacks of eddies on large-scale flows. However, such feedbacks are dynamically significant. Mesoscale variability often produces secondary eddy-induced patterns, controlled by the interplay between barotropic and baroclinic energy cascades (e.g., Rhines 1975, 1977; Salmon 1980; Panetta 1993). These dynamics are exemplified by spontaneous formation of quasi-zonal jets in eddy-rich areas in the presence of the planetary vorticity gradient (Maximenko et al. 2005, 2008; Berloff et al. 2009; Dritschel and McIntyre 2008; Kamenkovich et al. 2009 ,2015; Srinivasan and Young 2012; Radko 2016).

This study is focused on yet another source of large-scale, eddy-driven patterns: the interaction of mesoscale variability with bottom topography. The parameter regime chosen for this study is generally not conducive to the spontaneous formation of stationary zonal jets and therefore the observed time-mean patterns are attributed to large-scale topographic modulation. Topography is known to affect linear stability of shear flows (e.g., Hart 1975; Chen and Kamenkovich 2013), and eddy-induced fluxes of momentum and potential vorticity can be substantially altered by topographic features (Treguier and Panetta 1994; MacCready and Rhines 2001; Thompson 2010; Thompson and Sallée 2012). The interactions of eddies with topography generate a systematic stress that does not necessarily resist mean circulation but in some cases induce distinct secondary circulation patterns—the tendency that is aptly referred to as the Neptune effect (Holloway 1986, 1987). Holloway (1992) for instance suggested that eddies tend to drag the flow toward the equilibrium, high-entropy states and that the topographic eddy stresses could be parameterized by assuming viscous dissipation of the deviation from those equilibrium patterns. The flows induced by the interaction between eddies and topography are missing in numerical simulations that do not resolve mesoscales, and therefore realistic representation of mean circulation in such models may require proper adjustments (Holloway 1992, 2008; Frederiksen 1999). The implementation of simple parameterizations of eddy–topography interactions has led to improved simulations, in both global and regional models (Alvarez et al. 1994; Thompson 1995).

The general tendency of eddies to induce persistent flow patterns in the presence of topographic features was convincingly illustrated by Merryfield and Holloway (1999). This study examined randomly forced stratified flows, revealing the emergence of mean circulation patterns and interface displacements that are highly correlated with topography. These secondary circulations are maintained by eddy fluxes of layer thickness anomaly, supporting the key role of mesoscale variability in the process. Meridional ridges in eddying simulations can also lead to the formation of parallel jets due to the topographic beta effect (Vallis and Maltrud 1993) and induce nonlocal patterns in flat-bottom ocean regions (Chen et al. 2015). Boland et al. (2012) suggested that, in systems where the barotropic and layerwise potential vorticity (PV) gradients are not aligned, slowly drifting jets are parallel to the barotropic PV contours. Of direct relevance to our study is the analysis of Dewar (1998) who discussed the origin of the Zapiola anticyclone—the well-defined circulation pattern around a major bottom topographic feature (Zapiola drift) in the South Atlantic. Dewar (1998) related the formation and maintenance of the Zapiola anticyclone to the action of mesoscale eddies. He has shown that in the presence of topography, the lateral mixing of PV by eddies can induce intense time-mean circulation. These dynamics were subsequently confirmed by comprehensive numerical models (De Miranda et al. 1999).

The present study contributes to the subject by introducing a new and sufficiently general framework for the analysis of topographic/eddy-induced modulation of mean circulation. The striking ability of eddies to induce strong secondary flows is illustrated by the model in which the basic flow is restricted to the upper layer of a two-layer quasigeostrophic system. This flow is separated from gently varying topography by a thick, initially quiescent lower layer. Thus, topography does not directly affect the basic current, and the observed topographically induced generation of large-scale patterns is necessarily driven by mesoscale eddies. One of the key objectives of this study is to explain physical mechanisms of such interaction, and the analytical progress in this direction is achieved by employing methods of multiscale analysis.

Multiscale modeling is an actively developing field with numerous applications in various physical sciences (e.g., Mei and Vernescu 2010). Multiscale mechanics is based directly on governing equations and therefore it is free of empirical parameterizations and ad hoc assumptions, often used in other analytical approaches to eddy mean–flow interaction. Multiscale methods typically consider a homogeneous, small-scale pattern and analyze its interaction with a large-scale flow, which ultimately leads to explicit, large-scale equations. While multiscale models formally require asymptotically large separation among scales of interacting components, in practice, this condition does not effectively limit the model applicability (e.g., Radko 2011). The simplest primary pattern used in multiscale mechanics is represented by the Kolmogorov model, a parallel shear flow with sinusoidal velocity profile, maintained against viscous dissipation by external forcing (e.g., Meshalkin and Sinai 1961; Manfroi and Young 1999, 2002; Balmforth and Young 2002, 2005). In an attempt to introduce a more realistic representation of small-scale eddies, various two-dimensional background patterns—cellular, hexagonal, or dipolar—have also been considered (e.g., Gama et al. 1994; Novikov and Papanicolaou 2001; Radko 2011).

A significant technical problem with regard to the application of multiscale techniques is the high sensitivity of the obtained large-scale amplitude equations to the chosen small-scale background patterns. For instance, not only the magnitude but even the sign of the momentum transfer predicted by multiscale models depends on the assumed small-scale eddy patterns (e.g., Gama et al. 1994; Novikov and Papanicolaou 2001; Radko 2011). To make model predictions directly relevant to a particular eddying flow, a proposition has recently been made (Radko 2016) to construct background patterns for multiscale modeling using dynamically consistent eddy models. Such small-scale patterns can be easily obtained from numerical simulations, and the resulting multiscale solutions lead to transparent and unambiguous representation of eddy/mean–field interactions. This technique is referred to as the average eddy model, and in the present study it is applied to the problem of the combined topographic/eddy-induced modulation of large-scale flows.

The paper is organized as follows: In section 2, we describe the model configuration and present preliminary simulations that reveal salient features of the remote topographic/eddy-induced modulation. The observed phenomena are treated using multiscale techniques in section 3, where we formulate large-scale amplitude equations for the zonally oriented topography. In section 4, we compare the multiscale model predictions with the results of direct numerical simulations (DNS) and physically explain the key processes at play. The model is then generalized to include weak, zonal variation in topography (section 5). We summarize results and draw conclusions in section 6.

2. Preliminary simulations

Consider a baroclinically unstable zonal flow represented by the two-layer quasigeostrophic model (e.g., Vallis 2006):

 
formula

where the asterisks denote dimensional quantities; are the streamfunctions; are the potential vorticities; are the depths of the upper and lower layer, respectively; is the reduced gravity; is the reference value of the Coriolis parameter in the beta-plane approximation; ; is the lateral viscosity; is the bottom drag coefficient; and is the topography. The index 1 (2) represents the upper (lower) layer. The configuration used in this study is based on the classical model of baroclinic instability originally proposed by Phillips (1951). The net flow field is separated into the Phillips basic state , representing the laterally homogeneous zonal current of speed restricted to the upper layer and the perturbation . We assume that the basic state is sustained by an external source of energy associated with mechanical forcing by winds and/or differential surface heating. To be specific, we shall consider the eastward basic flow in the Northern Hemisphere . However, the following analysis can be readily reproduced for and/or .

The governing equations in (1) are expressed in terms of and, to reduce the number of governing parameters, nondimensionalized using as the horizontal length scale and as the velocity scale. The resulting nondimensional system takes the form

 
formula

where , , , and are the governing parameters; ; and are the nondimensional perturbation PV fields. In the following numerical experiments, we assume doubly periodic boundary conditions for (ψ1, ψ2) in x and y and integrate the governing system [(2)] using a dealiased, pseudospectral method based on the fourth-order, Runge–Kutta, time-stepping scheme.

Figure 1 presents a typical experiment in the flat-bottom regime (ηb = 0). We consider the configuration, which, in dimensional units, describes the basic current with = 0.05 m s−1, = 10−11 m−1 s−1, = 1 km, = 3 km, = 60 m2 s−1, = 10−6 s−1, and = 25 km—scales that are representative of typical midocean flows. The doubly periodic domain of size = (2500 km, 2500 km) is resolved by (Nx, Ny) = (256, 256) Fourier harmonics. The corresponding nondimensional parameters are

 
formula

The simulation was initialized from rest by a small-amplitude random (ψ1, ψ2) distribution. After the initial period of linear growth, active, statistically steady, eddying motion driven by baroclinic instability was established. A typical snapshot of the spatially homogeneous eddy field in the upper (lower) layer is shown for t = 2 × 105 in Fig. 1a (Fig. 1b). Figures 1c and 1d present the corresponding time-mean streamfunctions ([ψ1], [ψ2]), evaluated by averaging the instantaneous fields (ψ1, ψ2) over the interval 500 < t < 2 × 105. While the time-mean fields reveal no persistent large-scale patterns, the remnants of mesoscale variability remain clearly visible despite the long-term averaging. The magnitude of residual variability systematically reduces with the increasing averaging interval tav, and it is well described by the relations

 
formula

The lack of large-scale patterns in Figs. 1c and 1d indicates that eddy-driven jets that can form spontaneously in regions with active mesoscale variability (e.g., Berloff et al. 2009; Farrell and Ioannou 2003, 2007, 2008; Srinivasan and Young 2012; Radko 2016) are not present in the chosen parameter regime. In particular, their absence is attributed to a relatively large value of bottom drag, which has an adverse effect on jet formation (e.g., Berloff et al. 2011; Radko 2016)

Fig. 1.

(a),(b) The instantaneous (t = 2 × 105) patterns of the upper-layer and lower-layer streamfunctions in the flat-bottom experiment. (c),(d) The corresponding time-mean streamfunctions [ψ1] and [ψ2], where averages are computed over the interval 500 < t < 2 × 105. Note the dramatically reduced magnitude of variability in (c) and (d) associated with temporal averaging.

Fig. 1.

(a),(b) The instantaneous (t = 2 × 105) patterns of the upper-layer and lower-layer streamfunctions in the flat-bottom experiment. (c),(d) The corresponding time-mean streamfunctions [ψ1] and [ψ2], where averages are computed over the interval 500 < t < 2 × 105. Note the dramatically reduced magnitude of variability in (c) and (d) associated with temporal averaging.

Figures 2a–d show the analogous patterns realized in the presence of nonuniform topography with

 
formula

which corresponds to the dimensional height of 100 m. All other parameters of this simulation are identical to those in Fig. 1. While the intensity and general structure of the mesoscale field in both simulations are similar, topography introduces a detectable, large-scale modulation of the flow pattern in the upper layer. This modulation is clearly visible in Fig. 2c, where we plot the time-mean, upper-layer streamfunction. This signal is strongly correlated with topography (Fig. 2e). Assuming that the error in evaluating the time-mean streamfunctions associated with finite averaging interval is described by the empirical relations in (4), we conclude that [ψ1] in Fig. 2c (and in all subsequent calculations) is computed with the accuracy of at least 4%. Note the absence of the corresponding large-scale, time-mean flow in the lower layer (Fig. 2d), despite its direct contact with nonuniform topography.

Fig. 2.

(a)–(d) The counterparts of Fig. 1 for the experiment performed in the presence of meridionally varying topography. (e) Topography.

Fig. 2.

(a)–(d) The counterparts of Fig. 1 for the experiment performed in the presence of meridionally varying topography. (e) Topography.

The emergence of a persistent, large-scale, topographically induced pattern (Fig. 2c) in the configuration where the basic flow is contained in the upper layer is significant. It implies that this structure is generated and maintained by mesoscale variability, which, in turn, is modulated by topography. The following model attempts to examine this effect using techniques of multiscale analysis.

3. Topographic modulation as a multiscale problem

The intent of this section is to predict the topographic modulation of the upper layer by arbitrary large-scale ηb using properties of the flat-bottom homogeneous eddy field. The problem of the combined topographic/eddy-induced modulation of large-scale flows is treated using techniques of asymptotic multiscale analysis (Kevorkian and Cole 1996; Mei and Vernescu 2010). We start with the case of a purely meridional topographic slope and then (section 5) extend our investigation to two-dimensional patterns. Theoretical development generally follows Radko (2016), and therefore here we present an abbreviated description of the multiscale model, which summarizes key steps and emphasizes new effects brought by inclusion of topography.

To explore dynamic consequences of large-scale variation in topography, a new meridional coordinate Y is introduced:

 
formula

where ε ≪ 1 is a measure of the relative spatial scales of the mesoscale eddies and topography. The short and long length scales are treated as independent variables, and y derivatives in the governing equations [(2)] are replaced as follows:

 
formula

The resulting system is given in (A1), and the following calculation is based on the asymptotic expansion in ε. The leading-order component is represented by the fully developed and statistically homogeneous eddy field realized in the flat-bottom configuration. The presence of a gentle, large-scale variation in the bottom topography ηb(Y) is expected to produce large-scale, time-mean flow field pattern in the upper layer:

 
formula

For the uniform bottom drag model,1 the second layer experiences strong frictional decay on large scales, and therefore the large-scale component in the second layer does not appear at the leading O(1) order. This property is consistent with the DNS in Fig. 2 and can be rigorously rationalized using the asymptotic model ( appendix) as follows. If the large-scale, zero-order streamfunction were initially included in the model formulation, then at the second order in the expansion one encounters an additional solvability condition. This condition is revealed by spatially and temporally averaging the O(ε2) components of the second-layer PV equation [(A1)], which reduces to . Therefore, to streamline the exposition, in (8) we insist from the outset that the large-scale, zero-order flow is restricted to the upper layer.

Our main goal is to determine, in a self-consistent manner, how this large-scale pattern is related to the topography ηb(Y), assuming that the background solution is known. When (8) is substituted in the governing equations [(A1)] and terms of the same order are collected, we discover that the O(ε) correction to the background solution in the upper (i = 1) and lower (i = 2) layers takes the following form:

 
formula

Likewise, when (8) and (9) are substituted in (A1), we arrive at the O(ε2) perturbation:

 
formula

Variables , which vary on small scales only, are known as auxiliary functions, and they are linked to the primary solution through a series of PDEs given in (A3)(A6). Note that in (9) and (10) we retained only the terms that are linear in . The linearization was largely motivated by considerations of simplicity and transparency, since multiscale models can incorporate large-scale nonlinearities in a relatively straightforward manner (e.g., Gama et al. 1994; Novikov and Papanicolaou 2001). If nonlinearities in our case were not discarded, then the second-order components of ψ1,2 would also include terms proportional to and , which requires calculating additional auxiliary functions. However, the comparison with fully nonlinear DNS (section 4) shows that the simpler linear model adequately captures the topographic/eddy-induced modulation of large-scale flows in the considered parameter regime.

While the series (8)(10) can be extended to the third order in a similar fashion, the O(ε3) components play no role in the derivation of the closed large-scale amplitude equation. The latter is obtained by averaging the O(ε4) upper-layer balance in small-scale variables, resulting in

 
formula

where the expressions for K(j) (j = 1, 2) in terms of the primary homogeneous eddy fields and auxiliary functions are given in (A9)–(A10). The frictional term in (11) is generally small (ν/K(1) ~ 10−2) and can be neglected for most intents and purposes.

Both components on the left-hand side of (11) originate from the J(ψ1, q1) term in the PV equation. Therefore, its averaging in (x, y, t) represents the leading-order, large-scale divergence of the meridional eddy PV fluxes [ν1q1], where ν1 = ∂ψ1/∂x is the meridional velocity. While in the statistically steady state the meridional eddy PV flux is nearly nondivergent in the limit of weak friction, the multiscale theory offers a much deeper physical insight by identifying two distinct compensating components of this balance. The first component is determined by the large-scale current , and the second is controlled by the topography . Importantly, the transfer coefficients K(j) are fully determined by the properties of the background (flat bottom) solution. The large-scale equation (11) could be further simplified in cases when the topography is meridionally periodic as follows:

 
formula

We regard the simple analytical results in (11) and (12) as being of fundamental significance in eddy–topography interaction theory, as they explicitly connect the structure of secondary eddy-induced flows to topography. They are free from empirical parameterizations of eddy transfer, commonly used in earlier models (e.g., Alvarez et al. 1994; Dewar 1998), and are based entirely on the assumption of scale separation between eddies and topographic features of interest. However, obtaining more specific estimates of the strength of secondary, eddy-induced flows requires calibration of the eddy transfer coefficients K(j). It seems natural to attempt the evaluation of K’s by solving the auxiliary problems (A3)(A6) numerically and then use the resulting solutions to determine (A9)–(A10). However, a significant complication is met at this point, which is the instability of auxiliary problems, a common occurrence in multiscale models of this nature. Since (A3)(A6) are linear with respect to the auxiliary functions and therefore lack finite-amplitude mechanisms of equilibration, this instability can grow indefinitely, precluding the integration of auxiliary problems over long intervals of time. The proposition, however, was made recently (Radko 2016) that auxiliary problems can be integrated over finite periods and then, to ensure the accuracy of the model prediction, ensemble averaged over a large number of realizations. This approach, referred to as the average eddy model, was shown to be very effective in explaining the dynamics of large-scale, eddy-driven patterns, and it is employed in the present study as well.

The proposed procedure was implemented as follows: The governing equations [(2)] in the flat-bottom regime (ηb = 0) were integrated in time to generate the primary statistically homogeneous eddy field . The system evolved to the statistically steady regime by t00 = 500. Afterward, the calculations of primary fields were accompanied by the concurrent integrations of the auxiliary problems (A3)(A6), which were repeatedly solved over finite time intervals [t0, t0 + ΔT], where t0 = t00 + nΔT, n = 1, …, N, and ΔT = 40. Initial conditions for each auxiliary integration were taken to be , j = 1, …, 4. Such integrations were simultaneously performed with different (random) initial conditions for on np = 512 processors. The resulting Ntot = Nnp ≈ 8 × 105 time records of , where τ = tt0, obtained at each integration were averaged over all realizations, which effectively isolated the dynamically significant signal. Figure 3a presents the ensemble-averaged coefficients , which exhibit the tendency to converge in time to specific equilibrium values. As evident from (12), of particular significance for our model, is the ratio of the transfer coefficients K(2)/K(1) (Fig. 3b), which largely determines the large-scale circulation induced by the eddy–topography interaction. It is interesting that the convergence of with τ to the equilibrium value is visibly more rapid than that of the individual transfer coefficients. This ratio remains largely uniform after τ = 15, which instills confidence in the accuracy of the calibration procedure. For instance, its final value r(40) = 0.5577 differs from the average of r over the interval 20 < τ < 40 (rav = 0.5 587) by less than 0.2%. The final values of realized in the ensemble-averaged calculation (Fig. 3) were used as an estimate of K(j) in the model prediction [(12)].

Fig. 3.

(a) The temporal records of the ensemble-averaged instantaneous transfer coefficients of the multiscale model , j = 1, 2, and (b) their ratio.

Fig. 3.

(a) The temporal records of the ensemble-averaged instantaneous transfer coefficients of the multiscale model , j = 1, 2, and (b) their ratio.

4. Testing and interpretation

To validate the multiscale model, prediction (12) is now compared with the DNS results (section 2). Figure 4 presents the zonally and temporally averaged upper-layer streamfunction along with the corresponding theoretical estimate. Their apparent agreement indicates that the multiscale model adequately captures the dynamics of remote topographic/eddy-induced modulation of the large-scale flow. The explicit character of the model also makes it possible to identify the key effects involved in the formation of large-scale patterns and interpret the results accordingly.

Fig. 4.

The zonally and temporally averaged upper-layer streamfunction diagnosed from the DNS with meridionally varying topography (Fig. 2) is plotted (solid) as a function of y along with the corresponding theoretical estimate (dashed).

Fig. 4.

The zonally and temporally averaged upper-layer streamfunction diagnosed from the DNS with meridionally varying topography (Fig. 2) is plotted (solid) as a function of y along with the corresponding theoretical estimate (dashed).

As indicated in section 3, the amplitude equation [(11)] can be physically interpreted in terms of the large-scale meridional fluxes of PV in the upper layer. In particular, represents the divergence of the eddy-driven PV fluxes that is associated with the large-scale circulation in the upper-layer . The term , in turn, reflects the PV flux divergence induced by the variability in bottom topography. The equilibrium state can be maintained only when the two processes approximately balance each other, and we now attempt to illustrate how this balance is ultimately achieved.

Baroclinic instability of zonal flows is strongly affected by the meridional gradients of the large-scale PV. The meridional topographic slope can either stabilize or destabilize the flow by changing the PV gradient in the bottom layer (Pedlosky 1987; Chen and Kamenkovich 2013). This property can be linked to the topographic steering effect—the tendency of fluid columns, particularly in deep layers, to follow f/h contours (f here is the Coriolis parameter and h is the layer thickness), which restricts meridional displacements required for baroclinic instability. Also known is the critical role of the barotropic PV gradients in modulating the eddy intensity and setting the orientation of the associated eddy-driven patterns (e.g., Boland et al. 2012). Both barotropic and baroclinic (lower layer) PV arguments suggest that the topographic steering tendency would be reduced if the ocean depth increases northward . In this case, the northward increase in f is partially offset by the increase in h, and therefore f/h can be maintained for larger meridional displacements. Hence, the baroclinic instability is likely to be intensified, and mesoscale variability will be more energetic. On the other hand, if the ocean depth decreases to the north then meridional displacements will be more constrained. In this case, the baroclinic instability—and the resulting eddies—will be weaker.

Our simulations indicate that the eddy-induced PV flux in the upper layer is southward and thus negative:

 
formula

This is consistent with the tendency of eddies to transport PV downgradient, since the gradient of the basic PV is positive . Therefore, the direct consequence of the weakening of mesoscale activity for is the decrease in the magnitude of eddy-induced PV flux in the upper layer. Since the PV flux is still negative (southward), the reduction in its magnitude |Fq1| is equivalent to an increase in Fq1. Likewise, for we expect a decrease of Fq1. The schematic in Fig. 5 illustrates the associated modification of the flow pattern. The increase (decrease) in the PV flux for implies that the PV fluxes converge in the regions where ηb is large (e.g., y1 and y3) and diverge where ηb is small (e.g., y2). As a result, PV amplifies (reduces) in the regions of high (low) ηb. The large-scale streamfunction varies in the opposite sense:

 
formula

and therefore we expect the continuous decrease (increase) in ψ1 at high (low) ηb. Note that the approximation in (14) is fairly robust and not model dependent. It is based on two conditions: (i) the abyssal flows are generally much weaker than those in the main thermocline, and (ii) the scale of topographic features of interest exceeds the radius of deformation. Both assumptions are reasonable and are likely to be satisfied in much of the Word Ocean.

Fig. 5.

Schematic diagram illustrating the development and equilibration of remotely induced, large-scale patterns in the upper layer in response to the variation in bottom topography.

Fig. 5.

Schematic diagram illustrating the development and equilibration of remotely induced, large-scale patterns in the upper layer in response to the variation in bottom topography.

The ultimate equilibration of the system can be explained (Fig. 5) by focusing on the following negative feedback mechanism. The aforementioned topographic/eddy-induced changes in the streamfunction field correspond to the reduction (amplification) of zonal velocity in the regions where . As a result, the magnitude of the vertical zonal shear reduces for y2 < y < y3 and amplifies for y1 < y < y2. Since the intensity of baroclinic instability is controlled by the magnitude of shear (Pedlosky 1987), the strength of eddy transfer of PV is modulated accordingly. The circulation-induced PV fluxes reduce (increase) in the regions where , directly opposing the topographic tendencies. The system reaches the equilibrium state when the topographic and circulation-induced effects balance each other. A corollary of the proposed physical interpretation is that the equilibrium pattern of the streamfunction is expected to be opposite to that of topography: ψ1 should be high when ηb is low and vice versa. This prediction is readily confirmed by the inspection of both DNS (Fig. 2) and multiscale (Fig. 4) results. Note that the increase (decrease) in ηb is also accompanied by the increase (decrease) in the interfacial height . Thus, eddies act to moderate changes in the thickness of the bottom layer produced by variable large-scale topography. It should be emphasized, however, that the inverse relation between topography and time-mean streamfunction ([ψ1]) is realized only for the eastward Phillips flow assumed in this study (). If the background current were strong and negative (westward), the reinterpretation of the foregoing arguments would suggest a direct relation between [ψ1] and ηb.

To confirm the envisioned (Fig. 5) chain of events resulting in topographic/eddy-induced modulation, we now examine the adjustment of the upper-layer flow in DNS in greater detail. The following experiment is identical in all respects to the one in Fig. 2 (section 2) except for the zonal extent of the domain. The latter was increased to Lx = 800 in order to provide more statistically representative x averages. Figure 6 presents the Hovmöller (t, y) diagram of the key variables controlling the circulation in the upper layer. The evolution of the large-scale meridional PV flux is shown in Fig. 6a. The superscripts (sa) hereinafter represent smoothing in y, which retains scales ly > lsm = 30, and averaging in x. The first evolutionary stage representing the linear growth of baroclinic instability (0 < t < 100) is followed by the adjustment phase (100 < t < 250) in which the meridional PV fluxes are spatially nonuniform. In accord with the physical argument in Fig. 5, the highest fluxes occur in the vicinity of y ≈ 0, where the topographic gradient is at its largest. The convergence of fluxes (Fig. 6b) triggers the amplification of PV in the northern region (y > 0, ηb > 0) and its decrease in the southern (y < 0, ηb < 0), as indicated in Fig. 6c. This adjustment is largely completed by t ≈ 250, when the system reaches the quasi-equilibrium state characterized by the nearly nondivergent net meridional PV flux. The PV in this stage (t > 250) remains approximately uniform in time (Fig. 6d). The large-scale streamfunction (Fig. 6e) is strongly anticorrelated with PV (Fig. 6d) in agreement with (14). As a result, the streamfunction becomes anticorrelated with bottom topography (Fig. 6f), which is fully consistent with the prediction of the multiscale theory (12) and with the associated physical argument (Fig. 5).

Fig. 6.

Hovmöller diagrams of (a) the upper-layer PV flux , (b) its meridional convergence , (c) the rate of change of PV , (d) PV , and (e) streamfunction . (f) The topography pattern. The superscripts (sa) represent smoothing in y, which retains scales ly > lsm = 30, and averaging in x.

Fig. 6.

Hovmöller diagrams of (a) the upper-layer PV flux , (b) its meridional convergence , (c) the rate of change of PV , (d) PV , and (e) streamfunction . (f) The topography pattern. The superscripts (sa) represent smoothing in y, which retains scales ly > lsm = 30, and averaging in x.

5. Effects of zonal variation in topography

The multiscale methods can be applied to fully two-dimensional topographic patterns ηb(x, y) in a straightforward albeit algebraically extensive manner. However, for the sake of brevity and clarity, here we present a relatively simple case in which the bathymetric variation along the current is gentler than the variation across it. An explicit connection to the problem discussed in the previous section can be made by focusing on the asymptotic sector

 
formula

The specific choice of scaling the large-scale zonal coordinate X is dictated by our desire to concurrently incorporate both meridional eddy transfer, which scales as O(ε4), and the beta effect in large-scale amplitude equation. Another advantage of considering this strongly anisotropic (elongated in x direction) system is that the auxiliary problems (A3)(A6) are not modified by the inclusion of zonal variation in ηb. The zonal variability in the limit of (15) enters the formulation only at the final stage; averaging of the O(ε4) balances in small-scale variables. In this case, the counterpart of the two-dimensional, large-scale model [(11)] becomes

 
formula

where the coefficients K(j), j = 1, 2, remain the same as computed in section 3. In view of (15), it is apparent that (16) retains its form even when it is written using the original (small scale) spatial variables (x, y). The large-scale field is readily obtained for any given ηb pattern using the spectral method.

To determine whether the resulting large-scale solutions capture the essence of the remote topographic/eddy-induced modulation, we now compare our asymptotic predictions with the corresponding DNS results. Our first example (Figs. 7a–c) is the case in which topography varies downstream by periodically shifting in the meridional direction:

 
formula

where Lx = 800 and Ly = 50. The pattern [(17)] is shown in Fig. 7a, and the corresponding time-mean streamfunction [ψ1] obtained from DNS is plotted in Fig. 7b. Note that the large-scale flow is anticorrelated with topography, which is consistent with the physical arguments introduced earlier in section 4. Figure 7c presents the corresponding prediction based on the multiscale model (16).

Fig. 7.

(a),(d) Bottom topography. The zonally elongated topographic features in (a) and (d) remotely generate well-defined, time-mean patterns in the upper-layer streamfunction fields shown in (b) and (e). (c),(f) The corresponding solutions of the multiscale amplitude equation [(16)].

Fig. 7.

(a),(d) Bottom topography. The zonally elongated topographic features in (a) and (d) remotely generate well-defined, time-mean patterns in the upper-layer streamfunction fields shown in (b) and (e). (c),(f) The corresponding solutions of the multiscale amplitude equation [(16)].

In our second example (Figs. 7d–f), topography is represented by a zonally elongated Gaussian elevation:

 
formula

The resulting time-mean streamfunction is shown in Fig. 7e, and the corresponding multiscale estimate is in Fig. 7f. As in the previous case (Figs. 7a–c), the flow is strongly modulated by topography; the streamfunction has a maximum downstream of the seamount and exhibits the strongest zonal currents north and south of it. The comparison of DNS (Figs. 7b,e) and multiscale (Figs. 7c,f) predictions indicates that the multiscale model successfully captures large-scale modulations of the background flow by the topography. Such consistency lends credence to all elements on which the multiscale model is based. We also emphasize that the relatively accurate prediction in Fig. 7 was obtained for a rather modest scale separation between the topography and eddies. This finding supports earlier conclusions (e.g., Radko 2011) that the disparity in scales of interacting components—a formal requirement of multiscale methods—may not be critical for their applicability.

Our final set of experiments attempts to determine whether the remote modulation effect is sufficiently general and persists in other parameter regimes. Of particular concern are the low-drag systems, which are characterized by the spontaneous emergence of zonal jets (e.g., Berloff et al. 2011; Radko 2016). Figure 8 presents a series of simulations that are analogous in all respects to that in Fig. 7e, except for the bottom drag coefficient, which is reduced to γ = 0.25, 0.125, 0.1, and 0.08 in Figs. 8a–d, respectively. This reduction results in a systematic intensification of topographic/eddy-induced flows, a feature that can be readily attributed to the adverse effect of the bottom drag on the intensity of baroclinic instability. More significant dynamically, however, is the appearance of the quasi-zonal jets in the low γ regime (Fig. 8d). Despite the emergence of jets, the imprint of topography on the upper-layer flow remains clearly identifiable. Likewise, the topographically induced time-mean patterns exhibit limited sensitivity to the value of beta. For instance, its reduction from β = 0.2 to 0.1 resulted in the pattern of [ψ1], which is structurally similar to that in Fig. 7e, although the magnitude of the signal increased by 39%.

Fig. 8.

Sensitivity of the flows induced by the eddy–topography interaction to the bottom drag. The time-mean patterns of the upper-layer streamfunction for γ = (a) 0.25, (b) 0.125, (c), 0.1, and (d) 0.08.

Fig. 8.

Sensitivity of the flows induced by the eddy–topography interaction to the bottom drag. The time-mean patterns of the upper-layer streamfunction for γ = (a) 0.25, (b) 0.125, (c), 0.1, and (d) 0.08.

6. Discussion

This study examines the modulation of large-scale flows by spatially variable topography using a combination of direct numerical simulations and asymptotic models. Of particular interest is the action of eddies that emerge as a result of baroclinic instability of the broad background flow. Variation in topography introduces spatially nonuniform barotropic and lower-layer PV gradients, which modulate the intensity of eddies. Upon reaching finite amplitudes, these eddies produce spatially nonuniform fluxes of PV, and this internally generated forcing transforms the mean state. The significance of the presented results is twofold. We elucidate key mechanisms of the remote topographic/eddy-induced modulation of large-scale flows and concurrently introduce in our analysis a powerful analytical approach, which is based on the techniques of multiscale modeling.

The basic state considered in this study is characterized by the deep and motionless lower layer. Therefore, the modification of the upper-layer flow by the variation in topography, which is restricted to the lower layer, is necessarily eddy driven. The stationary structures identified in simulations are directly correlated with topography, and our major goal is to physically explain their origin, dynamics, and maintenance mechanisms. This is accomplished by formulating theoretical multiscale solutions, which explicitly capture the interplay between eddies, topography, and large-scale flows, thus leading to unambiguous interpretation of the numerical results. The proposed theory reflects the balance between changes in the meridional PV flux due to the variation in topography and due to the induced large-scale circulation. The multiscale model is validated by comparing the theoretical predictions with the corresponding DNS results.

The effects of remote topographic/eddy-induced modulation are surprisingly strong. When the model prediction in (12) is cast in the dimensional form, we arrive at

 
formula

Using (19), in turn, makes it possible to estimate the perturbation in sea surface height and in the thermocline depth :

 
formula

Thus, for instance, a seamount of only 500 m, which is separated from the thermocline by the initially inactive abyssal layer of 3000 m, is expected to produce a persistent eddy-induced depression in the sea surface height of and the reduction in thermocline thickness by . Such effects are easily detectable in observations and should be taken into account in the analysis of oceanographic data, particularly in regions with strong baroclinic instability and energetic eddy field. In more quiescent parts of the ocean, the upper-ocean currents can be effectively decoupled from the topography.

Our results also underscore some inherent limitations of coarse-resolution numerical models that do not resolve mesoscale variability. Such simulations cannot capture the topographic/eddy-induced modulation of large-scale flows unless they employ high-fidelity parameterization schemes. Development of such schemes (e.g., Holloway 1992) remains a major challenge in ocean and climate modeling, and therefore even the ability of coarse-resolution models to represent large-scale currents may be questionable. Another possible application of our findings is in the field of altimetric bathymetry, an active area of research that attempts to utilize satellite radar altimeter measurements of the ocean surface height to infer the presence of mountains below (e.g., Smith and Sandwell 1997). To the best of our knowledge, such studies have not yet incorporated the effects of remote eddy-induced modulation, which could be a source of a systematic bias in the reanalysis.

Equally important are the methodological advancements afforded by this study. Here, we present an interesting example of the successful utilization of multiscale techniques for problems involving the interaction between mesoscale variability, topography, and time-mean flows. While multiscale methods are well known and sufficiently widespread in most physical sciences (e.g., Mei and Vernescu 2010), their application to analyses of mesoscale variability (e.g., Manfroi and Young 1999, 2002; Radko 2011) has been mostly limited to highly idealized conceptual theories. The defining feature of the present model is its quantitative character. The proposed multiscale solutions are compatible with, and testable by, fully nonlinear simulations or observations. The key novel element of the formulation is the choice of a dynamically consistent time-dependent background eddy field, the approach that is referred to as the average eddy model (Radko 2016). Use of a realistic, numerically derived mesoscale field as a background does not adversely affect the model transparency, which makes it possible to trace and explain the chain of events leading to the remote topographic/eddy-induced modulation.

The present study is not meant to offer an exhaustive investigation of the remote modulation problem, and the multiscale model can be extended in a number of other directions. For instance, topographic variability should not necessarily be restricted to large spatial scales. Assuming that topography contains both large-scale and small-scale components [e.g., ] will make it possible to develop more physical and realistic topographic drag models. The application of our analysis to the actual bathymetric features of the world’s oceans may be complicated by the lack of clear scale separation between distinct scales of topography. In this regard, highly significant is our finding that the disparity in scales of interacting components, which is formally required by multiscale methods, may not be essential for their performance. The accuracy of the model predictions could also be improved in a straightforward manner by extending the asymptotic solutions beyond their leading-order predictions. The use of more general (nonquasigeostrophic and/or continuously stratified) governing systems opens another promising avenue of investigation. The analysis of nonzonal basic flows and the inclusion of large-scale nonlinearities in the multiscale formulation will undoubtedly bring new insights into the problem.

Nevertheless, even the present minimal model is highly suggestive. It reveals the basic dynamics of the remote topographic/eddy-induced modulation, concurrently serving as a proof-of-concept illustration of the power and utility of multiscale modeling for geophysical applications. Although the importance of topography for large-scale circulation may be intuitively obvious, the exact mechanisms that modulate the circulation in the upper ocean are complex and remain to be fully explained, which motivates further research into this problem.

Acknowledgments

The authors thank the editor Paola Cessi and the anonymous reviewers for helpful comments. Support of the National Science Foundation (Grants OCE-1155866 and OCE-1154923) is gratefully acknowledged.

APPENDIX

Multiscale Formulation

We are concerned by the ability of mesoscale eddies to modulate the large-scale, time-mean flow in the presence of variable topography. To quantify the dynamics of cross-scale interactions, we introduce the large-scale variable Y, representing the meridional extent of the modulating pattern, which is related to the small-scale variable y by (6). We treat as independent variables and replace y derivatives in the governing equations in (2) using transformation (7). The result is

 
formula

where and

 
formula

To examine the influence of the large-scale variation in bottom topography on the time-mean patterns, the solution is sought in terms of a power series in (8)(10). These series satisfy the governing equations, provided that

 
formula
 
formula
 
formula

and

 
formula

The linear, homogeneous, differential operator A in (A2)(A5) is defined as follows:

 
formula

where

 
formula

Since the operator A involves only the small-scale spatial variables, (A2)(A7) can be solved for , j = 1, …, 4. Note that the expansion in (8)(10) stops at the second order in ε. Although the third-order balance can be treated in a similar fashion, it plays no role in derivation of the closed, large-scale model. The final condition results from the fourth-order, upper-layer equation, which is averaged in small-scale variables (x, y, t), resulting in (11), where

 
formula
 
formula

The angular brackets in (A8) and (A9) represent averaging in x and y, and square brackets represent averaging in time over a period exceeding the typical eddy scale. The counterparts of K(j) in which the averages are taken in space only will be denoted by .

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Footnotes

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1

We have also explored the scale-dependent drag models in which frictional decay of large scales is less rapid than those of mesoscale, which allowed inclusion of the lower-layer, large-scale flows at the leading order. The results were qualitatively similar, and therefore here we present a more traditional γ = const model.