## 1. Introduction

The effect of transient eddies on the time-mean state is a fundamental problem in the theoretical studies of the circulation of the atmosphere and ocean. By redistributing momentum, heat, and vorticity in a systematic fashion, eddy flux divergences are capable of having an important impact on the large-scale, time-mean circulation. One way eddies can affect the larger-scale circulation is through the driving of mean motions. Here, we examine this phenomenon through a study of nonlinear rectification, the generation of nonzero mean flow from a forcing with zero mean, in an idealized setup. Specifically, we examine the time-mean flow response of a barotropic and equivalently barotropic fluid subject to a simple eddy vorticity forcing that is localized in space and oscillatory in time. The emergence of the mean flow is a result of nonlinear terms producing finite time-mean fluxes (Reynolds stresses) of momentum and vorticity, whose convergences and divergences act as a driving force for the time-mean flow.

Interest in the problem was originally motivated by its relevance to the dynamics of deep recirculation gyres observed with the eastward jet extensions of western boundary current (WBC) systems such as the Gulf Stream and Kuroshio (Fig. 1). One hypothesis for the driving of these abyssal recirculations originating in historic eddy-resolving ocean circulation studies (see, e.g., Holland and Rhines 1980) is through the action of energetic surface eddies in the thermocline, which act to provide localized sources and sinks of vorticity (“plungers”) to the deep ocean through fluctuating thickness fluxes. Understanding the details of the idealized rectification problem considered here will aid in evaluating the relevance of this mechanism to driving the deep recirculation gyres (Hogg 1992; Bower and Hogg 1996; Chen et al. 2007; Jayne et al. 2009).

The problem has more general application in our understanding of wave/eddy rectification from a localized forcing, serving as the contrasting limit to spatially broad (basin scale) forcing (Pedlosky 1965; Veronis 1966), and is important, given that many sources of ocean eddies are intermittent in space. For example, rectification from a localized forcing has relevance to the response of the ocean to a spatially localized wind stress pattern or a spatially localized concentration of eddy activity generated by mean flow instability. The radiation of Rossby waves by WBC jets and/or their recirculations (Flierl et al. 1987; Malanotte-Rizzoli et al. 1987; Hogg 1985) make these jet systems themselves potential localized wave sources.

Even more generally, the generation of mean zonal motion by the action of locally forced eddies has relation to the phenomenon of zonal jet formation, observed to occur spontaneously in turbulent *β*-plane flows (Rhines 1975, 1977; Williams 1979). This process of jet formation by the nonuniform stirring of potential vorticity (PV) resulting in an inhomogeneous distribution of PV gradients (the “PV staircase”) has been extensively discussed in the literature (Marcus 1993; Vallis 2006; Dritschel and McIntyre 2008). The topic has recently received renewed interest given the discovery of zonal jets in ocean observations (Maximenko et al. 2005) and ocean general circulation models (Nakano and Hasumi 2005; Richards et al. 2006). As discussed in Huang et al. (2007), a deeper understanding of the behavior of the zonal anisotropy in the mean velocities in these studies requires detailed studies based on the dynamical properties of the oceanic eddies/waves and jets, and investigating the scenario that zonal anisotropy is a result of the nonisotropic dispersion relation of Rossby waves may be fruitful.

The work presented here extends earlier studies of eddy-driven mean flow from localized forcing, specifically the pioneering laboratory experiments by Whitehead (1975) and numerical simulations by Haidvogel and Rhines (1983, hereafter HR83). The former was the first experimental demonstration of the generation of mean zonal currents from a localized periodic forcing, whereas the latter examined this phenomenon numerically through simulations of two-dimensional flow on a midlatitude *β* plane. In their analysis, HR83 make many insights into the rectification mechanism that forms the starting point for the present study. Notably they demonstrate the usefulness of the analytical solution in the form of a time-periodic Green’s function in understanding the forced wave field. Furthermore, from analysis of their numerical simulations, they demonstrate that the dynamics of the rectified circulation is related to the mean meridional eddy PV flux and that regions of counter (up) gradient PV fluxes provide the propulsion necessary for fluid particles to cross the mean quasigeostrophic contours.

The present work seeks to extend HR83 in several ways. Specifically the work presented here aims to

examine the rectification mechanism in more detail through examination of the eddy flux convergences of momentum and vorticity and the eddy enstrophy balance;

extend the study of parameter variations of the forcing and the flow and understand the mean flow dependence on these parameters in terms of the dynamics of the rectification mechanism;

explore and understand the effects of stratification;

explore and understand the effects of a background mean flow; and

extend these results from the weakly nonlinear to the fully nonlinear regime to examine which results from the small forcing amplitude limit breakdown.

This paper is organized as follows: In section 2, we describe insights into the rectification mechanism gained from an analytical expansion solution that is valid in the weakly nonlinear limit. In section 3, we describe numerical simulations and insights gained from the examination of the nonlinear eddy–mean flow interaction. In section 4, we discuss the results of numerical model parameter studies, describing the dependence of the strength of the rectified flow generated for various forcing and flow parameters, stratification, and mean background flows. In section 5, we extend our study from a weakly nonlinear regime to a strongly nonlinear regime. In the final section, we discuss the relevance of the study to oceanic applications.

## 2. Analytical solution

*β*plane forced by a spatially localized, oscillatory vorticity source. We add the effect of stratification by considering the dynamics in a reduced gravity configuration. This setup is an idealized model for motion in the deep ocean forced by a fluctuating thermocline above and is relevant to the driving of the deep recirculation gyres in WBC jet systems. The governing equation expressing the conservation of QGPV that we consider is

*q*, the QGPV, is given by

*x*is the zonal coordinate,

*y*is the meridional coordinate, and

*t*is time. The variable

*ψ*is the streamfunction such that the horizontal velocity components are given by

*u*= −(∂/∂

*y*)

*ψ*and

*υ*= (∂/∂

*x*)

*ψ*, and

*ζ*= ∇

^{2}

*ψ*is the relative vorticity. Here,

*r*

_{spin}is the barotropic spindown rate, the inverse time scale for vorticity decay due to linear (bottom) friction. We choose linear friction as our parameterization of dissipative effects here but note that numerical simulations with Laplacian (lateral) viscosity give qualitatively similar results in the weakly dissipative regime we consider. The variable

*R*is the Rossby deformation radius, and

_{D}*β*is the meridional gradient of planetary vorticity. The forcing

*AF*(

*x*,

*y*) cos(

*ωt*) is of the form of a localized oscillatory source/sink of vorticity (with zero time-mean vorticity input) fixed at the center of the basin with amplitude

*A*, an angular frequency

*ω*, and a spatial distribution described by

*F*(

*x*,

*y*). It can be interpreted as a localized and time-varying vertical velocity of the form (

*f*

_{0}/

*D*)

*w*(

*x*,

*y*,

*t*), where

*f*

_{0}is the reference Coriolis frequency,

*D*is the barotropic ocean depth, and

*w*is the vertical velocity at a layer interface.

*L*,

*T*, and

*U*. We scale length and time by the external forcing parameters (taking

*L*to be the characteristic length scale of the forcing and

*T*to be the inverse of the forcing frequency) and scale

*U*assuming a dominant balance of relative vorticity spinup and forcing (the expected high-frequency balance for Rossby waves) such that

*U*=

*AL*/

*ω*. The scaled governing equation can then be written in the form

*β** =

*βL*/

*ω*is the nondimensional planetary vorticity gradient and represents the relative importance of the planetary vorticity gradient to the gradients of relative vorticity; and

*μ*=

*U*/

*ωL*=

*A*/

*ω*

^{2}is the wave steepness and gives the degree of nonlinearity of the waves. Finally,

*R*=

*r*

_{spin}

*T*is the nondimensional linear friction coefficient, which is the inverse of the spindown time relative to the characteristic time scale.

*J*(

*ψ*, ∇

^{2}

*ψ*). If the forcing amplitude is small, however, one can make progress by expanding the solution in powers of the forcing amplitude. We set the nondimensional forcing amplitude such that

*μ*is a small parameter. We then take

*ψ*to have the form

*μ*then yields

*μ*equation is a linear equation whose solution is

*γ*

^{2}=

*B*

^{2}−

*S*

^{−1}is the radius of the Rossby wave dispersion circle,

*B*=

*β*/2

*ω*is the radius of the barotropic Rossby wave dispersion circle, and

*γ*replacing

*B*in the argument of the Hankel function. [We also note that we have corrected the sign error for the time-dependent term and the phase shift factor in the argument of the exponential to Eq. (3.2) of HR83.]

This more generalized solution immediately shows two important effects of stratification on the problem. First, stratification reduces the wavenumber of the response shifting the excited wave field to longer wavelengths than in the absence of stratification. Second, it introduces a new cutoff, as for sufficiently large values of the inverse Burger number (such that *γ* becomes imaginary) radiating solutions no longer exist. The appreciation of these effects aids in the interpretation of the results of the numerical parameter study varying stratification discussed in section 4.

*μ*

^{2}. Taking the time mean of (6) yields

*x*) of the time-mean eddy relative vorticity flux divergence of the first-order, linear forced wave field. It is important to note the first-order field has a zero time mean.

_{E}The first-order wave field, its time-mean eddy vorticity flux divergence, and the second-order time-mean flow it drives for the Green’s function solution (7) in the barotropic limit are shown in Fig. 2 (top). The Green’s function solution does not produce a time-mean rectified flow outside the immediate vicinity of the forcing, a consequence of the east–west antisymmetric pattern of vorticity flux divergence that, because of its exact zonal antisymmetry and the dependence of the rectified flow on the zonal integral of the flux divergence, produces zero rectified flow in the far field.

A comparison of properties of the analytical solutions for (top) a forcing function with a delta function spatial dependence (the Green’s function solution) (*β** = 0.05) versus (bottom) a forcing function with a Laplacian of a Gaussian spatial dependence (*β** = 0.05, *μ* = 0.25, and *L _{F}* = 0.1): (left) the wave field (as visualized by a snapshot of the instantaneous streamfunction), (middle) the time-mean eddy vorticity flux divergence that drives the time-mean rectified flow, and (right) the resulting time-mean rectified flow (as visualized by the time-mean streamfunction). Positive values are shaded light gray, and negative values are shaded dark gray (here and in all following contour plots).

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A comparison of properties of the analytical solutions for (top) a forcing function with a delta function spatial dependence (the Green’s function solution) (*β** = 0.05) versus (bottom) a forcing function with a Laplacian of a Gaussian spatial dependence (*β** = 0.05, *μ* = 0.25, and *L _{F}* = 0.1): (left) the wave field (as visualized by a snapshot of the instantaneous streamfunction), (middle) the time-mean eddy vorticity flux divergence that drives the time-mean rectified flow, and (right) the resulting time-mean rectified flow (as visualized by the time-mean streamfunction). Positive values are shaded light gray, and negative values are shaded dark gray (here and in all following contour plots).

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A comparison of properties of the analytical solutions for (top) a forcing function with a delta function spatial dependence (the Green’s function solution) (*β** = 0.05) versus (bottom) a forcing function with a Laplacian of a Gaussian spatial dependence (*β** = 0.05, *μ* = 0.25, and *L _{F}* = 0.1): (left) the wave field (as visualized by a snapshot of the instantaneous streamfunction), (middle) the time-mean eddy vorticity flux divergence that drives the time-mean rectified flow, and (right) the resulting time-mean rectified flow (as visualized by the time-mean streamfunction). Positive values are shaded light gray, and negative values are shaded dark gray (here and in all following contour plots).

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

*F*(

**x**) by numerically evaluating the convolution integral

*F*(

*x*,

*y*) were considered; however, for the mechanism that we describe here, it is important only that the forcing be sufficiently localized in space (sufficiently localized will be discussed later). To supply a localized forcing and yet still allow a large range of amplitudes to be considered, we favor a forcing function of the form

*r*is the radius from the origin of the forcing and

*L*is the forcing length scale. With this particular forcing, at any instant in time the basin integral of the PV input due to the forcing is approximately zero (it is exactly zero for infinite basin size), whereas the PV forcing input is concentrated within a radius

_{F}*L*of the plunger’s center. Note, however, that the results described remain robust for the forcing used by HR83.

_{F}As before, to compute the time-mean rectified flow associated with this wave field, we compute the zonal integral of the time-mean Jacobian using the particular solution *ψ*_{1P} in place of *ψ*_{1}. We show the wave field, its time-mean eddy vorticity flux divergence, and the time-mean flow that it drives in Fig. 2 (bottom). In contrast to the Green’s function solution, the particular solution for a forcing function with finite spatial extent does produce a pattern of vorticity flux divergence whose zonal integral is consistent with the two-gyre circulation pattern with an eastward jet at the latitude of the forcing and westward flow north and south of this. We conclude that, at least in the weakly nonlinear, inviscid limit, it is necessary for the forcing to have a finite length scale in order to generate rectified flow in the far field. The contrast between the solutions also implies that the mechanism that is generating the rectified flow is occurring inside the forcing region (where the Green’s function and particular solution differ) and not by the waves in the far field (where they do not).

## 3. Numerical simulations

To examine the effect of the full nonlinearities on the problem, we perform numerical simulations of the solution to (2) with the forcing specified by (11). We use the simulations to diagnose the nonlinear terms responsible for the driving of the time-mean flow and explore the dependence of the rectification mechanism on flow and forcing parameters. Integration in time and space is done using a scheme that is center differenced in the two spatial dimensions (an “Arakawa A grid”) and stepped forward in time using a third-order Adams–Bashforth scheme (Durran 1991). Advective terms are handled using the vorticity-conserving scheme of Arakawa (1966). Further details on the numerical method can be found in Jayne and Hogg (1999). The nondimensional grid spacing is 0.1 nondimensional length units and the number of grid points is 6001 (east–west) × 6001 (north–south). With the origin at the center of the domain, this puts the western boundary at *x* = −300 and the northern boundary at *y* = 300 nondimensional length units. The domain is closed with solid wall boundaries, and dissipative sponge layers, 2000 grid points wide, are placed next to all boundaries to absorb all waves leaving the domain. The width of these sponge layers is chosen to be sufficient to eliminate basin modes from the domain. In the interior, we examine the solution in the limit where the nondimensional linear friction coefficient *R* is small enough that dissipation is negligible in the time-mean vorticity balance. This near-inviscid limit is of interest both because it allows us to apply the theoretical considerations derived by HR83, and because, by minimizing the effect of friction, the dominant balances are simplified and emphasize eddy effects. In the “typical run” simulation (which we describe here in detail and around which parameter studies are varied), we set *S*^{−1} → 0 (the barotropic rigid-lid limit), *μ* = 0.25, *β** = 0.05, and *R* = 5 × 10^{−8}. These choices are consistent with, for example, dimensional scales *L* and *T* representing forcing length and time scales of *L _{F}* = 250 km and

*ω*= 2

*π*/60 days

^{−1}. They imply a forcing amplitude equivalent to a peak vertical velocity of ~2 × 10

^{−5}m s

^{−1}(here we assume a midlatitude value of

*β*= 2 × 10

^{−11}m

^{−1}s

^{−1}, a midlatitude value of

*f*= 1 × 10

^{−4}s

^{−1}, and a barotropic ocean depth of

*D*= 5000 m) and an inverse barotropic spindown time in the interior of ~(1/30 000) yr

^{−1}. As such, the forcing scales and nondimensional parameters are typical of oceanic synoptic scales in a midlatitude ocean. The very weak interior friction implies that a rectified flow would extend west to nearly infinity were it not for the western sponge layer, which closes the recirculations before the expected distance

*βL*

^{2}/

*R*(the distance traveled by a wave with speed

*βL*

^{2}in a spindown time of 1/

*R*) (Rhines 1983; Hsu and Plumb 2000).

The Reynolds stresses of vorticity diagnosed in the numerical simulations (shown for the typical set of parameters in Fig. 3) illustrate that the key to the eddy generation of mean flows is a systematic up-gradient (northward) eddy flux of potential vorticity that occurs in the vicinity of the forcing. This eddy transport produces a flux convergence in the northern half and a flux divergence in the southern half of the forced region. Based on the notion that the time-mean eddy flux divergence drives the mean rectified flow through the relation

Eddy vorticity transport *S*^{−1} → 0, *μ* = 0.25, *β** = 0.05, and *R* = 5 × 10^{−8}).

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Eddy vorticity transport *S*^{−1} → 0, *μ* = 0.25, *β** = 0.05, and *R* = 5 × 10^{−8}).

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Eddy vorticity transport *S*^{−1} → 0, *μ* = 0.25, *β** = 0.05, and *R* = 5 × 10^{−8}).

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Further insight comes from consideration of the eddy enstrophy budget. The eddy PV flux across time-mean contours of PV features in the steady-state eddy enstrophy equation, which can be rearranged to express the eddy vorticity flux relative to the mean PV gradient, *q*′ being positive. The importance of this positive correlation provides insight into why localized forcing is effective at producing rectification: a forcing scale smaller than the wavelength of the fluid response guarantees

(left) The eddy vorticity flux relative to the time-mean PV gradient

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

(left) The eddy vorticity flux relative to the time-mean PV gradient

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

(left) The eddy vorticity flux relative to the time-mean PV gradient

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Finally, Reynolds stresses of zonal momentum and their divergence (Fig. 5) provide further insight. In this framework, the mean zonal flows are driven by eddy “forces” that arise from a systematic eddy flux of zonal momentum toward the forcing region. This pattern of eddy zonal momentum flux is a result of the outward energy radiation of the Rossby waves emanating from the forcing, as first discussed by Thompson (1971), producing an eddy zonal momentum flux convergence in the forced zone and eddy zonal momentum flux divergences northwest and southwest of the forcing. Interpreting this eddy flux divergence quantity as a parameterization of an effective steady zonal eddy force, this pattern then translates to a positive (eastward) eddy force at the latitude of the forcing and negative (westward) eddy forces north and south of the forcing. In this way, eddies act to accelerate the eastward jet at the forced latitudes and drive the flanking westward flows. We note that the interpretation of the eddy effect on the time-mean flow in terms of momentum must be done with caution, because, in a nonzonally averaged time-mean framework, the Reynolds stress divergence of zonal momentum can act to accelerate zonal flows or balance the Coriolis torque of the time-mean residual (in this case ageostrophic) circulation. In addition, eddy momentum fluxes can be contaminated by a large rotational component that is balanced in the momentum budget by the time-mean pressure gradient and does not act to accelerate time-mean flows. Full treatment should consider a generalization of the Eliassen–Palm (E–P) flux appropriate for time-mean flows such as the E-vector approach of Hoskins et al. (1983), the radiative wave activity flux described by Plumb (1985, 1986), or the localized E-P flux developed by Trenberth (1986). In this case, however, the key features of a convergence of eastward momentum flux at the forcing latitudes and of westward momentum flux north and south of the forcing remain key features of the eddy zonal momentum forcing in these other more complete diagnostics. As such, there is heuristic value in the consideration of the eddy zonal momentum fluxes alongside the eddy vorticity fluxes (which are dependent only on the relevant divergent component of the eddy flux), because they provide additional insight into the rectification mechanism: namely connecting rectification to the process of wave energy radiation from the forcing. We hypothesize as a result that the properties of the eddy-driven mean flow will be closely tied to the properties of wave energy radiation.

Eddy zonal momentum transport

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Eddy zonal momentum transport

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Eddy zonal momentum transport

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

## 4. Dependence on system parameters

### a. The effect of forcing and flow parameters

In a suite of simulations, we vary the nondimensional *β* parameter and the dimensional forcing radius to explore how each affects the strength of the rectified flow (see Table 1 for a summary of the parameter ranges). The strength of the mean flow generated (the “rectification strength,” which we define as the maximum time-mean gyre transport, *k* and *l*) that are preferentially excited by the forcing in the spectrum of free Rossby waves available (a function of the forcing spectrum and hence of *L _{F}* and of the dispersion relation,

*ω*= −

*βk*/(

*k*

^{2}+

*l*

^{2}), and hence of

*β*and

*ω*, respectively) changes the properties of wave energy radiation away from the forcing through the dependence of the waves’ group velocity on wavenumber. This dictates the spatial distribution of wave activity, which in turn changes the rectified flow response because rectification depends directly on the spatial gradients of the wave/eddy terms. We find a resonant-like maximum response occurs when the length scale of the forcing is well matched to the spectrum of free Rossby waves available, in particular when

*L*is close to 1/

_{F}*B*(Fig. 6). The length scale

*L*= 1/

_{F}*B*is the most effective forcing scale for producing rectified flow because it preferentially excites the wave in the available Rossby wave spectrum with zero zonal group velocity and maximized meridional group velocity and as such is optimal for exciting waves that radiate meridionally.

Summary of numerical experiments. Here, *U _{o}* is the magnitude of a dimensionless constant background flow described in section 4.

Dependence of the rectification strength *β** parameter (circles) and the forcing radius *L _{F}* (× symbol). If not being varied, parameters are fixed at the typical run values of

*β** = 0.05 and

*L*= 250 km. In all cases,

_{F}*S*

^{−1}= 0. Rectification strength is normalized by the PV input supplied by the forcing in one-half period to make the ratio of the response to forcing strength equivalent for each set of runs. A resonant response occurs when the forcing length scale

*L*is equal to 1/

_{F}*B*.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Dependence of the rectification strength *β** parameter (circles) and the forcing radius *L _{F}* (× symbol). If not being varied, parameters are fixed at the typical run values of

*β** = 0.05 and

*L*= 250 km. In all cases,

_{F}*S*

^{−1}= 0. Rectification strength is normalized by the PV input supplied by the forcing in one-half period to make the ratio of the response to forcing strength equivalent for each set of runs. A resonant response occurs when the forcing length scale

*L*is equal to 1/

_{F}*B*.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Dependence of the rectification strength *β** parameter (circles) and the forcing radius *L _{F}* (× symbol). If not being varied, parameters are fixed at the typical run values of

*β** = 0.05 and

*L*= 250 km. In all cases,

_{F}*S*

^{−1}= 0. Rectification strength is normalized by the PV input supplied by the forcing in one-half period to make the ratio of the response to forcing strength equivalent for each set of runs. A resonant response occurs when the forcing length scale

*L*is equal to 1/

_{F}*B*.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

An illustration of how the selection of excited wavenumbers by the forcing length scale impacts rectification efficiency is shown in Figs. 7 and 8. We illustrate the excited wave field, the time-mean eddy kinetic energy distribution, and the eddy forcing terms for a fixed size plunger and three representative values of *β**: one such that *L _{F}* < 1/

*B*), and one with

*L*> 1/

_{F}*B*) (see Fig. 9). The effect of changing

*β** changes the character of the waves that are excited by the forcing consistent with expectations based on the intersection of the inverse forcing length scale 1/

*L*and the dispersion circle: for

_{F}*k*| > |

*l*|), whereas for

*k*| < |

*l*|) (Fig. 7, top). This change in the wavenumbers that are excited changes the pattern of the waves’ energy radiation through the Rossby waves’ group velocity dependence on wavenumber: as

*β** is varied from suboptimal to superoptimal, the waves that are excited change from waves with predominantly eastward group velocity to waves that have a predominantly westward zonal group velocity component. Waves with a significant meridional group velocity are included in the excited spectrum when

*β** is such to include waves near (

*k*,

*l*) = (−

*B*, ±

*B*) in the excited range. This impacts the time-mean eddy kinetic energy distributions for the three cases (Fig. 7, bottom). Finally, the changing properties of energy radiation impact rectification strength through changing the spatial distributions of the Reynolds stresses,

The effect of varying *β** from *β** corresponding to suboptimal, optimal, and superoptimal are 0.015, 0.025, and 0.035, respectively.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The effect of varying *β** from *β** corresponding to suboptimal, optimal, and superoptimal are 0.015, 0.025, and 0.035, respectively.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The effect of varying *β** from *β** corresponding to suboptimal, optimal, and superoptimal are 0.015, 0.025, and 0.035, respectively.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The effect of varying *β** from *β** corresponding to suboptimal, optimal, and superoptimal are as in Fig. 7.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The effect of varying *β** from *β** corresponding to suboptimal, optimal, and superoptimal are as in Fig. 7.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The effect of varying *β** from *β** corresponding to suboptimal, optimal, and superoptimal are as in Fig. 7.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A visualization of the relation between the forcing length scale *L _{F}* and the Rossby wave dispersion relation for fixed frequency for the three values of

*β** discussed in the text. Here, values of

*β** of 0.015, 0.025, and 0.05 are visualized as the suboptimal, optimal, and superoptimal

*β** values, respectively.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A visualization of the relation between the forcing length scale *L _{F}* and the Rossby wave dispersion relation for fixed frequency for the three values of

*β** discussed in the text. Here, values of

*β** of 0.015, 0.025, and 0.05 are visualized as the suboptimal, optimal, and superoptimal

*β** values, respectively.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A visualization of the relation between the forcing length scale *L _{F}* and the Rossby wave dispersion relation for fixed frequency for the three values of

*β** discussed in the text. Here, values of

*β** of 0.015, 0.025, and 0.05 are visualized as the suboptimal, optimal, and superoptimal

*β** values, respectively.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

### b. The effect of stratification

Numerical parameter studies varying the Burger number reveal that rectification strength has a nonmonotonic dependence on stratification (Fig. 10), and again one can understand the dependence as a result of changes in the waves that are selected to participate in the rectification process. In this 1½-layer configuration, the streamfunction and PV thickness contours are coincident and there is no advection of the stretching component of the PV. As such, rectification continues to result only from the advection of relative vorticity as in the barotropic case. Stratification does alter the barotropic problem, however, by changing the radius of the dispersion circle, and this has two important implications for the characteristics of the waves selected by the forcing. First, by reducing the spectrum of free wavenumbers available, increasing stratification concentrates the forcing amplitude into a narrower band of wavenumbers with increased wave amplitudes. Second, increasing stratification centers the excited spectrum more narrowly around |*k*| = *B*. Snapshots of the excited wave fields and the orientation of their energy radiation for various values of *S*^{−1} (Fig. 11) illustrate these effects. In this way, a finite deformation radius traps energy near the forcing and amplifies the meridional fluid velocity. The forced flow is more effective at stirring PV and the strength of the rectified flow increases as a result. A singularity arises at the critical value where only the wave (*k*, *l*) = (−*B*, 0) satisfies the dispersion relation, a wave with zero group velocity in both the zonal and meridional directions and as such no propagation of wave energy away from the forcing region. For larger stratifications, the radius of the dispersion circle becomes imaginary. No real pairs of wavenumbers (*k*, *l*) satisfy the dispersion relation, and, as suggested by the analytical solution, the forcing fails to radiate waves. The strength of the rectified flow goes to zero as a result.

Dependence of rectification strength on the inverse Burger number *S*^{−1}. The dependence on *S*^{−1} normalized by

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Dependence of rectification strength on the inverse Burger number *S*^{−1}. The dependence on *S*^{−1} normalized by

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Dependence of rectification strength on the inverse Burger number *S*^{−1}. The dependence on *S*^{−1} normalized by

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The effect of varying stratification by increasing *S*^{−1} toward the critical value on: (top) the waves excited (as visualized by a snapshot of the instantaneous streamfunction) and (bottom) the orientation of energy radiation from the forcing (as visualized by the time-mean eddy kinetic energy). Values of *S*^{−1} are 0.05, 0.10, and 0.15. Other forcing and flow parameters are set to the typical run values as in Fig. 3.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The effect of varying stratification by increasing *S*^{−1} toward the critical value on: (top) the waves excited (as visualized by a snapshot of the instantaneous streamfunction) and (bottom) the orientation of energy radiation from the forcing (as visualized by the time-mean eddy kinetic energy). Values of *S*^{−1} are 0.05, 0.10, and 0.15. Other forcing and flow parameters are set to the typical run values as in Fig. 3.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The effect of varying stratification by increasing *S*^{−1} toward the critical value on: (top) the waves excited (as visualized by a snapshot of the instantaneous streamfunction) and (bottom) the orientation of energy radiation from the forcing (as visualized by the time-mean eddy kinetic energy). Values of *S*^{−1} are 0.05, 0.10, and 0.15. Other forcing and flow parameters are set to the typical run values as in Fig. 3.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

### c. The effect of a mean background flow

We add a large-scale mean background flow to the setup by adding a constant, uniform, zonal background mean flow *J*(*ψ*, ∇^{2}*ψ*) to include advection also by the background flow *U _{o}*.

Results of these parameter studies (Fig. 12) again show a nonmonotonic dependence on *U _{o}*, and again the changing response can be understood by considering the rectification efficiency of the given waves that are selected by the parameters of the problem. Varying

*U*is a means of varying the spectrum of excited waves (by modifying the dispersion relation) but more critically is also a means to vary which wave in the excited spectrum is rendered stationary, and we find that rectification efficiency depends on the suitability of the stationary wave for rectification. Consideration of the zonal wavenumber of the stationary wave

_{o}*k*

_{stationary}, as a function of

*U*obtained from the condition

_{o}*k*

_{stationary}=

*ω*/−

*U*or equivalently

_{o}*k*

_{stationary}/

*B*= 2

*ω*

^{2}/

*U*(Fig. 13). Consistent with the fact that westward background flows cannot arrest Rossby waves’ westward phase propagation, for westward background flows (

_{o}β*U*< 0) no wave is rendered stationary (because negative values of

_{o}*k*/−

*B*are not permitted). Short waves with

*k*/−

*B*> 1 and eastward group velocities are arrested for weak eastward background flows (0 <

*U*< 2

_{o}*ω*

^{2}/

*β*), whereas long waves with

*k*/−

*B*< 1 and fast westward group velocities are arrested for strong eastward background flows (

*U*> 2

_{o}*ω*

^{2}/

*β*). The peak response occurs when the optimal wave with

*k*= −

*B*is rendered stationary by the background flow, a condition that is satisfied for

*U*for these two cases.

_{o}Dependence of rectification strength on the value of the constant zonal background flow *k*| = *B* is shown on the upper axis.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Dependence of rectification strength on the value of the constant zonal background flow *k*| = *B* is shown on the upper axis.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Dependence of rectification strength on the value of the constant zonal background flow *k*| = *B* is shown on the upper axis.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The dependence of the zonal wavenumber of the stationary wave *k*_{stationary}, normalized by the optimal wavenumber *B* on *U _{o}* obtained from the condition

*k*/−

*B*are not permitted because westward background flows cannot arrest the Rossby waves’ westward phase propagation (gray shading). The optimal response is achieved when

*U*arrests the optimal wave with |

_{o}*k*| =

*B*(dotted lines).

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The dependence of the zonal wavenumber of the stationary wave *k*_{stationary}, normalized by the optimal wavenumber *B* on *U _{o}* obtained from the condition

*k*/−

*B*are not permitted because westward background flows cannot arrest the Rossby waves’ westward phase propagation (gray shading). The optimal response is achieved when

*U*arrests the optimal wave with |

_{o}*k*| =

*B*(dotted lines).

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The dependence of the zonal wavenumber of the stationary wave *k*_{stationary}, normalized by the optimal wavenumber *B* on *U _{o}* obtained from the condition

*k*/−

*B*are not permitted because westward background flows cannot arrest the Rossby waves’ westward phase propagation (gray shading). The optimal response is achieved when

*U*arrests the optimal wave with |

_{o}*k*| =

*B*(dotted lines).

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

## 5. Extension to the strongly nonlinear regime

Results from numerical tests varying the wave nonlinearity *μ* via increasing the forcing amplitude *A* confirm the analytical prediction of a quadratic dependence of mean flow strength on *A* for small values of *A*. This dependence, however, is observed to break down as the forcing amplitude is increased (Fig. 14). Beyond this critical value, in a regime where the degree of nonlinearity is large enough that the weakly nonlinear classification is no longer valid, the mean flow response shows signs of saturation, increasing much more slowly with an increase in nonlinearity/forcing amplitude. This qualitative change in behavior in the mean flow response is consistent with the laboratory results of Whitehead (1975). Despite this systematic change of behavior however, examination of both the wave fields and the time-mean rectified flow for the weakly nonlinear and strongly nonlinear cases remain qualitatively similar in many respects (Fig. 15).

Dependence of rectification strength on wave steepness/forcing amplitude for the typical case. The transition between a quadratic dependence of rectification strength on *μ* in the vicinity of *μ* = unity to a much slower rate of increase defines the transition between weakly and strongly nonlinear regimes.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Dependence of rectification strength on wave steepness/forcing amplitude for the typical case. The transition between a quadratic dependence of rectification strength on *μ* in the vicinity of *μ* = unity to a much slower rate of increase defines the transition between weakly and strongly nonlinear regimes.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Dependence of rectification strength on wave steepness/forcing amplitude for the typical case. The transition between a quadratic dependence of rectification strength on *μ* in the vicinity of *μ* = unity to a much slower rate of increase defines the transition between weakly and strongly nonlinear regimes.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A comparison of (top) wave fields (as visualized by snapshots of the instantaneous streamfunction) and (bottom) their associated time-mean rectified flows (as visualized by the time-mean streamfunction) for (left) the weakly nonlinear case (*μ* = 0.25) versus (right) the strongly nonlinear case (*μ* = 2.5). Contours are an order of magnitude larger in the strongly nonlinear case fields.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A comparison of (top) wave fields (as visualized by snapshots of the instantaneous streamfunction) and (bottom) their associated time-mean rectified flows (as visualized by the time-mean streamfunction) for (left) the weakly nonlinear case (*μ* = 0.25) versus (right) the strongly nonlinear case (*μ* = 2.5). Contours are an order of magnitude larger in the strongly nonlinear case fields.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A comparison of (top) wave fields (as visualized by snapshots of the instantaneous streamfunction) and (bottom) their associated time-mean rectified flows (as visualized by the time-mean streamfunction) for (left) the weakly nonlinear case (*μ* = 0.25) versus (right) the strongly nonlinear case (*μ* = 2.5). Contours are an order of magnitude larger in the strongly nonlinear case fields.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A helpful way to understand rectification in the strongly nonlinear regime is via the same interpretation in terms of the rectification efficiency of a given spectrum of linear Rossby waves developed in the weakly nonlinear case with the addition of a mean flow and its associated mean flow effects. This latter addition is required as a consequence of the rectified mean flow now being sufficiently strong that the mean flow advection and the interaction between the waves and the mean flow are significant. These effects will be much more complicated than the case of a uniform mean background flow not only because the mean flow strength will be a direct function of the rectification efficiency, but also because the feedback it will have on the waves’ ability to rectify will have important spatial dependence. This may include making a significant contribution to the effective background PV gradient through which the waves are propagating, as well as competing with the waves’ intrinsic propagation. Despite this complexity, however, we find that the net effect of the mean flow’s role is to counteract or reduce the ability of the waves/eddies to rectify. It is this counteracting effect that results in the saturation in the mean flow strength at large forcing amplitudes that is observed.

To illustrate, first consider the time-mean PV balance for typical weakly nonlinear and strongly nonlinear cases (Fig. 16). In the weakly nonlinear case, consistent with the simulations of HR83, the balance is predominantly a two-term one between the eddy potential vorticity flux divergence

The time-mean potential vorticity balance, (individual terms shown left–right) *μ* for the weakly and strongly nonlinear cases are as in Fig. 15. Contours are an order of magnitude larger in the strongly nonlinear case fields.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The time-mean potential vorticity balance, (individual terms shown left–right) *μ* for the weakly and strongly nonlinear cases are as in Fig. 15. Contours are an order of magnitude larger in the strongly nonlinear case fields.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The time-mean potential vorticity balance, (individual terms shown left–right) *μ* for the weakly and strongly nonlinear cases are as in Fig. 15. Contours are an order of magnitude larger in the strongly nonlinear case fields.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

(left) Contours of the time-mean PV field for the most nonlinear simulation considered (*μ* = 4.0). (right) The meridional profile of time-mean PV at *x* = 0 for the weakly nonlinear (*μ* = 0.25) (gray) and most strongly nonlinear (*μ* = 4.0) (black) cases. The dashed–dotted lines indicate the meridional extent of the plunger radius.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

(left) Contours of the time-mean PV field for the most nonlinear simulation considered (*μ* = 4.0). (right) The meridional profile of time-mean PV at *x* = 0 for the weakly nonlinear (*μ* = 0.25) (gray) and most strongly nonlinear (*μ* = 4.0) (black) cases. The dashed–dotted lines indicate the meridional extent of the plunger radius.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

(left) Contours of the time-mean PV field for the most nonlinear simulation considered (*μ* = 4.0). (right) The meridional profile of time-mean PV at *x* = 0 for the weakly nonlinear (*μ* = 0.25) (gray) and most strongly nonlinear (*μ* = 4.0) (black) cases. The dashed–dotted lines indicate the meridional extent of the plunger radius.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

The increasing importance of mean flow advection for increasing nonlinearity is also seen in the eddy enstrophy balance for the two cases. In the weakly nonlinear case, an up-gradient eddy vorticity transport is achieved by a dominant enstrophy budget balance in the vicinity of the forcing between the up-gradient eddy flux of potential vorticity

As in Fig. 16, but for the time-mean eddy enstrophy budget: (left)–(right)

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

As in Fig. 16, but for the time-mean eddy enstrophy budget: (left)–(right)

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

As in Fig. 16, but for the time-mean eddy enstrophy budget: (left)–(right)

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

Finally, mean flow interaction in the strongly nonlinear case is also important in the eddy zonal momentum flux convergence field (Fig. 19). In the weakly nonlinear case (Fig. 19, left), the regions of zonal momentum flux divergence occur north and south of the forcing latitude and result from the waves radiating away from the forcing. As such, because of the separation in latitude of the regions of momentum flux divergence and convergence, the waves/eddies are effective at driving the westward recirculations. In the strongly nonlinear case (Fig. 19, right), however, similar to the case of a strong eastward mean background flow, the main source of momentum flux divergence now lies on the jet axis west of the forcing, here resulting from the mean jet “running into” the westward-propagating waves. We note that this change in the orientation of the eddy momentum flux is consistent with a change in the wave population in the vicinity of the forcing toward shorter horizontal wavelengths, as would be expected if the eddy-induced flow was now competing with the waves’ westward propagation. The eddy momentum flux divergence acts now not to accelerate westward flows but instead to decelerate the mean eastward jet, and rectification efficiency is reduced as a result.

A comparison of the time-mean zonal eddy force {the time-mean eddy zonal momentum flux convergence, *μ* are as in Fig. 15. Contours are an order of magnitude larger in the strongly nonlinear case field.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A comparison of the time-mean zonal eddy force {the time-mean eddy zonal momentum flux convergence, *μ* are as in Fig. 15. Contours are an order of magnitude larger in the strongly nonlinear case field.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

A comparison of the time-mean zonal eddy force {the time-mean eddy zonal momentum flux convergence, *μ* are as in Fig. 15. Contours are an order of magnitude larger in the strongly nonlinear case field.

Citation: Journal of Physical Oceanography 42, 3; 10.1175/JPO-D-11-060.1

## 6. Discussion

To close, we consider the relevance of the results described here to oceanic applications. It is challenging to obtain enough observational data to accurately calculate eddy statistics; as a consequence, diagnostic studies of the relation between the time-mean or low-frequency state and eddies using direct observations of the ocean circulation have been rare. Some attempts, however, have been made with limited data on regional scales, and some do observe signatures of eddy–mean flow interactions consistent with the mechanism discussed here. Thompson (1977, 1978), for example, suggests that eddies may be playing a role in the net driving of the mean Gulf Stream jet and its inshore counter current by transferring momentum between the jet and the countercurrent via a pattern of momentum flux convergences and divergences consistent with the process studied here. In an examination of eddy effects on abyssal ocean dynamics also in the region of the Gulf Stream, Hogg (1993) argues that the eddy relative vorticity flux divergence is of the right sign and order of magnitude to drive a recirculation of the observed strength. Other analyses in the Gulf Stream (e.g., Hall 1986; Dewar and Bane 1989; Cronin 1996) suggest a more complicated picture of the eddy forcing that appears at some times and locations consistent with the rectification mechanism considered here (e.g., Hall 1986) but at others is instead typical of baroclinic (e.g., Cronin 1996) or barotropic (e.g., Dewar and Bane 1989, at 68.8°W) instability processes. We conjecture that, in WBC jets, rectification will be important at locations and times where wave radiation dominates over instability processes and as such is more likely in the far field of jets (e.g., Malanotte-Rizzoli et al. 1987), in the downstream regions of WBC jet extensions (see Waterman and Jayne 2011), and at times not dominated by meander formation events (as in Cronin 1996).

The parameter values explored in this study are realistic and relevant to synoptic scales of, for example, a strong localized wind stress or a localized concentration of eddy activity as, for example, generated by mean flow instability. The magnitude of the mean flow generated, on the order of the dimensional equivalent of a few centimeters per second for small-amplitude forcing and up to the dimensional equivalent of tens of centimeters per second for large-amplitude forcing, could be significant to the oceanic circulation, especially below the thermocline where the mean velocities are small. It is sufficient for instance to account for the mean deep recirculation gyre velocities observed, typically 2–10 cm s^{−1} in the Gulf Stream (Schmitz 1980; Hogg et al. 1986) and 2–4 cm s^{−1} in the Kuroshio Extension (Jayne et al. 2009). Our results suggest that in order for rectification to be efficient, a very special relationship between the forcing and flow parameters must exist such that the forcing selects meridionally propagating waves; however, this is for the case in which only one forcing frequency is applied. In a more realistic setting, the forcing would likely include a broad range of frequencies and the rectification mechanism would then select the forcing power around the resonant frequency and use that to drive the mean rectified flow. In this way, we would expect the rectified flow to reflect the forcing energy available in the suitable selection of the frequency–wavenumber forcing spectra. Nevertheless, significant stratification or a background mean flow could still render the mechanism ineffective in practice. Hence, for this mechanism to be important in the ocean, it requires a sufficiently well-matched set of forcing and environmental parameters, as well as weak stratification and a weak or well-matched background flow. The latter two requirements make rectification to be likely more significant for abyssal as opposed to above-thermocline flows. The incomplete PV homogenization seen even in the strongly nonlinear cases illustrates that the range of forcing amplitudes considered in this study do not reach the point of violent stirring, which does occur in nature (Rhines and Schopp 1991, and references therein). As such, the relevance of larger forcing amplitudes and/or other processes leading to eddy mixing in the real ocean should be considered.

Mean flows driven by the mechanism of rectification would not be represented in a general circulation model with eddy effects parameterized simply as down-gradient diffusion. For recirculations in WBC jet systems, driven likely by a combination of inertial forcing and eddy driving via the mechanism proposed here, this could have significant impacts on WBC jet vertical structure and transport with important climatic implications for the oceanic meridional heat transport. We find in a subsequent study examining the downstream equilibration of an unstable jet (as would be relevant to a separated western boundary current) that this “plunger like” mechanism is indeed a useful model for the effect of the eddies downstream of jet stabilization, where eddies are responsible for driving weakly depth-dependent recirculations that add significantly to the jet’s time-mean transport. See Waterman and Jayne (2011) for further discussion.

## Acknowledgments

The authors would like to gratefully acknowledge useful discussions with Pavel Berloff, Joseph Pedlosky, and John Whitehead, as well as very insightful reviews provided by Peter Rhines. SW was supported by the MIT Presidential Fellowship and the National Science Foundation under Grants OCE-0220161 and OCE-0825550. The financial assistance of the Houghton Fund, the MIT Student Assistance Fund, and WHOI Academic Programs is also gratefully acknowledged. SRJ was supported by the National Science Foundation under Grants OCE-0220161 and OCE-0849808. The model integrations were performed at the National Center for Atmospheric Research, which is supported by the National Science Foundation.

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