## Abstract

The real transfer function and the phase shift between sea surface height (SSH) and sea surface buoyancy (SSB) were estimated from the output of a realistic eddy-resolving model of the Mediterranean Sea circulation. The analysis of their temporal evolution unveiled the existence of a clear seasonal cycle closely related to that of the mixed layer depth. The phase shifts between SSH and SSB attain their minimum for deep mixed layers, which is different from zero. Besides, the spectral slope of the transfer function at scales shorter than 100 km fluctuates between *k*^{−1} and *k*^{−2}. For deep mixed layers, it is close to *k*^{−1}, as predicted by the surface quasigeostrophic (SQG) solution. At longer wavelengths, it is approximately constant under the different environmental conditions in all of the subbasins analyzed with the exception of the Gulf of Lions. The capability to observe sea surface temperature (SST) from satellites motivated the extension of this analysis to SST and SSH. Results showed a similar qualitative behavior but with larger phase shifts. In spite of the presence of a phase shift, even for deep mixed layers, results revealed that it is still possible to reconstruct surface dynamics from SST using a transfer function, provided that the mixed layer is deep enough. For the present study, a threshold value of 70 m was enough to identify the appropriate environmental conditions. In addition, the results revealed that a precise estimation of the transfer function significantly improves the reconstruction of the flow in comparison with the application of the classical SQG solution.

## 1. Introduction

The Mediterranean Sea is a hot spot for climate change and plays a major role in the climate variability over Europe. This has led to efforts being made to better characterize its surface currents and to develop numerical models of its circulation. However, the dynamics in most of the Mediterranean are characterized by a Rossby radius of deformation of the order of 10–15 km, which requires spatial resolutions higher than the available observations of surface velocities (~100 km). Besides, current observations of sea surface temperature (SST) do have the appropriate spatial resolutions. However, the lack of a direct connection between SST and surface currents has proven to be an obstacle difficult to overcome for their quantitative estimation from SST satellite observations.

The diagnosis of surface velocities from SST images is based on the exploitation of the properties of upper-ocean dynamics. One of such properties is that small-scale patterns tend to be advected by larger-scale velocities in a turbulent flow. Then, mesoscale currents (~20 km) can be inferred, in principle, by tracking submesoscale patterns (1–10 km) in consecutive cloud-free SST images (e.g., Emery et al. 1986; Bowen et al. 2002). Another possibility consists of the exploitation of heat conservation and the inversion of the temperature equation to retrieve ocean currents using a sequence of SST images (e.g., Kelly 1989; Ostrovskii and Piterbarg 1995; Chen et al. 2008). A third approach, which has emerged during the last years, is based on the property that SST and surface currents have a similar complex phase in the upper ocean under the appropriate conditions (e.g., Isern-Fontanet et al. 2006a; LaCasce and Mahadevan 2006; Isern-Fontanet et al. 2008; Isern-Fontanet and Hascoët 2014). Indeed, Lapeyre and Klein (2006) put forward that potential vorticity (PV) in the upper ocean is correlated to the sea surface buoyancy (SSB), which locks the phase between SSB and the streamfunction. Contrary to the previous approaches, this property allows diagnosing velocities from a single SST field.

Indeed, Lapeyre and Klein (2006) and LaCasce and Mahadevan (2006) proposed that the mesoscale and submesoscale dynamics of the upper-ocean layers could be modeled using an effective version of the surface quasigeostrophic (SQG) equations. This implies that the surface streamfunction can be retrieved directly from SSB by convolving it with a kernel proportional to *k*^{−1}, where *k* is the modulus of the two-dimensional wavevector. This kernel is usually known as the transfer function in signal processing theory. Other solutions can be found depending on the geometrical conditions (finite depth vs semi-infinite ocean; Tulloch and Smith 2006), the stratification (constant vs exponential; LaCasce 2012), or the relative contribution of the interior PV to respect the surface boundary condition (Lapeyre and Klein 2006; Klein et al. 2010). All of these solutions are also characterized by the phase locking between the streamfunction and SSB. It is worth mentioning that although the relevant dynamical variable in the PV inversion problem is the SSB, the field that can be remotely observed is the SST. Therefore, some additional assumptions regarding the alignment between SST and buoyancy gradients (Klein et al. 1998; Ferrari and Paparella 2003) have to be made if the objective is to retrieve ocean dynamics from SST. Because these solutions imply that SSB (or SST) shares the phase with the streamfunction, it is necessary to verify this condition before applying these transfer functions and identify under which conditions the assumption of zero phase shift holds.

The validity of the SQG approach implies an energy spectrum with slopes of *k*^{−5/3} (Blumen 1978) and a spectrum of *k*^{−11/3} for sea surface height (SSH). This has motivated the study of its validity through the analysis of altimetric observations of SSH. Le Traon et al. (2008) showed that spectral slopes in the mesoscale band in high energy areas are significantly different from a *k*^{−5} law, predicted by quasigeostrophic (QG) turbulence, but they very closely follow the *k*^{−11/3} slope, which indicates that the surface QG is a much better dynamical framework than the QG turbulence theory to describe the ocean surface dynamics. However, results in low energy areas revealed shallower slopes (Xu and Fu 2011). Sasaki and Klein (2012) further investigated such areas using numerical simulations and showed that even in low energy areas SSH had spectra of the type *k*^{−4}, close to SQG predictions, and suggested that instrumental noise could hide such slopes. Furthermore, they showed that the scales for which the *k*^{−4} slope could be seen depend on the energy level. In particular, for low energy areas with sea level variance on the order of 25 cm^{2}, the *k*^{−4} slope can be observed at scales shorter than 150 km. These studies have been complemented by the analysis of high-frequency radar just off the U.S. West Coast, which found spectral slopes equivalent to a SSH *k*^{−4} slope for scales smaller than 100 km (Kim et al. 2011). Nevertheless, this does not necessarily imply that velocities can be diagnosed from SST. On one side, Armi and Flament (1985) pointed out that different dynamical regimes could have similar spectral slopes and stressed the importance of the complex phases. On the other side, Klein and Hua (1990) showed that the emergence and evolution of the SST mesoscale variability depends on the mixed layer (ML) dynamics, which can introduce a phase shift between SST and currents.

The Mediterranean Sea is a semienclosed basin with an excess of evaporation mainly compensated by the entrance of the relatively fresher waters from the Atlantic Ocean through the Strait of Gibraltar. Satellite images of SST have been widely used to investigate the circulation patterns in the Mediterranean Sea and track coherent vortices (e.g., Puillat et al. 2002; Hamad et al. 2005). However, the difficulty to quantitatively estimate surface velocities from SST patterns has limited its use to qualitative analysis. Consequently, quantitative studies have been based on altimetric (e.g., Larnicol et al. 2002; Pujol and Larnicol 2005; Isern-Fontanet et al. 2006c) and in situ measurements (e.g., Millot 1999; Malanotte-Rizzoli et al. 1999; Poulain et al. 2012, and references therein). The picture that emerges from both satellite and in situ observations reveals that incoming fresh waters circulate anticlockwise along the coast in the western Mediterranean Sea and the Levantine Basin (Millot and Taupier-Letage 2005; Poulain et al. 2012). The circulation in the Ionian Sea is still under debate, and different circulation schemata have been proposed (e.g., Millot and Taupier-Letage 2005; Pinardi et al. 2005; Poulain et al. 2012). The major discrepancies concern the existence of a jet that crosses the Ionian Sea, which is supported by the observations provided by surface drifters (Millot and Gerin 2010; Poulain et al. 2012), although some researchers consider it an artifact (Millot and Gerin 2010). In addition to these patterns, the instability of the inflow and local wind action often generate coherent vortices in several parts of the basin that tend to propagate following rather well-defined patterns in some areas of the Mediterranean Sea (Isern-Fontanet et al. 2006b). Although the use of SST has proven to be in good agreement with in situ measurements (e.g., Taupier-Letage et al. 2003; Isern-Fontanet et al. 2004) and very useful to get insight about ocean circulation in the area, there are some potential difficulties. By the end of summer, surface resident waters are warmer than incoming Atlantic water, while in winter the Atlantic water is warmer than the resident Mediterranean waters. This has two main implications for the diagnosis of surface velocities from SST: some features may not be visible during certain periods of the year, and the alignment between salinity and temperature gradients may vary (Taupier-Letage 2008).

The main objective of this study is to characterize the transfer function between SSB and SSH and better determine the conditions under which SSB shares the complex phase with SSH in the Mediterranean Sea. Because SSB cannot be determined from direct observations, the study has been extended to include the analysis of SST to clearly identify the capability to reconstruct surface velocities from satellite observations of SST. The paper is organized as follows. Section 2 defines the theoretical framework used in this study. Section 3 describes the numerical simulations used for this study and the characteristics of the areas analyzed. Section 4 describes our results and section 5 discusses them.

## 2. Theoretical framework

The principle of invertibility of PV (Hoskins et al. 1985) allows the diagnosis of the 3D dynamics of a balanced flow from the knowledge of PV in the ocean interior and density on the vertical boundaries. If we assume that the flow is in QG equilibrium, the problem consists of inverting the equation

where is the PV anomaly, is the streamfunction of the flow, *f*_{0} is the local Coriolis frequency, *N*(*z*) is the Brunt–Väisälä frequency, , and **∇** ≡ (∂_{x}, ∂_{y}). The appropriate boundary conditions can be derived from the hydrostatic equation. In particular, for the ocean surface they can be written as

where is the SSB and the subscript *s* stands for surface fields.

This problem can be split into two solutions as proposed by Bretherton (1966) and Lapeyre and Klein (2006): an interior solution obtained assuming zero surface buoyancy and nonzero interior PV, that is, *b*_{s} = 0 and *Q* ≠ 0; and a surface solution obtained assuming nonzero surface buoyancy and zero interior PV, that is, *b*_{s} ≠ 0 and *Q* = 0. Then, the total solution to the inversion problem is the sum of both contributions, that is,

### a. The transfer function formalism

Lapeyre and Klein (2006) showed that the large-scale forcing in density and PV can lead to the property that the interior PV mesoscale anomalies are correlated to the surface buoyancy anomalies in the upper ocean. In that case, the PV anomaly can be separated as

with *ξ*(*z*) being a function that specifies the amplitude of PV anomaly. Then, the total solution to the PV inversion problem [Eqs. (1) and (2)] has the form

where the hat stands for the Fourier transform, is the wavevector, and is a transfer function that depends on the modulus of the wavevector and the depth *z*. This transfer function can also be separated into interior and surface contributions, and , respectively.

A simple analytical solution for and can be obtained for a finite depth *H* with constant stratification *N*_{0} ≡ *f*_{0}*n*_{0}. Then, the surface transfer function is given by

(Tulloch and Smith 2006) from which the classical SQG solution

is recovered in the limit of infinite depth (Held et al. 1995). Notice that the surface behavior of Eq. (6) at short scales (large *k*) is similar to the classical SQG solution , while at large scales (small *k*) it is steeper, . Besides, the interior solution for constant stratification and finite depth is given by

(e.g., Klein et al. 2010), which corresponds to the baroclinic mode. At long scales, this solution tends to a constant, while at short scales it behaves as . The relative dominance of each solution can be separated by a critical wavelength that depends on the large-scale properties of the flow (Lapeyre 2009; Klein et al. 2010).

### b. The transfer function at the ocean surface

Equation (5) implies that the streamfunction and surface buoyancy are in phase. However, previous studies have shown that there may exist a phase shift between the streamfunction and surface buoyancy (Isern-Fontanet et al. 2008; Hausmann and Czaja 2012). To take this into account, we introduce a phase shift^{1 } into Eq. (5) and focus at *z* = 0. Then,

where *i* is the complex unit, is now the surface streamfunction, and *F*_{b}(*k*) is the total transfer function at the ocean surface, that is,

The transfer function *F*_{b}(*k*) is a real function that controls the amplitude of the resulting streamfunction and only depends on the modulus of the wavevector.

Besides, the geostrophic streamfunction at the ocean surface is proportional to the SSH , that is,

This implies that the transfer functions for surface buoyancy can be calculated as

where angle brackets with a subscript *k* indicate that the average is taken over those wavevectors with the same modulus. On the other side, the phase shift can be obtained as

Equivalently, a transfer function *F*_{T}(*k*) and phase shift can be defined for the SST anomaly . Then, Eq. (9) becomes

Notice that, if salinity would not contribute to surface buoyancy variability then, *F*_{T}(*k*) ∝ *F*_{b}(*k*) and .

## 3. Numerical simulations

### a. The Mediterranean Forecasting System

In this study, we used the nowcasts provided by the Mediterranean Forecasting System (MFS) running at the Istituto Nazionale di Geofisica e Vulcanologia (INGV) (http://gnoo.bo.ingv.it/mfs/). The model is an implementation of the Nucleus for European Modelling of the Ocean–Océan Parallélisé (NEMO–OPA), version 3.2 (Madec et al. 1998; Madec 2008) that covers the entire Mediterranean Sea and part of the Atlantic Ocean with a spatial resolution of ° × ° and 71 unevenly spaced vertical levels (Tonani et al. 2008; Oddo et al. 2009). In the Atlantic, the model is nested within the monthly-mean climatological fields computed from the daily output of the ¼° × ¼° global model (Drevillon et al. 2008). The model is forced by momentum, water, and heat fluxes interactively computed by bulk formulae using the 6-h, 0.5° horizontal-resolution operational analyses from the European Centre for Medium-Range Weather Forecasts (ECMWF) and the model-predicted surface temperatures (Tonani et al. 2008). In addition, evaporation is derived from the latent heat flux, precipitation is taken from monthly-mean Climate Prediction Center (CPC) Merged Analysis of Precipitation (CMAP) data (Xie and Arkin 1997), and river runoff is composed of monthly-mean climatological data for the seven major rivers in the area (Fekete et al. 1999; Raicich 1996). The flow through the Dardanelles Strait is parameterized as a river with monthly climatological net inflow rates taken from Kourafalou and Barbopoulos (2003).

The system assimilates sea level anomaly (SLA), sea surface temperature, in situ temperature profiles from Voluntary Observing Ship (VOS) expendable bathythermographs (XBTs), in situ temperature and salinity profiles from Argo floats, and in situ temperature and salinity profiles from CTD using the three-dimensional variational data assimilation (3DVAR) scheme (Dobricic and Pinardi 2008). The background error correlation matrix is estimated from the temporal variability of parameters in a historical model simulation. Background error correlation matrices vary seasonally and in 13 regions of the Mediterranean that have different physical characteristics (Dobricic et al. 2007). The mean dynamic topography used for the assimilation of SLA has been computed by Dobricic (2005). Satellite SST data are used for the correction of surface heat fluxes.

### b. Regions under study

This study has been focused on four regions of the Mediterranean Sea representative of its circulation: the Algerian Basin, the Ionian Sea, the Levantine Basin, and the Gulf of Lions (Fig. 1). The dynamics and environmental conditions present during the period analyzed were characterized by the sea level variance (SLV), the root-mean-square (RMS) of wind stress, and the RMS of the mixed layer depth (MLD; see Table 1). Here, the MLD was estimated using the classical difference criterion with a reference depth of 10 m, as in de Boyer Montégut et al. (2004) and D’Ortenzio et al. (2005). Notice that the SLV values shown in Table 1 correspond to low energy areas.

The Algerian Basin is a highly energetic area characterized by the presence of eddies with sizes ranging from 30 to 150 km generated by the instability of the fresh waters of Atlantic origin that flow along the Algerian coast. These eddies tend to follow well-defined patterns characterized by a recirculation loop approximately enclosed within the box used in our analysis (Isern-Fontanet et al. 2006b). The presence of such eddies produced the largest spatially averaged SLV (Table 1). On the contrary, the Ionian Sea had the lowest SLV of all four boxes, mainly concentrated in the southern part of the box. Its dynamics are characterized by a lower concentration of intense eddies (Isern-Fontanet et al. 2006b) and the presence of a jet crossing the box from east to west (e.g., Poulain et al. 2012). The Levantine Basin had dynamical characteristics that resemble those of the Algerian Basin. It is characterized by the presence of energetic eddies generated in part by the instability of the waters that circulate counterclockwise along the coast. However, the length of the time series here analyzed was not long enough to produce more homogeneous patterns of SLV. Indeed, the box in this area was dominated by the presence of a highly energetic eddy in its southwestern corner, which is evident in Fig. 1. Finally, the Gulf of Lions, which is an area of deep-water formation, was characterized by the strongest wind forcing, very deep MLD, and quite active dynamics, mainly in its southwestern part due to the contribution of Algerian eddies. In addition to these eddies, this area is characterized by the presence of smaller vortices (~50 km) generated by the instability of the Liguro–Provençal Current that follows the continental shelf (e.g., Rubio et al. 2005).

Salinity gradients play a major role in the dynamics of the Mediterranean Sea. This made it necessary to explore the contribution of salinity and temperature to surface buoyancy. In particular, the alignment and compensation in density between salinity and temperature gradients were analyzed using the complex ratio given by Ferrari and Paparella (2003):

where *α* < 0 is the thermal expansion coefficient, *β* > 0 is the haline contraction coefficient, and *S*_{s} is the sea surface salinity (SSS). The phase *ϕ* of the complex density ratio *R* quantifies the degree of alignment between SST and SSS gradients, while its magnitude quantifies the relative strength of the gradients. For *ϕ* = 0 rad or *ϕ* = ±*π*, there is a thermohaline alignment. Figure 2 shows the probability density functions (PDF) of the alignment between salinity and temperature gradients observed in the simulations. The resulting PDFs were bimodal with peaks at *ϕ* = 0 and *ϕ* = *π*, indicating a strong tendency toward the alignment. However, the relative importance of each peak depended on the area analyzed. In the western Mediterranean Sea (Algerian Basin and the Gulf of Lions), it was more likely to find alignments around *ϕ* = 0, while in the eastern Mediterranean Sea (Ionian Sea and Levantine Basin), it was more likely to find alignments around *ϕ* = *π*. The probability of finding a misalignment between salinity and density gradients larger than was estimated to be approximately 26%. If only MLs deeper than 70 m were considered, such probability reduced to values between 21% and 23%. Alignment properties in all boxes were similar, although in the Ionian Basin the probability of finding a misalignment was found to be slightly larger. The probability of having compensated fronts, that is, |*R*| = 1 and *ϕ* = 0 rad, was estimated to be less than 0.1%.

## 4. Results

### a. Phase shift between SSB and SSH

The temporal evolution of the phase shift between SSB and SSH for different *k* was investigated for all boxes using a spectral correlation coefficient *r*(*k*) based on Eq. (13):

Results revealed a clear seasonal cycle with the largest shifts between 30 and 200 km during summer (May–September), which was the period of the year with the shallowest ML and lower winds (see the example in Fig. 3). Because wind stress was highly variable, its similarity with the MLD was more evident once the daily winds were smoothed using a moving mean with an amplitude of 45 days. For those days that had smaller phase shifts, a higher degree of homogeneity across *k* in the western Mediterranean Sea (Gulf of Lions and Algerian Basin) than in the eastern Mediterranean (not shown) could be observed. Indeed, for the Ionian Sea and the Levantine Basin, it was evident that the phase shifts tended to be smaller for wavelengths larger than 100 km.

To complement the spectral analysis, a streamfunction was built combining the phases of SSB with the spectrum of SSH for the spectral band between 20 and 300 km, that is,

By construction, such streamfunction only differed from Eq. (11) by the relative position of flow patterns. Both streamfunctions were compared, once properly filtered with the same cutoff wavelengths, using the lineal correlation between vorticities as a measure (Fig. 4). Results also revealed a clear seasonal cycle characterized by higher correlations during winter. However, correlations never reached 1, suggesting the presence of a phase shift between SSB and SSH. This is in agreement with results shown in Fig. 3, which showed that *r*(*k*) < 1. Results also unveiled differences between basins and from year to year. Indeed, correlations were larger or equal to 0.9 in the Algerian and Levantine Basins between December and March, while in the Ionian Basin they were around 0.8 during winter 2009–10. In the Gulf of Lions, the period of correlations higher or equal to 0.9 was shorter. The interannual variability was evident, for example, in the Ionian Basin, where higher correlations were obtained from January to April 2009 than during the same period in 2010. Interestingly, a difference between these two periods was the depth of the ML, which was thicker in 2009.

The difficulty in reaching correlations higher than 0.9 still indicated the existence of a phase shift between SSB and SSH. To estimate it, daily phase shifts associated with spatial correlations larger or equal to 0.9 were averaged (Fig. 5). Results revealed that even for high spatial correlations, there was a phase shift between SSB and SSH of the order of . Although it was quite similar in all boxes for wavelengths shorter than 100 km, there were significant differences at longer wavelengths. In particular, the Ionian Sea and Levantine Basin exhibited smaller phase shifts between 100 and 250 km than the boxes in the western Mediterranean. Unsurprisingly, higher threshold values of correlation led to lower phase shifts.

### b. The transfer function between SSB and SSH

In addition to the phase shift, the characteristics and temporal evolution of the transfer function were investigated. The transfer function was computed from daily SSB and SSH fields using Eq. (12). Results shown in Fig. 3 revealed that it was characterized by negative slopes particularly for wavelengths shorter than 100 km, and a clear seasonal cycle characterized by steeper slopes for wavelengths shorter than 100 km during summer. A qualitative correspondence between the patterns observed in the transfer function and those observed for the temporal evolution of the phase shift could be found. In general, all boxes exhibited similar features with the exception of the Gulf of Lions, which had less marked patterns (not shown).

The transfer function associated with the smallest phase shifts was estimated by averaging the observations for those days with vorticity correlations larger or equal to 0.9 as before (Fig. 5). From results, two different bands in most basins could be clearly identified: one characterized by an SQG-like slope, that is, *F*_{b}(*k*) ~ *k*^{−1}, between 20 and 100 km (see Table 2); and another characterized by a plateau, that is, *F*_{b}(*k*) ~ *C*, for wavelengths larger than 100 km. Nevertheless, the situation was slightly different in the Gulf of Lions, where the *F*_{b}(*k*) ~ *k*^{−1} extended along the whole range of wavelengths analyzed.

### c. Dependence on environmental conditions

The strong seasonal dependence of the phase shift and transfer function pushed us to investigate which parameters better characterized such evolution. We focused on the spatial averages of the MLD , wind stress , and the stratification just below the ML *n*, because these parameters play a key role in the deepening of the ML (Klein and Hua 1988). Results (Fig. 6) showed a clear dependence on the MLD, while the dependence on wind was much less clear. Notice, however, that when averaging wind using a running mean with an amplitude of 45 days, results are similar to those for the MLD. It is worth mentioning that in the Gulf of Lions deep ML did not always imply high correlations, as is evident by the set of circles with MLD larger than 70 m and correlations lower than 0.8 seen in Fig. 6. On the other side, the results were less clear using the stratification than with the MLD.

The averaged phase shift and transfer function were explored using the MLD as a criterion to classify the daily results. In particular, the mean transfer function was obtained under different MLD ranges, as is shown in Fig. 7. For MLD deeper than 70 m, the results were similar to those obtained selecting correlations larger or equal to 0.9 (Fig. 5), except for the Gulf of Lions, which exhibited larger phase shifts and more of a departure from the mean transfer functions than that observed for the other boxes. The reduction of the MLD threshold value used to select which days were averaged lead to an increase of the phase shift and a change of the transfer function for wavelengths shorter than 100 km from *F*_{b}(*k*) ~ *k*^{−1} to *F*_{b}(*k*) ~ *k*^{−2}. The plateau for wavelengths longer than 100 km did not change.

The kinetic energy spectra estimated from SSH was compared to the spectra of surface buoyancy in order to further understand the variation of the transfer function. Notice that the kinetic energy spectra is related to SSH and SSB through

This implies that the kinetic energy spectra and surface buoyancy spectra will have the same slope if *F*_{b}(*k*) ~ *k*^{−1}. As for the transfer functions, spectra were averaged based on the MLD (Fig. 8). For MLD deeper than 70 m, surface buoyancy and kinetic energy had the same slopes at wavelengths shorter than 100 km, as expected by the *k*^{−1} slope of the corresponding transfer function (Fig. 7). For shallow MLD, the main difference corresponded to the kinetic energy spectra, which was steeper than for deep MLD at wavelengths shorter than 100 km approximately. In both situations the spectra of surface buoyancy was qualitatively similar, which originated the ~*k*^{−2} slope in the transfer function. On the contrary, the kinetic energy had a shallower spectrum at larger wavelengths for the different environmental conditions.

### d. Flow reconstruction from SST

The same analysis applied to surface buoyancy was applied to SST. Phase shifts and transfer function temporal evolutions were qualitatively similar to those observed for surface buoyancy with the main difference that, in general, phases at wavelengths shorter than 20 km tended to be orthogonal and the annual cycle stronger (not shown). On the contrary, the mean transfer function estimated for different MLD showed the same patterns as for buoyancy, that is, a *F*_{T}(*k*) ~ *k*^{−1} behavior for wavelengths shorter than 100 km and *F*_{T}(*k*) ~ *C* for wavelengths longer than 100 km. The observed transfer function in the Gulf of Lions also revealed a SQG-like behavior for the whole range of wavelengths analyzed. As before, a transfer function equivalent to Eq. (17) was defined:

Resulting vorticity correlations time series (Fig. 4) were systematically lower than for SSB, although both followed a similar seasonal cycle. Correlations in the Ionian Sea were quite small (less than 0.8) in comparison to the other boxes.

The capability to diagnose surface dynamics from SST was investigated, focusing on the reconstruction capabilities for the surface streamfunction, surface velocity , and vorticity and its dependence on the MLD. In particular, the surface streamfunction was derived from SST and SSH using Eq. (19), focusing on two different bands: the band with SQG-like behavior (20–100 km) and the band with a flat slope (100–300 km). Figure 9 shows the dependence of the averaged correlations with the MLD. Averaged correlations for all variables showed the same qualitative behavior as in Fig. 6: a rapid increase of correlation with the MLD until reaching a maximum value for MLD between 50 and 70 m (Fig. 9). In addition, the capability to reconstruct dynamical variables was better for wavelengths larger than 100 km than for shorter wavelengths, indicating a smaller phase shift at these wavelengths in agreement with the results reported in previous sections.

## 5. Discussion

It has been shown that the transfer function between the streamfunction and the surface buoyancy does not follow the classical SQG solution for large scales, with the exception of the Gulf of Lions, and the existence of a phase shift between the streamfunction and surface buoyancy has been evidenced. Indeed, results have shown that the amplitude of the mean transfer function for both, surface density and SST, has an SQG-like response for wavelengths shorter than 100 km and deep ML in all the regions analyzed. On the contrary, it has almost flat amplitude for longer wavelengths in most boxes. The qualitative comparison of these results with the analytical solution obtained assuming a finite depth and a constant stratification [Eqs. (6) and (8)] suggests a dominance of the interior solution at large scales and of the surface solution at smaller scales, in agreement with theoretical studies (Lapeyre and Klein 2006; Lapeyre 2009). The above results were observed mainly during winter, when a deep ML can be found. On the contrary, observations for shallow ML reveal a different response characterized by steeper slopes in the amplitude of the transfer function at short wavelengths, that is, *F*_{b}(*k*) ~ *k*^{−2}.

A recent study by Callies and Ferrari (2013) showed that the interior QG solution dominates the flow at scales larger than 20 km in highly energetic areas, such as the Gulf Stream ( cm^{2}). But in low energy areas, such as the eastern subtropical North Pacific ( cm^{2}), results were inconsistent with the dominance of both surface and interior solutions. The sea level variance of the boxes analyzed in this study ranges between cm^{2} and cm^{2} (Table 1), which is closer to the variance in the eastern subtropical North Pacific than in the Gulf Stream area. Consequently, we would expect that the dynamics here observed would be closer to that of the subtropical North Pacific. However, contrary to the observations done by Callies and Ferrari (2013), our results are statistically consistent with the predictions of the SQG solution with constant stratification, at least for scales shorter than 100 km. The slightly higher variability in the Mediterranean Sea could be at the origin of these differences. Besides, our results are based on the numerical simulation of the Mediterranean circulation, which may not include all the physical process that contributes to the energy. This implies the need to extend our study to real data in order to confirm the results here found and compare them with the observations of Callies and Ferrari (2013).

The deviation of the streamfunction from the SQG model, particularly at larger scales, has an impact on the performance of the flow reconstruction from satellite SST. The use of the SQG model as in Isern-Fontanet et al. (2006b, 2008) would imply an overestimation of the relative importance of the large scales to respect small scales. Indeed, the comparison of the performance of the SQG reconstruction with the full knowledge of the transfer function represented dramatic improvements. In particular, the correlation for vorticity obtained applying the SQG approach and using Eq. (19) for the range of wavelengths between 40 and 200 km were compared. Results shown in Fig. 10 confirm such an improvement, which can be so important that correlations can increase from values below 0.5 to values on the order of 0.8. These results suggest the introduction of an heuristic transfer function given by

and based on the family of two-dimensional fluid models proposed by Pierrehumbert et al. (1994), where *C*_{γ} is a constant, and *γ*_{1} and *γ*_{2} are the spectral slopes in the two bands separated by the wavevector *k*_{c}. The four parameters of this model could be estimated from simultaneous satellite observations of SST and SSH.

Indeed, current observations of SSH and SST have synergetic characteristics. On one side, along-track altimetric measurements of SSHs are very well suited to quantify across-track currents and have been widely used to estimate turbulence spectra in the ocean (e.g., Le Traon et al. 1990; Stammer 1997; Xu and Fu 2011), but their sampling geometry and the number of available platforms strongly constrain the reconstruction of two-dimensional streamfunctions (Pascual et al. 2006). On the other side, SST measurements provided by infrared radiometers are able to locate ocean structures with small phase shifts, if the environmental conditions are appropriate. This suggests to derive the transfer function between simultaneous SST and SSH measurements to obtain its best estimation. Furthermore, along-track SSH measurements could then be used to quantify if the reconstruction is good enough to be used for scientific or operational applications by correlating across-track velocities derived from SSH with those diagnosed from the combination of SSH and SST. Our results have shown that the different response in the transfer function originates from a different slope of SSH (Fig. 8). Such a result should be verified with satellite observations of SSH. However, the main difficulty is the low signal-to-noise ratio of available altimetric measurements in the Mediterranean Sea, which could hide such behavior.

Another approach to exploit the complementary behavior between SST and SSH is to use the information provided by each field to reconstruct the flow at different scales. Indeed, our results reveal that surface velocities can be reconstructed from SST fields in the band between 20 and 100 km with correlations on the order of 0.8, if the ML is deep enough (Fig. 9). Furthermore, the transfer function for this band is close to the SQG predictions (Fig. 7). This suggests that the spatial resolution of SSH observations could be improved by merging the low-resolution altimetric measurements (*λ* > 100 km) with the high-resolution (20 < *λ* < 100 km) streamfunctions derived from SST using the SQG model as a first approach or incorporating the along-track SSH spectrum. This idea is similar to the underlying approach proposed by Gaultier et al. (2013). In particular, their procedure is based on extracting high-resolution information from the SST gradients and finite size Lyapunov exponents computed from altimetry (e.g., d’Ovidio et al. 2009) to correct the low-resolution altimetric maps. The underlying concept to retrieve the high-resolution information about the velocity field has some similarities with the technique proposed by Turiel et al. (2005) and Isern-Fontanet et al. (2007). Nevertheless, the method to reconstruct the velocities from this information was different.

As it has been outlined in the introduction, there are alternative methods to derive surface currents from SST. Among different techniques, the maximum cross correlation (MCC) method is by far the most widely used (Emery et al. 1986; Bowen et al. 2002; Barton 2002; Notarstefano et al. 2008). The basic idea consists of computing spatial correlations over windows between consecutive images to estimate their displacement. In general, the MCC method acts poorly in regions of uniform SST and presents some difficulties to estimating the velocity along the front of the scalar field (e.g., Zavialov et al. 2002). Consequently, the resulting velocity fields tend to be sparse, which requires merging of the estimated velocities with other measurements such as altimetry to retrieve the two-dimensional velocity field (Wilkin et al. 2002). Other approaches used to estimate velocities from SST are the constrained optical flow methods to solve the heat conservation equation (e.g., Kelly 1989; Vigan et al. 2000) or variational filter and interpolation techniques (Afanasyev et al. 2002). All these techniques rely on the availability of cloud-free very high-resolution (~1 km) sequences of images over short enough time periods. This latter restriction is imposed by the lack of absolute conservation of SST. These requirements impose strong restrictions on their applicability and geographically and seasonally limit the regions over which velocities can be estimated. On the contrary, our approach based on the combination of SST and SSH is able to provide dense velocity fields from an SST snapshot.

The diagnosis of subsurface fields requires separating the interior and surface contributions due to their different vertical variations. Besides, the empirical determination of the transfer function would be only able to provide the total contribution implying the need to find a criterion to separate both contributions. A possible solution could be the method of Wang et al. (2013), who propose to first determine the surface contribution using surface buoyancy (or SST) through the inversion of the problem given by Eqs. (1) and (2) with *Q* = 0 and *b*_{s} ≠ 0 and then find the amplitude of the barotropic and first baroclinic mode by matching to the surface streamfunction (or SSH) resulting from the subtraction of the surface component.

The underlying assumption needed to reconstruct the flow from SST is that , that is, the surface streamfunction or the SSH has to be in phase with surface buoyancy or SST. However, our results have shown that this is not the case. Even under the most favorable environmental conditions there is a phase shift present, although it is small enough to allow the reconstruction of the flow with high correlations. A similar result can be deduced from the study of Isern-Fontanet et al. (2008), who showed that the spectral correlations were never equal to 1 (see their Fig. 2). Assuming that the correct transfer function can be deduced from satellite observations, the phase shift represents the major limitation to reconstruct the flow from SST measurements. Lapeyre and Klein (2006) suggested that PV and SSB are in phase [Eq. (4)], implying that the solution of the PV problem can be written as the convolution between a real transfer function and SSB. However, the failure of this hypothesis could introduce a phase shift between SSH and SSB. Following Lapeyre and Klein (2006), the linear correlation between PV below the ML and SSB was computed and then averaged for the different MLD used to analyze the results (Fig. 11). Results clearly revealed that correlations were higher for the deep ML than for the shallow ML, but even under the most favorable environmental conditions, values larger than 0.5 were constrained to the upper 100 m. Such observations could explain the phase shift between SSH and SSB observed in the numerical simulations because the former responds not only to SSB but also to the interior PV.

Some studies have investigated and characterized the phase shift between SST and SSH and the mechanisms associated with it. Hausmann and Czaja (2012) investigated the phase shift between SST and SSH associated with mesoscale eddies in the global ocean. They showed that SST anomalies tend to be westward shifted with respect to the eddy core for both cyclonic and anticylonic eddies and poleward (equatorward) for anticyclonic (cyclonic) eddies, recalling how vortices propagate in the ocean (Cushman-Roisin and Beckers 2011). A qualitative exploration of our simulations also revealed a systematic shift associated with eddies, although our analysis did not allow us to determine any preferred direction. Notice that the propagation paths of Mediterranean vortices form complex patterns due to the interaction between them and the constrains imposed by the topography (Isern-Fontanet et al. 2006b). Besides, the strong dependence on the MLD of the observed phase shifts suggests that it can play a nonnegligible role. Indeed, Klein and Hua (1990) investigated the role of the ML dynamics in the emergence of SST patterns and identified two different regimes. One regime is characterized by the ML deepening that generated the spatial variability of the SST that was a linear combination of subsurface temperatures and relative vorticity depending on the strength of the ML deepening. The other regime, which can be found after the wind stops, is characterized by the advection of the SST by the flow. This process generates fronts and energetic smaller scales.

Our results have shown that it is possible to reconstruct surface currents from SST observations in the Mediterranean Sea. However, the need to minimize the phase shift between SST and SSH implies that the reconstruction is constrained to winter, when less cloud-free images are available. This would force the use of the geostationary satellite to minimize the impact of clouds. Besides, our results have shown that the reconstruction in the Ionian Basin was not possible during most of the analyzed period. Because the performance of the reconstruction from SSB was similar to the results in the other areas, it points to the lower capability of SST to identify SSB patterns.

## 6. Conclusions

The conditions under which the SST can be used to diagnose surface velocities in the Mediterranean Sea have been identified. It has been shown that the reconstruction of upper-ocean dynamics from SST works better for larger scales than shorter scales and for ML deeper than approximately 70 m. In addition, results have confirmed that the transfer function has a SQG-like behavior for wavelengths shorter than 100 km. However, in most of the studied areas the transfer function differs from the SQG for longer wavelengths and was close to a constant. This implies that SST at these wavelengths can be considered a proxy of the streamfunction in some areas of the Mediterranean Sea. On the other side, our results have pointed to a new approach to improve the estimation of surface currents from satellite observations through the combination of simultaneous SST and SSH measurements.

## Acknowledgments

This study is a contribution to the MED3D Project funded by the Spanish R+D+i Plan (CTM2009-11020) and to the MICROVELS Project funded by the Fundación Ramón Areces (CIVP16A1819). JIF is funded through a Ramon y Cajal contract.

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## Footnotes

^{1}

In our notation, a phase shift with no imaginary part has been absorbed into the definition of the transfer function, that is, can take negative values.