## Abstract

Following recent studies based on altimetric data, this paper analyses the spectral characteristics of the sea surface height (SSH) using a new realistic simulation of the North Pacific Ocean with high resolution ( in the horizontal and 100 vertical levels). This simulation resolves smaller scales (down to ≈10 km) than altimetric data (limited to 70 km because of the noise level). In high eddy kinetic energy (EKE) regions (as in the western part), SSH spectral slope almost follows a *k*^{−4} (with *k* the wavenumber) or slightly steeper law in agreement with altimeter studies. The new result is that, unlike altimeter studies, such a *k*^{−4} slope is also observed in low EKE regions (as in the eastern part). In these regions, this slope mostly concerns scales not well resolved by altimetric data. Such *k*^{−4} SSH spectral slopes are weaker from what is expected from quasigeostrophic turbulence theory but closer to surface quasigeostrophic (SQG) turbulence theory. The consequence is that the small scales concerned by these spectral slopes, in particular in low EKE regions, may significantly affect the larger ones because of the inverse kinetic energy cascade. These results need to be confirmed using a longer numerical integration. They also need to be corroborated by high-resolution observations.

## 1. Introduction

Mesoscale geostrophic eddies [with scales of km and for which Coriolis force balances pressure gradient] represent the largest kinetic energy (KE) reservoir in the oceans (Ferrari and Wunsch 2009), but the mechanisms that distribute and control the eddy kinetic energy (EKE) are still an issue to address (Ferrari and Wunsch 2010). Most of the theories of EKE redistribution, such as quasigeostrophic (QG) and surface quasigeostrophic (SQG) turbulence theories, operate in wavenumber space, which requires global measurements with a broad spatial resolution. In that respect, satellite altimetry that measures sea surface height (SSH) has become the central global dataset for describing and understanding these mechanisms: the EKE wavenumber spectrum is retrieved from the SSH spectrum multiplied by *k*^{2}, where *k* is the wavenumber (defined in this study as , where *λ* is the wavelength). In this context, the new study by Xu and Fu (2011) is the first one that evaluates the SSH wavenumber spectrum on a global scale.

Altimetric data have so far provided spectral statistics at best for scales larger than 70 km (because of the signal to noise ratio), with smaller scales being assumed to be driven by the same dynamics as larger ones (Le Traon et al. 1990; Stammer 1997). Using the most recent altimetric datasets Le Traon et al. (2008) and Xu and Fu (2011) found in high EKE regions, like the Kuroshio, SSH spectral slopes close to or slightly steeper than *k*^{−4} for scales between 70 and 250 km. This is shallower than the *k*^{−5} law expected from QG turbulence (Hua and Haidvogel 1986; Smith and Vallis 2001) but closer to (although slightly steeper than) the law related to SQG turbulence (Held et al. 1995). Such finding suggests an inverse EKE cascade (Capet et al. 2008a), which agrees with Scott and Wang’s (2005) and Qiu et al.’s (2008) results (using similar altimetric datasets). On the other hand, Le Traon et al. (1990) and Xu and Fu (2011) found in low EKE regions a shallower SSH spectral slope, between *k*^{−2} and *k*^{−3}, in the scale range between 70 and 250 km. This result is quite puzzling in terms of EKE spectrum: if such slope persists for smaller scales, this would suggest very high EKE at small scales.

The aim of this paper is to explore the SSH wavenumber spectrum at wavelengths shorter than 100 km. For this purpose, the first results of a new realistic simulation of the North Pacific Ocean (NP) performed with high resolution ( in the horizontal and 100 vertical levels) are analyzed. Such high resolution is the best one used so far for the NP but is the minimum required to fully represent the mechanisms that maintain and redistribute EKE between scales smaller than 100 km and larger scales [as noted by Capet et al. (2008b) and Klein et al. (2008)]. Indeed, the classical resolution used in realistic ocean simulations (Boning and Budich 1992; Brachet et al. 2004) cannot capture the dynamics of scales smaller than 100 km (as demonstrated in Capet et al. 2008b).

The focus of the present study is on the SSH spectral characteristics and how they compare in high and low EKE regions with the existing datasets. A short description of the new NP simulation is presented in section 2, and main SSH results are detailed in sections 3 and 4. Conclusions are offered in section 5.

## 2. Numerical simulation of the North Pacific

A hindcast simulation of the North Pacific Ocean (20°S–66°N, 100°E–70°W) with sea ice has been conducted with a high resolution ( in the horizontal and 100 vertical levels) using the Ocean General Circulation Model (OGCM) for the Earth Simulator (OFES; Masumoto et al. 2004; Komori et al. 2005; Masumoto 2010). For the horizontal mixing of momentum and tracers, a biharmonic operator is applied just to reduce numerical noise, and the vertical mixing scheme developed by Noh and Kim (1999) is used. This realistic simulation has used as an initial state the output of a realistic simulation already in equilibrium: this last simulation was spun up for 30 yr using climatological forcing and then integrated from 1979 to 2003. Analysis of the last 10 yr of the simulation has revealed that the interannual and seasonal variations of the kinetic energy (not shown) do not change the spectral shape of the SSH around mesoscales (300 km) (it changes only the EKE by 20%). We have used, as the initial state for the new simulation, the output of the simulation that corresponds to 1 January 2000, when the kinetic energy is close to the mean value (over 10 yr) in the eastern and western Pacific. The simulation is forced by surface wind stress and heat flux estimated using 6-hourly Japanese 25-year Reanalysis (Onogi et al. 2007) with a 1° horizontal resolution.

So far the new simulation has been integrated on the Earth Simulator for 2 yr (longer integration is in progress). During the course of the integration, larger scales (>300 km) are almost unaffected. Only smaller scales become more energetic. This period is of course not long enough to study the interannual and seasonal variability of the EKE (as done in Stammer and Wunsch 1999; Brachet et al. 2004; Qiu et al. 2008). Use of such spinup strategy in previous studies (although in idealized domains), however, indicates that an equilibrium, in terms of spectral slope, is attained in the high-resolution simulation after a one-year integration (Klein et al. 2008). In the present realistic simulation, the SSH spectral characteristics, for scales smaller than 300 km, have been found to not change during the last 7 months of integration and in particular between summer and wintertime.

Figure 1a shows the EKE deduced from SSH using the geostrophy averaged over 7 months. It displays low EKE east of 170°W, between 20° and 50°N, and east of 160°E, between 40° and 50°N. Higher EKE is found in other regions. Figure 1b shows a snapshot of the relative vorticity field normalized by the Coriolis frequency (equivalent to a Rossby number) in the western part: this field is intensified in the Kuroshio around 35°N with values up to ≈0.5 but is much less intensified in low EKE regions north of 40°N. These characteristics are comparable with those reported by Ducet et al. (2000) and Chelton et al. (2011), although the time period is different.

## 3. SSH statistical characteristics in the North Pacific

SSH statistical characteristics analyzed in this study correspond to the period from June to December 2001. These characteristics are estimated in two-dimensional boxes with a size of 10°. This box size, similar to that chosen in previous studies (Scott and Wang 2005; Xu and Fu 2011), is large enough to contain the scales of interest (<400 km). To estimate the spectral characteristics, each two-dimensional box is made double periodic (i.e., periodic in both zonal and meridional directions) by doubling its size in the same manner as in Lapeyre (2009). This is done by considering the mirror of the original box in the north and then the mirror of the resulting box in the east (leading to a new box with a size of 20° × 20°). To focus on the mesoscale part of ocean turbulence, zonal mean values are removed. Note that these zonal mean values have been found to affect only scales larger than 300 km. In each new box, the rms SSH value and the SSH wavenumber spectrum are calculated with the spectral slope estimated down to 30 km. Surface velocity and also the surface relative vorticity are estimated from SSH using the geostrophic approximation. Figure 2 shows rms SSH values in 10° × 10° boxes. These values, which vary from ≈2.5 cm in the western part to less than 4–5 cm in the eastern part, are consistent with the EKE results (Fig. 1a) and match those directly estimated from altimetric data (Ducet et al. 2000; Chelton et al. 2011).

The major new result concerns the characteristics of the SSH wavenumber spectrum in low EKE regions (Fig. 2). In these regions, a well-defined *k*^{−4} slope is found, but only for scales between ≈150 and 30 km (as discussed below). In high EKE regions, on the other hand, a *k*^{−4} or slightly steeper slope is found for scales smaller than 200–300 km (as discussed below).

Figures 3a and 4a show SSH spectrum in a high (30°–40°N, 150°–160°E) and a low (20°–30°N, 140°–130°W) EKE region, respectively. To further characterize the differences between these regions, we have also plotted the surface velocity *U _{g}* spectrum (thick curves in Figs. 3b, 4b) as well as the surface relative vorticity

*ζ*spectrum (Figs. 3c, 4c). These two spectra allow to define two wavenumbers:

*k*(corresponding to a wavelength ) associated with the peak of the

_{e}*U*spectrum and

_{g}*k*(corresponding to a wavelength ) associated with the peak of the

_{υ}*ζ*spectrum. Because the

*ζ*spectrum is also the SSH spectrum compensated by

*k*

^{−4}, this means that SSH spectral slopes equal to or steeper than

*k*

^{−4}emerge only for scales smaller than

*λ*, whereas SSH slopes should be shallower for scales larger than

_{υ}*λ*. One characteristic is that

_{υ}*λ*is usually smaller than

_{υ}*λ*(see Figs. 3–5). Another important characteristic (see Fig. 5) is that

_{e}*λ*is significantly smaller in low EKE regions (<150 km) than in high ones (≈200–300 km). This explains why, in low EKE regions, a

_{υ}*k*

^{−4}or steeper spectral slope is observed only for scales smaller than 150 km, whereas, in higher EKE regions, such slope is observed on a larger-scale range.

The velocity spectrum at 250 m (thin curves in Figs. 3b, 4b) displays a slope (between *k*^{−3} and *k*^{−3.5}) significantly steeper than at the surface. This means that submesoscales at depth are less energetic than at the surface, which indicates a different dynamical regime. These spectral results are consistent with those previously reported from higher-resolution simulations of mesoscale and submesoscale turbulence in idealized domains (see Klein et al. 2008; Capet et al. 2008b), that highlighted a SQG-like regime near the surface and a QG-like regime at depth. This suggests for the NP a scenario close to that analyzed in Lapeyre and Klein (2006) and Tulloch and Smith (2009): a surface dynamics dominated by the frontal dynamics [consistent with the shallow (≈*k*^{−2}) SST spectrum observed in both low and high EKE regions; see Figs. 3d, 4d] and a classical QG regime at depths.

## 4. North Pacific dynamics: A nonlinear regime

One question raised by Le Traon et al. (1990) and Xu and Fu (2011) is whether the dynamics, for scales smaller than 400 km, in low EKE regions (the type-3 regions of Xu and Fu 2011) is characterized by a nonlinear regime (as in SQG or QG turbulence) or by a linear regime as would be expected from the response to wind forcing (Muller and Frankignoul 1981). We have addressed this question by calculating in each 10° × 10° box a nonlinear criterion NL that measures the contribution of the nonlinear terms relatively to the linear ones in the relative vorticity equation (see Rhines 1975; Pedlosky 1987; Vallis 2006). This implies to define a length scale, characteristic of the dynamics involved, to compare with the Rhines scale (equal to , where is the rms velocity deduced from SSH and *β* the planetary vorticity gradient; Rhines 1975). A natural choice is the length scale related to the peak of the relative vorticity spectrum *λ _{υ}*, because of its importance discussed in the preceding section. As a result,

Then, NL > 1 corresponds to a nonlinear regime and NL < 1 corresponds to a linear one (Vallis 2006). As expected, NL is large (>10) in high EKE regions, but in low EKE regions NL is still significantly larger than one (Fig. 5). This means that in all regions, for scales smaller than 400 km, the nonlinear terms dominate the linear ones with the consequence of an EKE redistribution among the wavenumbers. Note that the Rhines scale values in this simulation (equal to ) are close to those estimated by Scott and Wang (2005) from altimetric data.

To further identify the EKE redistribution by the nonlinear terms, we have calculated the kinetic energy flux Π_{A}. This flux arises through the horizontal advective term in the kinetic energy equation whose usual form in spectral space is (see, e.g., Klein et al. 2008)

where **u**_{g} is the horizontal velocity field estimated from SSH, **∇**_{H} is the horizontal gradient operator. Here, is the horizontal spectral transform and ()* stands for the complex conjugate, where Re() is the real part. The advective term comes from the nonlinear terms of the momentum equations. Because we focus only on the EKE redistribution estimated from SSH, other terms [involving in particular KE sources and sinks as detailed in Klein et al. (2008) and Capet et al. (2008c)] are not discussed. The spectral kinetic energy flux Π_{A} is defined as the integral of the local horizontal advective term from *k* to the largest wavenumber *k _{s}* (see also Scott and Wang 2005),

where *k _{s}* is the wavenumber corresponding to the grid size. Note that, for the double periodic box considered, the geostrophic approximation leads in physical space to the following constraint: . From the identity of the integrals in spectral and physical spaces, this leads to Π

_{A}(

*k*

_{0}) = Π

_{A}(

*k*) = 0, where

_{s}*k*

_{0}is the wavenumber corresponding to the box size.

Figures 6 and 7 show Π_{A}(*k*), averaged over 7 months for a high and a low EKE region, respectively. The spectral kinetic energy flux is mostly negative, which indicates a dominant inverse nonlinear KE transfer (i.e., a KE cascade from smaller to larger scales) within a large spectral range. A noticeable but small direct KE transfer (i.e., a KE cascade from larger to smaller scales) is present in the small-scale range (scales smaller than 30 km) in the low EKE region. The dominant inverse KE cascade is consistent with estimates using altimetric data (Scott and Wang 2005).

However one interesting characteristic of this inverse KE cascade is that it concerns a larger-scale range (including scales down to 30 km) than in Scott and Wang (2005) (Figs. 6, 7). This strongly emphasizes that, in both high and low EKE regions, mesoscale turbulence near the surface is significantly affected by scales not resolved by conventional altimetric data (<70–100 km). These scales are smaller than the first Rossby radius of deformation (i.e., in these regions, larger than 100 km in terms of wavelength; Chelton et al. 1998). As such this result is more consistent with the SQG dynamics (see Capet et al. 2008a) than with the QG dynamics (see Hua and Haidvogel 1986). A similar result has also been reported in recent high-resolution numerical studies (Klein et al. 2008; Klein and Lapeyre 2009; Capet et al. 2008c; Roullet et al. 2012). The integral surface density variance budget reveals that the advection term contribution (not shown) is positive, implying a direct nonlinear transfer of density variance (from large to small scales), which is expected because the action of eddies is to stir the surface density field to small scales (as displayed by the SST spectrum; Figs. 3d, 4d). All these results suggest that surface frontogenesis in the scale range not well resolved by altimetry (scales smaller than 100 km) has a strong impact on the mesoscale eddy field and therefore on the ocean dynamics.

## 5. Discussion and conclusions

A new high-resolution realistic simulation has been used to estimate the SSH spectral characteristics in the NP over a scale range larger than allowed by the previous realistic simulations or altimetric datasets. The purpose was to investigate these characteristics with a focus on scales smaller than 100 km. In all, high and low, EKE regions, the SSH spectral slope almost follows a *k*^{−4} or slightly steeper law. Such slopes and the associated inverse KE cascades, observed in particular for scales smaller than 100 km, highlight the dynamical impact of small-scale processes on the larger-scale ones.

This *k*^{−4} slope emerges only for scales smaller than *λ _{υ}* (associated with the peak of the relative vorticity spectrum) that can be significantly smaller than

*λ*(associated with the peak of the velocity spectrum). In high EKE regions,

_{e}*λ*is close to or larger than 200 km. So the

_{υ}*k*

^{−4}or slightly steeper law is close to the results from the altimeter-based analysis by Le Traon et al. (2008) and Xu and Fu (2011) that reveal similar SSH spectral slopes within the range 70–250 km in these regions. In low EKE regions, however,

*λ*is smaller (

_{υ}*λ*< 150 km). This means that in some low EKE regions the scale range between

_{υ}*λ*and 70 km can be small, which may lead to an underestimation of the SSH spectral slope between 250 and 70 km. However, the shallower SSH spectral slope (close to

_{υ}*k*

^{−3}) found between

*λ*and

_{υ}*λ*in some of these low EKE regions qualitatively agrees with those reported by Le Traon et al. (1990) and Xu and Fu (2011) in similar regions. Yet the spectral slopes found in other low EKE regions by Xu and Fu (2011) and also by Ray and Zaron (2011) in the 70–250-km range are even shallower than those suggested by the present realistic simulation.

_{e}The emergence of the scale *λ _{υ}* (that differs from

*λ*) in the spectral characteristics of the SSH (or surface velocity) does not appear to be new. Indeed Kim et al. (2011), using both altimetric data and high-frequency radar observations just off the U.S. West Coast, found similar results: their surface velocity spectrum (see their Fig. 7a) exhibits a

_{e}*k*

^{−1}slope between its spectral peak (at ≈300 km) and a scale of 100 km followed by a

*k*

^{−2}slope for smaller scales. This would correspond to a

*k*

^{−3}SSH spectral slope between ≈300 and 100 km and to a

*k*

^{−4}SSH slope for scales smaller than 100 km. In terms of the length scales defined in the present study, these observations should lead to

*λ*= 300 km and

_{e}*λ*= 100 km. These observations are in a region much closer to the coast than the low EKE regions examined in this study. However, the results are qualitatively consistent with ours.

_{υ}As already mentioned in section 2, the present results need to be corroborated using a longer integration period (which is in progress), in particular to take into account the interannual variability. With this caveat in mind, this new high-resolution NP simulation, however, emphasizes, both in high and low EKE regions, the dynamical importance of scales smaller than 100–150 km near the surface (whereas a different dynamical regime dominates at depths with a steeper velocity spectrum). The similar SST and surface velocity spectrum slope and the inverse KE cascade within a large-scale range at the surface indicate that mesoscale turbulence is significantly affected by scales not well resolved by altimetry. Such dynamics is consistent with the well-known mesoscale/submesoscale turbulence properties that have been recently confirmed by idealized oceanic simulations with high resolution (Klein et al. 2008, 2009; Capet et al. 2008b,c; Roullet and Klein 2010; Lévy et al. 2010).

As mentioned before, in some low EKE regions, there are still some discrepancies (in the 70–250-km-scale range) with altimeter-based studies (Xu and Fu 2011; Ray and Zaron 2011). These studies display, there, SSH spectral slopes shallower than *k*^{−3}. If such discrepancies persist for smaller scales, this would mean very high kinetic energy at small scales, which disagrees with the well-known mesoscale/submesoscale turbulence properties. Other dynamical processes need to be invoked. A possible one is that associated with internal tides (Arbic et al. 2010), not taken into account in the present simulation, and their interactions with mesoscale turbulence (Chavanne and Klein 2010; Ray and Zaron 2011). Thus, Ray and Zaron (2011) noted that, at some sites where mesoscale and submesoscale energy follows a *k*^{−3} (QG) or (SQG) spectral slope, much of the SSH signal could be masked by the nonstationary internal tide, which may affect both the SSH variance and the spectral slope. Such impact has been observed in some low EKE regions in recent numerical simulations where both mesoscale turbulence and internal tides are present (J. Richman and B. Arbic 2011, personal communication). The linear oceanic response to stochastic forcing is also invoked in some studies to explain the shallow SSH spectral slope in low EKE regions. However, the present simulation uses realistic atmospheric forcings and both the resulting SSH slopes displayed in section 3 and the nonlinear oceanic regime found in section 4 do not favor this assumption.

Thus, although many of these results from different observational and numerical studies appear qualitatively consistent, the differences that emerge in some low EKE regions remain to be addressed in particular in the small-scale range. This emphasizes the need to observe and monitor this small-scale dynamics on a global scale and, in that respect, future space missions such as the Surface Water Ocean Topography (SWOT) mission (Fu and Ferrari 2008) will allow to better highlight their impact in all regions. Further high-resolution in situ observations, such as the high-frequency radar observations, can complement satellite observations as demonstrated by Kim et al. (2011). Confrontation of these observations with realistic numerical simulations with an even higher resolution should allow us to better understand the physical mechanisms related to these small scales. In that respect, modeling studies that take into account both internal tides and mesoscale/submesoscale turbulence, such as the new one by J. Richman and B. Arbic (2011, personal communication), but with a much higher resolution, should be very useful.

## Acknowledgments

We thank Rob Scott and Pierre-Yves Le Traon for their insightful comments and stimulating discussions during this study, as well as Xavier Capet for pointing out the study by Kim et al. This study is supported by JAMSTEC (Earth Simulator Center, Yokohama) and also by IFREMER and CNRS (France). HS is partly supported by a Grand-in-Aid for Scientific Research (22106006 and that on Innovative Areas 2205) from MEXT of Japan. PK is partly supported by IFREMER, the French Agence Nationale pour la Recherche (Contract ANR-09-BLAN-0365-02) and CNES (Contracts DAR 4800000544 2010 and 4800000599 2011). Simulations reported here were done on the Earth Simulator (Yokohama, Japan) through an MOU signed between IFREMER and JAMSTEC. We warmly thank both reviewers who contributed to greatly improve the manuscript and in particular the discussion.