The ocean mesoscale regime of the reduced-gravity quasi-geostrophic model

Transcription

1 1 The ocean mesoscale regime of the reduced-gravity quasi-geostrophic model 2 R. M. Samelson, D. B. Chelton, and M. G. Schlax 3 CEOAS, Oregon State University, Corvallis, OR USA Corresponding author address: College of Earth, Ocean, and Atmospheric Sciences, 14 CEOAS Admin Bldg, Oregon State University, Corvallis, OR USA rsamelson@coas.oregonstate.edu Generated using v4.3.2 of the AMS LATEX template 1

2 ABSTRACT A statistical-equilibrium, geostrophic-turbulence regime of the stochastically forced, one-layer, reduced-gravity, quasi-geostrophic model is identified in which the numerical solutions are representative of global mean, midlatitude, open-ocean mesoscale variability. Solutions are forced near the internal deformation wavenumber and damped linearly and by high-wavenumber enstrophy dissipation. The results partially rationalize a recent semi-empirical stochastic field model of mesoscale variability motivated by a global eddy identification and tracking analysis of two decades of satellite altimeter seasurface height (SSH) observations. Comparisons of model results with observed SSH variance, autocorrelation, eddy, and spectral statistics place constraints on the model parameters. A nominal best fit is obtained for dimensional SSH stochastic-forcing variance production rate 1/4 cm 2 d 1, an SSH damping rate 1/62 wk 1, and a stochastic forcing autocorrelation timescale τ 1 wk. This ocean mesoscale regime is nonlinear and appears to fall near the stochastic limit, at which 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. Comparison of linearly-inverted wavenumberfrequency spectra shows that a strong effect of nonlinearity, the removal of energy from the resonant linear wave field, is resolved by the gridded altimeter SSH data. These inversions further suggest a possible signature in the merged altimeter SSH dataset of signal propagation characteristics from the objective analysis procedure. 2

3 29 1. Introduction The interior ocean mesoscale supports energetic velocity and sea-surface height (SSH) variability that is typically at least an order of magnitude greater than the long-term mean currents and SSH gradients, and is often described as a form of geostrophic turbulence (MODE Group 1978; Rhines 1975). Lateral eddy fluxes driven by this mesoscale variability are poorly understood and a major uncertainty in our understanding of the ocean s role in the Earth s climate system. Wavenumberfrequency spectra of mesoscale ocean sea-surface height (SSH) variability from satellite altimeter measurements typically show an apparent mix of wave and turbulent dynamics: a non-dispersive spectral structure with an effective phase speed that is loosely consistent with phase speeds of theoretical linear, gravest-mode, long-wave, planetary waves, but an apparent absence of the dispersion at scales near the deformation radius wavelength and larger that would be predicted by linear theory (Fig. 1). However, the rapid decline in spectral power near and beyond the deformation radius wavenumber lends ambiguity to interpretation of the spectral structure and the short-wavelength extent of the non-dispersive behavior, because it occurs close to the wavelength resolution of presently available SSH fields constructed from satellite altimeter data, and so may reflect limitations in the observations that obscure the possible signatures of dynamical processes. From the complementary point of view of eddy phenomenology, a global, quantitative, statistical characterization of mid-latitude mesoscale variability has been developed by Chelton et al. (211) 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. 2; Le Traon et al. 23; Pujol et al. 216). Recently, Samelson et al. (216) proposed a semi-empirical model of mesoscale variability that was able to reproduce nearly all of the basic global mean statistical characteristics of the Chelton et al. 3

4 (211) eddy dataset. However, the semi-empirical 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 semi-empirical, linear, stochastic field model of Samelson et al. (216) with the simplest standard dynamical model of meoscale ocean variability, the reduced-gravity quasi-geostrophic model (e.g., Pedlosky 1987). While the reduced-gravity model has a highly simplified vertical structure, it affords a consistent representation of the fundamental potential vorticity dynamics relevant to mesoscale variability and the associated advective nonlinearity, which the semi-empirical stochastic model lacks. This work may be seen also as an extension of the related modeling study of Early et al. (211), in which observed eddy variability was compared with reduced-gravity quasi-geostrophic 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 statistical equilibrium characteristics of geostrophic turbulence in the reduced-gravity quasigeostrophic model by Morten et al. (217), which provides a convenient quantitative framework and starting point for the numerical simulations described here. In addition, it is related in a more general way to many other previous studies of geostrophic turbulence and quasi-geostrophic models of mesoscale flows Formulation In standard dimensionless form, the reduced-gravity quasi-geostrophic potential vorticity equa- tion for the velocity stream function ψ is t ( 2 ψ F 1 ψ) + β ψ x + J(ψ, 2 ψ) = F + D, (1) 4

5 in which distance has been scaled by L, velocity by U, time by L/U, stream function by UL, and the forcing F and damping D by U 2 /L 2. The dimensionless parameters in (1) are F 1 = L 2 /L 2 R, β = β L 2 /U, where L R is the deformation radius and β is the meridional gradient of the Coriolis parameter, and the notation J(, ) denotes the Jacobian derivative. In the reduced-gravity interpretation, L R = (g H) 1/2 / f, where g = g ρ/ρ is the reduced gravity for a single active layer of thickness H overlying a deep, motionless layer with fixed fractional density difference ρ/ρ, g = 9.81 m s 2 is the acceleration of gravity, and f is the Coriolis parameter. For the case of stochastic forcing, (1) may be written as a stochastic differential equation for the potential vorticity q on the characteristics defined by the velocity field, where dq = σ F dw t + D dt on ( dx dt, dy ) ( = ψ dt y, ψ ), (2) x q = 2 ψ F 1 ψ + βy, (3) σ F is a constant and W t is the Wiener process (i.e., standard continuous Brownian motion). For numerical solution, the Wiener process is replaced here by σ F dw t = σ F F (x,y,t/τ) τ 1/2 dt = F dt, (4) where F (x,y,t/τ) is a smooth stochastic function with unit time-mean spatial standard deviation and auto-correlation timescale τ, and the second equality defines F in (1). The factor 1/τ 1/2 in (4) ensures that the expectation value of the time-integral of the variance of F is equal to σf 2 t for t τ, consistent with (2). The numerical implementation of F follows Morten et al. (217), as does the specification of D as the sum of a linear damping on ψ and a tri-harmonic term to smooth the small scales, D = r ψ ψ r 6 6 ψ, (5) 5

6 with constant damping coefficients r ψ and r 6. The dimensional scales L and U are arbitrary and may be chosen to reduce the number of di- mensionless parameters in (1)-(5). Setting L = L R, U = σ 2/3 F L R, (6) where σ F is the dimensional equivalent of σ F, effectively sets F 1 = 1 and σ F = 1, yielding (1) in the reduced form considered here: t ( 2 ψ ψ) + β ψ x + J(ψ, 2 ψ) = F (x,y,t/τ) τ 1/2 + r ψ ψ r 6 6 ψ. (7) In (7), there are three parameters of primary physical interest: β, r ψ, and τ. The spatial structure of the forcing function F must also be specified, but is here taken to have the same form as in Morten et al. (217), with forcing restricted to wavenumbers in an annulus of width δk = 1/3 around the deformation radius wavenumber K = 1, while the constant r 6 is chosen as a small value sufficient to maintain smoothness at small scales. If τ is sufficiently small relative to the intrinsic timescales 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. (217). Specification of a reference latitude φ determines the corresponding values of the dimensional Coriolis parameter f and its meridional gradient β. If a dimensional value of L R is also specified, then, for a given solution of (7), the value of the velocity scale U can be obtained from the given value of the dimensionless parameter β: U = β L 2 R β. (8) 6

7 18 This in turn allows determination of the dimensional SSH, η, from the dimensionless stream 19 function ψ, η = f g 1 ψ = f g 1 UL R ψ = β f gβ L3 Rψ, (9) where ψ is the dimensional stream function, as well as the dimensional linear damping and stochastic forcing rates, r ψ and σ F : r ψ = U r ψ = β L R L R β r ψ, (1) ( ) U 3/2 ( ) β L 3/2 R σ F = =. (11) β L R 112 The equivalent SSH stochastic forcing amplitude σ W is then σ W = f L2 R g σ F = f L7/2 R g ( ) 3/2 β. (12) β Values of σ W may be compared directly with those characterizing the forcing rate for the related semi-empirical model of Samelson et al. (216), which was based on a long-wave version of (1) and written in terms of the dimensional SSH. Numerical solutions of (7) were computed on a zonal-meridional grid with periodic boundary conditions, using a non-aliasing pseudo-spectral code originally derived from the twolayer channel model described by Samelson and Pedlosky (199) 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 t = 116 for the first set and t = 625 for the second set, corresponding to dimensional times of roughly 3 and 37.5 years, respectively. A dimensionless integration time of up to t 5, equivalent to a dimensional time of 1-3 years, was typically required for the numerical solutions to reach statistical equilibrium. The autocorrelation and spectral analyses were conducted on output-field time series of length 248, and the eddy analysis on output-field time series of length 125, for dimensionless output time-increments of.46 for the 7

8 first set and.25 for the second set, corresponding to dimensional output time-increments of 4.4 d and 5.5 d, respectively. For the eddy identification and tracking analysis, the model SSH field was tiled periodically and interpolated onto at grid with uniform zonal and meridional spacing x = y = km, identical to the domain used by Samelson et al. (216) for the eddy analysis of the linear stochastic model SSH fields The ocean mesoscale parameter regime 132 In the stochastic limit τ 1, an integration of (2) over the autocorrelation time τ gives q(t + τ) q(t) δ F + r ψ ψ(t)τ, (13) where δ F = F τ 1/2 is the stochastic increment with ensemble-averaged (or t-averaged) variance δf 2 = τ for σ F = 1, and the enstrophy dissipation term has been neglected for simplicity, so that D = r ψ ψ. If an appropriately weighted mean wavenumber K may be chosen so that q(t + τ) q(t) ( K 2 + 1) [ψ(t + τ) ψ(t)], the resulting ensemble-averaged variance equation may be further approximated as ( K 2 + 1) 2 ψ(t + τ) 2 ( K r ψ τ) 2 ψ(t) 2 + τ. (14) An equivalent result can be obtained from an integration of the energy equation for (7), under the assumption that ψ is effectively constant on the timescale τ except for a directly-forced component proportional to F t. At statistical equilibrium, the ensemble-averaged variance of ψ is stationary, and it then follows from (14) that for r ψ τ 1, the standard deviation σ ψ of ψ can be estimated as σ ψ 1 [2( K 2. (15) + 1)r ψ ] 1/2 Thus, in the stochastic limit, and under the assumption that the potential vorticity variance is concentrated near the deformation wavenumber K = 1, the dimensionless stream function standard 8

9 deviation σ ψ is approximately equal to one-half of the inverse square-root of the dimensionless damping rate. A second relation between σ ψ and r ψ, which depends upon dimensional parameters, may be derived and then combined with (15) to obtain an a priori estimate of the dimensionless parameter values appropriate for the ocean mesoscale regime. The dimensionless quantity σ ψ /β serves as a measure of nonlinearity: with the solution variance again assumed to be concentrated near the deformation wavenumber K = 1, the ratio of nonlinear to linear terms in (7) may be estimated as J(ψ, 2 ψ) β ψ x σ ψ β. (16) By (9), this ratio is proportional to the standard deviation σ η of dimensional SSH: σ ψ β = g β f LR 3 σ η. (17) 153 In turn, by (1), 154 and therefore r ψ β = r ψ β L R, (18) σ ψ r ψ = g f L 2 R σ η r ψ. (19) With L R = c 1 / f (Fig. 3), where c 1 3 m s 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 (17) and (19) are known, while σ η and r ψ must be estimated from observations. The majority of the eddies in the Chelton et al. (211) dataset that are not associated with western boundary currents or the Antarctic Circumpolar Current are found between latitudes of 2 N to 4 N or 2 S to 4 S (Fig. 2). In these subtropical interior regimes, the standard deviation σ η of SSH from the altimeter dataset used by Chelton et al. (211)) ranges roughly from.2 m to.7 m, with zonal averages that include the western boundary currents and their extensions lying 9

10 toward the upper end of that range (Figs. 2,3). Then, for example, taking σ η.7 m at φ = 35 gives σ ψ /β 7 and indicates strong nonlinearity, while taking σ η.2 m at φ = 24 gives σ ψ /β.9 and indicates moderate nonlinearity (Fig. 3). From their model fit to observations, Samelson et al. (216) estimate a damping rate for η or ψ as r ψ 1/25 wk 1. This rate could be used directly in (18) and (19), but an improved estimate is obtained for r ψ 1/5 wk 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. (216), all of the autocorrelation decay along long-wave characteristics must derive from the damping. With this nominal value for r ψ and with L R = c 1 / f as previously, the ratio r ψ /β depends only on φ, taking values of roughly.45 at φ = 35 and.28 at φ = 24 (Fig. 3). Combining these two estimates of σ ψ /β and r ψ /β, or substituting the representative values for σ η and r ψ into (19), gives the linear relations σ ψ (7/.45)r ψ 155r ψ at φ = 35 and σ ψ (.9/.28)r ψ 32r ψ at φ = 24. Each of these two equations may then be combined with (15) to obtain two estimates of the dimensionless parameters r ψ and β at which the solutions with the desired values of σ η and σ ψ /β may be found (Fig. 3). For σ η.7 m at latitude φ = 35, the estimated values are (β,r ψ ) (.53,.2), while for σ η.2 m at latitude φ = 24, the estimated values are (β,r ψ ) (2.4,.7). The corresponding estimated values of σ ψ are roughly 3.7 at φ = 35 and 2 at φ = 24. For these estimates, the value K =.75 has been used in (15). Along with the damping timescale R 1/25 wk 1, Samelson et al. (216) estimate a dimensional value of m s 1/2 for σ W. For L R = 4 at φ = 35, this corresponds by (11) to σ F = (g/ f LR 2)σ W s 3/2, and thus to U.6 m s 1 by (8) and to dimensionless parameters β.5, r ψ = RL R /U.4, τ = t U/L R.9, where the autocorrelation timescale is derived from the Samelson et al. (216) discrete time-step t = 1 wk. For L R = 6 km 1

11 at φ = 24, the same values of σ W, R, and t give σ F s 3/2, U =.66 m s 1, and β 1.1, r ψ.6, τ.67. This provides an alternative point of reference for the relevant model parameter values. A third point of reference is provided by the Run 3 simulation of Morten et al. (217), which was identified by Morten et al. (217) 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. (217), the Run 3 simulation had parameter values β MAF = 1, k d = F 1/2 1 = 6, τ MAF =.1, r MAF = 1.4. The equivalent parameter values for (7) are β = β MAF /(k d τ 1/3 MAF ) = 3.59, τ = τ4/3 MAF =.464, r ψ = r MAF /(kd 2 τ1/3 MAF ) =.84 (see Appendix). With the stochastic-limit estimate (15), the Run 3 simulation can be anticipated to have σ ψ 1.7 and σ ψ /β.5, and thus to be less nonlinear than the moderately nonlinear estimate σ ψ /β.9 obtained for σ η.2 m at latitude φ = 24. Guided by these estimates, two primary sets of simulations were conducted, in each of which β was held fixed while r ψ and τ were varied (Figs. 4,5). The first set, with β =.62, was directed toward reproducing a more strongly nonlinear regime, with dimensional SSH standard deviation σ η =.7 m for L R = 4 km at a nominal latitude φ = 35. For this set, the dimensional velocity and time scales are U.5 m s 1 and T = L R /U s 9.5 d, and the dimensional forcing amplitude is σ F = T 3/ s 3/2, equivalent to σ W m s 1/2. The second set, with β 2.4, was directed toward reproducing a more moderately nonlinear regime, with dimensional SSH standard deviation σ η =.2 m for L R = 6 km at a nominal latitude φ = 24. For this set, the dimensional velocity and time scales are U.3 m s 1 and T = L R /U s 22 d, and the dimensional forcing amplitude is σ F s 3/2, equivalent to σ W m s 1/2. A number of additional simulations were conducted, of which only the equivalent to the Morten et al. (217) Run 3 is described here. 11

12 21 4. Amplitude and auto-correlation For β =.62, the three simulations with (r ψ,τ) = {(.22,.28),(.22,.92),(.54,9.3)} are within 5% of the amplitude target σ ψ /β = 7, while for β = 2.4, the two simulations with (r ψ,τ) = {(.7,1),(.16,5)} are within 5% of the amplitude target σ ψ /β =.9 (Fig. 6). For β =.62, σ ψ has equal values for the two simulations with r ψ =.22 and the smallest stochastic timescales, τ = {.28,.93)}, but progressively smaller values for fixed r ψ and increasing τ, suggesting that the stochastic limit is approached for τ 1. For r ψ =.22, the value of σ ψ is reduced by half for τ = 9.3 from its value for τ < 1, suggesting that the stochastic limit is violated for τ 1, as the stochastic timescale becomes comparable to the intrinsic dynamical timescale and correlations can develop between the forcing and the dynamically evolving field. Because the associated transition occurs for τ 1, this correspondence also indicates that the dimensional timescale T = L R /U used in the scaling is an accurate estimate of the intrinsic dynam- ical timescale. Nonetheless, the dependence σ ψ r 1/2 ψ predicted by (15) for solutions with fixed τ and β is approximately reproduced even when τ 1 (Fig. 6). For β =.62 and r ψ =.22, the estimate (15) approximately matches the result from the numerical simulations with τ < 1 if the choice K.75 is made, consistent with a concentration of the solution variance at wavenumbers slightly smaller than the radian deformation wavenumber k d = L 1 R. For the linear stochastic models of Samelson et al. (214, 216), the damping coefficient r determined the temporal autocorrelation parameter α directly, according to α = 1 r < 1, where for the stochastic field model (Samelson et al. 216), the autocorrelation was computed along long-wave characteristics. Samelson et al. (214) computed values of α from the observed SSH along long-wave characteristics, and showed that the values so computed at mid-latitudes were approximately.95, for time units of weeks, and that these values were much larger than those 12

13 computed at fixed spatial points. The corresponding values of α computed for the models of Samelson et al. (214) and Samelson et al. (216) were.96 and.94, respectively. For the nonlinear quasigeostrophic model (7), the nonlinearity can alter the autocorrelation, and the parameter α cannot be computed directly from r ψ. Instead, α must be estimated from the empirical autocorrelation function A a (t) computed from the model fields, and is taken here as the maximum value of the effective α obtained along characteristics of the form dx/dt = ac R, where c R = β (with F 1 = 1) is the model linear long-wave speed and a is a parameter with.5 a 1.. The empirical autocorrelation function was assumed to have the form A a (t) = α t, and the model value α eff was computed as the mean of A a (t) 1/t for 1 wk t 2 wk, with the optimization using a-increments of.5 (Fig. 7). The resulting equivalent damping values r α = 1 α eff are roughly.2 wk 1 greater than the dimensional model damping r ψ = ULR 1 r ψ (Fig. 8). This additional decorrelation found for solu- tions of (7) relative to the linear stochastic model solutions is evidently the result of the nonlinear, advective, scrambling of the model fields, as the difference between r α and r ψ tends to increase with increasing nonlinearity σ ψ /β (Fig. 8). Two of the numerical solutions satisfy both an amplitude criterion and an autocorrelation crite- rion: both have nonlinearity measure σ ψ /β within 5% of the respective target value and both have empirical autocorrelation parameter α eff within the observed bounds,.94 < α eff <.96 (Fig. 8). These are the two solutions with β =.62 and r ψ =.22, for which r ψ =.16 wk 1 1/62 wk 1, and either τ =.92 or τ =.28. For β = 2.4, the solution with r ψ =.7,τ = 1 meets the nonlinearity criterion and nearly meets the autocorrelation criterion. A few additional solutions meet or nearly meet at least one criterion, while the remainder are relatively far from meeting either. 13

14 Eddy statistics Coherent eddies in the dimensional SSH field from the solution with β =.62, r ψ =.22, and τ =.92, which met both the amplitude and the autocorrelation criteria, were identified and tracked using the Chelton et al. (211) SSH-based procedure, after conversion of ψ to dimensional SSH η using the 35 -latitude scaling given in Section 3 for the β =.62 set of simulations. At each time point along an identified eddy track, the procedure produces an eddy amplitude, length scale, and rotational speed, as well as an index identifying the eddy as cyclonic or anticyclonic. The amplitude is defined as the difference between the maximum SSH within the eddy and the mean SSH around the outermost closed contour of SSH. The length scale is the radius of a circle with area equivalent to that contained by the closed SSH contour with maximum mean geostrophic speed, and the rotational speed scale is that mean geostrophic speed. Consistent with Chelton et al. (211), the amplitude, length scale, and rotational speed statistics discussed below were computed for eddies with lifetimes of at least 16 weeks. The basic structure of the mean normalized eddy amplitude, length-scale, and rotational-speed lifecycles found in the observations by Samelson et al. (214) and reproduced in the semiempirical stochastic models of Samelson et al. (214, 216) is found also in these simulations: symmetry under time-reversal, negative curvature with respect to t throughout the cycles, and near-universality over the lifetime range from 16 weeks to 8 weeks (Figs. 9,1,11). The cyclemean dimensional amplitude, length-scale, and rotational speed all 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. The model innovations the first differences of the corresponding quantities during the eddy life- 14

15 cycles have distributions that are similar to but systematically less normal than the observed innovation distributions, with smaller standard deviations and larger kurtoses. The structure of the simulated eddy number distribution vs. 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 (Fig. 12). While the relatively small number of model eddies gives rise to large scatter in the number distribution at long lifetimes, 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. 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 singleactive-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. For the β =.62, r ψ =.22, τ =.92 simulation, the distributions of amplitudes, lengthscales, and rotational-speeds for all eddy realizations for eddies with lifetimes of 16 weeks or greater roughly reproduce the observed distributions, with the rotational-speed distribution showing the largest differences (Fig. 13). The fraction of eddies with moderate, 8-15 cm amplitudes is modestly overestimated, while the fraction of eddies with relatively small or large amplitudes is underestimated. The distribution of model eddy length scales is more sharply peaked near 7 km than is the observed distribution, and the model produces no eddies with length scales larger than 18 km. The distribution of model eddy rotational speeds is approximately symmetric around a mean of roughly.2 m s 1, in notable contrast to the observed rotational speed distribution, which is approximately log-normal with a peak near.1 m s 1. 15

16 The eddy number distributions vs. lifetime change only modestly for solutions with the stochastic forcing timescale increased or decreased by a factor of 1, with the shorter forcing timescale resulting in a slight increase in the relative number of short-lifetime eddies, and the longer forcing timescale having the opposite effect, with similar magnitude (Fig. 14). Note that for this largerτ solution with β =.62, the damping parameter r ψ has been reduced by a factor of four, in order that the amplitude target is still matched (Fig. 6). The mean lifecycle statistics are likewise only modestly changed, with the fit to observations improving slightly for the shorter forcing timescale. The amplitude and rotational speed distributions are more substantially affected, with the amplitude distribution for the longer forcing timescale, τ = 9.3, closely matching the observed distribution for amplitudes less than 2 cm, and the rotational speed distribution for τ = 9.3 also shifting closer to the observed distribution (Fig. 15). In contrast, when the forcing timescale is decreased to τ =.93, the amplitude and rotational speed distributions both shift farther from the observed distributions. The length-scale distributions are relatively insensitive to these changes in τ, with a slight shift toward smaller scales for the larger τ and toward larger scales for the smaller τ. For the large-τ solution with β =.62, the reduced friction results in an excessively large autocorrelation scale α eff =.983, r α = 1 α eff =.17 (Fig. 8). Thus, while this solution meets the amplitude criterion and matches the overall eddy amplitude, length-scale, and rotational-speed statistics better than the smaller 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 property comparisons. In summary, changing the forcing timescale has mixed effects on the eddy statistics. Lengthening it so that τ 1, while decreasing the damping to maintain the amplitude, gives a substantially 16

17 improved match to the observed amplitude and rotational-speed distributions, has a mixed effect on the mean lifecycle statistics, and slightly increases the bias toward long-lifetime eddies. Shortening it so that τ 1 slightly reduces the bias toward long-lifetime eddies, and fits amplitude, length-scale, and speed lifecycles and mean amplitude vs. lifetime slightly better, but increases the mismatch in mean eddy speed vs. lifetime and in the eddy amplitude, length-scale, and speed distributions. The eddy statistics for the more moderately nonlinear solutions, with β = 2.4, φ = 24, and L R = 6 km, show much greater differences from the observed statistics. Perhaps surprisingly, the corresponding eddy number distributions vs. lifetime are even more biased toward long lifetimes than the more strongly nonlinear solutions, although there is some hint that they may better represent the long-lifetime distribution structure (Fig. 16). The eddy amplitudes and rotational speeds for the moderately nonlinear solutions are generally much smaller, and the length scales much larger, than observed, though again the model produces no eddies with length scales larger than 18 km (Fig. 17). The primary focus of the subsequent analysis is therefore on the more strongly nonlinear solutions with β.6. It should be noted, however, that the more moderately nonlinear solutions may be more relevant for comparison to observations in the tropics, within approximately 2 latitude of the equator Spectral description The advective nonlinearity for all solutions with β.6 spreads the model SSH spectral power over total wavenumber K from the stochastic forcing band near the deformation radius K = 1 to higher and lower wavenumbers, with the high-wavenumber spectrum following a K 5 power law, consistent with the classical theory of geostrophic turbulence (Fig. 18). Of the three solutions with β =.62 for which the eddy statistics were analyzed, the two with r ψ =.22, and τ < 1 have 17

18 very similar total-wavenumber SSH power spectra, while the third, with τ = 9.3 and r ψ =.54, has relatively more power at wavenumbers K <.5 and less power at wavenumbers K >.5 (Fig. 18). The spectral shape of this third solution, with longer stochastic forcing timescale and weaker damping, appears to match that of the observed mean AVISO radial wavenumber spectrum slightly better than the other two for.3 K.7, that is, for scales roughly one and a half to three times larger than the deformation radius wavelength (Fig. 18). For the model solution with β =.62, r ψ =.22, and τ =.92, zonal wavenumber - frequency power spectra of dimensionless stream function or, equivalently, SSH (Fig. 19) show nondispersive structure that is qualitatively similar to observed wavenumber-frequency spectra, with effective phase speed near the linear long-wave speed, and an apparent absence of short-wave dispersion with rapidly declining spectral power beyond the deformation radius wavenumber (Fig. 1). Some subtle differences in the spectral structure may be noted away from the non-dispersive line: for spectral power levels two or more orders of magnitude smaller than the maximum, the observed spectra (e.g., Fig. 1) tend to have isolines of constant spectral power that are roughly parallel to the non-dispersive line, while the model spectra tend to have constant-power isolines that are roughly parallel to lines of constant zonal wavenumber. The other solutions with β.6 have similar wavenumber-frequency spectra, with similar subtle differences from the observed spectrum. An alternate representation of the same wavenumber-frequency spectral information is available that more directly illustrates the influence of the nonlinearity. If the nonlinear term in (7) is moved to the right-hand side, the Fourier transform of the resulting equation is taken, and the power spectrum is then formed, the result is: L 1 ˆψ 2 = ˆ F ˆ J 2, (2) 18

19 37 where L is the transfer function defined by, L = { [ω(k 2 + 1) + βk ] 2 + r 2 } 1, (21) ˆ J is the Fourier transform of J(ψ, 2 ψ) and for simplicity the enstrophy dissipation term ˆD ens, which is negligible in the SSH dynamical balance, has been dropped. If (7) were linear, so that ˆ J =, and the forcing were known, then the transfer function L would give the power spectrum of the solution directly, ˆψ 2 = L ˆ F 2. (22) Conversely, if the spectrum ˆψ were known, then in the linear case the corresponding forcing spectrum could be obtained directly from (2) wth ˆ J = : F ˆ 2 = L 1 ˆψ 2. (23) For linear simulations conducted using the same numerical scheme as for the full equation (7) but with the nonlinear term neglected, this reconstruction (23) of the power spectrum of the forcing from that of the solution is accurate (Fig. 2). For the nonlinear numerical simulations, the power spectrum L 1 ˆψ 2 in (2) computed from the solution can be compared to the power spectrum F ˆ 2 of the forcing, and any differences between these two spectra can be attributed to the nonlin- earity J(ψ, 2 ψ) in (7). This gives a direct illustration of the influence of the nonlinearity on the spectral structure. For the nonlinear model solutions with β =.62, the inverted zonal wavenumber - frequency power spectrum L 1 ˆψ 2 (k,ω) shows two distinct effects of the nonlinearity (Figs. 21, 22). First, there is a dramatic reduction in power of the inverted spectrum around the linear inviscid dispersion relation, ω = βk/(k 2 + 1), which contrasts sharply with the symmetric wavenumber dependence of the forcing (Fig. 21). Second, away from the linear dispersion relation the spectral 19

20 power is spread beyond the interval of forcing wavenumbers to much larger wavenumbers, with the wavenumber range tending to increase with frequency, and to larger frequencies (Fig. 22). Thus, at statistical equilibrium, the nonlinearity removes energy from the resonant linear wave field and scatters it to other wavenumbers and frequencies. For all wavenumbers k > 1, outside the range of the forcing, there is a net input of energy from the nonlinearity, but along the dispersion relation this input is much reduced relative to its magnitude at higher and lower frequencies for the same wavenumber k <, and at similar frequencies for the corresponding positive wavenumber k (Fig. 22). The reduction in power around the dispersion relation is most dramatic at wavenumbers k <.5, corresponding to scales smaller than twice the deformation radius wavelength, but is still discernible at even larger scales, in the wavenumber range.5 k < (Fig. 21). For the observed SSH zonal wavenumber - frequency power spectrum from the gridded AVISO data (Fig. 1), an analogous inversion L 1 ˆη 2 can be computed (Fig. 21). For simplicity, and to avoid additional dependence on the degree of mismatch between the various theoretical linear dis- persion relations and the observed SSH spectrum, an empirical dispersion relation was constructed from the observed spectrum, with the form ω = c k /(λ 2 k 2 + 1), (24) where c is an effective long-wave speed, λ is an effective deformation-radius wavelength, and k is the dimensional zonal wavenumber. Any effect of meridional structure was incorporated only implicitly through the fit of the empirical parameters. The empirical inverse transfer function was then constructed from this empirical dispersion relation, taking the form L 1 = { ω [k 2 + (2π/λ ) 2 ] + βk } 2 + r 2, (25) 2

21 where r = 1/16 wk 1 is an estimated empirical damping rate that was deliberately chosen to be relatively large, in order not to underestimate the minimum of L 1 which would result in an overestimate of the effect of the nonlinearity. The structure of the resulting inverted spectrum L 1 along the dispersion relation, ˆη 2 is strikingly similar to the inverted model spectrum L 1 ˆψ 2 (k,ω) (Fig. 21). Most notably, the local minimum in power along the empirical dispersion relation is well defined and very similar to that in the model spectrum. The minimum in the observed inverted spectrum is clearly evident for all zonal wavenumbers k < cpkm and so extends to wavelengths as large as 5 km, which are much larger than the approximately 2-km wavelength resolution limit of the gridded AVISO data (Chelton et al. 211). This is strong evidence that nonlinearity controls the structure of the wavenumber- frequency spectrum of SSH in the ocean mesoscale regime. This result has been previously sug- gested on the basis of the non-dispersive structure of the SSH power spectrum (e.g., Early et al. 211), but the rapid decline in spectral power toward higher wavenumbers has been difficult to sep- arate from the decline in power associated with the approach to the spatial resolution limit of the AVISO data, and consequently it has been difficult to attribute the structure of the power spectrum in this range unambiguously to nonlinear effects. Although based on the same spectral infor- mation, the representation in terms of the inverted power spectrum (2) illustrates more clearly and definitively that a dramatic effect of nonlinearity on the wavenumber-frequency spectrum is resolved by the gridded altimeter data. The inverted model spectrum L 1 ˆψ 2 (k,ω) illustrates that the spreading of energy by the nonlinearity extends continuously in the model to the highest frequencies and smallest spatial scales, with the nonlinear removal of energy from the linear dispersive wave line extending also to the smallest spatial scales (Fig. 22). The structure of this inverted spectrum is dramatically different from the forcing spectrum, revealing the broad influence of nonlinearity. The structure is 21

22 not strongly dependent on the timescale of the forcing, except to the extent that the signature of the high-frequency forcing can be recognized in the inverted power spectra for the solutions with small stochastic timescales (Fig. 23). From comparisons of the power spectra and inverted power spectra, there is not a clear best match to the observed 35 S Indian-Ocean spectra (Figs. 1, 21) among the β.6 solutions, although the structure of the power spectrum for k > may suggest a preference for the τ = 9.3 solution among the three for which the eddy statistics were computed. A broader perspective on the wavenumber-frequency spectral structure of the observed SSH field is provided by a set of 16 spectra ˆη 2 (k,ω ) computed from the AVISO gridded dataset along longitudinal sections at various locations, for frequencies extending to the 1/14 cpd Nyquist frequency and zonal wavenumbers extending to cpkm, or wavelengths of 62 km, much smaller than the nominal 2-km spatial resolution limit of the dataset (Fig. 24). This set includes the 35 S spectrum discussed above (Fig. 1). The locations for these spectra were selected to avoid western boundary current extensions and the Antarctic Circumpolar Current, where advection by mean currents strongly modifies the spectral structure. Viewed from this perspective, all of the 16 spectra have similar qualitative structure: a maximum along a non-dispersive line corresponding to westward propagation, at an effective phase speed that is typically within a factor of two of theoretical long-wave phase speeds for the corresponding region, and a decay in spectral level to- ward higher frequencies and away from the fixed-frequency maximum toward higher wavenumber magnitudes. The corresponding set of 16 inverted power spectra (again including the 35 S spectrum, Fig. 21), computed as L 1 ˆη 2 for transfer functions from empirical dispersion relations estimated indi- vidually for each spectrum, show a similarly consistent, but more complex, structure (Fig. 25). In each, there is a reduced spectral level along the empirical dispersion relation, consistent with the removal of energy from the linear wave field by nonlinearity, and two other dominant fea- 22

23 tures: a maximum centered along the non-dispersive line and two lobes of variance centered near k = ±5 1 3 cpkm and increasing toward the Nyquist frequency. The maxima along the nondispersive line extend to higher frequency than the corresponding maxima in the power spectra, generally terminating either at the spatial resolution limit, for the higher latitudes where the nondispersive propagation is slower, or at the Nyquist frequency, for the lower latitudes where the non-dispersive propagation is faster. Both of these features extend continuously into frequencies and wavenumbers well beyond the nominal 3-d temporal and 2-km spatial resolution limits of the AVISO gridded dataset, and thus may in part reflect characteristics of the data processing procedures used to produce the merged dataset from the along-track altimeter data. The reduced spectral level along the dispersion relation is reproduced by the inverted power 466 spectra for the model solutions. However, the other two dominant 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 dominant maximum along the non-dispersive line (Figs. 22,23). The model 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 Space-time filtering A maximum along the non-dispersive line in the inverted power spectrum, which is present in the observed spectra but absent from the model spectra, can be reproduced by the model if the model output is smoothed with a space-time filter prior to the spectral analysis, for filter characteristics that are chosen to be relatively consistent with basic aspects of the data processing used to produce the AVISO merged, gridded dataset (Fig. 26). The space-time filter used here con- 23

24 sists of an initial zonal and meridional smoothing with half-power wavelength 65 km, followed by smoothing along zonal modified-long-wave characteristics with half-power wavelength 2 km, and finally smoothing in time with half-power period 3 d (see Appendix). Each smoothing step is carried out using a Parzen smoother. The Gaussian decay scales L G and T G that correspond to Parzen-smoother half-power wavelengths and periods L P and T P may be computed from eq. (C.5) of Chelton et al. (218): (L G,T G ) =.187(L P,T P ). (26) Thus, for Parzen-smoother half-power wavelength L P = 2 km and period T P = 3 d, the corresponding Gaussian decay scales are L G = 37.4 km and T G = 5.6 d. The twin lobes of high-frequency variance in the observed inverted spectra are not reproduced by the spectra of the filtered model output. However, the filtered output for the case β =.62, r ψ =.22, τ =.92 does better reproduce the slope of the low-power isolines in the observed spectra: for the filtered power spectra, these isolines are roughly parallel to the non-dispersive line even for k > (Fig. 26,1), while for the unfiltered inverted power spectra, these isolines are roughly independent of frequency for k > (Fig. 19). The filtered output also more closely resembles the AVISO total wavenumber power spectrum, with reduced power and steeper decay for wavenumbers K > 1 (Fig. 27). 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 (Fig. 28). These distributions are also similar to those obtained in the case with the long stochastic forcing timescale, β =.62, r ψ =.54, τ = 9.3 (Fig. 17). On the other hand, the agreement of the simulated eddy length-scale distribution (Fig. 28) and of the mean normalized eddy lifecycle statistics with the observations slightly 24

25 51 52 degrades with the filtering, with the latter again similar to the case with the long forcing timescale but the former showing an opposite effect Spectral description for β = The solution for β = 2.4, r ψ =.7, and τ = 1 was intended to represent a dimensional SSH standard deviation of.2 m near latitude 24 with internal deformation radius L R = 6 km. This solution has σ ψ /β =.85 and consequently is less nonlinear than the solutions with β.6 and σ ψ /β 7. This solution has power spectrum and inverted power spectrum structure (Fig. 29) that are intermediate between the more nonlinear β.6 cases (Figs. 19,22,21,23) and the linear example for which the inverted power spectrum exactly reproduces the forcing spectrum (Fig. 2). While some aspects of the β = 2.4 spectral structure may appear to match the observed spectra better, more detailed examination reveals further discrepancies. The low-level power isolines in the power spectrum for k > have a slope that is roughly parallel to the line of maximum power, in perhaps better agreement with the observed low-level spectral isolines, but the amplitude decay away from the line of maximum power is more abrupt than in the observations. In addition, the line of maximum power falls well below the long-wave speed. Similarly, although the effective damping r α computed from autocorrelation decay scale for this solution is.35, not far from the target range.4 r α.6, the characteristic along which this maximum autocorrelation is found has an effective speed that is only slightly more than half of the long-wave speed. These slow propagation speeds are inconsistent with the observed SSH signal propagation speeds, which are systematically no less, and typically larger, than the local theoretical long-wave speeds. In addition, the inverted power spectra show only a relatively weak reduction of power along the dispersion relation, in contrast with the sharp reductions evident in the observed inverted power spectra and in those for the β.6 solutions (Figs. 21,29). There is an amplification of the in- 25

26 verted power spectrum that is aligned approximately along the long-wave non-dispersive line though again at a slower effective propagation speed that bears some resemblance to the nondispersive feature in the observed inverted power spectra (Fig. 3). However, even for this more linear solution, there is broad spreading of energy at slightly lower power levels across positive and negative wavenumbers, with the wavenumber range increasing toward higher frequency. Consequently, the inverted model spectrum contains only a hint of the structure along the non-dispersive line that is a dominant but unexplained feature of the observed inverted power spectra, and does not provide a persuasive physical explanation for this structure. 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. 31). This spectrum has maximum levels along a non-dispersive line with effective propagation speed that, as anticipated, is equal to or greater than the local theoretical long-wave speed estimates. The corresponding zonalaverage observed SSH standard deviation is substantially greater than.2 m (Fig. 2), consistent with dynamics that are more nonlinear than those represented by the β = 2.4, r ψ =.7, τ = 1 solution. The corresponding observed inverted power spectrum shows a sharp reduction in power along the linear dispersion relation (Fig. 31), which would also be consistent with a greater degree of nonlinearity than in the β = 2.4 model solution. The basic structure of maximum spectral level along a non-dispersive 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 (24) directly from along-track SSH data from the Topex/POSEIDON altimeter (Fig. 32). At the lower spectral levels away from the maximum, the power isolines for k > for this observed spectral estimate are roughly independent of frequency, consistent with the unsmoothed model output (Fig. 19). There are other differences in the data processing and spectral estimate that could also 26

27 contribute to the different spectral structure, but the similarity between this feature of the Fu (24) spectral estimate and the model spectral structure would seem to be consistent with the inference that the effect of the space-time smoothing along long-wave characteristics in the data processing procedures may be visible in the wavenumber-frequency spectra computed from the AVISO data 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 K = 1, a restriction that is motivated partly by the need to simplify the problem and partly by an appeal to baroclinic instability of the mean field near the deformation radius as the likely dominant source of mesoscale energy (e.g., Gill et al. 1974). This leaves the stochastic timescale τ as the single model parameter describing physical variations in the statistics of the spatio-temporal 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 mid-latitude ocean mesoscale regime is characterized by a stochastic SSH forcing rate σ W m s 1/2, an SSH damping rate r ψ 1/62 wk 1, and a stochastic forcing autocorrelation timescale τ L R /U 1 wk. The SSH forcing rate σ W is perhaps more intuitively 568 accessible when expressed as the equivalent SSH variance accumulation rate, σ 2 W m 2 s 1 1/4 cm 2 d 1. This variance accumulates in the band of forcing wavenumbers, through an effectively stochastic process with autocorrelation timescale τ. 27

28 The model stochastic forcing thus evidently represents a physical process that supports a continuous production of 1 cm 2 of SSH variance every 4 days at wavelengths near 2π L R 2 4 km, with the fluctuations containing the generated variance having timescales of a week or longer. These estimates are derived under the approximation that the mesoscale variance is entirely contained in the equivalent gravest-mode or equvalent-barotropic structure represented by the single moving layer in the reduced-gravity model, and could be altered in various ways by exchanges and distributions of energy with other vertical-mode structures that are not represented in the model. The absence of these latter interactions and the associated baroclinic eddy instability mechanisms is one possible source of the discrepancy between the observed and modeled eddy number distributions vs. lifetime, with the model producing relatively more long-lived eddies and relatively fewer short-lived eddies than in the observations. The spatial scales of the forcing in these simulations were imposed, but the relatively long inferred forcing timescale, equal to or greater than the mesoscale advective timescale, appears independently consistent with the identification of baroclinic instability of the large-scale fields as the primary variance production mechanism. As the forcing timescale increases past the mesoscale advective timescale, 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 (15). 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. 28

29 The inferred SSH damping rate of r ψ 1/62 wk 1 is less than half that estimated by Samelson et al. (216), a decrease that is consistent with the inverse-square-root dependence (15) of amplitude response on damping rate and the 5% decrease in the inferred forcing rate σ W relative to that estimated by Samelson et al. (216). The physical processes represented by this simplified linear damping rate are not easily identified, but it is intriguing to note that the inferred value is roughly comparable to the estimated eddy damping rates from surface current effects on wind stress (Gaube et al. 215; Dewar and Flierl 1987). Thus, effective drag at the ocean surface, rather than the sea floor, may be a fundamental element of the mesoscale energy balance. Another candidate mechanism is the propagation of energy out of the ocean interior, to regions such as western boundaries where dissipation may be enhanced, a process that is not represented in the doubly-periodic geometry considered here. For example, the simulations of Early et al. (211), which were conducted in an effectively open domain with dissipation at the edges, show a westward-increasing energy level similar to that apparent in the observed mid-latitude SSH variance (Fig. 2). 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. (216) 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 stream function fields. On the other hand, the dominance of non-dispersive 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 ψ J(ψ, 2 ψ) vanishes identically in the doubly-periodic geometry, and the amplitude of the nonlinear solution is essentially set a priori in the stochastic limit τ 1 29

30 just as it is for the linear model, by the balance between the fixed stochastic variance production rate and the damping rate. 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 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 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 non-dispersive propagation characteristics and with the nonlinear enhancement of the autocorrelation decay Summary The primary goal of this study was to reconcile the semi-empirical stochastic model results of Samelson et al. (214, 216) 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 semi-empirical stochastic model results and the observations. The dynamical and semi-empirical models are both forced stochastically, and the linear wave propagation included in the semi-empirical model is consistent with the long-wave, linear limit of the dynamical model. From this point of view, the reduced-gravity model (7) can be seen as an extension of the semi-empirical model that incorporates the basic elements of relative vorticity 3

31 dynamics and advective nonlinearity while retaining the elements of stochastic forcing and longwave propagation. The general correspondence between the eddy statistics from the dynamical and semi-empirical models and from the observations is therefore persuasive evidence that the semi-empirical model the simplest of the three systems effectively extracts the essential, determining characteristics of the processes supporting the spatio-temporal 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 lifecycles, with their nearly exact time-reversal symmetry and universality for eddy lifetimes of 16 to 8 weeks. Evidently, the quasi-geostrophic 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 quasi-geostrophic model (7) as configured here, the simulations that nominally best reproduces the global-mean mid-latitude statistics derived from the AVISO merged dataset of satellite altimeter SSH measurements have parameters in the range β.6, r ψ =.2, τ 1, corresponding at 35 latitude and for L R = 4 km to a dimensional SSH stochastic-forcing variance production rate 1/4 cm 2 d 1, an SSH damping rate 1/62 wk 1, and a stochastic forcing autocorrelation timescale of 1 wk 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 reducedgravity quasi-geostrophic 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 2-km spatial and 3-d 31

32 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 timescale 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. 21) 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 ocean mesoscale variability. 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. 32

33 689 Acknowledgments. Support for this research was provided as part of the NASA (National Aero nautics and Space Administration) Ocean Surface Topography and Surface Water and Ocean To- pography Science Team activities through NASA grants NNX13AD78G and NNX16AH76G. 33

34 692 APPENDIX 693 Scaling transformations and space-time smoothing Scaling Let F = AF /τ 1/2 for some constant A, and let D have the form (5), so that (1) takes the form t ( 2 ψ F 1 ψ ) + β ψ x + J (ψ, 2 ψ ) = A τ 1/2 F + r ψψ r 6 6 ψ. (A1) The equation (A1) transforms to (7) through the re-scaling ψ = Pψ, t = Tt, (x,y ) = D(x,y), where P = A 2/3 F 1 1, T = A 2/3, D = F 1/2 1. Note that for T = D/U A, this gives U A = A 2/3 F 1/2 1 and P = U A D, so this is a consistent re-scaling of the original nondimensionalization. Two cases are of interest. First, the equation (A1) takes the form used by Morten et al. (217) if the choice A = τ 1/2 MAF, 71 F 1 = k 2 d is made. The equivalent dimensionless parameters in the unprimed-variable equation (7) may then be computed from those in (A1) by the formulas given in the text. Second, with F 1 = 1 the re-scaling is determined by the forcing amplitude alone. The equivalent dimensionless parameters in (7) are then obtained from those in (A1) according to β = A 2/3 β, τ = A 2/3 τ, (r ψ,r 6 ) = A 2/3 (r ψ,r 6 ). After the analysis of the β.6 simulations were completed, a minor error in the normalization of F was found, the effect of which was equivalent to setting A.9 in (A1), rather than the value A = 1 that would yield (7) directly. This re-scaling of the β.6 simulations was subsequently used to obtain the equivalent dimensionless parameter and variable values for (7), with a subset of the simulations also repeated with the correct normalization of F to confirm the re-scaling. The equivalent, re-scaled values have been used in the text to describe the β.6 simulations by reference to (7). 34

35 Smoothing Consider a space-time smoothing function W X (x,t) in the zonal spatial coordinate x and time t, W X (x,t) = 1 { ( } x 2 δ(x c t)exp, (A2) πl L) where δ(x) is the Dirac-δ distribution over x and c is a constant wave speed, with c < for westward propagation. Smoothing in space, with scale L, is then carried out along the characteristic X(t) = c t, on which x c t =. For this spatial smoothing operation, the distance along the characteristic is computed according to the exponential in (A2) as the spatial distance between points on the characteristic, independent of differences in the time coordinate t; in effect, the characteristic is projected from the (x,t) plane onto the x-axis for the purposes of computing this distance. The corresponding filter transfer function Ŵ X (k,ω) is the Fourier transform of W X (x,t), Ŵ X (k,ω) = e ikx+iωt W X (x,t)dxdt, (A3) where the signs of the transforms are chosen opposite so that ω = kc gives a Fourier component propagating with speed c (dx = cdt). For the function W X from (A2), the transform (A3) gives { Ŵ X (k,ω) = exp 1 4 L2 (k c 1 }. ω)2 (A4) 724 The pure time smoothing at fixed x may also be represented as a space-time smoothing function 725 W t (x,t), where W t (x,t) = 1 { ( t ) 2 } δ(x)exp, (A5) πt T 726 with transform Ŵ t (k,ω) = exp { 14 } T 2 ω 2, (A6) 727 which has no dependence on k. 35

36 The x-t space-time smoothing function used for filtering the simulation output can be written as a single function W Xt (x,t;x,t ), where W Xt (x,t;x,t ) = 1 { ( } { ( x 2 t πlt δ(x c t)exp δ(x L) ) } 2 )exp, (A7) T so that W Xt (x,t;x,t ) is essentially the product of the two smoothing functions W X (x,t) and W t (x,t), with the second set of space-time variables (x,t ) introduced to allow the second smooth- ing operation. The filter transfer function is then obtained by two sequential space-time Fourier transforms, Ŵ Xt (k,ω) = e ikx +iωt e ikx+iωt W Xt (x,t;x,t )dxdt dx dt (A8) 734 and can be obtained as the product of (A4) and (A6), { Ŵ Xt (k,ω) = Ŵ X (k,ω)ŵ t (k,ω) = exp 1 4 L2 (k c 1 ω)2 1 4 T 2 ω }. 2 (A9) A meridional smoothing factor exp( y 2 /L 2 ) can be included but, as long as the characteristics are purely zonal, is independent of t and results in a transformed factor exp( 1 4 L2 l 2 ), where l is meridional wavenumber. In order to allow eddy analysis of the smoothed fields, the filtering of the simulation output fields was conducted in space-time coordinates using the numerical equivalent of the functions (A7) with the additional meridional smoothing, and the spectra of the smoothed fields were then computed. The numerical implementation used a Parzen filter in place of a Gaussian, which the former closely approximates (see, e.g., Appendix C of Chelton et al. 218). 36

37 743 References Chelton, D. B., R. A. de Szoeke, M. G. Schlax, K. E. Naggar, and N. Siwertz, 1998: Geographical variability of the first-baroclinic rossby radius of deformation. Journal of Physical Oceanogra- phy, 28, Chelton, D. B., M. G. Schlax, and R. M. Samelson, 211: Global observations of nonlinear mesoscale eddies. Progress in Oceanography, 91 (2), , doi: doi.org/1.116/j.pocean , URL S Chelton, D. B., M. G. Schlax, R. M. Samelson, J. T. Farrar, M. J. Molemaker, J. C. McWilliams, and J. Gula, 218: Prospects for future satellite estimation of small-scale variability of ocean surface velocity and vorticity. Progress in Oceanography, accepted Dewar, W. K., and G. R. Flierl, 1987: Some effects of the wind on rings. Journal of Physical Oceanography, 17 (1), , doi:1.1175/ (1987) :SEOTWO 2..CO;2, URL :SEOTWO 2..CO;2, https: //doi.org/1.1175/ (1987) :SEOTWO 2..CO; Ducet, N., P. Y. Le Traon, and G. Reverdin, 2: Global high-resolution mapping of ocean circulation from topex/poseidon and ers-1 and -2. Journal of Geophysical Research: Oceans, 15 (C8), , doi:1.129/2jc963, URL 2JC Durand, M., L.-L. Fu, D. P. Lettenmaier, D. E. Alsdorf, E. Rodriguez, and D. Esteban-Fernandez, 21: The surface water and ocean topography mission: Observing terrestrial surface water and oceanic submesoscale eddies. Proc. IEEE, 98,

38 765 Early, J. J., R. M. Samelson, and D. B. Chelton, 211: The evolution and propagation of quasigeostrophic ocean eddies. Journal of Physical Oceanography, 41 (8), , doi: /211JPO461.1, URL Fu, L.-L., 24: Latitudinal and frequency characteristics of the westward propaga- tion of large-scale oceanic variability. Journal of Physical Oceanography, 34 (8), , doi:1.1175/ (24)34 197:LAFCOT 2..CO;2, URL https: //doi.org/1.1175/ (24)34 197:LAFCOT 2..CO;2, (24)34 197:LAFCOT 2..CO; Gaube, P., D. B. Chelton, R. M. Samelson, M. G. Schlax, and L. W. O Neill, 215: Satellite observations of mesoscale eddy-induced ekman pumping. Journal of Physical Oceanography, 45 (1), , doi:1.1175/jpo-d , URL 1, Gill, A., J. Green, and A. Simmons, 1974: Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies. Deep Sea Research and Oceanographic Abstracts, 21 (7), , doi: URL com/science/article/pii/ Le Traon, P. Y., Y. Faugère, F. Hernandez, J. Dorandeu, F. Mertz, and M. Ablain, 23: Can we merge geosat follow-on with topex/poseidon and ers-2 for an im- 783 proved description of the ocean circulation? Journal of Atmospheric and Oceanic Technology, 2 (6), , doi:1.1175/ (23)2 889:CWMGFW 2..CO;2, URL 889:CWMGFW 2..CO;2, / (23)2 889:CWMGFW 2..CO;2. 38

39 MODE Group, T., 1978: The mid-ocean dynamics experiment. Deep Sea Research, 25 (1), , doi: URL com/science/article/pii/ x. 79 Morten, A. J., B. K. Arbic, and G. R. Flierl, 217: Wavenumber-frequency analysis of single-layer shallow-water beta-plane quasi-geostrophic turbulence. Physics of Fluids, 29 (1), 16 62, doi:1.163/ , URL / Oh, S., J. Pedlosky, and R. M. Samelson, 1993: Linear and finite-amplitude localized baroclinic instability. Journal of the Atmospheric Sciences, 5 (16), Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed., Springer Berlin Heidelberg Pujol, M.-I., Y. Faugère, G. Taburet, S. Dupuy, C. Pelloquin, M. Ablain, and N. Picot, 216: Duacs dt214: The new multi-mission altimeter data set reprocessed over 2 years. Ocean Sci., 12 (167 19) Rhines, P. B., 1975: Waves and turbulence on a beta-plane. Journal of Fluid Mechanics, 69 (3), , doi:1.117/s Samelson, R. M., and J. Pedlosky, 199: Local baroclinic instability of flow over variable topog- raphy. Journal of Fluid Mechanics, 221, Samelson, R. M., M. G. Schlax, and D. B. Chelton, 214: Randomness, symmetry, and scaling of mesoscale eddy life cycles. Journal of Physical Oceanography, 44 (3), [Corrigendum 44 (9), , doi: /JPO D ], URL JPO-D

40 Samelson, R. M., M. G. Schlax, and D. B. Chelton, 216: A linear stochastic field model of midlatitude mesoscale variability. Journal of Physical Oceanography, 46 (1), , doi:1.1175/jpo-d , URL http: //dx.doi.org/1.1175/jpo-d

41 LIST OF FIGURES Fig. 1. Power spectrum of SSH variability along 35 S in the South Indian Ocean vs. zonal wavenumber and frequency, computed from the approximately 16-year period October 1992 through December 28 of the AVISO Delayed-Time Reference Series. Also shown are: three theoretical dispersion relations (black solid, dotted, dashed, respectively) corresponding to standard Rossby wave theory, the rough bottom topography theory of Tailleux and McWilliams (21), the vertical shear-modified theory of Killworth et al. (1997), and the extension of the latter to the case of nonzero zonal wavenumber (Fu and Chelton 21); and an empirical dispersion relation (white line) of the form ω = c R k /(λ 2 k 2 + 1), with c R.4 m s 1 and λ /(2π) 32 km. The raw spectral values have been smoothed in both dimensions, with 3 times more smoothing in the frequency dimension than in the wavenumber dimension Fig. 2. Fig. 3. (Upper panel) AVISO SSH standard deviation and (middle and lower, respectively) eddy tracks for eddies with lifetimes of at least 16 and 26 weeks, with the latitude ranges 2 N - 4 N and 2 S - 4 S indicated (white boxes). After Chelton et al. (211) Model parameters. In each panel, the points corresponding to the target values for the dimensional comparisons at latitudes φ = 35 and φ = 24 are indicated ( ). (Upper left panel) Deformation radius 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). (Lower left) Contours (labeled, solid and dotted lines) of σ/β from (15) for L R = c 1 / f with c 1 = 3.4 m s 1 vs. latitude φ and specified SSH standard deviation σ η. The zonally averaged AVISO SSH standard deviation in the northern and southern hemispheres are also shown (dashed lines). (Upper right) r ψ /β from (18) vs. for L R = c 1 / f with c 1 = 3.35 m s 1 vs. latitude φ. (Lower right) Graphical illustration of the joint solution of (15) and (19) for the dimensionless parameters corresponding to the target values for the dimensional comparisons at latitudes φ = 35 and φ = Fig. 4. Summary of values of (τ,r ψ ) for simulations with β =.62 ( with enclosing ), β = 2.4 ( with enclosing ), and for the Run 3 simulation ( with enclosing ) of Morten et al. (217). Simulations with σ ψ /β within 5% of the respective target values σ ψ /β = {7,.9} for β = {.62,2.4} (blue, large ), more than 5% greater than the target values (red), and less than 95% of the target values (black) are indicated Fig. 5. Fig. 6. Spatial standard deviation of dimensionless sea-surface height σ η (t) (blue) and forcing amplitude σ F (t) (green) vs. dimensionless time t from simulations with (upper panels) β =.62,τ =.92,r ψ =.22 and (lower) β = 2.4,τ = 5.,r ψ = Time-mean dimensionless stream function standard deviation σ ψ (markers as in Fig. 4) for simulations with β = {.62,2.4} vs. (left panel) τ, with and size proportional to rψ 1/2, and (right) r ψ, with and size proportional to τ 1/4. The target values σ ψ = {7,.9}β {4.3, 2.2} for the simulations with β = {.62, 2.4}, respectively, are indicated (thick and thin dashed lines, respectively). The relation ψ rψ 1/2 is also shown (right panel, dotted lines) for several values of the proportionality constant Fig. 7. Autocorrelation function A.8 (t) computed for the model solution with β =.62,r ψ =.22,τ =.92 along the characteristic dx/dt =.8c R of maximum autocorrelation (thick solid black line), and along the stationary characteristic dx/dt = (dashed). The theoretical autocorrelation function α t for the inferred value α eff =.958 wk 1 or r α = 1 α eff =.42 wk 1 is also shown (blue), along with the functions α t for values α = 1 r α based on the explicit model damping r α = r ψ = ULR 1 r ψ =.16 wk 1 (magenta) and the stochastic 41

42 Fig. 8. model values r α = {.94,.96} wk 1 (green, red) from Samelson et al. (214, 216). The model autocorrelation function is interpolated from domain-averaged values computed at the times marked ( ) along the t-axis Empirical autocorrelation parameter r α = 1 α eff (wk 1 ) vs. dimensional model damping parameter r ψ = ULR 1 r ψ for simulations as in Fig. 4, with and size proportional to (left panel) τ 1/4 and (right) nonlinearity measure σ ψ /β. The range.94 r α.96 (solid lines) of estimates from from mid-latitude observations, the relation r α = r ψ (thick dashed) satisfied by the linear stochastic models of Samelson et al. (214, 216), and the relation r α = r ψ +.2 wk 1 (thin dashed) are indicated Fig. 9. Normalized eddy amplitude lifecycles for solution with β =.62,r ψ =.22,τ =.92. (Upper left panel) Mean lifecycles vs. normalized time for each lifetime class with lifetime between 16 weeks and 8 weeks. (Upper right) Mean lifecycle for all eddies with lifetimes between 16 and 8 weeks. The mean and standard deviation lifecycles (labeled) are shown for the model (green solid lines) and observations (black and blue dashed), with the model mean and standard deviation shown separately only over the sub-intervals normalized time t <.5 and t >.5. The symmetric and anti-symmetric parts under time-reverseal are shown separately, with all of the anti-symmetric parts being negligibly small relative to the symmetric parts. (Lower left) Mean dimensional amplitude vs. eddy lifetime for the model (green) and observations (black), for eddies with lifetimes of at least 4 weeks. (Lower right) Normalized distributions of innovations (first-differences in time) for the normalized eddy lifecycles for all model (green) and observed (black) eddies with lifetimes of at least 16 weeks, with mean, standard deviation, skewness, and kurtosis of the (model, observed) distributions as shown Fig. 1. As in Fig. 9, but for normalized eddy length-scale lifecycles Fig. 11. As in Fig. 9, but for normalized eddy rotational speed lifecycles Fig. 12. Number distributions of eddies for the model solution with β =.62,r ψ =.22,τ =.92 (green dots) and for the AVISO observations (black). The analytical function fit by Samelson et al. (216) to the observed distribution is also shown (dashed magenta) Fig. 13. Fig. 14. Fig. 15. Probability distributions of eddy amplitudes, length-scales, and rotational speeds for the solution with β =.62,r ψ =.22,τ =.92 (green) and for the AVISO observations (black). 57 Number distributions of eddies for solutions β =.62,r ψ =.22,τ =.93 (red dots), β =.62,r ψ =.22,τ =.92 (green), β =.61,,r ψ =.54,τ = 9.3 (blue), and for the AVISO observations (black) (upper panel) Amplitude, (middle) length-scale, and (lower) rotational speed distributions of eddies for solutions with β =.62,r ψ =.22,τ =.93 (red lines), β =.62,r ψ =.22,τ =.92 (green), β =.62,r ψ =.54,τ = 9.3 (blue), and for the AVISO observations (black) Fig. 16. Number distributions of eddies for solutions β = 2.4,r ψ =.7,τ = 1 (red dots), β = 2.4,r ψ =.1,τ = 5 (blue), β =.62,r ψ =.22,τ =.92 (green), and for the AVISO observations (black)

43 Fig. 17. Fig. 18. Fig. 19. (upper panel) Amplitude, (middle) length-scale, and (lower) rotational speed distributions of eddies for solutions with β = 2.4,r ψ =.7,τ = 1 (red lines), β = 2.4,r ψ =.1,τ = 5 (blue), β =.62,r ψ =.22,τ =.92 (green), and for the AVISO observations (black) Power spectra of dimensionless model streamfunction ψ or sea-surface height η (solid) and stochastic forcing F (dashed) vs. dimensionless wavenumber magnitude K from simulations with β =.62,r ψ =.22,τ =.93 (red); β =.62,r ψ =.22,τ =.92 (green); and β =.61,r ψ =.54,τ = 9.3 (blue). Also shown is the approximate AVISO SSH wavenumber-magnitude spectrum (solid black) used by Samelson et al. (216) to constrain the wavenumber structure of the linear stochastic field model Power spectrum of dimensionless streamfunction ψ or sea-surface height η vs. dimensionless zonal wavenumber k and frequency ω from a simulation with β =.62,r ψ =.22, τ =.92. The linear dispersion relation is also shown (white solid line), along with the non-dispersive lines corresponding to the long-wave speed c R = β (white solid) and the maximum-autocorrelation effective phase speed dx/dt = ac R (dashed white), with a =.8 for this simulation Fig. 2. Power spectrum of dimensionless (upper panel) stream function ψ or sea-surface height η, (middle) stochastic forcing F, and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from a linear simulation with β =.62,r ψ =.22,τ =.92. Dispersion relation and non-dispersive lines are as in Fig. 19, with a =.55 for the maximum-autocorrelation characteristic Fig. 21. Dimensionless (upper panel) power spectrum of stochastic forcing F and (middle) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from the simulation with β =.62,r ψ =.22,τ =.92. Dispersion relation and non-dispersive lines are as in Fig. 19. The k-ω power spectrum of dimensionless stream function ψ for this simulation is shown in Fig. 19. (lower) Inverted power spectrum L 1 ˆη 2 of SSH variability along 35 S in the South Indian Ocean vs. zonal wavenumber (1 3 cpkm) and frequency (cpd), computed from the observed spectrum in Fig. 1 using (25), for an empirical linear dispersion relation (white line) of the form (24) with c.4 m s 1 and λ /(2π) 32 km, and dimensional damping r = 1/16 wk Fig. 22. Power spectrum of dimensionless (upper panel) stream function ψ or sea-surface height η, (middle) stochastic forcing F, and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from the simulation with β =.62,r ψ =.22,τ =.92. Dispersion relation and non-dispersive lines are as in Fig Fig. 23. Power spectrum of dimensionless (upper panel) stream function ψ or sea-surface height η, (middle) stochastic forcing F, and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from the simulation with (left) β =.62,r ψ =.22,τ =.93 and (right) β =.62,r ψ =.54,τ = 9.3. Dispersion relation and nondispersive lines are as in Fig Fig. 24. Power spectra of SSH variability computed from the AVISO gridded dataset vs. zonal wavenumber (1 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 along (upper panel; all rows left to right) South Pacific 45 S, 15 W-11 W; South Indian 4 S, 8 E-11 E; South Pacific 38 S, 13 W-1 W; South Indian 35 S, 4 E- 85 E; (upper middle) North Pacific 33 N, 18 E-13 W; South Atlantic 33 S, 5 W- W; North Atlantic 3 N, 7 W- 4 W; South Pacific 3 S, 17 E-12 W; (lower middle) South Indian 3 S, 6 E-1 E; North Pacific 24 N, 125 E-165 E; South Pacific 24 S, 16 E-135 W ; 43

44 North Atlantic 24 N, 6 W- 3 W; (lower) South Atlantic 24 S, 4 W- 1 W; South Indian 24 S, 5 E-1 E; North Pacific 21 N, 13 E-17 W; South Indian 14 S, 7 E-12 E. Dispersion relations are shown as in Fig Fig. 25. As in Fig. 24, but for inverted power spectra L 1 ˆη 2 of SSH variability Fig. 26. Fig. 27. Fig. 28. Fig. 29. Fig. 3. Fig. 31. Fig. 32. Power spectrum of dimensionless space-time smoothed (upper and lower-middle panels) stream function ψ or sea-surface height η and (upper-middle and lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from the simulation with β =.62,r ψ =.22,τ =.92. Dispersion relation and non-dispersive lines are as in Fig Power spectrum of space-time smoothed (blue line) and unsmoothed (green) dimensionless model streamfunction ψ or sea-surface height η vs. dimensionless wavenumber magnitude K from a simulation with β =.62,r ψ =.22,τ =.92. Also shown is the approximate AVISO SSH wavenumber-magnitude spectrum (solid black) used by Samelson et al. (216) to constrain the wavenumber structure of the linear stochastic field model, and the spectral structure of the model forcing (dashed blue and green) (upper panel) Amplitude, (middle) length-scale, and (lower) rotational speed distributions of eddies for solutions with β =.62,r ψ =.22,τ =.92, with (blue lines) and without (green) space-time smoothing, and for the AVISO observations (black) Dimensionless (upper panel) power spectrum ˆψ 2 or ˆη 2 of stream function ψ or seasurface height η and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from the simulation with β = 2.4,τ = 5,r ψ =.24. Dispersion relation and non-dispersive lines are as in Fig. 19, with a =.65 for the maximumautocorrelation characteristic Power spectrum of dimensionless (upper panel) stream function ψ or sea-surface height η, (middle) stochastic forcing F, and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from the simulation with β = 2.4,τ = 5,r ψ =.24. Dispersion relation and non-dispersive lines are as in Fig. 19, with a =.65 for the maximum-autocorrelation characteristic (Upper panel) Power spectrum ˆη 2 and (lower) inverted power spectrum L 1 ˆη 2 of SSH variability along 24 N in the western North Pacific vs. zonal wavenumber (1 3 cpkm) and frequency (cpd), where L 1 is computed using an empirical linear dispersion relation (white line) of the form ω = c R k /(λ 2 k 2 + 1), for dimensional zonal wavenumber k and frequency ω, with c R.9 m s 1 and λ /(2π) 63 km. Upper panel after Fig. 4 of Early et al. (211), with increased wavenumber range Power spectrum of SSH at 24.5 N in the Pacific. The solid curve represents the dispersion relation for the first-mode baroclinic Rossby waves based on the traditional theory. The dashed curve represents the dispersion relation based on the theory of Killworth et al. (1997). From Fu (24)

45 FIG. 1. Power spectrum of SSH variability along 35 S in the South Indian Ocean vs. zonal wavenumber and frequency, computed from the approximately 16-year period October 1992 through December 28 of the AVISO Delayed-Time Reference Series. Also shown are: three theoretical dispersion relations (black solid, dotted, dashed, respectively) corresponding to standard Rossby wave theory, the rough bottom topography theory of Tailleux and McWilliams (21), the vertical shear-modified theory of Killworth et al. (1997), and the extension of the latter to the case of nonzero zonal wavenumber (Fu and Chelton 21); and an empirical dispersion relation (white line) of the form ω = c R k /(λ 2 k 2 +1), with c R.4 m s 1 and λ /(2π) 32 km. The raw spectral values have been smoothed in both dimensions, with 3 times more smoothing in the frequency dimension than in the wavenumber dimension. 45

46 991 F IG. 2. (Upper panel) AVISO SSH standard deviation and (middle and lower, respectively) eddy tracks for 992 eddies with lifetimes of at least 16 and 26 weeks, with the latitude ranges 2 N - 4 N and 2 S - 4 S 993 indicated (white boxes). After Chelton et al. (211). 46

47 L R (km) Latitude ( o ) -1 r Latitude ( o ) * (m) Latitude ( o ) = = r FIG. 3. Model parameters. In each panel, the points corresponding to the target values for the dimensional 989 FIG. 3. Model parameters. In each panel, the points corresponding to the target values for the dimensional comparisons at latitudes φ = 35 and φ = 24 are indicated ( ). (Upper left panel) Deformation radius L R = c 1 / f 99 comparisons at latitudes f = 35 and f = 24 are indicated ( ). (Upper left panel) Deformation radius 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). (Lower left) Contours (labeled, solid vs. latitude f for c 1 = 3.35 m s 1 (solid line) and c 1 = 3ms (dashed). (Lower left) Contours (labeled, solid and dotted lines) of σ/β from (15) for L R = c 1 / f with c 1 = 3.4 m s 1 vs. latitude φ and specified SSH standard and dotted lines) of s/b from (15) for L R = c 1 / f with c 1 = 3.4ms vs. latitude f and specified SSH standard deviation σ η. The zonally averaged AVISO SSH standard deviation in the northern and southern hemispheres 993 deviation s h. The zonally averaged AVISO SSH standard deviation in the northern and southern hemispheres are also shown (dashed lines). (Upper right) r ψ /β from (18) vs. for L R = c 1 / f with c 1 = 3.35 m s 1 are also shown (dashed lines). (Upper right) r y /b from (18) vs. for L R c 1 with m s 1 vs. latitude 994 vs. latitude φ. (Lower right) Graphical illustration of the joint solution of (15) and (19) for the dimensionless parameters 995 f. (Lower right) Graphical illustration of the joint solution of (15) and (19) for the dimensionless parameters corresponding to the target values for the dimensional comparisons at latitudes φ = corresponding to the target values for the dimensional comparisons at latitudes f 35 and φ = 24 and f

48 r FIG. 4. Summary of values of (τ,r ψ ) for simulations with β =.62 ( with enclosing ), β = 2.4 ( with enclosing ), and for the Run 3 simulation ( with enclosing ) of Morten et al. (217). Simulations with σ ψ /β within 5% of the respective target values σ ψ /β = {7,.9} for β = {.62,2.4} (blue, large ), more than 5% greater than the target values (red), and less than 95% of the target values (black) are indicated. 48

49 mean( sd,f sd )=(4.3,1) t F t 2.5 mean( sd,f sd )=(1.8,.45) t 1 F t FIG. 5. Spatial standard deviation of dimensionless sea-surface height s h (t) (blue) and forcing amplitude FIG. 5. Spatial standard deviation of dimensionless sea-surface height σ η (t) (blue) and forcing amplitude s F (t) (green) vs. dimensionless time t from simulations with (upper panels) b =.62,t =.92,r y =.22 and σ F (t) (green) vs. dimensionless time t from simulations with (upper panels) β =.62,τ =.92,r ψ =.22 and (lower) b = 2.4,t = 5.,r y =.24. (lower) β = 2.4,τ = 5.,r ψ =

50 r -1/ r FIG Time-mean dimensionless stream function standard deviation σs ψ y (markers as as in infig. Fig. 4) 4) for for simulations withb β = {.62,2.4} vs. (left panel) t, τ, with and size proportional to tor 1/2 ry 1/2, and, (right) r ψ r, y with, with and and size size proportionaltototτ 1/4.. The target values sσ y ψ = {7,.9}β {7,.9}b {4.3,2.2} for the simulations with β b = = {.62,2.4}, respectively, are indicated (thick and thin dashed lines, respectively). The relation ψy µ r 1/2 r 1/2 is is also also shown (right (right panel, dotted lines) for several values of the proportionality constant. ψ ψ y 5 49

51 1 OP3aRK, r qg* =.16: c/c R =.8.8 autocorrelation =.984, r =.16 =.958, r =.42 =.96, r =.4: SFM =.94, r =.6: LC t (wk) FIG. 7. Autocorrelation function A.8 (t) computed for the model solution with β =.62,r ψ =.22,τ =.92 along the characteristic dx/dt =.8c R of maximum autocorrelation (thick solid black line), and along the stationary characteristic dx/dt = (dashed). The theoretical autocorrelation function α t for the inferred value α eff =.958 wk 1 or r α = 1 α eff =.42 wk 1 is also shown (blue), along with the functions α t for values α = 1 r α based on the explicit model damping r α = r ψ = UL 1 R r ψ =.16 wk 1 (magenta) and the stochastic model values r α = {.94,.96} wk 1 (green, red) from Samelson et al. (214, 216). The model autocorrelation function is interpolated from domain-averaged values computed at the times marked ( ) along the t-axis. 51

52 r = 1 - eff (wk -1 ) r = 1 - eff (wk -1 ) r * (wk -1 ) r * (wk -1 ) FIG. 8. Empirical autocorrelation parameter r α = 1 α eff (wk 1 ) vs. dimensional model damping parameter r ψ = ULR 1 r ψ for simulations as in Fig. 4, with and size proportional to (left panel) τ 1/4 and (right) nonlinearity measure σ ψ /β. The range.94 r α.96 (solid lines) of estimates from from mid-latitude observations, the relation r α = r ψ (thick dashed) satisfied by the linear stochastic models of Samelson et al. (214, 216), and the relation r α = r ψ +.2 wk 1 (thin dashed) are indicated. 52

53 dimensionless amplitude and std dev normalized life cycles Mean Std dev wks wks wks 64-8 wks dimensionless time dimensionless amplitude and std dev mean mean normalized life cycles model mean std dev model std dev dimensionless time 15 mean dimensional amplitude.6 amplitude innovation distribution mean amplitude (cm) 1 5 i P(increment) i = (.,-.) s i = (-.,-.) i = (.16,.31) k i = (5.49,3.72) lifetime (wks) dimensionless increment/ i FIG. 9. Normalized eddy amplitude lifecycles for solution with β =.62,r ψ =.22,τ =.92. (Upper left panel) Mean lifecycles vs. normalized time for each lifetime class with lifetime between 16 weeks and 8 weeks. (Upper right) Mean lifecycle for all eddies with lifetimes between 16 and 8 weeks. The mean and standard deviation lifecycles (labeled) are shown for the model (green solid lines) and observations (black and blue dashed), with the model mean and standard deviation shown separately only over the sub-intervals normalized time t <.5 and t >.5. The symmetric and anti-symmetric parts under time-reverseal are shown separately, with all of the anti-symmetric parts being negligibly small relative to the symmetric parts. (Lower left) Mean dimensional amplitude vs. eddy lifetime for the model (green) and observations (black), for eddies with lifetimes of at least 4 weeks. (Lower right) Normalized distributions of innovations (first-differences in time) for the normalized eddy lifecycles for all model (green) and observed (black) eddies with lifetimes of at least 16 weeks, with mean, standard deviation, skewness, and kurtosis of the (model, observed) distributions as shown. 53

54 dimensionless scale and std dev a) normalized life cycles mean std dev wks wks wks 64-8 wks dimensionless time dimensionless scale and std dev b) mean mean normalized life cycles std dev dimensionless time scale (km) c) mean dimensional scale i P(increment) d) scale innovation distribution i = (-.,.) s i = (.21,.1) i = (.15,.3) k = (13.74,5.75) i lifetime (wks) dimensionless increment/ i FIG. 1. As in Fig. 9, but for normalized eddy length-scale lifecycles. 54

55 dimensionless rotational speed and std dev normalized life cycles Mean wks wks wks 64-8 wks Std dev dimensionless time dimensionless rotational speed and std dev mean mean normalized life cycles model mean std dev model std dev dimensionless time rotational speed (cm s -1 ) i P(increment) speed innovation distribution i = (.,-.) s = (.1,-.1) i i = (.8,.18) k = (6.6,5.69) i lifetime (wks) dimensionless increment/ i FIG. 11. As in Fig. 9, but for normalized eddy rotational speed lifecycles. 55

56 1 1 fraction of eddies fraction of eddies lifetime (wks) lifetime (wks) FIG. 12. Number distributions of eddies for the model solution with β =.62,r ψ =.22,τ =.92 (green dots) and for the AVISO observations (black). The analytical function fit by Samelson et al. (216) to the observed distribution is also shown (dashed magenta) 56

57 P(A) Eddy amplitude, A (cm).15 P(L s ) Eddy scale, L s (km) P(U) Eddy speed, U (cm s -1 ) FIG. 13. Probability distributions of eddy amplitudes, length-scales, and rotational speeds for the solution FIG. 13. Probability distributions of eddy amplitudes, length-scales, and rotational speeds for the solution with b.62,r y.22,t =.92 (green) and for the AVISO observations (black). with β =.62,r ψ =.22,τ =.92 (green) and for the AVISO observations (black). 57

58 Fraction of eddies Fraction of eddies Lifetime (wks) Lifetime (wks) FIG. 14. Number distributions of eddies for solutions β =.62,r ψ =.22,τ =.93 (red dots), β =.62,r ψ =.22,τ =.92 (green), β =.61,,r ψ =.54,τ = 9.3 (blue), and for the AVISO observations (black). 58

59 P(A) Eddy amplitude, A (cm).2.15 P(L s ) Eddy scale, L s (km).8.6 P(U) Eddy speed, U (m s -1 ) FIG. 15. (upper panel) Amplitude, (middle) length-scale, and (lower) rotational speed distributions of eddies FIG. 15. (upper panel) Amplitude, (middle) length-scale, and (lower) rotational speed distributions of eddies for solutions with b.62,r y =.22,t =.93 (red lines), b =.62,r y =.22,t =.92 (green), b = for solutions with β =.62,r ψ =.22,τ =.93 (red lines), β =.62,r ψ =.22,τ =.92 (green), β =.62,r y.54,t 9.3 (blue), and for the AVISO observations (black)..62,r ψ =.54,τ = 9.3 (blue), and for the AVISO observations (black)

60 Fraction of eddies Fraction of eddies Lifetime (wks) Lifetime (wks) FIG. 16. Number distributions of eddies for solutions β = 2.4,r ψ =.7,τ = 1 (red dots), β = 2.4,r ψ =.1,τ = 5 (blue), β =.62,r ψ =.22,τ =.92 (green), and for the AVISO observations (black). 6

61 P(A) Eddy amplitude, A (cm).2.15 P(L s ) Eddy scale, L s (km) P(U) Eddy rotational speed, U (cm s -1 ) FIG. 17. (upper panel) Amplitude, (middle) length-scale, and (lower) rotational speed distributions of eddies for solutions with b = 2.4,r y =.7,t = 1 (red lines), b = 2.4,r y =.1,t = 5 (blue), b =.62,r y = FIG. 17. (upper panel) Amplitude, (middle) length-scale, and (lower) rotational speed distributions of eddies for solutions with β = 2.4,r ψ =.7,τ = 1 (red lines), β = 2.4,r ψ =.1,τ = 5 (blue), β =.62,r ψ =.22,t =.92 (green), and for the AVISO observations (black)..22,τ =.92 (green), and for the AVISO observations (black). 61 6

62 PSD, F, AVISO SSH K FIG. 18. Power spectra of dimensionless model streamfunction ψ or sea-surface height η (solid) and stochastic forcing F (dashed) vs. dimensionless wavenumber magnitude K from simulations with β =.62,r ψ =.22,τ =.93 (red); β =.62,r ψ =.22,τ =.92 (green); and β =.61,r ψ =.54,τ = 9.3 (blue). Also shown is the approximate AVISO SSH wavenumber-magnitude spectrum (solid black) used by Samelson et al. (216) to constrain the wavenumber structure of the linear stochastic field model. 62

63 FIG. 19. Power spectrum of dimensionless streamfunction ψ or sea-surface height η vs. dimensionless zonal wavenumber k and frequency ω from a simulation with β =.62,r ψ =.22,τ =.92. The linear dispersion relation is also shown (white solid line), along with the non-dispersive lines corresponding to the long-wave speed c R = β (white solid) and the maximum-autocorrelation effective phase speed dx/dt = ac R (dashed white), with a =.8 for this simulation. 63

64 164 FIG. 2. Power spectrum of dimensionless (upper panel) stream function y or sea-surface height h, (midd 165 stochastic forcing F, and (lower) inverted power spectrum L 166 FIG. 2. Power spectrum of dimensionless (upper panel) stream function ψ or sea-surface height η, (middle) stochastic forcing F, and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k 1 ŷ 2 vs. dimensionless zonal wavenumber and frequency w from a linear simulation with b =.62,r y =.22,t =.92, with dispersion relation a 167 and frequency ω from a linear simulation with β =.62,r ψ =.22,τ =.92. Dispersion relation and nondispersive lines are as in Fig. 19, with a =.55 for the maximum-autocorrelation characteristic. non-dispersive lines as in Fig. 19, with a =.55 for the maximum-autocorrelation characteristic. 64

65 FIG. FIG Dimensionless (upper panel) power powerspectrum spectrumofof stochastic stochastic forcing forcing F Fand and (middle) (middle) inverted inverted power power spectrum L 1 1 ŷ ˆψ 22 vs. dimensionless zonal wavenumber k kand andfrequency ω from w from the the simulation simulation with with β = b =.62,r y ψ = =.22,t.22,τ =.92. The Dispersion k-w power relation spectrum and non-dispersive of dimensionless linesstream are asfunction in Fig. 19. y for The this k-ωsimulation power is shown spectrum in Fig. of dimensionless 19. (lower) Inverted streampower function spectrum ψ for this L simulation 1 ĥ 2 of SSH is shown variability in Fig. along 19. (lower) 35 S Inverted in the South power Indian Ocean spectrum vs. zonal L 1 wavenumber ˆη (1 3 2 of SSH variability cpkm) along and35 frequency S in the(cpd), Southcomputed Indian Ocean fromvs. thezonal observed wavenumber spectrum(1in 3 Fig. 1 using cpkm) (25), andfor frequency an empirical (cpd), linear computed dispersion from the relation observed (white spectrum line) in offig. the 1form using (24) (25), with for can empirical.4 mlinear s 1 and l dispersion /(2p) 32 relation km, and (white dimensional line) of thedamping form (24) r with = 1/16 c wk.4 1. m s 1 and λ /(2π) 32 km, and dimensional damping r = 1/16 wk 1. 65

66 FIG. 22. Power spectrum of dimensionless (upper panel) stream function ψ or sea-surface height η, (middle) stochastic forcing F, and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from the simulation with β =.62,r ψ =.22,τ =.92. Dispersion relation and non-dispersive lines are as in Fig

67 FIG. 23. Power spectrum of dimensionless (upper panel) stream function ψ or sea-surface height η, (middle) stochastic forcing F, and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k FIG. 23. Power spectrum of dimensionless (upper panel) stream function y or sea-surface height h, (middle) and frequency ω from the simulation with (left) β =.62,r ψ =.22,τ stochastic forcing F, and (lower) inverted power spectrum L 1 ŷ 2 =.93 and (right) β =.62,r ψ = vs. dimensionless zonal wavenumber k.54,τ = 9.3. Dispersion relation and non-dispersive lines are as in Fig. 19. and frequency w from the simulation with (left) b =.62,r y =.22,t =.93 and (right) b =.61,r y =.54,t =

68 FIG. 24. Power spectra of SSH variability computed from the AVISO gridded dataset vs. zonal wavenumber (1 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 along (upper panel; all rows left to right) South Pacific 45 S, 15 W-11 W; South Indian 4 S, 8 E-11 E; South Pacific 38 S, 13 W-1 W; South Indian 35 S, 4 E- 85 E; (upper middle) North Pacific 33 N, 18 E-13 W; South Atlantic 33 S, 5 W- W; North Atlantic 3 N, 7 W- 4 W; South Pacific 3 S, 17 E-12 W; (lower middle) South Indian 3 S, 6 E-1 E; North Pacific 24 N, 125 E-165 E; South Pacific 24 S, 16 E-135 W ; North Atlantic 24 N, 6 W- 3 W; (lower) South Atlantic 24 S, 4 W- 1 W; South Indian 24 S, 5 E-1 E; North Pacific 21 N, 13 E-17 W; South Indian 14 S, 7 E-12 E. Dispersion relations are shown as in Fig

69 FIG. 25. As in Fig. 24, but for inverted power spectra L 1 ˆη 2 of SSH variability. 69

70 FIG. 26. Power spectrum of dimensionless space-time smoothed (upper and lower-middle panels) stream FIG. 26. Power spectrum of dimensionless space-time smoothed (upper and lower-middle panels) stream function y or sea-surface height h and (upper-middle and lower) inverted power spectrum L 1 ŷ 2 vs. dimensionless zonal wavenumber k and frequency w from the simulation with b =.62,r y =.22,t =.92. function ψ or sea-surface height η and (upper-middle and lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from the simulation with β =.62,r ψ =.22,τ =.92. Dispersion relation and non-dispersive lines are as in Fig

71 PSD, F, AVISO SSH K FIG. 27. Power spectrum of space-time smoothed (blue line) and unsmoothed (green) dimensionless model streamfunction ψ or sea-surface height η vs. dimensionless wavenumber magnitude K from a simulation with β =.62,r ψ =.22,τ =.92. Also shown is the approximate AVISO SSH wavenumber-magnitude spectrum (solid black) used by Samelson et al. (216) to constrain the wavenumber structure of the linear stochastic field model, and the spectral structure of the model forcing (dashed blue and green). 71

72 P(A) Eddy amplitude, A (cm) P(L s ) Eddy scale, L s (km) P(U) Eddy speed, U (m s -1 ) FIG. 28. (upper panel) Amplitude, (middle) length-scale, and (lower) rotational speed distributions of eddies 112 FIG. 28. (upper panel) Amplitude, (middle) length-scale, and (lower) rotational speed distributions of eddies for 113 solutions for solutions with b with = β.62,r =.62,r y = ψ.22,t =.22,τ =.92, with (blue lines) lines) and and without without (green) (green) space-time space-time smoothing, smoothing, and114 forand thefor AVISO the AVISO observations (black). 72

73 FIG. 29. Dimensionless (upper panel) power spectrum ŷ 2 or ĥ of stream function y or sea-surface heigh 116 h and (lower) inverted power spectrum L 117 FIG. 29. Dimensionless (upper panel) power spectrum ˆψ 2 or ˆη 2 of stream function ψ or sea-surface height η and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from 1 ŷ 2 vs. dimensionless zonal wavenumber k and frequency w from the simulation with β = 2.4,τ = 5,r ψ =.24. Dispersion relation and non-dispersive lines are as in Fig. 19, the simulation with b = 2.4,t = 5,r y =.24. Dispersion relation and non-dispersive lines are as in Fig with a =.65 for the maximum-autocorrelation characteristic. with a =.65 for the maximum-autocorrelation characteristic. 73

74 FIG. 3. Power spectrum of dimensionless (upper panel) stream function ψ or sea-surface height η, (middle) stochastic forcing F, and (lower) inverted power spectrum L 1 ˆψ 2 vs. dimensionless zonal wavenumber k and frequency ω from the simulation with β = 2.4,τ = 5,r ψ =.24. Dispersion relation and non-dispersive lines are as in Fig. 19, with a =.65 for the maximum-autocorrelation characteristic. 74

75 FIG. FIG (Upper (Upper panel) panel) Power spectrum ĥ ˆη 2 and (lower) inverted power spectrum L 1 ĥ 2 2 and (lower) inverted power spectrum L 1 ˆη 2 of SSH vari- of SSH vari ability ability along along N in N the in the western North Pacific vs. zonal wavenumber (1 3 ( cpkm) cpkm) and frequency and frequency (cpd), where (cpd), where L 1 Lis 1 computed is computed using using an an empirical linear dispersion relation relation (white (white line) line) of theof form theωform = cw R k = /(λ c R 2 kk /(l 1), 2 k 2 + 1), for dimensional zonal wavenumber k for dimensional zonal wavenumber and frequency ω and frequency, with c w, with R.9 m s c R and m s 1 λ /(2π) 63 km. Upper and l /(2p) 63 km. Upper 1117 panel after Fig. 4 of Early et al. (211), with increased wavenumber range. panel after Fig. 4 of Early et al. (211), with increased wavenumber range. 75

76 FIG. 1.(a)Thetransferfunctionofthespatiallow-passfiltering.(b)Samplezonal(dashedline)andmeridional(solidline)wavenumber spectra after the filtering. FIG. 2.SpectrumofSSHat24.58N inthepacific.thesolidcurverepresentsthedispersion relation for the first-mode baroclinic Rossby waves based on the traditional theory. The dashed curve represents the dispersion relation based on the theory of Killworth et al. (1997) FIG. 32. Power spectrum of SSH at 24.5 N in the Pacific. The solid curve represents the dispersion relation for the first-mode baroclinic Rossby waves based on the traditional theory. The dashed curve represents the dispersion relation based on the theory of Killworth et al. (1997). From Fu (24). 76