## 1. Introduction

Wang et al. (2013, hereinafter W13) proposed a method for projecting surface density and height downward in the water column. The method requires simultaneous observations of surface density (or temperature, in the absence of salinity) and height. The density projection is made using the surface quasigeostrophic (SQG) approximation (Blumen 1978; Held et al. 1995; Lapeyre and Klein 2006; LaCasce and Mahadevan 2006; Tulloch and Smith 2006; Isern-Fontanet et al. 2008). The height is then used to deduce the two gravest baroclinic modes. W13 found that the SQG portion was most important in the near-surface region, while the baroclinic modes dominated at depth.

Here we simplify the method by using analytic solutions for the vertical projection. This obviates determining the SQG and baroclinic mode solutions numerically. We also consider a solution with a surface “mixed layer,” allowing us to gauge the latter’s effect on the construction.

## 2. Method

*N*(

*z*) is the Brunt–Väisälä frequency,

*ψ*=

*p*/(

*f*

_{0}

*ρ*

_{0}) is the geostrophic streamfunction, and

*Q*is the potential vorticity (PV). Because the PV equation is linear, the solution can be written as a superposition. The homogeneous solution is the SQG streamfunction

*ψ*

_{s}, while the particular solution is the “interior” streamfunction

*ψ*

_{i}(Lapeyre and Klein 2006). These have different surface boundary conditions:where

*b*

_{s}is the surface buoyancy. Thus, only the SQG solution is directly linked to the surface density.

*Q*is unknown, but the two gravest baroclinic modes can be deduced if the surface pressure is also known (W13). The baroclinic modes are solutions of the Sturm–Liouville problem:givenwith the same boundary conditions as for

*ψ*

_{i}. Here

*R*

_{n}is the

*n*th deformation radius and the

*γ*

_{n}are the modal coefficients.

^{1}As such, there is only a single unknown, the amplitude of the first baroclinic mode

*γ*

_{1}, which can be determined from the surface elevation

*η*:

### a. Exponential stratification

*N*=

*N*

_{0}

*e*

^{z/h}. Using this in (1) yieldsHere

*I*

_{n}are modified Bessel functions and

*L*

_{e}≡

*N*

_{0}

*h*/

*f*

_{0}is a deformation-like scale associated with the exponential stratification. The baroclinic modes, on the other hand, satisfyThe solution that decays with depth has the formwhere

*J*

_{1}is a Bessel function of the first kind. Imposing the surface condition [(2)] yieldsSo, the

*R*

_{n}are determined from the zeros of

*J*

_{0}. The first is 2.4048, soNotice that the eigenvalue problem is solved

*without*a transcendental equation. This is because of the choice of lower boundary condition.

### b. Mixed layer

*z*= −

*D*, and generally

*N*

_{m}≪

*N*

_{D}. The stratification

*N*is thus discontinuous at

*z*= −

*D*.

*z*= −

*D*, which are then matched at

*z*= −

*D*. The streamfunction is matched, so that the horizontal velocities are continuous. We also match

*z*= −

*D*.

It will be seen, however, that this condition yields unrealistic solutions, particularly when the mixed layer stratification is weak, so we tested matching the buoyancy instead at the mixed layer base (as would be done in the absence of a discontinuity in *N*). This produced more realistic density variations both near and below the mixed layer base.

## 3. Results

We evaluate the analytical solutions using fields from the same North Atlantic simulation discussed by W13. Full model details are given therein. The three regions lie in the western and eastern Atlantic, and in the subpolar gyre (Fig. 1). In each region we average the density laterally to obtain a profile for *N*(*z*) and use the result to fit the analytical *N* curves. We also calculate the rms density and vorticity as functions of depth, for comparison with the solutions.

The three stratification profiles, with the two idealized fits, are plotted in the left column of Fig. 2. The exponentials were obtained by fitting the deeper portion of *N*, below the region of rapid variation in the upper several hundred meters. For the mixed layer solution, the stratification *N*_{m} was obtained for the shallowest, weakly stratified layer, while the depth *D* was determined from the position of the maximum of *N* (about 50 m in regions 1 and 2 and 400 m in region 3). All parameters are listed in Table 1. The values of the deformation radius *R*_{1} range from 22.6 km (region 3) to 32 km (region 2).

Parameters for the two *N*^{2} fits. The depth range for the fitting is [380, bottom]. EXP and WML represent the cases with a pure exponential and with a mixed layer overlying an exponential, respectively. The deformation radii come from (12) (EXP) and (A12) (WML). The second estimate for WML (in parentheses) stems from using the exponential estimate [(12)], with *N*_{m} replacing *N*_{0}.

The results in Fig. 2 correspond to the *second* mixed layer solution in the appendix, in which ∂*ψ*/∂*z* is matched at *z* = −*D*. This requires solving a transcendental equation, given in (A12). However, the first root varies little for reasonable values of the mixed layer stratification and depth, as indicated in Table 1. Thus, *R*_{1} is well approximated by the pure exponential result [(12)], with the deep stratification *N*_{D} replacing the surface value *N*_{0}. As such, neither analytical profile requires a numerical solution for the baroclinic mode.

Shown in the other columns of Fig. 2 are the standard deviations of the density and vorticity plotted against depth. The results for the exponential and mixed layer solutions are shown, as are the curves obtained by W13. The latter derive from a numerical solution for the baroclinic modes, using the actual stratification shown in the left panels.

In region 1, the vorticity deviations (Fig. 2c) are of similar magnitude for all three solutions, down to roughly 700 m. The density deviations (Fig. 2b) are also similar, and the solutions capture the subsurface maximum seen near 250-m depth. The exponential and mixed layer solutions behave much the same, though the latter is better in the mixed layer itself; below that, the two yield very similar results. Moreover, both analytical solutions are as successful as the numerical solution of W13.

Similar comments apply in region 2 (Figs. 2d–f). The vorticity variations are somewhat better for the simple exponential solution, although the differences from the other solutions are small. Interestingly, the two analytical solutions perform better than the full numerical solution in terms of density, as the latter yields much greater variations with depth. Evidently, the analytical fit smooths out the small-scale structures in *N*, which have little impact on the density variations. Again, the mixed layer solution is most successful in the mixed layer itself.

Region 3 differs because the mixed layer is substantially deeper, extending to roughly 400 m. The mixed layer solution accordingly performs better. The predicted density variations are nearly depth independent in the mixed layer itself, as in the model. The exponential solution instead yields density variations that decay monotonically with depth. The full numerical solution varies even more with depth, but in this case the variations are realistic. It performs best below 1000 m, where the analytical solutions are less accurate. However, the differences for the vorticity deviations are much less. All three solutions yield reasonable estimates in the upper 500 m.

Thus, the analytical solutions are generally as good as the reconstructions of W13, despite their idealizations. Including the surface mixed layer improves the density variations in the mixed layer itself but has relatively little impact on the horizontal velocities. Below the mixed layer the simpler exponential fit yields equally good fits for both density and vorticity.

As noted, the mixed layer solution uses an improper matching condition on ∂*ψ*/∂*z* at the base of the mixed layer. The result is that the density is continuous at *z* = −*D* but the vertical velocity is not. Matching *w* instead yields the solution in (A6). The two mixed layer solutions are compared in Fig. 3, using the region 3 fields.

Matching (∂*ψ*/∂*z*)/*N*^{2} instead of ∂*ψ*/∂*z* produces a discontinuity in the density variations at the mixed layer base. The variations in the mixed layer in the former solution are smaller than in the latter, and they are much larger below. The vorticity deviations in the mixed layer are also better captured by the solution with a continuous density; the other solution grossly overestimates the variations. But while both solutions produce too large deviations below the mixed layer, the continuous *w* solutions are much greater, producing the appearance of a discontinuity at the mixed layer base. The curve is actually continuous, but the deviations increase greatly over a small depth range.

A third solution was also tested, in which *N* was assumed to increase exponentially to the base of the mixed layer and decay exponentially below (see the appendix). While the results (not shown) were better than with the continuous *w* solution above, they were still significantly worse than with the continuous density solution.

Thus, we retain the mixed layer solution that matches ∂*ψ*/∂*z* at *z* = −*D*. This is obviously a practical choice rather than a rigorous one, as the solution implies discontinuous vertical velocities. The unrealistic element is the discontinuity in *N*, as the model profiles are instead continuous. We retain the continuous density solution in the interest of having a relatively simple analytical profile.

Further comparisons are shown in the subsequent figures. Snapshots of the density anomaly in region 1 are shown in Fig. 4. The model fields are in the left column and the exponential and mixed layer solutions are in the middle and right columns, respectively. Figures 4a–c show plan views of the density anomalies at 520-m depth, and Figs. 4d–f show density cross sections taken along the dashed line in the upper panels, near 40.5°N.

Both analytical solutions capture the horizontal structure and amplitude of the model anomalies (Figs. 4a–c), beyond relatively minor regional differences. Note that 520 m is well below the mixed layer depth (roughly 50 m). The vertical structure (Figs. 4d–f) is also very similar. The analytical solutions decay more slowly with depth, a consequence of using only a single baroclinic mode in the decomposition. Nevertheless, the overall picture is very similar. The mixed layer solution is slightly better in the mixed layer itself, capturing, for example, the vertical variations near 43°W, but the exponential solution is basically as good.

The comparisons in region 2 are very similar and are not shown. The fields for region 3 are contoured in Fig. 5. The depth (520 m) for the plan views is just below the mixed layer (400 m). Again, the structures and amplitudes are very similar, outside of regional variations. Differences are more apparent in the vertical though, with the mixed layer solution clearly superior at capturing the variations in and just below the mixed layer. Again, the decay below the mixed layer is too gradual in both solutions because of having only a single baroclinic mode.

The vorticity fields compare similarly. In region 1 (Fig. 6), the two solutions produce realistic structures with reasonable amplitudes. The vertical slices too are very similar, above 800 m. However, the model anomalies extend deeper than in the solutions, which is unsurprising given that the barotropic component is absent in the solutions. But the result is good in the upper part of the water column, and there is no appreciable improvement with the mixed layer model.

Similar comments apply in region 3 (Fig. 7). The solutions are comparable in the upper 1000 m, but the model vorticity anomalies extend deeper. And here too, the mixed layer yields only minor changes from the pure exponential.

The similarities between the solutions and the model are quantified in Fig. 8, which shows the correlations between the analytical solutions and the model density (left) and vorticity (right) as functions of depth. In region 1 (top panels), the correlations for the density for both solutions (solid contours) are between 0.8 and 1.0 in the upper 1000 m. The correlations for the vorticity are similarly high. Moreover, the two solutions are not greatly different.

W13 found that the SQG contribution to the full solution was generally less than that of the baroclinic modes. We examined this by calculating the correlations for the SQG portions alone (dashed contours in Fig. 8). The correlations for the density are comparable to those for the full solutions in the upper 50 m. This is as expected, since the SQG solution matches the model density at the surface, but the correlations decrease below the mixed layer; at 500 m they are nearer to 0.5 for both analytical profiles. Of course, the correlations do not reflect the amplitude of the SQG contribution, and the latter decrease even more rapidly in comparison to the model’s (not shown). The correlations for the SQG vorticity (right panel) are lower even at the surface, being slightly less than 0.6. This reflects that the surface density is not always aligned with the surface pressure (W13). Thus, the baroclinic mode contribution is more important in this regard.

The results are qualitatively the same in the other two regions (middle and bottom panels). It is striking in particular that the two idealized profiles yield such similar correlations. Despite that the mixed layer improves the amplitude of the density variations in the mixed layer, it changes the structures little.

## 4. Summary and conclusions

We extended the study of W13 for predicting subsurface velocities and density from surface buoyancy and sea surface height. We use analytical solutions to make the vertical projection, assuming either an exponential *N* profile or an exponential with a mixed layer at the surface. The solution is a combination of an SQG component and the first baroclinic mode.

The solutions perform remarkably well in comparison with subsurface fields from a North Atlantic simulation. Both the density and vorticity fields are realistic, above about 1000 m depth. Moreover, the comparisons are generally as good as those of W13, who used numerical solutions to obtain the vertical dependence.

Several additional points are of interest. Including a surface mixed layer improves the density fields in the mixed layer itself but has little effect below. Also, both solutions yield similar vorticity fields. This suggests that a simple exponential profile may suffice in many applications, and this can be easily determined from climatological density.

Second, the reconstructions require only a single baroclinic mode. We have dispensed with the barotropic mode and accordingly miss variability at great depths, but this is to be expected as the fields are reconstructed solely from surface data. Nevertheless, if a barotropic mode is desired, a flat bottom condition can be applied, as with the exponential *N* solution of LaCasce (2012). Then the two unknown modal amplitudes would be determined as in W13. The price would be having to solve a transcendental equation for the baroclinic modes, something we have avoided here.

The mixed layer solution has a discontinuity at the base of the mixed layer, and the solutions require matching conditions there. Matching the density rather than the vertical velocity yields better results, despite that the latter is theoretically preferable. The reasoning is that the weakly stratified mixed layer is always joined to the stratified interior via a transition region, so that the density is continuous. Including such a layer in the theoretical model is possible but would defeat the purpose of having a simplified solution.

We found too that the SQG portion of the solution is most important in the mixed layer, where the stratification is weak and the PV near zero. Thus, the SQG construction is probably most valuable as an idealized representation of mixed layer flow.

## Acknowledgments

It is with pleasure and sadness that we acknowledge Bach Lien Hua, a good friend and an inspiration. She made major contributions to oceanography, in problems like the one considered here. Thanks to two anonymous reviewers, Paola Cessi, Bill Young, and Jörn Callies for diverse comments. The work was supported in part by NSF Grant OCE-1234473 (Wang) and by the Norwegian Research Council Grant 221780, NORSEE (LaCasce).

## APPENDIX

### The Mixed Layer Solutions

*L*

_{D}≡

*N*

_{D}

*h*/

*f*

_{0}. We obtain

*A*

_{1}and

*A*

_{2}by matching

*z*= −

*D*. As noted, this guarantees continuity of the horizontal and vertical velocities, respectively. The result iswith

*L*

_{m}≡

*N*

_{m}

*D*/

*f*

_{0}. In the limit of vanishing surface stratification,

*ψ*is nearly barotropic in the mixed layer and proportional to the surface buoyancy.

*z*= −

*D*yields a transcendental equation for the

*R*

_{n}:We solve (A6) numerically, using Newton’s method.

*γ*

_{1}is determined again from the surface elevation. As this only involves the mixed layer solution, the result is particularly simple:

The solution obtained by matching ∂*ψ*/∂*z* instead at *z* = −*D* is very similar. Having a continuous buoyancy at the interface with a discontinuous *N* is equivalent to having a delta-function PV sheet at the mixed layer base. Such a discontinuity can support buoyancy anomalies (Bretherton 1966; Plougonven and Vanneste 2010; Smith and Bernard 2013), which in turn can permit Eady-type instabilities in the mixed layer (Pedlosky 1987). For the present solutions, however, the density anomaly is assumed confined to the upper surface, precluding instability.

*J*

_{0}. As such, one can approximate

*R*

_{1}by

*L*

_{D}/2.4048 in the full solution.

*R*

_{1}:The constant

*γ*

_{1}is again determined by (A8).

*z*= −

*D*. But the results were as unsatisfactory, as with the first solution described above. The density variations increased below the surface to unrealistically large values near the mixed layer base, and the vorticity variations were overly large at depth, so we chose to focus on the second solution given above.

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^{1}

Over steep topography the barotropic mode is replaced by a bottom-intensified topographic wave mode, which generally cannot be deduced from surface information alone.