1. Introduction
Vertical velocities associated with mesoscale O(100) km and submesoscale O(10) km processes in the upper ocean are important for vertical exchanges of heat, salinity, and tracers, such as nutrients and dissolved organic and inorganic carbon, between the surface mixed layer (ML) and the ocean interior (Klein and Lapeyre 2009). These vertical velocities are usually weaker than 100 m day−1 (1 mm s−1) and are therefore very difficult to measure directly. Instead, their balanced components have often been estimated indirectly from observations of density and horizontal velocity using the omega equation (Hoskins et al. 1978), but such estimations require in situ observations obtained from high-resolution ship surveys (e.g., Pollard and Regier 1992; Rudnick 1996; Legal et al. 2007). A decade ago, Lapeyre and Klein (2006) and LaCasce and Mahadevan (2006) proposed a method to diagnose the three-dimensional (3D) balanced motions in the upper ocean from a single snapshot of surface buoyancy or height anomalies, combined with a knowledge of the background stratification, allowing to estimate vertical velocities in the upper ocean from satellite observations (Isern-Fontanet et al. 2006).
The method developed by Lapeyre and Klein (2006) is based on the principle of invertibility of potential vorticity (PV; Hoskins et al. 1985), which states that given a balance condition and a reference state, a knowledge of the global distribution of PV and of buoyancy at the boundaries is sufficient to diagnose all the other adiabatic dynamical fields, including vertical velocities. Using the fact that in the quasigeostrophic (QG) balance condition, PV is a linear operator, Lapeyre and Klein (2006) decomposed the PV problem into a surface problem (with a nonzero surface buoyancy and zero interior PV) and an interior problem (with zero surface buoyancy and nonzero interior PV). The total solution is then the sum of the surface and interior solutions. For the surface problem, the special case of an infinitely deep ocean with buoyancy anomalies vanishing at great depths has received much attention and was dubbed surface quasigeostrophic (SQG) dynamics by Held et al. (1995) (see also Blumen 1978).
Lapeyre and Klein (2006) showed that for a baroclinic unstable flow forced by a large-scale mean PV gradient including a nonzero surface buoyancy gradient, PV anomalies in the interior are correlated with surface buoyancy anomalies when large-scale buoyancy gradients at the surface and in the interior are correlated. As a result, the interior and surface solutions are correlated, and the total solution can be approximated in the upper ocean by an “effective” SQG (eSQG) solution, using a constant reference stratification N(z) = N0 that must be determined empirically. While Lapeyre and Klein (2006) obtained spatial correlations greater than 0.85 between vertical velocities from a primitive equation (PE) numerical simulation and those diagnosed by the eSQG inversion in the first 500 m of their model, Klein et al. (2009) obtained much poorer correlations (<0.6) in the ML of another PE simulation with higher horizontal and vertical resolution, in which the ML was forced by spatially homogeneous high-frequency winds. Using the PE simulation of Klein et al. (2009), Ponte et al. (2013) showed that correlations for vertical velocities significantly increased at 40-m depth (within the ML) when adding diabatic vertical velocities due to turbulent vertical mixing of momentum (Garrett and Loder 1981) to the adiabatic vertical velocities diagnosed by the eSQG inversion.
However, the eSQG model, by considering a constant background stratification, does not take into account the fact that the stratification within the ML is usually much weaker than that in the ocean interior. The discontinuity between the ML and interior stratification leads to different physics for the ML than for the ocean interior by allowing interactions within the ML between edge waves associated with buoyancy anomalies at the ocean surface and ML base (Flierl et al. 2014). Such interactions lead to baroclinic instabilities as in the Eady problem (Eady 1949) that captures their main features in the QG approximation. These ML instabilities (MLIs; Stone 1971; Haine and Marshall 1998; Boccaletti et al. 2007; Fox-Kemper et al. 2008) have recently been shown to dominate vertical velocity variance in deep winter MLs in high-resolution numerical simulations of the northwestern Atlantic (Mensa et al. 2013) and North Pacific (Sasaki et al. 2014) Oceans. Observations in the western subtropical North Atlantic confirm this seasonal cycle in the ML submesoscale energy (Callies et al. 2015).
Two questions therefore arise: What are the relative contributions of adiabatic and diabatic dynamics to vertical velocities in the ML? Can vertical velocities associated with MLIs be diagnosed solely from surface observations? To address these questions, we develop the simplest possible diagnostic QG model that explicitly takes into account the presence of the ML in the ocean. We show that under the assumption of zero PV anomalies in the ML, buoyancy anomalies at the ML base can be recovered if observations of both surface buoyancy and height anomalies are taken into account. This enables a diagnosis, based solely on surface observations, of vertical velocities associated with MLIs in dynamical regimes where the Rossby number Ro ≪ 1. We also consider diabatic ML processes by including turbulent vertical mixing of momentum and buoyancy in the QG equations. We diagnose diabatic vertical velocities within the ML using a diabatic omega equation instead of the approximate solution of Garrett and Loder (1981) used by Ponte et al. (2013). The model, which we call MLQG hereinafter, is developed in section 2 and is then tested in section 3 against the numerical simulation reported in Klein et al. (2009) and Ponte et al. (2013). The originality and limitations of the MLQG model are discussed in section 4. The main model equations are summarized in the appendix.
2. Quasigeostrophic model with a surface mixed layer
Schematic of the MLQG model. Bold symbols represent the model inputs.
Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-14-0178.1
(a) Potential density and (b) buoyancy frequency (normalized by f) in the PE simulation (gray line represents the horizontal average and shading represents the standard deviation) and in the MLQG (solid black line) and SQG (dashed black line) models. Dotted horizontal line represents the ML base in the MLQG model. Buoyancy N0 in the MLQG and SQG models is lower than values of N in the PE simulation between 100 and 500 m to take into account the effects of interior PV anomalies on the total solution [see text in introduction and Lapeyre and Klein (2006)]. (c) Turbulent vertical diffusivity prescribed in the MLQG model.
Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-14-0178.1
(a) Streamfunction, (b) buoyancy anomalies, and (c) normalized interface displacements in the PE simulation. The black square delimits the area shown in Fig. 5.
Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-14-0178.1
The equations required to diagnose the 3D circulation in the upper ocean using the MLQG model developed in this section are recalled in the appendix in dimensional form.
3. Numerical results
a. Numerical simulation
We apply the MLQG model to data from a PE numerical simulation of a turbulent flow field in a zonal β-plane channel centered at 45°N with a size of 1000 km by 2000 km and a depth of 4000 m. This numerical simulation has been described in detail by Klein et al. (2008, 2009), Danioux et al. (2011), and Ponte et al. (2013). Here, we only briefly describe its main features. The numerical code is the Regional Oceanic Modeling System (ROMS; Shchepetkin and McWilliams 2005), with the KPP parameterization for vertical mixing (Large et al. 1994) and a bi-Laplacian parameterization for horizontal mixing (with Ah = 2 × 106 m4 s−2). The domain is zonally periodic and has vertical walls at the northern and southern boundaries with free-slip boundary conditions. The spatial resolution is 2 km in the horizontal and 100 levels in the vertical (with vertical grid spacing ranging from ~3 m near the surface to ~200 m near the bottom). The horizontally averaged vertical profile of buoyancy frequency N (Fig. 2b) features a permanent pycnocline centered at 600-m depth (not shown) that corresponds to a first Rossby radius of deformation Rd = 23 km. The mesoscale turbulence (Fig. 3) is forced by using a relaxation to a basic state that corresponds to a large-scale meridional density gradient that is surface intensified. An active ML with a mean depth of ~60 m and a ML deformation radius O(100) m, forced by spatially uniform but temporally varying winds and surface heat fluxes, caps a seasonal pycnocline. The data used here has been averaged over 18 h to filter out near-inertial oscillations.
b. Diagnosis of balanced 3D circulation in the upper ocean
The resting depth of the interface between the two layers of the MLQG model was chosen to be the depth of maximum background buoyancy frequency of the PE simulation (H = 70 m; Fig. 2b). Background buoyancy frequencies for the MLQG model were chosen to be Nm ~ 3f0 and N0 ~ 30f0 (as in Klein et al. 2009) to approximate the vertical profile used in the PE simulation in the upper 500 m. The value of N0 is lower than the values of N in the PE simulation between 100 and 500 m to take into account the effects of interior PV anomalies on the total solution (Lapeyre and Klein 2006), as discussed in the introduction. The buoyancy jump between the ML and the interior was chosen to be ΔB = 5 × 10−3 m s−2 to approximate the mean density profile of the PE simulation (Fig. 2a). The maximum vertical viscosity and diffusivity coefficients in the ML of the MLQG model were chosen to be A0 = K0 = 1.5 × 10−2 m2 s−1 (Fig. 2c), so that the ratio of the peak in MLQG vertical velocity variance in the ML over that in the interior matches that of the PE simulation (Fig. 4b). The corresponding Ekman depth is hE = 
(a) Spatial (horizontal) correlations as a function of depth between MLQG (solid lines) or SQG (dashed lines) and PE buoyancy anomalies (black), relative vorticities (red), and vertical velocities (blue). (b) Their standard deviations are shown for the PE simulation (thick solid lines) and MLQG (thin solid lines) and SQG (dashed lines) models. Horizontal dotted lines indicate the ML base in the MLQG model.
Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-14-0178.1
Figures 3a and 3b show the surface streamfunction and buoyancy from the numerical simulation at an arbitrary time (after the simulation has reached statistical equilibrium). The background state (zonally averaged fields) was removed to obtain surface streamfunction and buoyancy anomalies. These 2D fields were Fourier transformed, and the 3D circulation was diagnosed with the MLQG model described in section 2. To assess the validity of the assumption behind the MLQG model, namely, that interface displacements are small compared to the ML depth [i.e., η/H = (h − H)/H = O(Ro) ≪ 1], the ML depth h(x, y) was determined as the depth of the maximum value of buoyancy frequency at each grid point. Normalized interface displacements η/H are shown in Fig. 3c. Although interface displacements can be as large as the average ML depth along some fronts, they remain less than ~0.25H over most of the domain. Therefore, the QG assumption behind the MLQG model seems reasonable, except at strong fronts.
Figure 4a shows spatial (horizontal) correlation coefficients r as a function of depth between buoyancy anomalies, relative vorticities, and vertical velocities of the PE simulation and those diagnosed with the MLQG model. Relative vorticities are well correlated in the first 300 m (r > 0.85), but the correlation decreases quickly below 300 m. Buoyancy anomalies are almost perfectly correlated (r ~ 1) in a large part of the ML because of the use of surface buoyancy anomalies as input to the MLQG model [Eq. (22)]. Correlation decreases to 0.7 at the interface between the two layers. The MLQG model is unable to reproduce the local enhancement of buoyancy anomalies in the seasonal pycnocline (Fig. 4b, black lines). Correlation increases again above 0.9 between 100 and 500 m, but variance is underestimated. Vertical velocities are well correlated near the surface (r ~ 0.9), in contrast to the SQG-diagnosed vertical velocities for which r < 0.5 (dashed blue line in Fig. 4a), but correlation decreases to less than 0.7 (less than half the variance explained) in the seasonal pycnocline, below which it increases again to 0.77 between 100 and 500 m. Even at the depth where the correlation is a minimum, the main patterns of vertical velocities are qualitatively well reproduced by the MLQG model (Fig. 5). The local enhancement of vertical velocity variance in the ML is also reproduced by the MLQG model but not by the SQG model (Fig. 4b, blue lines).
Vertical and horizontal (at the ML base) sections of vertical velocities (color) and buoyancy anomalies (contours, spaced by 10−3 m s−2, solid for positive and dashed for negative values) are shown for (a) the PE simulation and (b) the MLQG model. The horizontal area is indicated by a black square in Fig. 3.
Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-14-0178.1
To understand the MLQG model strengths and limitations, it is helpful to look at correlation in the spectral domain or coherence c (Fig. 6, bottom panels). For buoyancy anomalies (Fig. 6g), coherence is close to 1 at all horizontal scales in the ML because of the use of surface buoyancy anomalies as input to the MLQG model as mentioned above. However, coherence quickly drops in the seasonal pycnocline for wavelengths smaller than 100 km, for which the validity of the QG approximation becomes questionable (2πRd = 145 km) but remains high for wavelengths between 100 and 350 km. For relative vorticity (Fig. 6h), coherence is large at horizontal scales and depths where most of the energy resides (Fig. 6b), explaining the high spatial correlation coefficients above 300 m. However, as horizontal scales decrease, the depth down to which coherence is large gets shallower, while energy remains fairly uniform with depth above 500 m in the PE simulation (Fig. 6b), explaining the decrease of the spatial correlation coefficient below 300-m depth (Fig. 4a). For vertical velocity (Fig. 6i), coherence is large at all wavelengths smaller than 350 km in the ML, and at wavelengths between 60 and 350 km below the ML, where most of the energy resides (Fig. 6c).
Spectral distributions of (top) variance in the PE simulation, (middle) variance in the MLQG model, and (bottom) coherence between MLQG and PE for (left) buoyancy anomalies, (center) relative vorticities, and (right) vertical velocities. Dotted horizontal lines indicate the ML base in the MLQG model.
Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-14-0178.1
To assess the relative contributions of adiabatic and diabatic dynamics to vertical velocities in the ML, the omega equation [Eq. (23)] is also solved by setting 
(top) Spectra of vertical velocities at 40-m depth for the PE simulation (black) and the MLQG (magenta) and SQG (cyan) models. The adiabatic (blue) and diabatic (red) contributions to the MLQG vertical velocities are also shown. (bottom) Squared coherence between the diagnosed and PE vertical velocities.
Citation: Journal of Physical Oceanography 46, 1; 10.1175/JPO-D-14-0178.1
4. Discussion
A QG model with a ML (MLQG model) has been developed to diagnose the 3D circulation, including vertical velocities, in the upper ocean from only surface observations of buoyancy and height anomalies and knowledge of the background stratification. Diabatic dynamics (namely, turbulent vertical mixing) are included in the ML.
The formulation for the adiabatic component departs from the classical SQG framework since it takes into account the presence of the ML by considering two constant stratification layers: a finite-thickness weakly stratified ML and an infinitely deep interior layer. PV is assumed to be constant within each layer. So the solution for the adiabatic dynamical component within the ML is similar to the Eady solution and that in the interior to the SQG solution. This configuration allows us to diagnose vertical velocities associated with MLIs within the limitations of the QG approximation. As pointed out by Haine and Marshall (1998), although the QG analysis of Eady (1949) is formally inapplicable to the oceanic ML because of its finite Richardson number, it is yet qualitatively correct, and a similar qualitative success may be expected for the diagnostic model presented here. The main problems to be solved are the boundary conditions at the interface between the ML and the interior. Because the streamfunction is continuous at the ML base at zeroth order in Ro, knowledge of surface streamfunction and buoyancy anomalies are sufficient to determine the streamfunction at all depths, without requiring knowledge of buoyancy anomalies at the ML base. This is a key point of our model.
Recently, LaCasce and Wang (2015) also used a simplified QG model incorporating a surface ML to diagnose subsurface horizontal velocities and buoyancy anomalies from surface fields. Their model differs from ours in two aspects. First, they explicitly solve the interior problem by projecting interior PV anomalies on the first baroclinic mode. Second, they use a matching condition at the ML base not only for streamfunction (which they require to be continuous as we do), but also for buoyancy anomalies, so that the surface problem can be solved using only surface buoyancy anomalies. This leaves sea surface height, minus the contribution from the surface solution, to determine the interior solution. They tested two different matching conditions for buoyancy anomalies at the ML base: b/N2, which guarantees continuity of w, and b itself, which leads to a discontinuity in w, an embarrassing feature when attempting to diagnose vertical velocities as we do here. Although matching b/N2 guarantees continuity of w for adiabatic dynamics, it is not a necessary condition since what needs to be matched is actually D(b/N2)/Dt [see Eq. (4)], which is done implicitly in our model through the omega equation.
The MLQG diagnosis was tested against high-resolution PE simulations of a turbulent flow field in a zonal β-plane channel. The main improvement over the classical SQG diagnosis is obtained for vertical velocities in the ML, which are well correlated with the PE velocities at all scales (Figs. 6i, 7b). This improvement is due to the diabatic vertical velocities, whose magnitude is larger than that of the adiabatic vertical velocities for wavelengths shorter than 30 km (Fig. 7a). However, the ML depth in the PE simulation used here is moderate (60 m), and consequently MLIs should be weak and are probably not even resolved with the 2-km horizontal resolution used here. For deeper MLs, such as in winter in large parts of the ocean, MLIs are much more energetic, and they dominate vertical velocity variance (see Fig. 5b of Sasaki et al. 2014). Whether the MLQG model would provide a good diagnostic for deep MLs remains to be verified.
There are many limitations to the MLQG model. One is the determination of the interior solution to the PV inversion problem. We have chosen to neglect the contribution of the interior solution in the ML under the premise that QG PV is close to zero in the weakly stratified ML. Below the ML, we have chosen to follow the approach of Lapeyre and Klein (2006) for its simplicity since the focus here was on the dynamics in the diabatic ML. More elaborate approaches have been proposed (e.g., Wang et al. 2013; Ponte and Klein 2013) and could possibly be adapted to the MLQG framework. Another limitation is the assumption of small Ek, which prevents resolving the surface Ekman layer. The effects of Ekman dynamics would be to modify the surface boundary condition for 
Acknowledgments
CC thanks Glenn Flierl for a useful discussion on the matching condition [Eq. (16)]. We thank both reviewers who significantly contributed to the improvement of the original version of this work. This work was supported by an individual discovery grant to CC from the Natural Sciences and Engineering Research Council of Canada. Numerical simulations reported here were done at IDRIS (France) and on the Earth Simulator (Yokohama, Japan) through a memorandum of understanding signed between IFREMER, CNRS, and JAMSTEC. PK acknowledges the support of IFREMER (through the MOU IFREMER-JAMSTEC), CNRS (France), and the Agence Nationale pour la Recherche [Contracts ANR-05-CIGC-010, ANR-09-BLAN-0365-02 and ANR-10-LABX-19-01 (LabexMER)].
APPENDIX
Dimensional MLQG Equations
We recall here the main steps and equations required to diagnose the 3D circulation in the upper ocean from the sole knowledge of surface streamfunction ψs and buoyancy bs anomalies [and background stratification N(z)] using the MLQG model developed in section 2. The equations are given in dimensional form for easier application to oceanographic data.
The second step consists of computing the 2D Fourier transforms 
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