## 1. Introduction

And we are born and born again

Like the waves of the sea

(P. Simon, “Señorita with a Necklace of Tears”)

The interior ocean mesoscale supports energetic velocity and sea surface height (SSH) variability (Fig. 1) that is typically at least an order of magnitude greater than the long-term mean currents and their geostrophic SSH signatures, and is often described as a form of geostrophic turbulence (The MODE Group 1978; Rhines 1975; see also references in Samelson et al. 2014, 2016). Lateral eddy fluxes driven by this mesoscale variability are poorly understood and a major uncertainty in our understanding of the ocean’s role in Earth’s climate system. Wavenumber–frequency spectra of mesoscale ocean SSH variability from satellite altimeter measurements typically show an apparent mix of wave and turbulent dynamics: a nondispersive spectral structure (e.g., Zang and Wunsch 1999; Fu and Chelton 2001; Fu 2004; Wunsch 2009; Chelton et al. 2011a) with an effective phase speed that is loosely consistent with phase speeds of theoretical linear, gravest-mode, long-wave, planetary waves (Chelton and Schlax 1996), but an apparent absence of the dispersive decrease in frequency that would be predicted by linear theory (e.g., Killworth et al. 1997; Fu and Chelton 2001; Tailleux and McWilliams 2001) at scales near and smaller than the deformation radius wavelength (Fig. 2a). A number of recent modeling studies have explored the possible origins of this spectral structure (e.g., Early et al. 2011; Berloff and Kamenkovich 2013; Wortham and Wunsch 2014; Morten et al. 2017; LaCasce 2017), and have shown it to be qualitatively consistent both with a geostrophically turbulent eddy field and with isolated, coherent, nonlinear eddy propagation. However, the rapid decline in spectral power with decreasing scale (increasing wavenumber magnitude) and the limited wavelength resolution of presently available SSH fields constructed from satellite altimeter data (see, e.g., appendix A of Chelton et al. 2011b) have hindered determination of the short-wavelength extent of the nondispersive behavior and prevented unambiguous identification of this apparent signature of mesoscale nonlinearity.

(a) Power spectrum ^{2} cpd^{−1} cpkm^{−1}) and (b) inverted power spectrum ^{2} week^{−2} cpd^{−1} cpkm^{−1}) of SSH variability along 35°S in the south Indian Ocean vs zonal wavenumber and frequency, computed from the approximately 16-yr period October 1992–December 2008 of the AVISO Delayed-Time Reference Series. Also shown are a mean regional deformation radius wavenumber ^{−1} and ^{−1}.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

(a) Power spectrum ^{2} cpd^{−1} cpkm^{−1}) and (b) inverted power spectrum ^{2} week^{−2} cpd^{−1} cpkm^{−1}) of SSH variability along 35°S in the south Indian Ocean vs zonal wavenumber and frequency, computed from the approximately 16-yr period October 1992–December 2008 of the AVISO Delayed-Time Reference Series. Also shown are a mean regional deformation radius wavenumber ^{−1} and ^{−1}.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

(a) Power spectrum ^{2} cpd^{−1} cpkm^{−1}) and (b) inverted power spectrum ^{2} week^{−2} cpd^{−1} cpkm^{−1}) of SSH variability along 35°S in the south Indian Ocean vs zonal wavenumber and frequency, computed from the approximately 16-yr period October 1992–December 2008 of the AVISO Delayed-Time Reference Series. Also shown are a mean regional deformation radius wavenumber ^{−1} and ^{−1}.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

From the complementary point of view of eddy phenomenology, a global, quantitative, statistical characterization of midlatitude mesoscale variability has been developed by Chelton et al. (2011b, 2007) from the AVISO (Archivage, Validation, Interpretation des donnees des Satellite Oceanographiques) Delayed-Time Reference Series merged, gridded dataset of two decades of satellite altimeter measurements (Ducet et al. 2000; Le Traon et al. 2003; Pujol et al. 2016). Because it identifies and tracks coherent features, this eddy-based analysis effectively retains much of the phase information that is discarded when wavenumber–frequency power spectra are computed, and so provides an independent basis of comparison. Recently, Samelson et al. (2016) proposed a linear, stochastic model of mesoscale variability that was able to reproduce nearly all of the basic global mean statistical characteristics of an updated version of the Chelton et al. (2011b) eddy dataset, as well as the nondispersive spectral structure. However, the semiempirical nature of that model made its relation to the physical dynamics unclear, and left a fundamental gap between the theory and the SSH observations.

The goal of the study described here is to take a first step toward reconciling the semiempirical stochastic field model of Samelson et al. (2016) with the simplest standard dynamical model of mesoscale ocean variability, the reduced-gravity quasigeostrophic model (e.g., Pedlosky 1987). While the reduced-gravity vertical structure is highly simplified, it affords a consistent representation of the fundamental potential vorticity dynamics relevant to mesoscale variability and the associated advective nonlinearity, which the semiempirical stochastic model lacks. This work may be seen also as an extension of the related modeling study of Early et al. (2011), in which observed eddy variability was compared with reduced-gravity quasigeostrophic model simulations seeded with random distributions of Gaussian eddies having statistical characteristics drawn from the observed eddy distributions, and of the recent, much more broadly aimed study of the nondispersive spectral structure and statistical equilibrium characteristics of geostrophic turbulence in the reduced-gravity quasigeostrophic model by Morten et al. (2017), which provided the quantitative framework and starting point for the numerical simulations described here. In addition, this work is related in a more general way to many other previous studies of geostrophic turbulence and quasigeostrophic models of mesoscale flows.

## 2. Formulation

*ψ*is

*L*and speed

*U*, time by

*H*overlying a deep, motionless layer with fixed fractional density difference

^{−2}is the acceleration of gravity, and

*f*is the Coriolis parameter.

A stochastic form of the forcing

*q*on the characteristics defined by the velocity field,

*τ*, and the second equality defines

*ψ*and a triharmonic term to smooth the small scales,

*L*and

*U*may be chosen to reduce the number of dimensionless parameters in (1)–(5). Setting

*β*,

*τ*. The spatial structure of the forcing function

*τ*is sufficiently small relative to the intrinsic time scales of the flow, the solution should be insensitive to its value. However,

*τ*is a fundamental descriptor of the physical processes represented in the model by the stochastic forcing, and as such is retained here as an independent parameter, rather than held constant as in Morten et al. (2017).

*ϕ*determines the corresponding values of the dimensional Coriolis parameter

*f*and its meridional gradient

*U*can be obtained from the given value of the dimensionless parameter

*β*:

*ψ*,

Numerical solutions of (7) were computed on a 256 × 128 zonal–meridional grid with periodic boundary conditions, using a nonaliasing pseudospectral code originally derived from the two-layer channel model described by Samelson and Pedlosky (1990) and Oh et al. (1993). Guided by the considerations described in section 3, two primary sets of simulations were conducted, with integrations generally to dimensionless

Spatial standard deviation of dimensionless sea surface height *t* from simulations with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Spatial standard deviation of dimensionless sea surface height *t* from simulations with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Spatial standard deviation of dimensionless sea surface height *t* from simulations with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Coherent eddies in the model fields were identified and tracked, after conversion of *ψ* to dimensional SSH

## 3. The ocean mesoscale parameter regime

### a. Analytical estimates

The original starting point for these simulations was provided by the “Run 3” simulation of Morten et al. (2017), which was identified by those authors as likely to be the most representative of the ocean mesoscale among their set of simulations. For the scaling of (1) used by Morten et al. (2017), the Run 3 simulation had parameter values

*τ*gives

*t*-averaged) variance

*ψ*is effectively constant on the time scale

*τ*except for a directly forced component proportional to

*ψ*is stationary, and it then follows from (16) that for

*ψ*can be estimated as

^{−1}is a fixed, nominal value of the gravest-mode internal gravity wave speed (e.g., Chelton et al. 1998), the first factors on the right-hand sides of (19) and (21) are known, while

Ocean mesoscale regime parameters. The points corresponding to the target values for the dimensional comparisons at latitudes *L _{R}* =

*c*

_{1}/

*f*vs latitude

*ϕ*for

*c*

_{1}= 3.35 m s

^{−1}(solid line) and

*c*

_{1}= 3 m s

^{−1}(dashed). (b) Contours (labeled, solid and dotted lines) of

^{−1}vs latitude

*ϕ*and specified SSH standard deviation

*r*

_{ψ}/

*β*from (20) for

^{−1}and

^{−1}vs latitude

*ϕ*. (d) Graphical illustration of the joint solution of (17) and (21) for the dimensionless parameters (

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Ocean mesoscale regime parameters. The points corresponding to the target values for the dimensional comparisons at latitudes *L _{R}* =

*c*

_{1}/

*f*vs latitude

*ϕ*for

*c*

_{1}= 3.35 m s

^{−1}(solid line) and

*c*

_{1}= 3 m s

^{−1}(dashed). (b) Contours (labeled, solid and dotted lines) of

^{−1}vs latitude

*ϕ*and specified SSH standard deviation

*r*

_{ψ}/

*β*from (20) for

^{−1}and

^{−1}vs latitude

*ϕ*. (d) Graphical illustration of the joint solution of (17) and (21) for the dimensionless parameters (

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Ocean mesoscale regime parameters. The points corresponding to the target values for the dimensional comparisons at latitudes *L _{R}* =

*c*

_{1}/

*f*vs latitude

*ϕ*for

*c*

_{1}= 3.35 m s

^{−1}(solid line) and

*c*

_{1}= 3 m s

^{−1}(dashed). (b) Contours (labeled, solid and dotted lines) of

^{−1}vs latitude

*ϕ*and specified SSH standard deviation

*r*

_{ψ}/

*β*from (20) for

^{−1}and

^{−1}vs latitude

*ϕ*. (d) Graphical illustration of the joint solution of (17) and (21) for the dimensionless parameters (

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

The majority of the eddies in the updated Chelton et al. (2011b) dataset that are not associated with western boundary currents or the Antarctic Circumpolar Current are found between latitudes of 20° and 40°N or between 20° and 40°S (Fig. 1). In these subtropical interior regimes, the standard deviation

From their model fit to observations, Samelson et al. (2016) estimate a damping rate for ^{−1}. This rate could be used directly for ^{−1}; effectively, this assumes that the nonlinearity in the present model contributes equally with the damping to the autocorrelation decay, whereas in the linear model of Samelson et al. (2016), all of the autocorrelation decay along long-wave characteristics must derive from the damping. With this nominal value for *ϕ*, taking values of roughly 0.045 at

Combining these two estimates of *β* at which the solutions with the desired values of

Along with the damping time scale ^{−1}, Samelson et al. (2016) estimate a dimensional value of ^{−1/2} for ^{−1} by (8) and to dimensionless parameters *R*, and ^{−1}, and

Guided by these considerations, two primary sets of simulations were conducted, in each of which *β* was held fixed while *τ* were varied (Fig. 5). The first set, with ^{−1} and ^{−3/2}, equivalent to ^{−1/2}. The second set, with ^{−1} and ^{−3/2}, equivalent to ^{−1/2}. A number of additional simulations were conducted, of which only the equivalent to the Morten et al. (2017) Run 3 is described here. With the stochastic-limit estimate (17), the Morten et al. (2017) Run 3 simulation can be anticipated to have

Summary of values of

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Summary of values of

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Summary of values of

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### b. Eddy phenomenology

Tracked eddies in these simulations show several distinctive characteristics that may be taken to define their overall phenomenology, including dominance of advective rearrangement over direct forcing of the potential vorticity field, sustained westward propagation modulated by modest departures from the linear long-wave trajectory, periods of relative stability punctuated by abrupt change, and continuous interaction of eddy structures with their surroundings. The resulting phenomenological picture differs distinctly from that of a single isolated eddy dominated by linear planetary-wave propagation and nonlinear self-interaction, and as such is much closer, for example, to the random eddy-seeded simulations of Early et al. (2011) than to the isolated monopole simulations in that same study or others (e.g., Larichev and Reznik 1976; McWilliams and Flierl 1979). In its complexity, it resembles qualitatively many previous simulations of two dimensional and geostrophic turbulence (e.g., Lilly 1969; Rhines 1979; McWilliams and Chow 1981; Smith and Vallis 2001, 2002; Arbic and Flierl 2004; Morss et al. 2009; Venaille et al. 2011; Morten et al. 2017; and numerous references therein). Likewise, it is distinct in these ways from the classical baroclinic-growth and barotropic-decay life cycles found in studies of midlatitude synoptic disturbances in the atmosphere (e.g., Simmons and Hoskins 1978).

A useful illustrative example of the complex life cycle of a simulated eddy identified by the tracking procedure is furnished by eddy 4386 from a simulation conducted for ^{−1} over most of the eddy lifetime. The length scale changes infrequently but more abruptly, with some correlation to the amplitude history but some notable deviations, and remains between 70 and 100 km over most of the eddy lifetime, with extreme values near 50 and 150 km. The eddy propagates zonally at a speed that, in the mean over eddy lifetime, is slightly slower than the linear long-wave speed, but exceeds that speed continuously for three months beginning near week 50. The SSH forcing ^{−1} and fluctuates stochastically on the time scale

Time (weeks) history of (a) amplitude, (b) speed, (c) length scale, (d) relative zonal (thick solid line), meridional (thin solid), and theoretical linear long-wave (dashed) displacements, and (e) equivalent-SSH stochastic forcing

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Time (weeks) history of (a) amplitude, (b) speed, (c) length scale, (d) relative zonal (thick solid line), meridional (thin solid), and theoretical linear long-wave (dashed) displacements, and (e) equivalent-SSH stochastic forcing

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Time (weeks) history of (a) amplitude, (b) speed, (c) length scale, (d) relative zonal (thick solid line), meridional (thin solid), and theoretical linear long-wave (dashed) displacements, and (e) equivalent-SSH stochastic forcing

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

The structural evolution of eddy 4386 shows several distinct stages that are related to the amplitude history. These stages can be described in terms of the dimensionless thickness [

Scaled dimensionless quasigeostrophic (top) thickness ^{−1}), at times *y* axes for the five sets of panels after time step 1 are offset relative to the step 1 axes by

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Scaled dimensionless quasigeostrophic (top) thickness ^{−1}), at times *y* axes for the five sets of panels after time step 1 are offset relative to the step 1 axes by

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Scaled dimensionless quasigeostrophic (top) thickness ^{−1}), at times *y* axes for the five sets of panels after time step 1 are offset relative to the step 1 axes by

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

## 4. Amplitude and autocorrelation

A basic statistical characterization of each simulation that may be compared directly with observations and with the linear field model results of Samelson et al. (2016) is provided by the domain-mean amplitude response and autocorrelation scale. For

Time-mean dimensionless streamfunction standard deviation *τ*, with open circle and open square size proportional to

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Time-mean dimensionless streamfunction standard deviation *τ*, with open circle and open square size proportional to

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Time-mean dimensionless streamfunction standard deviation *τ*, with open circle and open square size proportional to

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

For *τ* and *β* is approximately reproduced even when

For the linear stochastic models of Samelson et al. (2014, 2016), the damping coefficient *r* determined the temporal autocorrelation parameter *α* directly, according to *α* from the observed SSH along long-wave characteristics, and showed that the values so computed at midlatitudes were approximately 0.95, for time units of weeks, and that these values were much larger than those computed at fixed spatial points. The corresponding values of *α* computed for the models of Samelson et al. (2014) and Samelson et al. (2016) were 0.96 and 0.94, respectively, in close agreement with the observed values.

For the nonlinear quasigeostrophic model (7), the nonlinearity can alter the autocorrelation, and the parameter *α* cannot be computed directly from *α* must be estimated from the empirical autocorrelation function *α* obtained along characteristics of the form *a* is a parameter with *a* increments of 0.05 (Fig. 9).

Autocorrelation function ^{−1} or ^{−1} is also shown (red), along with the functions ^{−1} (magenta) and the stochastic model values ^{−1} (green, blue) from Samelson et al. (2014, 2016). The model autocorrelation function is interpolated from domain-averaged values computed at the times marked (×) along the *t* axis.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Autocorrelation function ^{−1} or ^{−1} is also shown (red), along with the functions ^{−1} (magenta) and the stochastic model values ^{−1} (green, blue) from Samelson et al. (2014, 2016). The model autocorrelation function is interpolated from domain-averaged values computed at the times marked (×) along the *t* axis.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Autocorrelation function ^{−1} or ^{−1} is also shown (red), along with the functions ^{−1} (magenta) and the stochastic model values ^{−1} (green, blue) from Samelson et al. (2014, 2016). The model autocorrelation function is interpolated from domain-averaged values computed at the times marked (×) along the *t* axis.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

The resulting equivalent damping values ^{−1} greater than the dimensional model damping

Empirical autocorrelation parameter ^{−1}) vs dimensional model damping parameter ^{−1} (thin dashed) are indicated.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Empirical autocorrelation parameter ^{−1}) vs dimensional model damping parameter ^{−1} (thin dashed) are indicated.

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Empirical autocorrelation parameter ^{−1}) vs dimensional model damping parameter ^{−1} (thin dashed) are indicated.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Two of the numerical solutions satisfy both an amplitude criterion and an autocorrelation criterion: both have nonlinearity measure ^{−1} ≈ 1/62 week^{−1}, and either

## 5. Eddy statistics

Coherent eddies in the dimensional SSH field from the solution with *ψ* to dimensional SSH *t* throughout the cycles, and near-universality over the lifetime range from 16 to 80 weeks (Figs. 11a,b, 12a,b, 13a,b). The cycle-mean model dimensional amplitude and rotational speed increase with lifetime in a manner that is roughly consistent with the observed distributions for lifetimes up to half a year, but continue to increase for longer lifetimes, for which the observed scales are roughly constant, so that the model overestimates the amplitudes and speeds of the longer-lived eddies (Figs. 11c,13c). The cycle-mean model dimensional length scale tends to modestly underestimate the observed length scale, except for lifetimes longer than a year (Fig. 12c). The model innovations—the first differences of the corresponding quantities during the eddy life cycles—have distributions that are similar to but systematically less normal than the observed innovation distributions, with smaller standard deviations and larger kurtoses (Figs. 11d, 12d, 13d).

Normalized eddy amplitude life cycles for solution with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Normalized eddy amplitude life cycles for solution with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Normalized eddy amplitude life cycles for solution with

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As in Fig. 11, but for normalized eddy length scale life cycles.

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As in Fig. 11, but for normalized eddy length scale life cycles.

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As in Fig. 11, but for normalized eddy length scale life cycles.

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As in Fig. 11, but for normalized eddy rotational speed life cycles.

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As in Fig. 11, but for normalized eddy rotational speed life cycles.

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As in Fig. 11, but for normalized eddy rotational speed life cycles.

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The structure of the simulated eddy number distribution versus lifetime is generally similar to that of the observed distribution: there are many more short-lived eddies than long-lived eddies, with a power-law dependence of number on lifetime for all but the longest lifetimes (Figs. 14a,b). There are 1781 model eddies with lifetimes longer than 16 weeks, roughly 1/12 of the observed midlatitude number, with the difference arising primarily because the tiled model eddy-tracking domain area is an order of magnitude smaller than the midlatitude ocean area. At the longest lifetimes, the relatively small number of model eddies limits the statistical reliability of the estimated number distribution, but the latter nonetheless appears qualitatively consistent with the exponential dependence found in the observations. Relative to the observed distribution, however, the model eddy distribution is biased toward long eddy lifetimes, with a smaller fraction of model eddies having shorter lifetime and a larger fraction having longer lifetime. The restricted midlatitude observed-eddy statistics show a similar tendency relative to the global observed-eddy statistics, so the model distribution matches the observed midlatitude distribution better than the observed global distribution (Figs. 14a,b). A bias toward long eddy lifetimes was found for all the simulations for which the eddy analysis was performed, and may be a consequence of the single-active-layer formulation, which excludes baroclinic instability processes that might otherwise lead to more frequent decay of eddy structures. Another possible contributor to this discrepancy is the effect of SSH mapping errors on the identification and tracking of observed eddies.

Tracked-eddy number distributions vs lifetime (weeks) on (a),(c),(e) log–log and (b),(d),(f) log–linear axes for the model solutions with (a)–(f)

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Tracked-eddy number distributions vs lifetime (weeks) on (a),(c),(e) log–log and (b),(d),(f) log–linear axes for the model solutions with (a)–(f)

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Tracked-eddy number distributions vs lifetime (weeks) on (a),(c),(e) log–log and (b),(d),(f) log–linear axes for the model solutions with (a)–(f)

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

For the ^{−1}, in notable contrast to the observed rotational speed distribution, which is approximately lognormal with a peak near 0.1 m s^{−1}.

Probability distributions of (a),(b) eddy amplitudes, (c),(d) length scales, and (e),(f) rotational speeds for the solutions with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Probability distributions of (a),(b) eddy amplitudes, (c),(d) length scales, and (e),(f) rotational speeds for the solutions with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Probability distributions of (a),(b) eddy amplitudes, (c),(d) length scales, and (e),(f) rotational speeds for the solutions with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

The eddy number distributions versus lifetime change only modestly for solutions with the stochastic forcing time scale increased or decreased by a factor of 10, with the shorter forcing time scale resulting in a slight increase in the relative number of short-lifetime eddies, and the longer forcing time scale having the opposite effect, with similar magnitude (Figs. 14c,d). The mean life cycle statistics are likewise only modestly changed, with the fit to observations improving slightly for the shorter forcing time scale. The amplitude and rotational speed distributions are more substantially affected, with the amplitude distribution for the longer forcing time scale, *τ*, with a slight shift toward smaller scales for the larger *τ* and toward larger scales for the smaller *τ*. Note that for this larger-*τ* solution with *τ* solution, it does not meet the autocorrelation criterion, and it exaggerates the bias toward long lifetimes in the eddy number distribution. Consequently, the selection of a single optimal solution among this set would depend strongly on the relative importance ascribed to the various criteria and comparisons.

In summary, changing the forcing time scale has mixed effects on the eddy statistics. Lengthening it so that

The eddy statistics for the more moderately nonlinear solutions, with

## 6. Spectral description

The advective nonlinearity for all solutions with *K* from the stochastic forcing band near the deformation radius

Power spectra of dimensionless model streamfunction *ψ* or sea surface height *η* (solid) and stochastic forcing *K* from simulations with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectra of dimensionless model streamfunction *ψ* or sea surface height *η* (solid) and stochastic forcing *K* from simulations with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectra of dimensionless model streamfunction *ψ* or sea surface height *η* (solid) and stochastic forcing *K* from simulations with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

For the model solution with

Dimensionless (a) power spectrum of streamfunction *ψ* or sea surface height *η* and (b) inverted power spectrum *k* and frequency *ω* from a simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Dimensionless (a) power spectrum of streamfunction *ψ* or sea surface height *η* and (b) inverted power spectrum *k* and frequency *ω* from a simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Dimensionless (a) power spectrum of streamfunction *ψ* or sea surface height *η* and (b) inverted power spectrum *k* and frequency *ω* from a simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

For linear simulations conducted using the same numerical scheme as for the full form of (7) but with the nonlinear term neglected, this reconstruction (25) of the power spectrum of the forcing from that of the solution is accurate (Fig. 18). For the nonlinear numerical simulations, the power spectrum

Power spectrum of dimensionless (a) streamfunction *ψ* or sea surface height *η*, (b) stochastic forcing *k* and frequency *ω* from a linear simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectrum of dimensionless (a) streamfunction *ψ* or sea surface height *η*, (b) stochastic forcing *k* and frequency *ω* from a linear simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectrum of dimensionless (a) streamfunction *ψ* or sea surface height *η*, (b) stochastic forcing *k* and frequency *ω* from a linear simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

For the nonlinear model solutions with

Power spectrum of dimensionless (a) streamfunction *ψ* or sea surface height *η*, (b) stochastic forcing *k* and frequency *ω* from the simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectrum of dimensionless (a) streamfunction *ψ* or sea surface height *η*, (b) stochastic forcing *k* and frequency *ω* from the simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectrum of dimensionless (a) streamfunction *ψ* or sea surface height *η*, (b) stochastic forcing *k* and frequency *ω* from the simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

^{−1}is an estimated empirical damping rate that was deliberately chosen to be relatively large, in order not to underestimate the minimum of

The structure of the resulting inverted spectrum

The inverted model spectrum

A broader perspective on the wavenumber–frequency spectral structure of the observed SSH field is provided by a set of 16 spectra

Power spectra of SSH variability (cm^{2} cpd^{−1} cpkm^{−1}) computed from the AVISO gridded dataset vs zonal wavenumber (10^{−3} cpkm) and frequency (cpd). The maximum frequency is the 1/14 cpd Nyquist frequency for the weekly dataset, and the spectra are computed from observations (from left to right) along (top) South Pacific 45°S, 150°–110°W; South Indian 40°S, 80°–110°E; South Pacific 38°S, 130°–100°W; and South Indian 35°S, 40°–85°E; (top middle) North Pacific 33°N, 180°E–130°W; South Atlantic 33°S, 50°W–0°W; North Atlantic 30°N, 70°–40°W; and South Pacific 30°S, 170°E–120°W; (bottom middle) South Indian 30°S, 60°–100°E; North Pacific 24°N, 125°–165°E; South Pacific 24°S, 160°E–135°W ; and North Atlantic 24°N, 60°W–30°W; (bottom) South Atlantic 24°S, 40°–10°W; South Indian 24°S, 50°–100°E; North Pacific 21°N, 130°E–170°W; and South Indian 14°S, 70°–120°E. Dispersion relations are shown as in Fig. 2, for the parameter values listed in the appendix.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectra of SSH variability (cm^{2} cpd^{−1} cpkm^{−1}) computed from the AVISO gridded dataset vs zonal wavenumber (10^{−3} cpkm) and frequency (cpd). The maximum frequency is the 1/14 cpd Nyquist frequency for the weekly dataset, and the spectra are computed from observations (from left to right) along (top) South Pacific 45°S, 150°–110°W; South Indian 40°S, 80°–110°E; South Pacific 38°S, 130°–100°W; and South Indian 35°S, 40°–85°E; (top middle) North Pacific 33°N, 180°E–130°W; South Atlantic 33°S, 50°W–0°W; North Atlantic 30°N, 70°–40°W; and South Pacific 30°S, 170°E–120°W; (bottom middle) South Indian 30°S, 60°–100°E; North Pacific 24°N, 125°–165°E; South Pacific 24°S, 160°E–135°W ; and North Atlantic 24°N, 60°W–30°W; (bottom) South Atlantic 24°S, 40°–10°W; South Indian 24°S, 50°–100°E; North Pacific 21°N, 130°E–170°W; and South Indian 14°S, 70°–120°E. Dispersion relations are shown as in Fig. 2, for the parameter values listed in the appendix.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectra of SSH variability (cm^{2} cpd^{−1} cpkm^{−1}) computed from the AVISO gridded dataset vs zonal wavenumber (10^{−3} cpkm) and frequency (cpd). The maximum frequency is the 1/14 cpd Nyquist frequency for the weekly dataset, and the spectra are computed from observations (from left to right) along (top) South Pacific 45°S, 150°–110°W; South Indian 40°S, 80°–110°E; South Pacific 38°S, 130°–100°W; and South Indian 35°S, 40°–85°E; (top middle) North Pacific 33°N, 180°E–130°W; South Atlantic 33°S, 50°W–0°W; North Atlantic 30°N, 70°–40°W; and South Pacific 30°S, 170°E–120°W; (bottom middle) South Indian 30°S, 60°–100°E; North Pacific 24°N, 125°–165°E; South Pacific 24°S, 160°E–135°W ; and North Atlantic 24°N, 60°W–30°W; (bottom) South Atlantic 24°S, 40°–10°W; South Indian 24°S, 50°–100°E; North Pacific 21°N, 130°E–170°W; and South Indian 14°S, 70°–120°E. Dispersion relations are shown as in Fig. 2, for the parameter values listed in the appendix.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

The corresponding set of 16 inverted power spectra (again including the 35°S spectrum shown in Fig. 2), computed as

As in Fig. 20, but for inverted power spectra ^{2} week^{−2} cpd^{−1} cpkm^{−1}).

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

As in Fig. 20, but for inverted power spectra ^{2} week^{−2} cpd^{−1} cpkm^{−1}).

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

As in Fig. 20, but for inverted power spectra ^{2} week^{−2} cpd^{−1} cpkm^{−1}).

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

The reduced spectral level along the dispersion relation is reproduced by the inverted power spectra for the model solutions (Figs. 17b, 19c). However, the two additional high-frequency features of the inverted power spectra of observed SSH are not recognizably reproduced by the model solutions. The model inverted power spectra generally lack a maximum along the nondispersive line at high frequencies (Fig. 19c). The model high-frequency spectral structure is more similar to the second feature, with its symmetry in positive and negative wavenumber at fixed frequency, but generally tends to decay toward higher frequency, in contrast to the symmetric spectral lobes in the observed spectra, which tend to increase in amplitude toward higher frequency.

## 7. Space–time filtering

A high-frequency maximum along the nondispersive line in the inverted power spectrum, which is present in the observed spectra (Fig. 21) but absent from the model spectra (Fig. 19c), can be reproduced by the model if the model output is smoothed with a moving space–time filter prior to the spectral analysis (Figs. 22a,b). The filter characteristics for this smoothing are chosen (see appendix) to be relatively consistent with basic aspects of the data processing used to produce the AVISO merged, gridded dataset (Fig. 22). This result is consistent, in a general sense, with previous studies showing that the correspondence between results obtained from numerical models and from AVISO gridded observations sometimes improves if the model results are smoothed (Arbic et al. 2013).

Power spectrum of dimensionless (a),(c) streamfunction *ψ* or sea surface height *η* and (b),(d) inverted power spectrum *k* and frequency *ω*, for the (left) space–time smoothed simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectrum of dimensionless (a),(c) streamfunction *ψ* or sea surface height *η* and (b),(d) inverted power spectrum *k* and frequency *ω*, for the (left) space–time smoothed simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Power spectrum of dimensionless (a),(c) streamfunction *ψ* or sea surface height *η* and (b),(d) inverted power spectrum *k* and frequency *ω*, for the (left) space–time smoothed simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

The second high-frequency feature found in the observed inverted spectra (Fig. 21), the twin lobes of variance centered near

The eddy analysis was also conducted on the filtered model output for this case. The resulting distributions of eddy amplitude and rotational speed are in better agreement with the observed distributions than the unfiltered model output (Figs. 15a,c,e). These distributions are also similar to those obtained in the case with the long stochastic forcing time scale,

A closer resemblance to the observed linear-inverted spectral structure (Fig. 21) can be achieved if the moving space–time filtered fields are blended with fields that have been smoothed only spatially. The inverted model spectrum for these blended fields from the

## 8. Spectral description for *β* = 2.4

The solution for

Dimensionless (a) power spectrum *ψ* or sea surface height *η* and (b) inverted power spectrum *k* and frequency *ω* from the simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Dimensionless (a) power spectrum *ψ* or sea surface height *η* and (b) inverted power spectrum *k* and frequency *ω* from the simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

Dimensionless (a) power spectrum *ψ* or sea surface height *η* and (b) inverted power spectrum *k* and frequency *ω* from the simulation with

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

While some aspects of the

In addition, the inverted power spectrum (Fig. 23b) shows only a relatively weak reduction of power along the dispersion relation, in contrast with the sharp reductions evident in the observed 35°S inverted power spectrum (Fig. 2b) and in those for the

Comparison of this nominal latitude 24° simulation may be made, for example, with the observed SSH power spectrum along 24°N in the North Pacific (Fig. 24). This observed spectrum has maximum levels along a nondispersive line with effective propagation speed that, as anticipated, is equal to or greater than the local theoretical long-wave speed estimates. The corresponding zonal-average observed SSH standard deviation is substantially greater than 0.02 m (Fig. 1), consistent with dynamics that are more nonlinear than those represented by the

As in Fig. 2, but along 24°N between 125° and 165°E in the western North Pacific, and with ^{−1} and

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

As in Fig. 2, but along 24°N between 125° and 165°E in the western North Pacific, and with ^{−1} and

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

As in Fig. 2, but along 24°N between 125° and 165°E in the western North Pacific, and with ^{−1} and

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-18-0260.1

The basic structure of maximum spectral level along a nondispersive line with effective propagation speed equal to or greater than the local theoretical long-wave speed estimates is evident also in the wavenumber–frequency spectrum at 24.5°N in the North Pacific computed by Fu (2004) directly from along-track SSH data from the TOPEX/Poseidon altimeter. At the lower spectral levels away from the maximum, the power isolines for

## 9. Discussion

A fundamental challenge to understanding the dynamics of ocean mesoscale variability is that neither the effective forcing rate—the flux of energy into the mesoscale field from other scales or sources—nor the effective damping rate is known. A third element of uncertainty is that the space and time scales of the effective forcing are not known. In the configuration explored here, the effective forcing is constrained for all simulations to a band of wavenumbers near the dimensionless deformation radius wavenumber *τ* as the single model parameter describing physical variations in the statistics of the spatiotemporal structure of the effective forcing.

The comparisons of the solutions of (7) with amplitude, autocorrelation, eddy, and spectral statistics derived from the AVISO SSH dataset suggest that, away from boundary currents and their extensions, the midlatitude ocean mesoscale regime is characterized by a stochastic SSH forcing rate ^{−1/2}, an SSH damping rate ^{−1}, and a stochastic forcing autocorrelation time scale ^{2} s^{−1} ≈ 1/4 cm^{2} day^{−1}. This variance accumulates in the band of forcing wavenumbers, through an effectively stochastic process with autocorrelation time scale

The model stochastic forcing thus evidently represents a physical process that supports a continuous production of roughly 1 cm^{2} of SSH variance every 4 days at wavelengths near

The spatial scales of the forcing in these simulations were imposed, but the relatively long inferred forcing time scale, equal to or greater than the mesoscale advective time scale, appears independently consistent with the identification of baroclinic instability of the large-scale fields as the primary variance production mechanism. As the forcing time scale increases past the mesoscale advective time scale, broader correlations begin to develop between the forcing and the dynamically evolving field, beyond those between the forcing and the immediate, locally generated, forced response that must always exist to support the stochastic variance production. The variance, or energy, production then begins to depend not only on the stochastic forcing rate but also on the correlation of the forcing and the dynamically evolving field, and the equilibrium amplitude response can no longer be computed accurately from the forcing and damping rates alone, as in (17). The ocean mesoscale regime appears therefore to fall at the edge of, or just outside, the stochastic limit, so that wave-mean interaction is just strong enough to begin to reduce the local mesoscale variance production, but is still weak relative to the overall nonlinearity.

The inferred SSH damping rate of ^{−1} is less than half that estimated by Samelson et al. (2016), a decrease that is consistent with the inverse-square root dependence (17) of amplitude response on damping rate and the 50% decrease in the inferred forcing rate

The different apparent effects of the nonlinearity on the energy balance, the propagation characteristics and the autocorrelation structure of the model fields are nearly paradoxical. On the one hand, the reduction of the damping rate relative to the linear model estimate from Samelson et al. (2016) implies that part of the autocorrelation decay in the numerical simulations must arise from the nonlinearity in (7), that is, from the advection of relative vorticity and the associated advective scrambling of the potential vorticity and streamfunction fields. On the other hand, the dominance of nondispersive propagation in the nonlinear simulations suggests that the local time rate of change of the relative vorticity must evidently be negligible in the mean, as it should otherwise lead to wave dispersion. In either case, the nonlinearity has no direct influence on the energy balance, as the spatial integral of

A speculative resolution of this apparent paradox, partly informed by visual examination of the propagating *ψ* and *q* fields, is that the relative vorticity behaves differently in the regions inside and outside of the dominant, propagating eddy structures. The eddies are approximately axisymmetric and may be assumed to have approximately constant SSH curvature with respect to the eddy-relative radial coordinate, giving an approximate local proportionality between *q* and *ψ* that removes the short-wave dispersion from the linear dynamics, in local analogy to isolated coherent eddy theories (e.g., Larichev and Reznik 1976). In the regions between the eddies, and during eddy growth and decay events, relative vorticity gradients and the consequent nonlinearity may be large. Such a splitting of the relative vorticity field would appear potentially consistent both with the nondispersive propagation characteristics and with the nonlinear enhancement of the autocorrelation decay.

## 10. Summary

The primary goal of this study was to reconcile the semiempirical stochastic model results of Samelson et al. (2014, 2016) with a consistent dynamical model of the ocean mesoscale. The comparisons with eddy statistics show that a correspondence between the numerical simulations from the dynamical model (7) and the observations can be obtained that is broadly similar to the correspondence between the semiempirical stochastic model results and the observations. The dynamical and semiempirical models are both forced stochastically, with the forcing in both taken to represent internal dynamical interactions, while the linear wave propagation included in the semiempirical model is consistent with the long-wave limit of the dynamical model. From this point of view, the reduced-gravity model (7) can be seen as an extension of the semiempirical model that incorporates the basic elements of relative vorticity dynamics and advective nonlinearity while retaining the elements of stochastic forcing and long-wave propagation.

The general correspondence between the eddy statistics from the dynamical and semiempirical models and from the observations is therefore persuasive evidence that the original stochastic amplitude model of Samelson et al. (2014)—the simplest of the three systems—effectively extracts the essential, determining characteristics of the processes supporting the evolution of mesoscale eddy variability both in the ocean and in the dynamical model. Among these characteristics is the remarkable simplicity of the normalized mean eddy amplitude, length scale, and rotational-speed life cycles, with their nearly exact time-reversal symmetry and universality for eddy lifetimes of 16–80 weeks. Evidently, the quasigeostrophic nonlinearity produces, in the mesoscale ocean regime, essentially the same stochastic eddy evolution dynamics that are encoded explicitly in the linear stochastic model as a minimal representation of the observed eddy evolution characteristics.

For the reduced-gravity quasigeostrophic model (7) as configured here, the simulations that nominally best reproduce the global-mean midlatitude statistics derived from the AVISO merged dataset of satellite altimeter SSH measurements have parameters in the range ^{2} day^{−1}, an SSH damping rate 1/62 week^{−1}, and a stochastic forcing autocorrelation time scale of 1 week or greater. The nonlinear removal of energy from the resonant linear wave field is directly apparent in the linearly-inverted SSH spectra for these simulations and is also clearly resolved by the AVISO SSH dataset. This ocean mesoscale regime of the reduced-gravity quasigeostrophic model is identified as a specific candidate for further theoretical and numerical study of ocean mesoscale dynamics.

Further study of the details of the wavenumber–frequency spectral characteristics of the altimeter data is also indicated. The mesoscale regime exists near the nominal 200-km spatial and 30-day temporal resolution limits of the gridded altimeter dataset, where the smoothing and interpolation inherent in development of a gridded dataset from the individual altimeter measurements is likely and, toward smaller scales, eventually certain to affect the observed fields to which the simulations are to be compared. For example, space–time smoothing of the model output prior to analysis was seen to affect the comparisons in ways that were in part similar to increasing the stochastic time scale *τ*, leading to ambiguities in estimation of the best-fit parameters. Similar ambiguities and effects may arise from the filtering used in the altimeter data processing. Some spectral features apparent in the linear-inverted observed spectra seemed possibly to retain a signature of the propagating space–time filtering from the objective analysis procedure used to produce the merged, gridded, SSH dataset, suggesting the possibility that additional information relevant to the comparisons might be present in the unfiltered altimeter data. If so, it is possible that this information might be extracted by other data-analytical methods and used to further constrain the dynamical model. SSH measurements from future advanced, high-resolution satellite altimeters such as that of the planned Surface Water and Ocean Topography (Durand et al. 2010) mission will likely be of great value in definitively constraining these subtle but important characteristics of mesoscale ocean variability.

More broadly, this study is intended as a step toward a future, more complete understanding of the dynamics of the ocean mesoscale. Of special importance in this broader context are the mechanisms governing mesoscale eddy transport and diffusion processes, which remain poorly understood. The parameterization of their effect on larger-scale fields is known to be an important source of uncertainty in projections of the trajectory of Earth’s future climate. Further study of a well-constrained, quasi-equilibrium dynamical model of the ocean mesoscale should yield new qualitative and quantitative insights into these processes and their role in ocean circulation and climate. These future extensions and refinements will need to address the systematic regional variations in ocean mesoscale variability, including the distinct eddy fields associated with eastern and western boundary current systems, which are here subsumed into the single, global midlatitude mean that is taken as the statistical definition of the ocean mesoscale regime.

## Acknowledgments

We are grateful to Brian Arbic and an anonymous reviewer for constructive comments that helped to refine the presentation. Support for this research was provided as part of the National Aeronautics and Space Administration (NASA) Ocean Surface Topography and Surface Water and Ocean Topography Science Team activities through NASA Grants NNX13AD78G and NNX16AH76G.