A linear stochastic field model of mid-latitude mesoscale variability

Transcription

1 1 A linear stochastic field model of mid-latitude mesoscale variability 2 R. M. Samelson, M. G. Schlax, D. B. Chelton 3 CEOAS, Oregon State University, Corvallis, OR USA Corresponding author address: College of Earth, Ocean, and Atmospheric Sciences, 104 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 semi-empirical model of mid-latitude sea-surface height (SSH) variability is formulated and tested against two decades of weekly global fields of merged altimeter data. The model is constrained to match approximately the observed SSH wavenumber power spectrum, but predicts the spatio-temporal SSH field structure as a propagating, damped, linear response to a stochastic forcing field. An objective, coherent-eddy identification and tracking procedure is applied to the model and altimeter SSH fields. The model eddy dataset reproduces the basic global-mean characteristics of the altimeter eddy dataset, including the structure of mean amplitude and scale life cycles, the number distributions vs. lifetime, the increases of mean amplitude and scale with lifetime, and the distributions of week-to-week amplitude and scale increments, and, approximately, of all amplitude and scale realizations for eddies with lifetimes of 16 weeks or more. The model underpredicts the numbers of eddy realizations with large amplitudes and large scales, overpredicts the growth of mean amplitude and scale with lifetime, and modestly overpredicts the curvature of the mean amplitude lifecycle and the number of eddies with intermediate lifetimes. The stochastic forcing evidently represents nonlinear dynamical interactions, implying that eddy splitting and merging events are equally likely, that mesoscale nonlinearity is weaker than long-wave linearity but as strong as short-wave dispersion, and that mesoscale eddy potential vorticity fluxes are, to leading order, independent of the mean field. These theoretical inferences have potential implications for eddy flux parameterizations. 2

3 29 1. Introduction The ocean mesoscale, away from boundary currents, is characterized by energetic variability and fluctuating horizontal currents that are typically at least one order of magnitude faster than the long-term mean (Dantzler 1977; Wyrtki et al. 1976; MODE Group 1978; Wunsch 1981; McWilliams et al. 1983; Robinson 1983; Schmitz et al. 1983). Recently (Chelton et al. 2011), a new, quantitative, global description of ocean mesoscale variability has become available through the development of an automated eddy identification and tracking procedure and its application to a nearly two-decade global record of merged satellite altimeter observations of sea-surface height (SSH). The time histories of the resulting eddy amplitudes were studied systematically for the first time by Samelson et al. (2014), who showed that the amplitude-based eddy life cycles had striking and unexpected global-mean statistical characteristics, including time-reversal symmetry and approximate self-similarity, and, further, that the basic qualitative and quantitative statistical properties of these series could be reproduced with a simple stochastic model, in which the SSH increments between successive time points were random numbers, and the eddy life cycles were represented by excursions exceeding a given threshold. The stochastic model was found also to predict accurately the empirical autocorrelation structure of the underlying observed SSH field itself, when the autocorrelations were computed along long planetary (Rossby) wave characteristics. The stochastic amplitude model of Samelson et al. (2014) is generalized here to a stochastic field model, which allows the model SSH field to be analyzed using the same automated eddy identification and tracking procedure that is applied to the merged satellite altimeter SSH observations. The previous model (Samelson et al. 2014) produced only eddy amplitude time series, and did not include nor provide any information about spatial structure; consequently, that previous 3

4 model can be described as a zero-dimensional (in space) model. The stochastic field model, in contrast, produces a time-evolving SSH field on an effectively unbounded (doubly-periodic) spatial domain, and so is properly described as two-dimensional (in space). This stochastic field model is shown to reproduce the primary statistical characteristics of the altimeter-derived eddy dataset even more accurately than the stochastic amplitude model, including statistics related to eddy horizontal scale that were not predicted by the amplitude model Stochastic field model 59 The model sea-surface height η(x,y,t) is taken to satisfy a linear, stochastic, field equation, η t + c η R x = Rη + df(x,y,t), (1) where c R is a constant linear wave speed, R is a constant damping coefficient, and df(x, y,t) is a stochastic increment field. For simplicity, the model is posed in Cartesian coordinates, with x zonal, y meridional, and t representing time. The equation (1) may be integrated from t = t p to t = t p+1 = t p + t along the long-wave characteristic x(t) = x 0 + c R t, y(t) = y 0, to obtain tp+1 η(t p+1 ;x 0,y 0 ) = η(t p ;x 0,y 0 ) R η[t;x(t)] + δp F dt (2) t p (1 R t)η(t p ;x 0,y 0 ) + δ F (x,y,t p ). (3) In (3), η(t;x 0,y 0 ) = η[x(t),y(t),t)], an Euler-forward step has been used to approximate the inte- gral of the damping term, and δ F is the random increment from the integral of df. The funda- mental model equation is (3), which can be written in the first-order autoregressive (AR1) form η(x 0 + c R t p+1,y 0,t p+1 ) = αη(x 0 + c R t p,y 0,t p ) + δ F (x 0 + c R t p,y 0,t p ), (4) where the AR1 parameter α = 1 R t, and the stochastic increment is registered to the initial point [x(t p ),y(t p )]. For definiteness, c R can be specified or interpreted physically as a linear, long 4

5 69 planetary wave speed, c R = βλ 2 (5) where the constants β and λ represent, respectively, the meridional gradient of the Coriolis parameter and an effective internal deformation radius for reduced gravity, quasi-geostrophic, β-plane dynamics. The model equation (4) was solved numerically on a fixed (x,y) grid, with isotropic grid spacing x = y = c R t, where the time increment t = 1 wk was chosen to match the weekly time interval of the observed, gridded altimeter SSH data. The field η was initially set to zero, and updated at each time step by a three-step implementation of (4): (1) the field η was multiplied by α, (2) the field δ F of random increments was added to αη, and (3) the resulting sum field, αη + δ F, was shifted one grid interval westward, equivalent to following the characteristic equation according to x(t 1 ) = x 0 + c R t = x 0 x. The process of scaling plus random-field increment followed by westward shift was iterated for a total of 1024 weeks, or approximately two decades, comparable to the temporal length of the gridded altimeter dataset. The total domain variance of η reached statistical equilibrium after roughly 20 weeks. For the simulations analyzed here, the model grid consisted of 512 zonal points and 128 meridional points, which corresponded, with c R = m s 1 so that x = 15.1 km, to a total domain size of 7726 km 1920 km. Consistent with the results of Samelson et al. (2014), the random-field increments δ F (x,y,t p ) were taken to be uncorrelated in time on the weekly time interval, corresponding to a white (i.e., independent of frequency ω) frequency spectrum. The spatial characteristics of the random-field increments were set by matching, approximately, the horizontal wavenumber spectrum of the gridded altimeter data. An SSH power spectrum S K (K) was constructed that depended only on total wavenumber K = (k 2 +l 2 ) 1/2, where k and l are zonal and meridional wavenumbers, such that the one-dimensional, zonal wavenumber spectrum S(k) derived from S K (K) matched the mean zonal- 5

6 wavenumber SSH power spectrum of the gridded altimeter data (Chelton et al. 2011, Fig. A3a). The explicit form of S K (K) is given in the Appendix. The Fourier transform ˆδ F of the stochastic increment field was then defined as the square-root of S K multiplied by a normalization constant C 0 and a random phase function, ˆδ F (k,l;t p ) = C 0 S K (K) 1/2 exp[iθ kl (t p )], k > 0, (6) where θ kl (t p ) was a random number drawn independently for each (k,l) pair at each time t p from a uniform distribution on [0,2π). The supplemental condition ˆδ F (k,l;t p ) = [ ˆδ F ( k, l;t p )] or θ kl (t p ) = θ k, l (t p ), k < 0, (7) was imposed, where indicates complex conjugate, so that the inverse Fourier transform of the complex random field ˆδ F (k,l;t p ) gave a real stochastic increment field δ F (x,y,t p ). With this specification, the ensemble-averaged (or time-averaged) wavenumber power spectrum of the solution η of (4) is directly proportional to S K (see Sec. 4). A modification to this increment spectrum was then introduced, in which S K (K) in (6) was replaced by S K (γk), where γ is a spatial scaling constant. The constant C 0 was chosen to give the stochastic increment field a mean spatial standard deviation equal to a specified value, denoted by σ S, which can be compared to the stochasticincrement amplitude parameter σ of Samelson et al. (2014). The eddy analysis was performed by analyzing the weekly model SSH fields η(x,y,t) on the full model domain, using an updated version of the numerical eddy detection and tracking code that was applied to altimeter data by Chelton et al. (2011), modified to use an SSH-based version of the more efficient eddy-identification algorithm of Williams et al. (2011). The model results were compared to an eddy dataset essentially equivalent to that described by Samelson et al. (2014), using the updated eddy detection and tracking code and the same 19.5-year (October 1992 through April 2012) version of the AVISO Reference Series merged satellite SSH anomaly dataset con- 6

7 structed by SSALTO/DUACS using the approach summarized by Ducet et al. (2000). The amplitude and scale life-cycle analysis for both the model and the altimeter eddy datasets followed the approach described in detail by Samelson et al. (2014), in which tracked eddies were first grouped by lifetime, and statistical descriptors of each lifetime class were then computed, with mean and standard deviation life cycles calculated after normalization of each eddy amplitude or scale time series to unit mean amplitude or scale and to unit lifetime. The combination of the characteristic integration to obtain (4) and the spectral specification of δ F means that the model boundary conditions were effectively doubly-periodic in space, but the eddy analysis was carried out on the finite, fixed 7726 km 1920 km model domain. With the wavenumber spectrum S K (K) chosen to match the mean altimeter spectrum, the model has only three free parameters: the AR1 constant α, the forcing amplitude σ S, and the spatial scaling constant γ. The parameter α is constrained to values near 0.95 by the autocorrelation structure of the observed SSH field (Samelson et al. 2014, Fig. 10). The forcing amplitude parameter σ for the previous, zero-dimensional stochastic model was computed directly by Samelson et al. (2014) from the ratio of modeled and observed mean eddy amplitudes, but it is not straightforward to compute the equivalent parameter σ S for the stochastic field model in that way, because the eddy tracking procedure uses fixed dimensional SSH disturbance criteria. Instead, an optimal, or approximately optimal, set of values of the three parameters was found by iteration. Numerous model solutions were examined in the course of this iteration, including many for which the model eddy statistics differed substantially from the observations. For this simple, semi-empirical model, the result of primary interest is the existence of a solution that reproduces accurately the observed eddy statistics. Consequently, rather than exploring the dependence of the model solutions on parameters, much of the discussion to follow focuses on a standard solution of the stochastic field model with α = 0.96, σ S = 1.96 cm and γ = 1.2, which gave, by interactive, manual search and 7

8 subjective judgment, an approximately optimal fit to the various quantities of interest. The use of the term standard solution is intended to emphasize that different parameter choices can produce model solutions with eddy statistics that deviate dramatically from the observations, and that the choice of proper parameter values is therefore essential Results: Model eddy statistics 142 a. Number distribution The observed fractional number (number normalized by total number) distribution N e (L) of tracked eddies vs. lifetime L has a transition in structure near L = 50 wk, with power-law behavior for, approximately, L < 50 wk and exponential behavior for L > 50 wk (Fig. 1). The standard solution of the stochastic field model, with a = 1.96 cm and γ = 1.2, reproduces this transition (Fig. 1). The slopes of the observed distributions in the respective regimes are approximately, though not exactly, reproduced by the model. In contrast, the previous, zero-dimensional, stochastic amplitude model did not clearly reproduce this transition near L = 50 wk, tending instead to have an exponential character for all L (Samelson et al. 2014, Figs. 4b,8d). The observed fractional number distribution can be approximated as, N e (L) = a(l1 2 + L b ) 1 + bexp(l/l 2 ), (8) where a = 4, b = 1/4, L 1 = 75, and L 2 = 30, and L is lifetime in weeks (Fig. 1), giving analytical expression to the transition from power-law to exponential structure. The accurate reproduction of the observed number distribution, including this transition in structure with lifetime, is the first significant new result of the stochastic field model. 8

9 156 b. Length scale The second significant new result of the stochastic field model, relative to the previous, zerodimensional, stochastic field model (Samelson et al. 2014), is the prediction of the statistics of eddy length (radius) scale (Fig. 2). Such a prediction could not be made from the previous model, which simulated only the eddy amplitude life cycles. A separate, zero-dimensional model of the length-scale statistics was presented by Samelson et al. (2014), but this was simply an independent twin of the stochastic amplitude model, with different model parameters and with amplitude reinterpreted as length scale. The stochastic field model instead provides a complete spatio-temporal SSH field, with eddy amplitude and scale determined simultaneously through the objective eddy identification and tracking procedure. The length scale that is determined by the eddy identification procedure is an equivalent radius, for which the circular area would be equal to the area enclosed by the SSH contour around which the contour-averaged geostrophic speed is maximum [L s in the nomenclature of Chelton et al. (2011)]. The standard solution of the stochastic field model predicts the mean life cycle of this speed-based eddy length scale with uncanny accuracy, as the modeled and observed values are nearly indistinguishable in a typical plotting format (Fig. 2, upper right). The standard solution also closely reproduces the life cycle of length-scale standard deviation, with only a modest underestimate of the observed mid-cycle values (Fig. 2, upper right). The lifetime-dependent, mean dimensional eddy length scale from the standard solution is roughly in agreement with the observed values, near 90 km for eddies with lifetimes L > 60 wk as observed, but km smaller than observed for L < 25 wk (Fig. 2, lower left). The mean dimensional length-scale standard deviation, which is determined by the same normalization as the mean length scale, shows a similar pattern of better agreement at longer than at shorter lifetimes. 9

10 Like the mean scale life cycle, the length-scale innovation distribution the distribution of weekto-week differences in normalized length-scale values for each tracked eddy with lifetime greater than 16 wk for the standard solution is almost indistinguishable from the observed distribution (Fig. 2, lower right). These distributions are distinctly non-normal, with kurtosis near 6. The relation of this non-normality to the structure of the stochastic forcing field δ F (x,y,t p ) involves the coherent-structure identification and tracking procedure, and is not immediately obvious. The standard solution also reproduces much of the structure of the distribution of all weekly, dimensional, tracked-eddy length-scale values for eddies with lifetimes greater than 16 wk, especially over the dominant length-scale range from 35 km to roughly 125 km (Fig. 3). The model distribution lacks the prominent tail of larger length scales that is found for the observations, and consequently also has a slightly sharper peak near 60 km than the observed distribution. 190 c. Amplitude The mean amplitude life cycle for the standard solution of the stochastic field model (Fig. 4, upper panels) is nearly identical to that from the standard solution of the previous, zero-dimensional, stochastic amplitude model (Samelson et al. 2014, Fig. 2b). This mean life cycle is also symmetric in time about the cycle mid-point t = 0.5, where t denotes the normalized time, so that the mean evolution is the same in forward and backward time. This symmetry is directly evident when the cycle is decomposed into components that are symmetric and anti-symmetric about t = 0.5, as the anti-symmetric components are essentially zero (Fig. 4, upper right). As previously, the basic structure of the observed mean cycle is reproduced but the predicted curvature is more uniform over the cycle than observed, with slower initial growth and final decay, and a mid-cycle structure that is less flat than observed. The amplitude standard deviation cycle from the stochastic field model matches the observations markedly better than that from the previous, zero-dimensional 10

11 model, with values near, rather than only half of, the observed values, and as before shares the time-reversal symmetry property with the observations (Fig. 4, upper right). The lifetime-dependent, mean dimensional amplitude from the standard solution is roughly in agreement with the observed values, near 8-10 cm for eddies with lifetimes L > 40 wk as observed, but 1-2 cm smaller than observed for L < 35 wk (Fig. 4, lower left). The mean dimensional amplitude standard deviation, which is determined by the same normalization as the mean amplitude, shows a similar pattern of better agreement at longer than at shorter lifetimes. This pattern is consistent with that described above for the length scale. The amplitude innovation distribution the distribution of week-to-week differences in normalized amplitude values for each tracked eddy with lifetime greater than 16 wk for the standard solution reproduces the basic structure of the observed distribution (Fig. 4, lower right). These distributions are modestly non-normal, with kurtosis near 3.4 for the model and 3.7 for the observations. As previously for the length scale, the relation of this non-normality to the structure of the stochastic forcing field δ F (x,y,t p ) involves the coherent-structure identification and tracking procedure, and is not immediately obvious. The standard solution only crudely reproduces the distribution of all weekly, dimensional, tracked-eddy amplitude values for eddies with lifetimes greater than 16 wk (Fig. 5). Both distributions have a single peak at an intermediate amplitude, and steeper decline from the peak toward smaller than toward larger amplitude. However, the model distribution has its peak near 7 cm, approximately twice the value at which the observed peak is found, and also lacks the prominent, observed, large-amplitude tail. In the subjective optimization of the amplitude parameter σ S, the matching of the overall mean amplitude was emphasized. Agreement with the small-amplitude peak of the observed distribution could be improved by reducing σ S, but this would degrade the 11

12 agreement with the observed overall mean, and exaggerate the disagreement with the observed large-amplitude tail. 227 d. Tracks Consistent with the meridionally symmetric dynamics of the linear model, and in contrast to the behavior of observed eddies (Chelton et al. 2011, Fig. 20), the model eddy tracks show no systematic meridional deflection for either cyclonic or anticyclonic polarities (Fig. 6), an effect that has been attributed theoretically to weakly nonlinear eddy dynamics (Early et al. 2011; McWilliams and Flierl 1979). The model tracks are generally shorter than the observed tracks, which extend up to 50 of longitude, but this is largely because of the fixed, m s 1 propagation speed imposed in the model. Many of the observed eddies have faster propagation speeds, giving longer tracks for the same eddy lifetimes. The number distribution of eddies vs. lifetime is captured well by the standard model solution (Fig. 1), and would give a track-length distribution similar to the observed if faster propagation speeds were also modeled Results: Model SSH field statistics The main results of this study are the comparisons of eddy statistics described in the previous section, which depend upon the instantaneous, evolving, spatial structure of the model SSH field. To complement and supplement these main results, basic statistical and spectral properties of the model SSH field, prior to the eddy analysis, can be obtained by solving the linear, stochastic, field equation (1) directly by Fourier transform in space and integration in time, or by Fourier transform in space and time. The former gives the transformed solution t ˆη(k,l;t) = ˆη(k,l,0)e (R+ikcR)t ˆF(k,l;s)e (R+ikc R)(s t) ds. (9)

13 245 For ˆη(k,l,0) = 0 or t 1/R, this gives the ensemble-averaged (denoted by ) SSH power spec- 246 trum ˆη ˆη = t 0 ˆF ˆF e 2R(s t) ds, (10) 247 which simplifies further to ˆη ˆη ˆF ˆF 2R (11) for an ensemble-averaged forcing power spectrum that is independent of time and for t 1/R. Alternatively, the space-time Fourier transform gives the power spectrum, η η (k,l,ω) = F F (ω + ir c R k), (12) 250 which, for a white (ω-independent) forcing power spectrum, gives the result η η dω = F F 2R. (13) An equivalent result follows for the stochastic difference equation (4). The solution and random forcing are uncorrelated in ensemble average, so ηη (x 0 + c R t p+1,y 0,t p+1 ) = α 2 ηη (x 0 + c R t p,y 0,t p ) + δ F δ F (x 0 + c R t p,y 0,t p ), (14) 253 and stationarity of forcing and solution then imply that (1 α 2 ) ηη = δ F δ F (15) 254 or, for t 1/R, ηη δ F δ F. (16) 2R t Thus, the model SSH power spectrum inherits its wavenumber dependence directly from the stochastic forcing spectrum, and so is constrained to agree with the mean observed spectral shape, shifted by the spatial scaling factor γ. 13

14 The results (11), (13), and (16) show also that, for sufficiently small R (i.e., α 1), the model SSH power spectral level and variance are inversely proportional to the damping parameter R. Consequently, R, and thus also the AR1 parameter α = 1 R t, could in principle be determined by fitting the model power spectral level to the observed level. However, because of the geographical variability in the observed spectral levels, and the effective weighting of this variability by the geographic distribution of observed tracked eddies, this fitting is not straightforward. Instead, the parameter R was used to fit the observed eddy amplitude and number distributions. For the best-fit solution here, with α = 0.96 (R = 1 α = 0.04), forcing amplitude parameter a = 1.92 cm, and spatial scaling constant γ = 1.2, the resulting space-time mean standard deviation of model SSH is 27 cm. This value is within the range of observed mid-latitude SSH standard deviations in 1 1 latitude-longitude bins and roughly three times the zonal average of the observed mid-latitude 1 1 bin values (Chelton et al. 2011, Fig.10). The model wavenumber-frequency power spectrum (12), with forcing spectrum given by (A1), has a local maximum along the non-dispersive line ω = c R k. When integrated over meridional wavenumber l, this maximum appears as a peak, or ridge, with spectral width across the maximum controlled by the damping parameter R, and slope along the ridge controlled by the forcing spectrum (Fig. 7, left panel). This structure is consistent with the observed zonal wavenumber and frequency spectrum of the altimeter SSH fields (e.g., Fig. 7, right panel). The full threedimensional wavenumber-frequency spectrum for the model SSH field may be visualized using a combined contour-isosurface plot (Fig. 8), which illustrates the decay in spectral level away from the maximum also as a function of meridional wavenumber l. This visualization of the model spectrum (12) may be compared, for example, to that of the empirical spectral form proposed by Wortham and Wunsch (2014) and shown in their Figure 7. This comparison shows the similarity 14

15 of the spectral representation given by (12) and (A1) to that of eq. (30) of Wortham and Wunsch (2014), despite differences in the corresponding functional forms. The fundamental model equation (4) shows that, along any long-wave characteristic x(t) = x 0 + c R t, y(t) = y 0, the model SSH field will have the simple autocorrelation function α t, where the integer t = {1,2,3,...} corresponds to time in weeks. The value α = 0.96 used for the standard model solution gives an autocorrelation that is close to that computed by Samelson et al. (2014) directly from the altimeter SSH observations along approximate long planetary wave characteristics (Fig. 9). This value is slightly larger than both that observational estimate and the autocorrelation for α = 0.94 from the reference solution of the previous, zero-dimensional stochastic amplitude model of Samelson et al. (2014). As in that previous case, it is emphasized that the observed estimate of this autocorrelation, like the other model and observed SSH statistics discussed in this section, does not depend on the eddy identification and tracking algorithm, but is computed directly from the altimeter SSH fields. This correspondence exhibits a basic consistency between the underlying stochastic representation of the time-evolving model SSH field and the altimeter SSH observations, independent of any properties of the identified eddy structures Discussion The standard solution (obtained for model parameter values α = 0.96, σ S = 1.96 cm and γ = 1.2; see Sec. 2) of the stochastic field model reproduces many aspects of the observed global-mean eddy statistics with remarkable accuracy, including the normalized life cycle of eddy length scale, a quantity for which a prediction could not even be obtained from the previous, zero-dimensional, stochastic amplitude model. The stochastic field model eddy statistics also show some significant departures from observed statistics. For the standard solution, these differences are most apparent in the distribution of eddy amplitudes, and in the dependence of mean dimensional amplitude 15

16 and length scale on eddy lifetime. One possible reason for these differences is that the structure and coherence of ocean eddy features may be affected by nonlinear dynamics that are absent from the linear model. More speculatively, an influence of nonlinearity might perhaps also be indicated by the departure of the spatial scaling factor γ from unity: the implied shift toward longer wavelengths, relative to the full SSH spectrum, of the eddy energy distribution might be taken as effectively representing an upscale energy transfer that manifests more strongly in the energetic, coherent eddy field than in the residual quasi-linear ambient variability. Despite these differences, the overall agreement of the modeled and observed eddy and SSH-field statistics indicates that the linear stochastic field model captures most of the essential, leading-order characteristics of the dynamical evolution of the observed SSH field. It is possible also that some of the differences may arise from imperfections in the extraction of mesoscale eddy statistics from the altimeter observations, such as the long tail of large observed eddy scales (Fig. 3), at least part of which may arise from large-scale SSH signals not fully removed by the spatial high-pass filtering applied prior to the eddy identification and tracking procedure. Similarly, the spatial scaling factor may indicate a departure of the effective mean SSH spectrum reflected in the global eddy statistics from the particular observed mean zonal spectrum used to generate the nominal model spectrum S K. Theoretical rationalization of the stochastic field model requires reconciliation of the linear, long-wave and stochastic forcing assumptions with the nonlinear, dispersive, deterministic, quasi-geostrophic dynamics that are broadly accepted as a first-order description of mid-latitude mesoscale ocean variability [e.g., Pedlosky (1987); Gill (1987)]. A typical form of the latter equations is q t + β ψ x = J(ψ,q) + F + D, (17) 16

17 325 where the potential vorticity q is related to the stream function ψ by q = 2 ψ + z [ f 2 N 2 (z) ] ψ. (18) z In (17)-(18), J(a,b) = ( a/ x)( b/ y) ( a/ y)( b/ x) and 2 = 2 / x / y 2 are, re- spectively, the Jacobian and Laplacian operators with respect to the horizontal coordinates x and y, F and D represent forcing and dissipation processes, f is the Coriolis parameter, N(z) is the buoyancy frequency, and z is a vertical coordinate. The elliptic equation (18) must be supple- mented by suitable boundary conditions, which in the standard theory are derived from vertical velocity conditions at the domain surface and bottom. The stochastic forcing field df(x, y,t) in (1), like the random increment in the previous, stochas- tic amplitude model of Samelson et al. (2014), is too large in magnitude to be interpreted consis- 334 tently as a representation of the external forcing F in (17). By the same argument given in Samelson et al. (2014), the term df must therefore instead primarily represent the nonlinear self- interactions J(ψ, q) of the near-geostrophic motion field. The representation of the nonlinear interactions of a turbulent field by an imposed stochastic forcing is a relatively standard theoret- ical approach; while still lacking rigorous support, this element of the rationalization can at least appeal to that familiarity. It should be noted, however, that such stochastic representations are typically invoked for the homogeneous, inertial-range dynamics of fully developed turbulence, a regime of doubtful direct relevance to the ocean mesoscale. Somewhat less clearly motivated is the absence of the linear dispersive terms from the field equation (1), which would generally arise from the rate of change of relative vorticity in (17)-(18), ( 2 ψ)/ t. In general, these linear dispersive terms cannot be neglected on the basis of theo- retical scaling arguments when the disturbance scales are comparable to the internal-deformation scale for a given vertical mode [e.g., Pedlosky (1987); Gill (1987)]. The dominant model eddy 17

18 radius scales are of order km, roughly comparable to the typical observed, first-internalmode deformation radii of order 50 km (Chelton et al. 1998) that gives a wave speed c R in (5) roughly consistent with observed propagation speeds. Consequently, the absence of the linear dispersive terms from (1) cannot easily be justified by appeal to a long-wave argument. Similarly, a simple appeal to nonlinearity dominating the linear dispersive terms also cannot be easily justified, because the linear long-wave terms must evidently dominate the leading-order dynamical balance, to support the observed propagation. A more subtle ordering is evidently required, in which the nonlinearity occurs at intermediate order, between the linear long-wave and linear dispersive terms, or perhaps at the same order as the dispersive terms, as in classical solitary wave theory for weakly nonlinear waves of permanent form, such as soliton solutions of the Korteweg-deVries equation (Drazin 1983). An appeal to the approximate axial symmetry of eddy structures (Chelton et al. 2011, Fig. 15) could motivate the neglect of the advection of relative vorticity by the eddy velocity, and even a functional dependence of the potential vorticity on the stream function, but this too is not enough on its own to motivate the neglect of the local rate of change of relative vorticity. Alternatively, one might hope to appeal directly to the approximately quadratic median (in distinction to mean) structure of the observed eddies (Chelton et al. 2011, Fig. 15), which would imply a uniform relative vorticity in the eddy interior, and apparently then the vanishing of its rate of change; however, a time-variation of the spatially constant vorticity must be allowed, as well as a local rate of change associated with the transition from the ambient value to the eddy-interior value as the eddy propagates. The present results have implications for the related problem of parameterizing mesoscale eddy fluxes in large-scale ocean models. If the model equation (1) can be interpreted as a potential vorticity evolution equation, then the stochastic forcing will represent, primarily, the divergence of eddy potential vorticity fluxes. To the extent that the model solutions reproduce the statistics of 18

19 the observed coherent-eddy variability, it may be inferred that the stochastic forcing function used here reproduces in a statistical sense the mesoscale potential vorticity flux divergence field. A potential function for the divergent part of the associated potential vorticity fluxes could be obtained by inverting a Poisson equation, with the stochastic flux divergences as the source term. The stochastic forcing function, however, is independent of the state of the flow, so that there is no deterministic relation between the model potential vorticity flux divergence and the local mean field. This would suggest a flux parameterization that is, at least to leading order, also independent of the mean field, and purely stochastic: evidently, the mesoscale fluxes missing from a large-scale model might best be represented stochastically, as in (1), rather than as quantities that depend in any way on the mean flow. On the other hand, these inferences remain speculative in the absence of a complete physical understanding of the processes represented by the stochastic forcing, which may plausibly include, for example, a hidden, implicit role for the mean flow as a source of energy to the mesoscale. In addition, the motions associated with, or driven by, these stochastic fluxes might still be inferred to have a deterministic effect on larger-scale property fields that may have been filtered out of the present description and model by the spatial high-pass filtering that was used to focus on the mesoscale SSH fields (Chelton et al. 2011). The results do imply that such larger-scale mean fluxes are evidently not required to support or explain the mesocale motion: by analogy with molecular fluxes, the present model might be thought of as representing or explaining the thermal molecular dynamics necessary to support the underlying Brownian motion, rather than the ensemble-averaged correlations of those molecular motions with property anomalies that result in mean down-gradient fluxes by molecular diffusion. From this point of view, it might be possible to construct an effective diffusivity based on the quantitative characteristics of the stochastic forcing, which could then be used in a form of downgradient parameterization. To extend the analogy, 19

20 this might be similar to attempting to construct a molecular diffusivity from the statistics of the momentum exchange distributions rather than potential vorticity flux divergences that describe the molecular collisions associated with Brownian motion Conclusions The stochastic field model (1) proposed here generalizes the stochastic amplitude model of Samelson et al. (2014) to simulate a full two-dimensional, evolving SSH field, to which the objec- tive eddy identification and tracking procedure of Chelton et al. (2011) can be applied directly. The model successfully reproduces many aspects of the global-mean spatio-temporal statistics of the observed eddy dataset. These results suggest the possibility that the mesoscale field of ocean po- tential vorticity fluxes may be appropriately represented, to leading order, as a stochastic forcing, in which the eddy fluxes are independent of the mean flow. This is quite different from parameter- izations such as that proposed by Green (1970), Gent and McWilliams (1990) and others, in which the mesoscale fluxes are assumed to be down-gradient with respect to the mean field, as in the case of molecular diffusion. The perspective is similar in some ways to the fluctuation-dissipation theorem approach explored for atmospheric circulation by, for example, Gritsun et al. (2008), but there are also differences: in the present case, the deterministic dynamics consist of just the non-dispersive, damped long-wave dynamics represented by (1) with df = 0, and the stochastic forcing is not a small perturbation, but contains most of the mesocale signal. A partial dynami- cal rationalization of this perspective may be possible through appeal to the stochastic properties of mesoscale turbulent flow, and suggests an ordering in which the nonlinearity is weaker than the long-wave linearity but at least as strong as the short-wave dispersion. More directly, the re- sults might be taken simply as quantitative evidence that the ocean mesoscale flow is genuinely turbulent, in this specific, stochastic sense. 20

21 A natural extension of this work would be to explore these statistical and dynamical hypotheses in a set of deterministically forced, baroclinic, mesoscale-turbulent, quasi-geostrophic model simulations. Results from reduced-gravity, quasi-geostrophic models have previously been compared to the altimeter-observed eddy statistics (Early et al. 2011), but these simulations have been forced directly with potential vorticity disturbances derived as representations of the observed eddy fields, rather than developing their own internal variability through baroclinic or other related instability processes. In simple channel or basin geometries, with steady boundary conditions and forcing, quasi-geostrophic models are well known to have a wide variety of equilibrium statistical responses, ranging from regular spatial structures dominated by direct forcing-dissipation balances, inertially recirculating gyres or zonal jets to more disordered, turbulent states. A natural and necessary next step in the extension of the present work through quasi-geostrophic dynamical modeling would be to identify a configuration and parameter regime in which statistical equilibria of the given quasi-geostrophic model are able to represent approximately the observed eddy statistics. If such a regime can be identified, then direct examination of the associated quasi-geostrophic dynamical processes in the model should be straightforward, and, in combination with the present results, should yield fundamental insights into the dynamics of the ocean mesoscale. 434 Acknowledgments. This research was funded as part of the NASA (National Aeronautics and Space Administration) Ocean Surface Topography Mission through NASA grant NNX13AD78G. The manuscript was completed during a sabbatical visit of RMS to the Universitet i Stavanger; he is grateful to Prof. O. T. Gudmestad and the teknisk-naturvitenskapelige fakultet of the UiS for their support of this visit and their generous hospitality. 21

22 439 APPENDIX 440 Model wavenumber spectrum 441 The basic model wavenumber power spectrum S K (K) in (6) was specified analytically as S K (K) = exp( ak2 ) 1 + (K 2 /K 2 b )b S ρ(k), (A1) 442 where the true spectrum S ρ (K) was given by, [ S ρ (K) = 1 π s s 0 1 xy ( s s 1 s 0 xy + K s 1s 0 ) 1/s1 ] S low (K), (A2) 443 with 1, K K low S low (K) =. (A3) (K/K low ) 4, K < K low The parameter values were chosen as a = (ln2)/ka, 2 K a = 2π/150 km 1, K b = 2π/70 km 1, b = 30, s xy = 1/45 km 1, s 0 = 4, s 1 =3, K low = 2π/1000 km 1. This form for S K gives, after integration over meridional wavenumber l, an accurate fit to the shape of the mean zonal AVISO spectrum over the entire resolved mesoscale range (Fig. A1). The factor multiplying S ρ in (A1) was intended as a crude representation of the effective spatial smoothing of the true SSH spectrum arising from the altimeter sampling characteristics and the optimal interpolation procedure of Ducet et al. (2000), so that the spectrum S ρ defined in (A2) might be loosely interpreted as a model spectral representation of the true spectrum of the SSH field, which essentially extrapolates the power-law behavior at the smallest observed scales toward smaller, unobserved scales. The scaled spectrum S K (K ), where K = γk and γ = 1.2 (so γ > 1) for the standard solution, effectively shifts the spectral form toward smaller wavenumber (longer wavelength), so that a nominal spectral value S K (K 1 ) will be found instead at wavenumber K = K 1 /γ 0.83K (wavelength λ = γλ 1 = 1.2λ 1, where λ 1 = 2π/K 1 ). 22

23 457 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, 2011: Global observations of nonlinear mesoscale eddies. Progress in Oceanography, 91 (2), , doi: doi.org/ /j.pocean , URL S Dantzler, H. L., 1977: Potential energy maxima in the tropical and subtropical north atlantic. Journal of Physical Oceanography, 7 (4), , doi: / (1977) : PEMITT 2.0.CO;2, URL :PEMITT 2.0. CO; Drazin, P. G., 1983: Solitons. No. 85, London Mathematical Society Lecture Note Series, Cam- bridge University Press. 471 Ducet, N., P. Y. Le Traon, and G. Reverdin, 2000: Global high-resolution mapping of ocean circulation from topex/poseidon and ers-1 and -2. Journal of Geophysical Research: Oceans, 105 (C8), , doi: /2000jc900063, URL JC Early, J. J., R. M. Samelson, and D. B. Chelton, 2011: The evolution and propagation of quasigeostrophic ocean eddies. Journal of Physical Oceanography, 41 (8), , doi: /2011JPO4601.1, URL 23

24 Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. Journal of Physical Oceanography, 20, Gill, A., 1987: Atmosphere-Ocean Dynamics. Academic Press Green, J. S. A., 1970: Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Quarterly Journal of the Royal Meteorological Society, 96, Gritsun, A., G. Branstator, and A. Majda, 2008: Climate response of linear and quadratic func- tionals using the fluctuation dissipation theorem. Journal of the Atmospheric Sciences, 65 (9), , doi: /2007jas2496.1, URL McWilliams, J. C., and G. R. Flierl, 1979: On the evolution of isolated, nonlinear vortices. Journal of Physical Oceanography, 9 (6), , doi: / (1979) : OTEOIN 2.0.CO;2, URL :OTEOIN 2.0. CO; McWilliams, J. C., and Coauthors, 1983: Eddies in Marine Science, chap. The Local Dynamics of Eddies in the Western North Atlantic, Springer Berlin Heidelberg, Berlin, Heidelberg, doi: / , URL MODE Group, T., 1978: The mid-ocean dynamics experiment. Deep Sea Research, 25 (10), , doi: URL com/science/article/pii/ x. 496 Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed., Springer Berlin Heidelberg Robinson, A. R., 1983: Eddies in Marine Science, chap. Overview and Summary of Eddy Science, Springer Berlin Heidelberg, Berlin, Heidelberg, doi: / , URL

25 Samelson, R. M., M. G. Schlax, and D. B. Chelton, 2014: Randomness, symmetry, and scaling of mesoscale eddy life cycles. Journal of Physical Oceanography, 44 (3), [Corrigendum 44 (9), , doi: /JPO D ], URL JPO-D Schmitz, W. J., W. R. Holland, and J. F. Price, 1983: Mid-latitude mesoscale variability. Reviews of Geophysics, 21 (5), , doi: /rg021i005p01109, URL /RG021i005p Williams, S., M. Petersen, P.-T. Bremer, M. Hecht, V. Pascucci, J. Ahrens, M. Hlawitschka, and B. Hamann, 2011: Adaptive extraction and quantification of geophysical vortices. IEEE T. Vis. Comput. Gr., 17, Wortham, C., and C. Wunsch, 2014: A multidimensional spectral description of ocean variability. Journal of Physical Oceanography, 44 (3), , doi: /jpo-d , URL http: //dx.doi.org/ /jpo-d Wunsch, C., 1981: The Evolution of Physical Oceanography, chap. Low-frequency variability of the sea, MIT Press, Cambridge, MA Wyrtki, K., L. Magaard, and J. Hager, 1976: Eddy energy in the oceans. Journal of Geophysical Research, 81 (15), , doi: /jc081i015p02641, URL JC081i015p

26 LIST OF FIGURES Fig. 1. Fractional number distributions of model (green) and altimeter-based (black) tracked eddies vs. lifetime, with lifetime on (left panel) logarithmic and (right) linear axes. The analytic approximation (8) to the observed distribution is also shown (dashed) Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Normalized eddy scale lifecycle statistics. (a) Model symmetric mean scale (shown for normalized times 0 < t < 1/2) and standard deviation of scale (shown for 1/2 < t < 1) lifecycles for lifetime classes 16 wk L 80 wk, as indicated in the legend; the corresponding anti-symmetric lifecycles are also shown, and are all near zero. (b) Mean model (green; components as in upper left panel) and altimeter-based (black, blue; both symmetric components shown for 0 < t < 1) scale and standard deviation of scale lifecycles for lifetime classes 16 wk L 80 wk, weighted by the number of eddies in the respective lifetime classes. (c) Mean model (green) and altimeter-based (black) dimensional scale (solid circles) and standard deviation of scale (squares) vs. lifetime class. (d) Distributions of model (green) and altimeter-based (black) innovations, from normalized scale lifecycles, with first four (model, altimeter-based) moments (mean, standard deviation, skewness, kurtosis) as indicated Distributions of all weekly scale realizations for model (green) and altimeter-based (black) eddies with lifetimes L 16 wk Normalized eddy amplitude lifecycle statistics. (upper left panel) Model symmetric mean amplitude (shown for normalized times 0 < t < 1/2) and standard deviation of amplitude (for 1/2 < t < 1) lifecycles for lifetime classes 16 wk L 80 wk, as indicated in the legend; the corresponding anti-symmetric lifecycles are also shown, and are all near zero. (upper right) Mean model (green, new mean and new std dev ; components as in upper left panel) and altimeter-based (black mean and blue std dev ; both symmetric components shown for 0 < t < 1) amplitude and standard deviation of amplitude lifecycles for lifetime classes 16 wk L 80 wk, weighted by the number of eddies in the respective lifetime classes. (lower left) Mean model (green) and altimeter-based (black) dimensional amplitude (solid circles) and standard deviation of amplitude (squares) vs. lifetime class; the corresponding mean initial ( ) and final ( ) values are also shown. (lower right) Distributions of model (green) and altimeter-based (black) innovations, from normalized amplitude lifecycles, with first four (model, altimeter-based) moments (mean µ i, standard deviation σ i, skewness s i, kurtosis k i : model value, observed value) as indicated Distributions of all weekly amplitude realizations for model (green) and altimeter-based (black) eddies with lifetimes L 16 wk Eddy tracks relative to initial location for all model cyclonic (upper panel) and anticyclonic (lower panel) eddies with final and initial longitudes separated by at least 500 km Power spectrum of (left) model and (right; from Early et al. (2011)) SSH vs. zonal wavenumber k and frequency ω. The frequency axis limits on the left panel have been chosen so that the slopes of the non-dispersive line ω = c R k are similar in the two panels. In this figure, the absolute model spectral level is arbitrary Three-dimensional visualization of the model wavenumber-frequency power spectrum of SSH, vs. zonal and meridional wavenumbers, k and l, and frequency, ω, for the standard solution described in the text. The base-10 logarithm of the SSH power spectrum is contoured on three orthogonal planes, and an isosurface of constant SSH spectral power (red) is shown, which is localized around the nondispersive surface ω = c R k. In this figure, the 26

27 Fig. 9. absolute model spectral level is arbitrary. This figure may be compared to Fig. 7 of Wortham and Wunsch (2014) Mean autocorrelation at 5-week lag of altimeter sea-surface height along linear planetary wave characteristics (filled circles) and at fixed longitudes (open circles) for the North (blue) and South (red) Pacific vs. latitude, for longitudes 180 W-130 W (North Pacific) and 170 W-120 W (South Pacific), from Samelson et al. (2014). Also shown are the autocorrelations at 5-week lag α 5 = {0.96 5, } {0.84,0.73} for AR1 processes with α = {0.96,0.94} (green solid and dashed lines, respectively) Fig. A1. One-dimensional, zonal wavenumber power spectra of SSH: AVISO (black), the model spectrum S K (green solid), and the model true spectrum S ρ (green dashed). The absolute spectral level is arbitrary. The AVISO spectrum was computed as the mean of the zonal wavenumber spectra in Fig. A3 (upper left panel) of Chelton et al. (2011). The model one-dimensional spectra were computed from the radial spectra S K (A1) and S ρ (A2) by numerical integration over wavenumber l

28 10 0 tracks eta rinc21.v0.1.ascii.nc n L = a (L -2 +L 1-2 )(1+b)/[1+b exp(l/l2 )] fraction of eddies fraction of eddies (a,b)=(4,0.25), (L 1,L 2 )=(75,30) wks lifetime (wks) lifetime (wks) FIG. 1. Fractional number distributions of model (green) and altimeter-based (black) tracked eddies vs. lifetime, with lifetime on (left panel) logarithmic and (right) linear axes. The analytic approximation (8) to the observed distribution is also shown (dashed). 28

29 dimensionless scale and std dev a) normalized life cycles mean std dev wks wks wks wks dimensionless scale and std dev b) mean mean normalized life cycles std dev dimensionless time dimensionless time scale (km) c) tracks eta rinc21.v0.1.ascii.nc σ i P(increment) scale innovation distribution µ = (-0.00,0.00) i s = (0.01,0.01) i σ = (0.25,0.30) k = (6.12,5.75) i i d) lifetime (wks) dimensionless increment/σ i FIG. 2. Normalized eddy scale lifecycle statistics. (a) Model symmetric mean scale (shown for normalized times 0 < t < 1/2) and standard deviation of scale (shown for 1/2 < t < 1) lifecycles for lifetime classes 16 wk L 80 wk, as indicated in the legend; the corresponding anti-symmetric lifecycles are also shown, and are all near zero. (b) Mean model (green; components as in upper left panel) and altimeter-based (black, blue; both symmetric components shown for 0 < t < 1) scale and standard deviation of scale lifecycles for lifetime classes 16 wk L 80 wk, weighted by the number of eddies in the respective lifetime classes. (c) Mean model (green) and altimeter-based (black) dimensional scale (solid circles) and standard deviation of scale (squares) vs. lifetime class. (d) Distributions of model (green) and altimeter-based (black) innovations, from normalized scale lifecycles, with first four (model, altimeter-based) moments (mean, standard deviation, skewness, kurtosis) as indicated. 29

30 0.02 tracks eta rinc21.v0.1.ascii.nc P(L s ) Eddy scale, L s (km) FIG. 3. Distributions of all weekly scale realizations for model (green) and altimeter-based (black) eddies with lifetimes L 16 wk. 30

31 dimensionless amplitude and std dev normalized life cycles Mean Std dev wks wks wks wks dimensionless amplitude and std dev mean new mean mean normalized life cycles std dev new std dev dimensionless time dimensionless time 15 tracks eta rinc21.v0.1.ascii.nc 0.5 amplitude innovation distribution µ i = (-0.00,-0.00) σ i = (0.31,0.31) s i = (-0.01,-0.00) k i = (3.41,3.72) amplitude (cm) 10 5 σ i P(increment) lifetime (wks) dimensionless increment/σ i FIG. 4. Normalized eddy amplitude lifecycle statistics. (upper left panel) Model symmetric mean amplitude (shown for normalized times 0 < t < 1/2) and standard deviation of amplitude (for 1/2 < t < 1) lifecycles for lifetime classes 16 wk L 80 wk, as indicated in the legend; the corresponding anti-symmetric lifecycles are also shown, and are all near zero. (upper right) Mean model (green, new mean and new std dev ; components as in upper left panel) and altimeter-based (black mean and blue std dev ; both symmetric components shown for 0 < t < 1) amplitude and standard deviation of amplitude lifecycles for lifetime classes 16 wk L 80 wk, weighted by the number of eddies in the respective lifetime classes. (lower left) Mean model (green) and altimeter-based (black) dimensional amplitude (solid circles) and standard deviation of amplitude (squares) vs. lifetime class; the corresponding mean initial ( ) and final ( ) values are also shown. (lower right) Distributions of model (green) and altimeter-based (black) innovations, from normalized amplitude lifecycles, with first four (model, altimeter-based) moments (mean µ i, standard deviation σ i, skewness s i, kurtosis k i : model value, observed value) as indicated. 31

32 0.18 tracks eta rinc21.v0.1.ascii.nc P(A) Eddy amplitude, A (cm) FIG. 5. Distributions of all weekly amplitude realizations for model (green) and altimeter-based (black) eddies with lifetimes L 16 wk. 32

33 Northward or poleward (km) Northward or poleward (km) Number of cyclonic eddies = Distance eastward (km) Number of anticyclonic eddies = Distance eastward (km) FIG. 6. Eddy tracks relative to initial location for all model cyclonic (upper panel) and anticyclonic (lower panel) eddies with final and initial longitudes separated by at least 500 km. 33

34 FIG. 7. Power spectrum of (left) model and (right; from Early et al. (2011)) SSH vs. zonal wavenumber k and frequency ω. The frequency axis limits on the left panel have been chosen so that the slopes of the non-dispersive line ω = c R k are similar in the two panels. In this figure, the absolute model spectral level is arbitrary. 34

35 FIG. 8. Three-dimensional visualization of the model wavenumber-frequency power spectrum of SSH, vs. zonal and meridional wavenumbers, k and l, and frequency, ω, for the standard solution described in the text. The base-10 logarithm of the SSH power spectrum is contoured on three orthogonal planes, and an isosurface of constant SSH spectral power (red) is shown, which is localized around the nondispersive surface ω = c R k. In this figure, the absolute model spectral level is arbitrary. This figure may be compared to Fig. 7 of Wortham and Wunsch (2014). 35