The fundamental principle highlighted in this study is that the balance between the wind stress and the surface shear, expressed in the standard surface boundary condition (τ0 = ρ0ν ∂u/∂z), requires consideration of both the geostrophic and ageostrophic components of the shear. In horizontally homogenous regions, the near-surface geostrophic (thermal wind) shear is negligible and thus the surface shear that balances the wind stress is entirely ageostrophic, as in the balances of Ekman (1905) and Stommel (1960). In frontal regions, however, the geostrophic shear can be substantial, particularly in the tropics where the 1/f factor is large. When winds blow along a front, wind stress may be partially or entirely balanced by the geostrophic shear. As a consequence, the ageostrophic shear required to make the total shear balance the wind stress may be much smaller than expected.
Two models were introduced to describe the mean near-surface shear in the cold tongue front region at 2°N, 140°W: a frontal Ekman model (5) and a generalized Ekman model (7). Both models assume steady, linear flow. As with the classical Ekman model (4), the frontal Ekman model assumes that viscosity is vertically uniform, but differs from (4) by allowing a vertically uniform density front and, thus, a vertically uniform geostrophic shear. At the bottom of the frictional layer, the ageostrophic flow is assumed to be zero; while at the surface, the total (geostrophic and ageostrophic) shear is assumed to be proportional to the wind stress. The top boundary condition for the ageostrophic shear thus depends not only upon the wind stress, but also upon the strength and orientation of the front. The resulting solution for the ageostrophic shear is that of a classical Ekman model forced by the portion of the wind stress that is out of balance with the surface geostrophic shear.
At 2°N, 140°W, warm water lies to the north and west and the surface geostrophic shear was thus oriented to the southwest. With a viscosity estimated at 16 × 10−3 m2 s−1, the magnitude of the stress associated with surface thermal wind was comparable to that of the southeasterly trade wind stress. Consequently the westward component of the geostrophic shear nearly balanced the westward wind stress so that the effective surface stress that forced the ageostrophic shear was roughly northward. The resulting near-surface ageostrophic shear is primarily northward and rotates to the right (east) with depth, much like a classical Ekman spiral forced by a northward wind stress. When added to the geostrophic shear associated with the cold tongue front, the resulting total shear was more to the left (i.e., more westward) than the wind stress, as observed (Figs. 2a,c).
The generalized Ekman model is the vertical derivative of the steady, linear equations of motion (1), with shear written in terms of stress according to (2a). The model requires the prescribed viscosity to decay to zero at the level of no stress, but does not require the prescribed horizontal density gradient to be zero (as required by the classical Ekman model), nor be vertically uniform (as required by the frontal Ekman model). Furthermore, the generalized model is valid on the equator, while both the classical and frontal Ekman models are not.
For the generalized Ekman model at 2°N, 140°W (Fig. 2d), the front was assumed to be vertically uniform and viscosity was assumed to decay exponentially to a value of zero by twice the decay scale (set to 125 m, the top of the thermocline). When forced by the observed mean wind stress, the generalized Ekman model, like the frontal Ekman model, predicted currents relative to 25 m that were to the left of the mean wind stress, roughly in agreement with the observed currents relative to 25 m (Figs. 2a,d). Although the data clearly show that the classical Ekman model (4) is inappropriate for this region (Figs. 2a,b), the simulations from the frontal and generalized Ekman models are quite similar and both agree well with the observations. Indeed, with a layer depth of 250 m, similar to that used in the generalized model, and with viscosity and a buoyancy gradient similar to that used in the frontal model, the Bonjean and Lagerloef (2002) model also produced currents relative to 25 m that were to the left of the wind stress (not shown). These models differ primarily in their bottom boundary conditions and thus their differences would likely become more apparent with deeper data. The shallow observations studied here cannot definitively distinguish between these modified Ekman models.
A composite tropical instability wave illustrates the sensitivity of the ageostrophic shear to the orientation of the front relative to the winds (Fig. 4). Although winds were relatively steady throughout the cycle, the observed shear was less so, and the ageostrophic shear was not. During the transition from the cold phase to warm phase, when the front was aligned with the winds with warm water to the right, the near-surface ageostrophic shear was also aligned with the wind. During the transition from the warm phase to cold phase, when the front was oriented southwestward, oblique to the southeasterly winds, the near-surface shear had a large northward ageostrophic component that countered the southward component of the geostrophic thermal wind shear.
This is not the first study to recognize that in frontal regions the near-surface geostrophic shear contributes to the frictional stress. The shear equation derived by BL02 explicitly treats shear as a whole, rather than as composed of geostrophic and ageostrophic components. As shown by Garrett and Loder (1981), Thompson (2000), Flament and Armi (2000), Nagai et al. (2006), and others, the geostrophic contribution to the frictional stress produces an Ekman transport that is to the left of the geostrophic shear (i.e., toward the cold side of the front). Since this transport is maximum at the center of the front, it implies convergence and downwelling on the cold side of the front and divergence and upwelling on the warm side of the front. The wind stress thus tends to enhance this secondary circulation when the winds blow up a front, and tends to oppose it when the winds blow down a front, as is the case at 2°N, 140°W. Using a semigeostrophic model, Thompson (2000) showed that the resultant secondary circulation tends to spin down the front when the winds blow up a front, and tends to maintain the front when winds blow down the front.
The poleward currents observed in this study were substantially weaker than Johnson et al. (2001)'s mean poleward currents that were based upon nine years of shipboard ADCP sections between 95° and 170°W. A southwestward oriented front, such as observed at 2°N, 140°W, reduces the expected northward current in two ways: by causing the southward geostrophic current to be surface intensified and by causing a westward geostrophic frictional stress that tends to balance the westward wind stress. If these frontal effects do, in fact, cause a local minimum in the surface poleward flow near 2°N, 140°W, this may not be evident in the Johnson et al. (2001) shipboard ADCP section due to the considerable temporal, meridional, and zonal averaging done in that analysis. If this is the case, then the cold tongue front would likely be associated with a secondary circulation that had downwelling on the cold side of the front and upwelling on the warm side. With a single mooring, however, this deduction is speculative. Likewise, with only five current meters on a single mooring, we cannot directly measure the ageostrophic transport, nor its heat transport. We note however that, because the ageostrophic Ekman currents depend upon the orientation of the front, care must be taken in diagnosing the Ekman heat transport.
Viscosity is generally not known and must be parameterized. In our analysis we found that a value of 16 × 10−3 m2 s−1 produced shears in the Ekman models (Figs. 2b–d) with similar magnitude to the observed near-surface shear. The value is more than three times larger than the Santiago-Mandujano and Firing (1990) wind speed–based parameterization. However, those measurements were made during the 1982/83 El Niño, when the cold tongue front was absent or not well developed, the thermocline was anomalously deep, and the surface mixed layer conditions were quite different than normal (Cronin and Kessler 2002). We note also that the Peters et al. (1988) turbulence measurements at 0°, 140°W indicate that viscosity averaged from 23- to 49-m depth have nighttime values of approximately 10 × 10−3 m2 s−1 (see their Fig. 16). Nighttime convective mixing likely causes the viscosity values above 23 m to be significantly higher than this. Although our study has no current measurements below 25 m, the observed mean Richardson number within the upper 25 m increases with depth (Fig. 7), indicating that viscosity in all likelihood decays with depth.
The influence of even weak stratification on the magnitude of the near-surface shear could be seen both in the tropical instability wave and the diurnal cycle composites. As shallow stratification forms during the day, a diurnal jet develops. The afternoon (1600 local time) mean diurnal jet was ∼12 cm s−1 faster in the direction of the wind at 5 m than at 25 m, while nighttime mean shears were very weak (less than 4 cm s−1 over 20 m). Because the winds were relatively steady, these changes must be due to changes in the viscosity associated with nighttime mixing and daytime restratification. The composite analysis also showed that, as the diurnal warm layer cooled and thickened during the late afternoon–early evening, the wind-trapped momentum can be seen at ever greater depths. The resulting downward propagation of shear-induced mixing is evident in the diurnal composite of the Richardson number and is consistent with the model study of Danabasoglu et al. (2006).
Wijffels et al. (1994) showed that a diurnal jet can form even for stratification that by many criteria would be considered mixed, and that a slab layer was observed only when a stringent definition of the mixed layer was used. In our case, the mean afternoon stratification between 1 and 25 m was ∼0.18°C, which is considered weakly stratified by their criteria. At nighttime, when the mean stratification was near zero, shears were very weak and the layer was more slablike. Although the nighttime shear is near the instantaneous measurement error of the Sontek current meters, the ensemble mean is significant if the errors are random. Nighttime mixing is not 100% efficient at removing shears, in part because the eddy viscosity, while large, is not infinite. The overall mean stratification was extremely weak (0.07°C over 27.3 m) and met the Wijffels et al. slab layer criteria. Yet, while the mean meridional currents were vertically uniform, the zonal currents were sheared (Fig. 1). Mixing should act on the total shear, not simply on the meridional component. Therefore, the uniform poleward currents on their own cannot be interpreted as evidence of a slab layer.
Finally, it remains an open question as to how the diurnal cycles of stratification, shear, and mixing influence the observed Ekman spiral. On the one hand, since mixing is a one-way process and the inertial time scale at 2°N is much longer than a day, nighttime mixing could set the overall effective viscosity. In that case, the viscosity that determines the Ekman spiral would be larger and extend deeper than might be expected based upon the mean stratification. The Danabasoglu et al. (2006) downward propagation of shear-instability mixing beginning in the late afternoon provides another mechanism by which the influence of wind forcing is felt below near-surface stratification.
Alternatively, the observed strong diurnal jet (Figs. 4–6) suggests that the Ekman spiral at 2°N, 140°W may be dominated by the afternoon stratification. Aspects of stratified Ekman spirals have been observed in the midlatitudes by Price et al. (1987) and Chereskin (1995), near the equator by Santiago-Mandujano and Firing (1990), along 10°N in the Pacific by Wijffels et al. (1994), along 11°N in the Atlantic by Chereskin and Roemmich (1991), and in the Arabian Sea by Chereskin et al. (1997). None of these studies, however, were in strong frontal regions. Indeed, metrics used to identify a stratified Ekman layer may need to be reconsidered for frontal regions. Price and Sundermeyer (1999) argued that at mid and high latitudes, currents in a stratified Ekman spiral are more strongly surface intensified than if the spiral occurred fully within a well-mixed layer. That is, a stratified Ekman spiral at these latitudes is flatter with thinner, faster layers near the surface and slower, thicker layers in the lower portion of the spiral. At low latitudes, Price and Sundermeyer hypothesized that the diurnal stratification would have less importance. Because our deepest measurement was at 25 m, the lower portion of the spiral at 2°N was not observed here. Thus, our study is unable to differentiate between the contrasting effects on the Ekman spiral of momentum trapping by near-surface stratification and leakage of momentum into the deeper water through nighttime and late-afternoon mixing.
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