1. Introduction

Ocean variability is broadband and, in general, dominated by mesoscale fluctuations on timescales between 20 and 150 days and spatial scales between 50 and 500 km (Wyrtki et al. 1976; Danzler 1977; Richman et al. 1977). Because eddies transport heat and momentum and interact with the mean flow field, it is important to understand eddy dynamics, their transport properties, and their impact on climate.

A detailed understanding of eddy dynamics and related eddy mixing in the World Ocean is missing, partly because of the previous lack of adequate ocean observations. Earlier field experiments during the mid-1970s suggested that eddy energy is generated primarily by instability processes of intense boundary currents (MODE Group 1978; McWilliams et al. 1983; Robinson 1983) and is radiated subsequently into the interior ocean by Rossby waves (Pedlosky 1977; Talley 1983; Hogg 1988). Direct atmospheric forcing by variable wind stress (Müller and Frankignoul 1981) was considered to be another important eddy source term in areas remote from intense currents. In addition, small-scale topography can transfer large-scale (barotropic) energy, enhanced by surface wind forcing or Helmholz-type shear instability, to smaller baroclinic scales through mode–mode coupling (Trèguier and Hua 1988). The geographical variation of eddy amplitude and eddy scales (Mercier and Colin de Verdière 1985; Krauss et al. 1990; Stammer and Böning 1992) presents a particularly formidable challenge to the understanding of ocean eddy dynamics and eddy source terms, and requires a systematic regional investigation of observed eddy characteristics. Comprehensive reviews on oceanic variability based on in situ time series of moored current meter and thermistor records were published by Wunsch (1981) and Schmitz and Luyten (1991).

In the last century, enhanced insight was gained into the generation and distribution of the oceanic mesoscale through the application of modern measurement technologies or the advanced developments in numerical ocean simulations. Stammer and Böning (1996, SB96 henceforth) give a recent review on that subject, in which they refer to satellite altimetry as a vital observational element in gaining further insight into oceanic low-frequency variability through its unique sampling characteristics.

A full discussion of near-surface variability involves an analysis of underlying dynamical principles, and this attempt is intimately connected to the compilation of frequency–wavenumber spectra. From two years of high quality TOPEX/POSEIDON (T/P) data, Wunsch and Stammer (1995, henceforth WS95) constructed the first global frequency–wavenumber spectrum of sea surface height (SSH) variability and associated one-dimensional frequency and wavenumber spectra for SSH and sea surface slope. These spectra are important references against which to measure local variations. For a comprehensive dynamical interpretation, however, a detailed regional study is required.

In this paper, based on three years of high quality altimeter observations now available from the T/P mission, various aspects of regional eddy characteristics are analyzed over the World Ocean. Our main objective is to summarize characteristics common to dynamically similar parts of the ocean, thereby simplifying the otherwise overwhelming amount of detail inherent in the T/P observations. Results will be used as the basis for a dynamical interpretation of the observed eddy field, an effort which naturally leads to a discussion of universal aspects of ocean variability. Regional details will emerge subsequently as deviations from the references given here. Although some of the provided estimates are readily available from previous Geosat data (e.g., Fu 1983; Fu and Zlotnicki 1989; Wunsch 1991; Le Traon et al. 1990; Le Traon 1991; Stammer and Böning 1992), their intrinsic uncertainties made previous dynamical interpretation speculative and controversial (see Le Traon 1993 and Stammer and Böning 1993).

The geographical distribution of eddy characteristics and their relation to mean flow properties can shed important light onto questions regarding eddy generation mechanisms and eddy–mean flow interactions. Here we will demonstrate a correlation of the observed geographical distribution of eddy variability with mean flow horizontal density gradients (which are related to the mean available potential energy) and discuss a relation of eddy length scales inferred from T/P data, with a first-mode Rossby radius of deformation and a Rhines scale, both of dynamical significance. In a second part of this study (Stammer 1997), present results on eddy statistics are interpreted in the context of the theory of a baroclinically unstable flow field as pioneered by Green (1970) and Stone (1972) for atmospheric conditions. T/P eddy statistics will also be used there to provide an estimate of a horizontally varying field of eddy diffusivity, as well as resulting eddy heat and salt transports.

TOPEX/POSEIDON data from the period 1 December 1992 through 27 November 1995 (repeat cycles 8–117) were edited and corrected as discussed by King et al. (1994) and Stammer and Wunsch (1994) with the following two exceptions that significantly improved the accuracy of the data (Shum et al. 1997, Tapley et al. 1997): first, we used the Version 3.0 ocean tide model provided by the University of Texas/Center for Space Research Group (Ma et al. 1994) to remove the ocean tide component from the SSH observations and, second, the standard orbits based on the JGM-2 geoid model (Nerem et al. 1994) were replaced by the improved estimates based on the recent JGM-3 geoid model (Tapley et al. 1996). No data from the French instrument were used because of enhanced noise levels in the Centre National d’Etudes Spatiales (CNES) altimeter (see also Fig. 8); earlier TOPEX cycles were omitted because of lingering questions about initial pointing errors of the T/P instrument. In practice, however, results with TOPEX cycle 2–7 included are indistinguishable from the ones shown here. Fu et al. (1994) provide details on the mission characteristics and its performance. Numerous recent studies demonstrated the high accuracy and precision of the T/P instrument, which are now at the 2–3 cm level (see the T/P special issues of Journal of Geophysical Research from December 1994 and December 1995). The spacecraft provides SSH observations with global coverage every 9.91 days (a nominal “10-day” “exact repeat” cycle) and with a horizontal alongtrack resolution of about 6.2 km.

Only the time variable SSH component is considered during this study, thus eliminating any uncertainties related to the geoid and the mean ocean state. A 3-yr alongtrack (local) mean was subtracted from each individual repeat cycle to produce its time variable part ζ(t) and its horizontal slope δ(t) = ∂ζ/∂l, with l being the alongtrack distance. The data are subsequently split into areas spanning 10° in latitude and longitude on an overlapping 5° grid (Fig. 1) for which one-dimensional ensemble averages of ζ and δ spectra are computed in both the frequency and wavenumber domains. The spectra and their related autocorrelation functions are used subsequently to infer scales of ocean variability.

The paper is organized as follows in section 2 we discuss estimates of eddy kinetic energy derived from T/P data and its sensitivity to the applied filter length scale. Spectra are analyzed in section 3, and covariances and scales of oceanic variability are discussed in section 4. Their relation to the first-mode Rossby deformation radius L_Ro and an estimate of a “Rhines” scale L_Rh of the mean flow field are the basis of a dynamical interpretation given in section 5. A global description of surface variability in terms of universal wavenumber spectra and correlation functions is provided in section 6.

2. Eddy kinetic energy

In mid and high latitudes, T/P results are qualitatively similar to those from earlier studies (e.g., Shum et al. 1990; Le Traon et al. 1990; Beckmann et al. 1994) in that maximum amplitudes of K_E are associated with the paths of energetic current systems. In the interior ocean, the field exhibits zonally banded structures with areas of enhanced eddy activity following major mean fronts. T/P data, however, reveal regional details that previously were hidden in the noise or were removed by filters. In the present estimate, amplitudes of the background variability are of the order of 50 cm² s⁻² in the eastern North Atlantic as compared to 100 cm² s⁻² previously from Geosat, about 40 cm² s⁻² in the South Pacific, and as low as 20 cm² s⁻² in the northern North Pacific. However, exact numbers depend on details of the filtering, as discussed below.

Because Fig. 2 is dominated by the low-latitude bulk of the high eddy kinetic energy, much more detail in spatial structures can be seen from a field of an equivalent sea surface slope variability K_sl = K_Esin²(ϕ), shown in Fig. 3a. This field largely coincides in its spatial structure with those from small-scale SSH variability (potential energy) in which contributions from steric and other large-scale variability on wavelength exceeding 1000 km were removed by bandpass filtering (see Wunsch and Stammer 1995; Stammer 1997). Note that there are clear bands of enhanced surface slope variability present in the Indian Ocean and along the South Pacific intertropical convergence zone, roughly along 25°S. A structure similar to that in the Indian Ocean is likewise present in the global Semtner and Chervin model (Stammer et al. 1996), suggesting a major frontal, yet undocumented, structure at that location.

In Figs. 2 and 3a, regions of enhanced eddy kinetic energy appear to coincide with locations of mean frontal structures. This is most clearly suggested in the fundamental difference between the mean flow pattern and the variability of the North Pacific and North Atlantic. The North Atlantic Current and the associated enhanced eddy energy reaches far into the European basin; equivalent structures in the mean flow field and eddy variability are completely missing in the North Pacific because of its different thermohaline circulation.

To emphasize an intriguing relation between the geographical distribution of K_E and horizontal density gradients embedded in the mean flow field, we show in Fig. 3b the kinetic energy K_M = (ū² + ῡ²)/2 as it results from the mean thermal wind in 50-m depth relative to 1000 m in the Levitus climatology;

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Results are plotted in Fig. 3b after scaling by sin²(ϕ), similar to the field in Fig. 3a. Not surprisingly, amplitudes of K_M are about an order of magnitude smaller than K_E (see, e.g., Wyrtki et al. 1976). But otherwise most of the areas characterized by enhanced eddy variability in T/P observations are characterized by enhanced mean horizontal density gradients and related mean kinetic energy in the upper thermocline. This holds also in the Southern Hemisphere along the aforementioned eddy variability ridges in the Indian Ocean and western Pacific along 25°S.

There are a few noteworthy discrepancies from this general tendency, especially in the vicinity of the East Australia Current, Brazil–Malvinas confluence, and the Agulhas Retroflection. There only weak mean baroclinic currents can be found, suggesting primarily barotropic fluctuations. Along the Antarctic Circumpolar Current (ACC), differences in K_E and K_M are primarily located close to topographic structures such as the Bollons Seamounts southeast of New Zealand (about 50°S, 183°E), or the Southwest Indian Ridge south of Africa (50°S, 30°E).

The close relation between the observed eddy variability and the mean horizontal density gradients is quantified in Fig. 4, showing a comparison of related amplitudes of the mean and the eddy velocity fields, V_M = K_M and V_E = K_E, respectively. Both fields were previously averaged over 5° by 5° areas and result in a significant correlation of r = 0.60. A similar relation between the path of mean frontal structures and locations of enhanced eddy energy was previously recognized by Wyrtki et al. (1976) from ship drift data. Those authors discuss also the apparent decorrelation of high eddy activity with patterns of high atmospheric variability. These two findings are important ingredients for a dynamical interpretation of observed eddy activity and its source mechanisms and strongly suggest internal instability processes (primarily baroclinic instability) to be a primary eddy source term as opposed to direct atmospheric forcing (see also the discussion in SB96). As will be discussed below, the latter term becomes important only where the general level of variability is low, for example, in the eastern North Pacific.

For many purposes a zonally averaged description of ocean variability appears useful. In Fig. 5, zonal averages between 0° and 360°E of K_E, K_sl, and SSH variance are provided as a function of latitude. In all three curves, high variability near 40°N and 40°S is associated with strong currents at those latitudes. But otherwise, the fields show interesting differences in their general latitudinal distribution. In terms of K_E, values decrease gradually from maximum amplitudes near the equator to minimum amplitudes in high latitudes. The opposite is true for K_sl, which is minimum in low latitudes and increases toward high latitudes. The SSH variability, on the other hand, is highest in mid latitudes [partly related to maximum contributions from seasonal steric effects in that latitude range (see Stammer 1997)], decreasing toward the poles and the equator. Note that both K_sl and SSH variance show a clear asymmetry with respect to the equator in that minimum amplitudes are located south of the equator, with considerably higher variability present at similar latitudes of the Northern Hemisphere associated with the North Equatorial Current (NEC) and the North Equatorial Countercurrent (NECC).

3. Spectral analysis

Maps of ocean variability call one’s attention to the pronounced inhomogeneous nature of the mesoscale and highlight regions of intense eddy variability and eddy mixing. But because oceanic variability is broadband, a detailed spectral analysis is required to fully understand its characteristics. Ultimately, regional two-dimensional frequency–wavenumber spectra are sought, equivalent to the global frequency–wavenumber spectrum in WS95. Because one-dimensional spectra are more common in the literature (and therefore easier to compare with previously published results), we will start here with a discussion of regional and globally averaged one-dimensional spectra that are obtained separately in the frequency and wavenumber domains. A similar discussion of ocean variability—but in its full two-dimensional form—is in preparation.

To compute frequency spectra of SSH variability ζ(t) and cross-track geostrophic velocity υ_s(t), local (alongtrack) time series from both ascending and descending arcs were Fourier analyzed to produce corresponding functions in frequency domain, ζ̂(σ) and υ̂_s(σ), and related periodograms; for example, |ζ̂(σ)|². Here σ has dimensions of cycles per second. Time series were tapered previously by using a cosine squared function over 10% of the number of observations. Gaps smaller than 20 days in time and 30 km in space were closed by interpolation, and data with larger gaps were rejected. To enhance statistical independency of results, periodograms were computed at only every second alongtrack position (every 12 km). Because in many cases the spectra of surface slope δ seems to be more appropriate than those of velocity, we show primarily slope spectra, which differ from the velocity spectra by a latitudinal dependent coefficient (but see Fig. 13).

Estimates of power spectral density of ζ and δ, Γ_ζ(σ) and Γ_δ(σ), were computed subsequently by ensemble averaging all individual periodograms present in individual 10° by 10° geographical areas. The number N of time series present in each region is typically about 500, which would result in a formal degree of freedom around N_dof = 1000, if all periodograms were truly independent.

Wavenumber spectra, Γ_ζ(k), were computed along those fractions of ascending and descending arcs that completely span individual 10° regions meridionally, with k being the alongtrack wavenumber. The length of each individual arc segment was required to be 180 × 6.2 km = 1110 km. The number N of valid data series, which is now the number of arcs times repetitions, is typically of the order of 500 in the interior oceans; close to boundaries, however, N becomes as low as 100.

As with computations in the frequency domain, wavenumber spectra can be computed for SSH and surface slope. Because of the geostrophic relation, slope and velocity wavenumber spectra are readily obtained from the height spectra as

_δkπk²_ζk(4)

and

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respectively, with g being the gravity constant and f, as before, the Coriolis parameter. Results are similar in detail, and most of what follows is based on the latter equation, with the advantage that filtered slope spectra are available once filtered versions of Γ_ζ(k) are provided (see below).

An alongtrack spectrum can be converted into an isotropic scalar wavenumber spectrum (e.g., Fu 1983; see also Le Traon et al. 1990). However, because alongtrack spectra are more common in the literature, we will continue to work with that form. To the extent that spectral relations in Fu (1983) did not change significantly upon the transformation, conclusions drawn here appear insensitive to the specific representation.

c. Wavenumber spectra

Examples of regional ζ and υ_s wavenumber spectra, Γ_ζ(k) and Γ_u(k), are shown in Fig. 13 from several 10° by 10° areas in the latitude band 30° to 40° N across the North Atlantic. As can be expected from a nearly isotropic eddy field (see Krauss and Böning 1987; Schäfer and Krauss 1995), spectra computed individually from ascending and descending arcs do not differ significantly from each other and are therefore not shown here separately.

Generally, the regional wavenumber spectra are consistent with the global estimates, but depict the pronounced geographical variation of eddy variability in the ocean. As a result, the energy in Γ_ζ(k) and Γ_u(k) on wavelengths longer than about 100 km substantially increases from the eastern and central low-energy subtropical gyre toward the eddy-active boundary current regime of the North Atlantic. But apart from this, the general shape of individual spectra appear strikingly similar across the entire basin. All Γ_ζ(k) curves exhibit a long-wavelength plateau and a spectral break at about 400-km wavelength, followed by a drop in energy close to a k⁻⁴ relation. Associated velocity spectra have a maximum in energy at the above cutoff wavelength, with roughly k^+1.5 and k⁻² relations toward longer and smaller wavelength, respectively.

Most remarkable in the figure is that both types of spectra lead into a unique high-wavenumber tail, which in terms of Γ_ζ(k) can be described by a k^−2/3 behavior. As indicated in Figs. 8c,d this finding holds over the entire World Ocean, and there are strong reasons to believe that this short wavelength tail of the spectrum is dominated by noise rather than ocean signal. Apart from the suspicious universal character over a highly inhomogeneous ocean, the breakdown of a geostrophic assumption at those very short spatial scales is another reason. Miscalibration is also an important source of uncertainty on wavelength smaller than 60 km (Rodriquez and Martin 1994). The “noise” in T/P data is also made up from a variety of complex and unknown contributions from environmental and geophysical corrections. At the present level of measurement accuracy, it also includes physical ocean processes that are not the subject of this study, such as the expression of internal wave and internal tide signals at the sea surface (Radok et al. 1967; Wunsch and Gill 1976). In terms of surface velocity and sea surface slope, the noise tail is responsible for their unphysical “blue” wavenumber energy distribution, which, on global average, leads to the peculiar bimodal energy in the slope spectrum (Fig. 8d).

1) Optimal filter design

Because our present attempt is to relate observed energy distributions to underlying physical processes, it is essential to obtain reliable estimates of spectral slopes. To reduce noise contamination, we apply a filter to the wavenumber spectra prior to any further analysis. Press et al. (1992) give a useful introduction into the concept of an optimal filter.

Our goal is to analyze a spectrum Γ̆_ζ(k) of the true signal ζ(t). Our task is then to find a filter Φ(k), which when applied to the corrupted spectrum Γ_ζ(k), produces a best estimate Γ̃_ζ(k) that is as close as possible to the true spectrum:

Γ̃_ζ(k)_ζkk(6)

The underlying assumption is that the signal ζ and the noise n are uncorrelated. An optimal least-squares filter solution is given by (Press et al. 1992)

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To construct the filter, some prior knowledge of the noise spectrum N(k) is required, but adequate information on the noise statistics is not now available from the T/P Project. Under such conditions, a noise spectrum is usually determined empirically from the data itself. This was accomplished here by fitting a smooth curve to the high-wavenumber part of Γ_ζ obtained from the area of global minimum variance, centered at 15°S, 0°E in the South Atlantic (dashed line in Fig. 14). As before, we assume that all short-scale energy is purely noise related in this specific minimum variance location—a reasonable assumption under present conditions. Future refined applications clearly call for detailed information on the T/P error spectrum.

Using the dashed curve in Fig. 14a as a noise estimate of Ñ(σ) in (7), (6) then leads to the spectral estimate Γ̃_ζ(k) (bold line), which is unaltered at long wavelengths, but shows an increased decay close to k⁻² toward short scales as compared to k⁻¹ previously. In this specific example, the filter process reduces the total variance by about 50% from (2.8 cm)² to (2.0 cm)², leaving (1.9 cm)² as the noise estimate, a number consistent with the noise estimates in section 2 and with Fu et al. (1994). On the global average, about 7% of the variance is removed from the raw data by the filter process; but in high energy areas, the noise component amounts only to 1% of the observed variance. Note that the associated surface slope spectrum, Γ_δ(k), is changed by the filter process from completely “blue” into “white,” with all short-scale energy reduced by more than a decade (Fig. 14b).

2) Filtered wavenumber spectra

To produce filtered spectra on the complete global grid, Γ̃_ζ(k) follows from (6) with (7) evaluated in each of the 10° areas and with the above Ñ from the minimum variance area used as a general noise estimate. The resulting global average filtered spectrum Γ̃_ζ(k) (bold line in Fig. 8c) shows a steepened spectral decay of about k^−3.3 between the 350-km cutoff and 20-km wavelength. In the mean slope spectrum Γ̃_δ(k), the “blue” short-wavelength part has been completely removed; instead the energy follows now a k⁻¹ relation at wavelengths between 400 and 150 km, and a k⁻² relation at shorter wavelengths.

To facilitate a comparison of wavenumber spectra from various dynamically distinct regimes of the World Ocean, filtered spectra Γ̃ were subsequently normalized by their local variance γ² to produce spectra in normalized form

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They are shown in Fig. 15 from all 10° areas on the 5° grid along various latitude bands spanning the complete sphere in both hemispheres, thus sampling all ocean basins. The figure reveals the unexpected result that, with the geographical variations in eddy energy removed from the spectra, all resulting curves of Γ̂_ζ(k) are to first order similar in shape. Some regional deviations from this general behavior are evident but are mostly confined to the eastern North and South Pacific (Figs. 15a,b), where a relative high long-wavelength energy component is apparent that can be associated with barotropic (wind driven) fluctuations (see Chao and Fu 1995 and Fu and Davidson 1995).

In contrast to extratropical conditions, spectra from the tropical oceans are “red” over the full scale range and show a k^−2.5 relation at all wavelength exceeding 100 km. However, two limiting factors of the present analysis have to be considered. First, it is likely that residual noise effects are still present in low-energy regions. More important, though, is that the geographical extent over which spectra are evaluated is becoming too small to resolve the long scales in low latitudes. Going from a 10° to 20° box size indeed leads to a spectral plateau at the longest wavelength, even in the Tropics (not shown, but see Stammer and Böning 1992). Some tropical locations exhibit enhanced energy at wavelengths around 110 km (Atlantic and Indian Ocean) and 150 km (western Pacific). The origin of those features—which are apparent north and south of the equator and which has been observed previously in Geosat data (Stammer and Böning 1992)—is not obvious and needs further attention. Because the Tropics are largely governed by different physics and because our main focus is on midlatitudes, we will not go into further details of low-latitude spectra. Instead we will limit the following discussion on dynamics to regions poleward of 15° latitude.

To emphasize the meridional variation in spectral shape, Fig. 16a shows mean 〈Γ̂_ζ〉 spectra that result from a zonal averaging around each latitude band on the 5° grid poleward of 20° latitude in both hemispheres. The figure illustrates the striking similarity among extratropical wavenumber spectra. All 〈Γ̂_ζ〉 curves show a close to universal k^−1/2 long-wavelength plateau and a steep decay of roughly k⁻⁴. However, the cutoff wavenumber k₀ exhibits a latitudinal dependency by decreasing from high latitudes toward the equator. This meridional shift in the cutoff wavenumber is even more obvious from the slope spectra (Fig. 16b), which show a maximum energy at the wavenumber k₀ followed by a drop in energy similar to a k⁻³ relation.

Note that this general result would still hold without the application of the filter. However, it would not be as clean as it appears from Fig. 16, partly because the observed variance in low-energy regions is primarily effected by the short-scale noise and normalized spectra Γ̂ would then appear blurred at the long-wavelength part, which is of interest here.

In boundary current regions, baroclinic instability is believed to be the primary source of eddy kinetic energy. In an ideal geostrophically turbulent flow field (Kraichnan 1967; Charney 1971; Rhines 1979), the resulting kinetic energy should be close to k^−5/3 and k⁻³ relations at wavenumbers smaller and larger than the wavenumber of maximum instability, and should exhibit a maximum in eddy kinetic energy around a scale, usually referred to as the “β arrest” or “Rhines” scale, L_Rh = U/β with U being a typical velocity scale and β = ∂f/∂y, at which the “red” energy cascade comes to a halt as a result of Rossby wave dispersion. In terms of SSH spectra, this rule translates into k^−11/3 and k⁻⁵ power laws in Γ_ζ(k); but it is not clear to what extent isotropic and homogeneous geostrophic turbulence is fully developed in the oceans in the sense that regions are bounded and other dynamics might come into play (e.g., mode coupling through bathymetry).

If the turbulent energy cascade were localized in space, as is advocated for the oceans, then, as the energy evolves into a wavelike form, it should simply disperse by radiating Rossby waves. The nonlinear interaction of energy in the wavenumber domain then would largely cease, resulting in spectra of vastly different shape. Instead, the findings from this analysis can be summarized in a strikingly universal shape of wavenumber spectra over the entire extratropical World Ocean, suggesting that interior ocean dynamics are not significantly different from those near boundary currents. Note, that present velocity spectra are similar to previous finding from both the atmosphere and numerical ocean models (e.g., Julian et al. 1970; Frankignoul and Müller 1979; McWilliams and Chow 1981; Böning and Budich 1992).

4. Covariance functions and variability scales

There are various purposes, including the estimation of integral scales of variability or surface mapping, for which the covariance functions C, of ζ and δ are useful. Because the covariance function of a process is intimately related to its spectrum by the Fourier transform

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the descriptions are mathematically equivalent, but complementary. In this section we will briefly describe covariance functions as they emerge from ζ and δ in space and time, after averaging over the same spatial extent as the corresponding frequency and wavenumber spectra discussed in the previous section. Similar to the close agreement of individual spectra, curves of individual estimates of C from 10° regions deviate only little from results shown.

a. Temporal covariances and timescales

The temporal correlation functions C_ζ(τ) that correspond to the four averaged frequency spectra Γ_ζ(σ) given in Fig. 10a are displayed in Fig. 17a. Like those spectra, all autocorrelation functions are similar in their general shape. Variations in the first zero-crossing of individual curves are minor (between 70 and 80 days), with the smallest values appearing in the high-energy regime. At small lags, the smallest correlations are associated with the area of minimum variability, that is, smallest signal-to-noise ratio (area 4). In terms of eddy velocity or surface slopes (Fig. 17b), their correlation drops rapidly below 0.5 everywhere within only a few repeat cycles. Both height and slope autocorrelation functions suggest an exponential decay with e-folding timescales of about 25–30 days for SSH and roughly 10–15 days for surface velocity measurements.

A measure of characteristic timescales of ocean variability is usually obtained from an integral over the small-lag positive fraction of the autocovariance function

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with T₀ being the first zero-crossing (e.g., Richman et al. 1977). A corresponding timescale T of SSH variability was computed from the autocovariance functions C_ζ in all individual 10° by 10° areas from the T/P data (Fig. 18). In the presence of significant negative lobes of the autocovariance functions it is more useful to compute the integral scale in its quadratic form

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where T_max is the maximum time lag, usually taken to be one-half the length of the time series. Because this estimate appears more sensitive to small-lag noise effects, results are not shown, which generally agree with those from T.

In the Atlantic, the scales are qualitatively similar to previous Geosat results (Stammer 1992), but values are larger by about 30% because fluctuations at long periods (annual and interannual variability) were significantly removed from the Geosat analysis by the data processing scheme (orbit error correction). Overall, the emerging spatial pattern shows the largest timescales in the less energetic centers of the ocean, while energetic current systems are associated with shorter timescales. Interestingly, this finding does not apply to the Kuroshio.

Because of the presence of the strong seasonal and interannual (ENSO related) fluctuations in the T/P data, values in Fig. 18a are biased toward long scales. Removing all energy at annual and longer periods from the data prior to the computation of the integral timescale leads therefore to a noteworthy change in the amplitudes and geographical distribution of the resulting eddy timescale (Fig. 18b). Although the central subtropical gyres still show the longest scales, both boundary currents and the tropical ocean are now characterized by short-period variability. As a general result, eddy timescales vary by a factor of 5 between maxima in the subtropical gyres and high-latitude minima.

5. Relation of eddy scales to L_Ro and L_Rh

Previous results suggest that physical processes leading to the observed mesoscale variability are surprisingly homogeneous geographically. Although direct wind forcing could be a significant eddy source term, its contribution to observed mesoscale variability seems to be limited to those locations where the general variability level is low and fluctuations in atmospheric forcing are high, notably the eastern North and South Pacific (Fu and Davidson 1995). Instead, results provided here strongly favor baroclinic instability as the dominant eddy source in the extratropical oceans. Supporting evidence for this hypothesis is obtained from the close relation between mean horizontal density gradients—equivalent to mean available potential energy—and from the spectral characteristics that are consistent with the theory of geostrophic turbulence. An analysis of ocean observations and numerical models in the North Atlantic led Beckmann et al. (1994) and SB96 to come to the same conclusion on a regional basis. Here we are able to show its truly global character.

Under uniform dynamical conditions, we should be able to find a close relation between eddy characteristics inferred from T/P data and mean flow properties. Such relations are advocated by theories describing amplitudes and transport properties of eddies in the presence of a baroclinically unstable flow field that were pioneered by Green (1970) and Stone (1972). Figures 3 and 4 indeed lend strong support to those theories in which baroclinic instability is the dynamical link between the ambient available potential energy of the mean flow field and observed eddy amplitudes.

A discussion of present results on eddy characteristics in the context of the theory of a baroclinically unstable flow field and an estimate of a related field of eddy diffusivity inferred from T/P data is provided in Stammer (1997). Here we restrict ourselves to a discussion of the relation of T/P eddy scales to those following from the first-mode baroclinic Rossby radius of deformation L_Ro, and the “Rhines” scale, L_Rh. The Rossby radius of deformation is of fundamental importance in atmosphere–ocean dynamics, both for the geostrophic equilibrium solution as well as transient ones. In the theory of baroclinic instability, it is closely related to the length scale of maximum eddy growth rates. In a strictly linear system both scales would be similar; however, in reality scales of the most unstable wave can be significantly larger (Simmons 1974).

A correlation of altimetric eddy length scales inferred from Geosat with the first-mode baroclinic Rossby radius was discussed previously by Stammer and Böning (1992) from a study in the Atlantic Ocean. A similar relation had been inferred before from local ensembles of eddies in selected (geographically limited) areas—for example, at the MODE and TOURBILLON sites (see Mercier and Colin de Verdiere 1985)—and from analysis of satellite sea surface temperature images and drifter data (Krauss et al. 1990).

For the present global comparison, we estimated L_Ro on a 1° grid over the global ocean from a local solution of the eigenvalue problem

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subject to the boundary condition∂F/∂z = 0 at z = 0, −H. Here N denotes the annual mean buoyancy frequency computed from the Levitus et al. (1994) climatology. The resulting field L_Ro = λ^−1/2₁ (Fig. 23c) is consistent with previous publications by Emery et al. (1984) and Houry et al. (1987), but shows substantially more finestructure in the presence of a spatially variable topography H(λ, ϕ).

We also find a linear relation between L_Ro and the scale of the first zero-crossing of the slope covariance C_δ. Poleward of 30° latitude it results in L_δ0 = 0.8L_Ro + 43 km. However, relevant for high latitudes, scales from C_δ are biased toward larger values because the applied extra filter step by which all energy at wavelength smaller than 60 km is completely removed from Γ̃_δ prior to the computation of C_δ (see above). On the other hand, there is still a considerable noise effect present in C_δ in low latitudes that leads to a decrease in L_δ0 equatorward of about 30° latitude. Because a factor of ∼2 in scales between those from ζ and δ follows from Fig. 20, the results based on SSH should be closely related to those of sea surface slope (up to a scale factor).

A universal relation like (13), obtained from surface pressure data, suggests that first-mode processes dominate observed SSH fluctuations, globally. For a generalization of the result, a partitioning of SSH fluctuations into barotropic and baroclinic components would be necessary, which is not now generally available. An analysis of data from moored current meters, however, led Wunsch (1997) to the conclusion that the near-surface flow field in the interior ocean is primarily baroclinic and that barotropic contributions are becoming important only in the recirculation regime of boundary currents (high-latitude regions were not adequately sampled).

The high-resolution Community Modeling Effort of the North Atlantic, which also yielded a linear relation between eddy scales and L_Ro similar to (13), allowed more insight into the the effect of the barotropic energy on this relation (Beckmann et al. 1994). It was shown there that SSH spatial scales are increased as a result of a barotropic SSH component and that the purely baroclinic motion reflected in the sea surface variability indeed shows a direct proportionality with the deformation radius. Consequently, the offset between L₀ and L_Ro in Fig. 24 must be considered an artifact because of a barotropic (long wavelength) component in T/P observations.

A truly linear relation between L_ζ and L_Ro can be expected only from SSH signal associated with first-mode processes. It appears therefore straightforward to derive scale estimates directly from the cutoff wavenumber k₀ present in the wavenumber spectra Γ_ζ. In Fig. 25a, the associated spatial scale L_c = (2πk_o)⁻¹ is plotted (bold dots) as it results from the zonal-average wavenumber spectra given in Fig. 16. Indeed L_c shows a close relation to the zonal average L_Ro (open circles), to which it is similar in low latitudes. However, L_c is a factor of 3 larger in high latitudes (Fig. 25b).

For an interpretation of this peculiar result, it should be recalled that L_c is associated with the wavelength of maximum slope energy. Therefore, L_c either indicates the scale of instability, or, in the presence of geostrophic turbulence, it should be associated with L_Rh = U/β at which (on a β plane) one expects the energy cascade from smaller to larger scales to be arrested (Rhines 1975). The occurrence of the related spectral peak at this transfer arrest wavenumber (2πL_Rh)⁻¹ has indeed been seen in a number of turbulence calculations (e.g., Haidvogel and Held 1980; McWilliams and Chow 1981; Held and Larichev 1996).

An estimate of a zonally averaged L_Rh scale is included in Fig. 25a, which follows from the assumption that U = K_E, with K_E taken from Fig. 3a. Most noticeably, L_Rh is about 100 km poleward of 20° latitude and closely resembles both L_c and L_Ro in low latitudes. Moreover, the ratio L_Rh/L_c is nearly identical to L_c/L_Ro, while L_Rh is one order of magnitude larger than L_Ro in high latitudes.

The finding that L_c is much closer to L_Ro than to L_Rh over most of mid and high latitudes does suggest that a “red” energy cascade is not strongly developed there. Instead, the energy released by baroclinic instability from the mean-flow field appears to be kept near the input wavelength. This interpretation is consistent with the wavenumber spectra in Fig. 16b that reveal a k^−5/3 regime of a “red” energy cascade to be almost nonexistent in the high-latitude range. However, both Figs. 16 and 25 suggest the presence of a broader k^−5/3 regime toward low latitudes. McWilliams (1989) lists various physical mechanisms that could cause the inverse cascade to fail including too small a domain size, excessive Rossby wave influence, or a highly dissipative regime. [Because L_Rh ≈ L_Ro in the Tropics, higher-mode (smaller scale) processes are likely to dominate instability processes there (see also Wunsch 1997).]

6. Global spectral relations

The similarity of wavenumber spectra from the global extratropical oceans discussed above suggests the existance of a universal description of wavenumber spectra of sea surface height and surface geostrophic velocity. In the following we will provide an estimate of such a relation. To do so we transform the alongtrack wavenumber k to a normalized wavenumber k̃ = k/k₀, where k₀ is the wavenumber of maximum energy in Γ̂_δ. This was done in Fig. 26a, showing 〈Γ̂_ζ〉 and 〈Γ̂_δ〉 curves as a function of k̃. In addition, individual curves were normalized by their energy at k₀ to accomplish an alignment also in the vertical. The results indicate a first-order description of wavenumber spectra in the general form

View Expanded

where γ²_ζ and γ²_δ denote the local variances of both fields, k̃ is a wavenumber, normalized by the local eddy scale k₀ = (2πL_c)⁻¹, and Γ₀ is a universal spectrum. Figure 4, in principle, allows one to relate observed eddy energy γ²_δ to the available potential energy embedded in the mean-flow field. From Fig. 25 a least squares solution for k₀ as a function of L_Ro and latitude ϕ reads

k₀L_RoϕπϕL_Ro⁻¹(16)

From Fig. 26a an approximate analytic fit of Γ₀ follows as

View Expanded

Another approximation to the spectrum for the wavenumber range 0.18 ≤ k̃ ≤ 4.54, which is continuous in its first derivatives, is the Padé form (compare also WS95)

View Expanded

with coefficients a₀ = 99.3, a₁ = 5.2 × 10², a₂ = 1.5 × 10², and b₁ = 9.8, b₂ = 13.6, b₃ = 7.6, and b₄ = 1.6 (specific coefficients are sensitive both to the order chosen and the way in which the fitting was weighted, but the general shape appears stable). It is noteworthy that, while the steep decay of k^−4.6 is close to the theoretical k⁻⁵ relation, the intermediate regime is less steep than predicted by a k^−5/3 power law for velocity.

Similar to the spectrum, a universal analytic description of the correlation functions for ζ and δ can be specified in terms of a normalized spatial lag r = l/L₀, where L₀, as before, is the first zero-crossing of C_ζ; its relation to L_Ro follows from (13). An approximation of C_ζ(r) and C_δ(r) by higher-order polynomials leads to

View Expanded

which are shown in Fig. 26b together with individual curves of 〈C_ζ〉(r) and 〈C_δ〉(r). The SSH covariance is related to those from the geostrophic alongstream and cross-stream velocity components as

View Expanded

respectively (Bretherton et al. 1976; see also Freeland and Gould 1976). Assuming that S is describing the cross-track covariance, the corresponding relation holds only approximately here. As before, deviations are most likely due to the extra filter step involved in the computation of C_δ.

7. Concluding remarks

A major outcome from this study is that over most of the World Ocean, variability of sea surface height and surface velocity show strikingly universal characteristics. The largest variations in frequency and wavenumber spectra appear to be related to the geographically varying amplitude of eddy energy. This outcome appears to hold for both sea surface elevation ζ and surface geostrophic velocity or sea surface slope δ.

Frequency spectra can be described by three basic types representing (i) the tropical interior oceans, (ii) the bulk of the extratropical basins, and (iii) the energetic boundary currents. Apart from the tropical regime, spectra change to first order only in their local variance, and their shapes are always surprisingly close to the σ^−3/4 relation found in numerical simulations of geostrophic turbulence (McWilliams and Chow 1981).

Extratropical characteristics of wavenumber spectra were found to basically depend only on a poleward shift of its cutoff wavenumber, with spectral shapes otherwise being close to uniform. All sea surface height spectra show a plateau at long wavelengths and a pronounced cutoff followed by a steep spectral decay close to k⁻⁵ toward smaller wavelengths. In the low-latitude band, the energy distribution is “red” over a larger wavenumber range but leads also into a long-wavelength plateau when computed from lags spanning 20° latitudinally.

In terms of surface slope spectra, maximum energy is found at a wavenumber that corresponds to the cutoff wavenumber of SSH spectra. Energy decays toward smaller wavelengths in a manner consistent with the theory of baroclinic instability and geostrophic turbulence. This result is found basically over the entire extratropical ocean, where the variation of the eddy scale with latitude is significantly correlated with that of the first-mode Rossby deformation radius.

All those individual findings fit into the general hypothesis that baroclinic instability is the major eddy generation mechanism, not only near western boundary currents, but on a broad basis. Other eddy generation mechanisms such as direct atmospheric forcing are of some significance, but spatially limited to areas of low-level eddy energy with high atmospheric forcing. A barotropic, wind-induced, component in oceanic fluctuations should be enhanced in high latitude because of a decreasing vertical stratification and the associated increase in vertical penetration scale there (Philander 1978). This is consistent with findings from Fu and Davidson (1995), who report a significant correlation between changes in sea level and the wind stress curl only from the eastern North and South Pacific, and to some extent from the North Atlantic. It is those locations where we find the evidence of enhanced barotropic energy from this study (see Fig. 21b and Fig. 15).

There are various reasons to classify the present results as preliminary, including the existing uncertainties about the high wavenumbers in the data. But as time goes on, the T/P dataset will extend and improve in quality. In particular, more information will become available on T/P error characteristics. Another route to gain further confidence is to test numerical high-resolution ocean circulation models with respect to statistical relations and dynamical principles specified here from T/P data. Results will shed light on the validity of universal relations and allow more complete insight into dynamics than can be inferred from data alone.

It can be anticipated that the existence of universal eddy characteristics is of value for combining and understanding observations from diverse geographical locations. For that purpose, present results provided here only for near-surface conditions need to be extended into a full three-dimensional form. Ultimately analytic solutions are needed that combine eddy characteristics observed by altimetry and in situ data into one single theoretical framework.