Baroclinic Frontal Instabilities and Turbulent Mixing in the Surface Boundary Layer, Part II: Forced Simulations

Transcription

1 1 2 Baroclinic Frontal Instabilities and Turbulent Mixing in the Surface Boundary Layer, Part II: Forced Simulations Eric D. Skyllingstad *, Jenessa Duncombe, Roger M. Samelson CEOAS, Oregon State University, Corvallis, OR USA Submitted to the Journal of Physical Oceanography *Corresponding author address: College of Earth, Ocean, and Atmospheric Sciences, 14 CEOAS Admin Bldg, Oregon State University, Corvallis, OR USA skylling@coas.oregonstate.edu

2 Abstract Coherent ocean boundary layer roll instabilities are examined in the context of winddriven and geostrophic shear associated with horizontal density gradients using a large-eddy simulation model. Numerical experiments over a range of surface wind forcing and horizontal density gradient strengths indicate that the dominant instability mechanism is related to shear instability of the Ekman layer, with symmetric instability having minimal influence. Sheared inertial currents control the growth of instabilities through differential advection of the background horizontal density gradient, producing an oscillating upper-ocean stratification that periodically traps momentum near the surface. Rapid disturbance growth is produced when the stratification is reduced by the inertial cycle. The interpretation of the Ekman-instability source for the roll structures is supported by results from a companion linear instability analysis. A simple scaling model based on wind speed and horizontal temperature gradient is developed that accurately predicts the importance of these instability processes in ocean frontal zones

3 Introduction Mixing in the surface ocean boundary layer is typically produced by a combination of processes related to surface fluxes and the interaction of surface waves. Basic turbulence growth mechanisms common to both the atmosphere and ocean include shear instabilities generated by surface stress and buoyant instabilities produced by surface cooling of the ocean surface. Unique to the ocean is the generation of Langmuir circulation resulting from the interaction of surface wave Lagrangian motion and wind-forced upper ocean current shear (Craik and Leibovich 1976). In recent years, the global importance of Langmuir circulation in defining the ocean boundary layer thickness has been recognized (Belcher et al. 12). Improved representations of mixing including the effects of Langmuir circulation have been developed (e.g., McWilliams et al. 12) and are being adapted in global ocean applications (e.g., Noh et al., 16). In addition to surface-forced turbulent mixing in the ocean, internal ocean dynamics can also inject turbulence energy through current shear and buoyancy advection associated with surface fronts. The most direct way surface fronts can affect mixing is through differential advection by wind-forced Ekman transport, which has long been recognized as a potential source of mechanically-driven convection in both the coastal (e.g., Foo, 1981; Samelson and de Szoeke, 1988) and open-ocean environments (e.g., Samelson and Paulson, 1988; Thomas and Lee, 5). Ekman velocities are concentrated at the surface and directed to the right of the wind direction in the northern hemisphere. If winds are aligned with a front such that denser water is to the left of the wind direction and less dense water is to the right, then (in the northern hemisphere) Ekman current shear will be destabilizing, as it will transport denser water over less dense water, toward a convectively unstable state. This advective buoyancy flux divergence is sometimes

4 characterized as an effective surface, or Ekman, buoyancy flux; for strong fronts, it can lead to effective fluxes equivalent to 1 s of watts of surface cooling (Thomas et al., 13), which would be sufficient to cause vigorous convective mixing. Surface fronts are also prone to baroclinic instability and the formation of submesoscale features that stir the ocean as frontal zones restratify (Samelson, 1993; Samelson and Chapman, 1995, Capet, et al., 8). In Part I of this study (Skyllingstad and Samelson, 12), the evolution of a small-scale baroclinic instability in the surface boundary layer was studied in the absence of surface forcing. The baroclinic development caused frontogenesis and a transition to turbulence through Kelvin-Helmholtz instability at the intensified front (Skyllingstad and Samelson, 12; Samelson and Skyllingstad, submitted). The original intent of the present study was to revisit this development in the presence of wind forcing, and this is the initial case analyzed below. It is found that when the wind is aligned along the frontal axis, so that the steady wind-driven Ekman transport would be directed across the front, the baroclinic development is strongly affected by the wind-driven turbulent and mean advective responses. In addition, coherent roll structures form within the turbulent surface layer when the Ekman transport is directed toward the warm side of the front. Evidence of similar coherent structures in the ocean boundary layer has been previously reported, for example, by Sundermeyer et al. (15) from observations of dye releases in a mixed layer front along with related LES modeling experiments, and it has been previously suggested that observations of strong frontal mixing events along the Gulf stream (Thomas et al. 13; Thomas et al. 16) and Kuroshio current (D Asaro et al., 11) may also be associated with coherent-roll-structure instabilities within the boundary layer. Most of the simulations and analysis here therefore focus on the turbulent response and the coherent roll structures, and their dependence on the strengths and relative

5 alignment of the frontal gradients and wind forcing. Various candidate dynamical mechanisms for formation of the roll structures are considered, including Ekman instability (Lilly, 1966; Asai and Nakasuji, 1973), symmetric instability (Thorpe and Rotunno 1989; Taylor and Ferrari, 1; Thomas et al. 13; Thomas et al. 16), and Langmuir circulations (Hamlington et al.,14). Relatively few modeling studies have examined both the action of surface wind and wave forcing along with frontal dynamics. Taylor and Ferrari (1) first looked at this problem using a large-eddy simulation (LES) model for a growing boundary layer and found instances of disturbance growth consistent with symmetric instability. Hamlington et al. (14) presents work that examines how Langmuir cells interact with frontal spin down suggesting that Langmuir turbulence greatly enhances the mixing when acting on surface fronts. Wind forcing in their experiments was relatively weak (e.g. wind stress of.25 N m -2, equivalent to wind speeds of ~ 4 m s -1 ), but was still strong enough to dominate over symmetric instability in some cases. Simulation results presented in Thomas et al. (16) for a strong storm system acting on the Gulf stream front suggest that symmetric instabilities can drive strong mixing, especially when wind forcing is reduced. In general, the role that symmetric instability has in upper-ocean mixing when surface fluxes are large is not well understood. We address the frontal turbulence problem through a modeling study spanning a range of surface forcing and mixed layer frontal strength. We employ a high-resolution large-eddy simulation model with a constant background horizontal density gradient following Taylor and Ferrari (1). Our experiments differ from previous studies by assuming an existing mixed surface boundary layer and applying moderate to strong wind and wave forcing with relatively short spin up periods, to represent the onset of storm conditions. For the simulations focusing on the frontal-zone turbulence, we used a configuration with a single region of uniform horizontal

6 gradient, representing the local structure within a frontal zone, in order to focus on the mechanisms of the local turbulent response. We employ a three-dimensional domain that extends horizontally ~3 times the mixed layer depth in both directions, permitting large-scale perturbations in both the cross- and along-stream directions The main goal of our research is to determine what processes control the dynamics of mixed layer turbulence in areas with significant horizontal gradient and geostrophic shear. A description of the model and initial conditions is presented in Section 2 along with a range of surface forcing conditions and horizontal temperature gradients used in the study. Section 3 presents results from the parameter study, focusing on the behavior of boundary layer mixing with imposed horizontal temperature gradients and the turbulent kinetic energy (TKE) budget associated with coherent circulations. Results from the model are then used in Section 4 to test a simple scaling argument relating shear production from geostrophic and ageostrophic shear to surface wind forcing and horizontal temperature gradient. The paper ends in Section 5 with a discussion of our main conclusions Model description and Experimental design Experiments are conducted using a three-dimensional large-eddy simulation model based on the Deardorff (198) equation set with subgrid scale turbulence parameterization from Ducros et al. (1996). The model configuration is similar to that described in Skyllingstad et al. () and Skyllingstad and Samelson (12). Two configurations are discussed, the first essentially identical to that of the warm-filament simulations of Skyllingstad and Samelson (12), but with along-front wind forcing, and the second with an imposed uniform horizontal temperature gradient, to focus on the local response within the frontal zones.

7 The warm-filament simulation used the initial temperature distribution described in Skyllingstad and Samelson (12), with a temperature anomaly of.8 o C relative to the ambient fluid, in a filament with depth of 8 m, zonal width of order 5 km, a north-south extent that was effectively infinite in the doubly-periodic domain, and frontal zones of order 1 m width on each side of the filament. A northerly wind forcing (southward wind stress) was applied, which under steady conditions would drive warm-to-cold (stabilizing) Ekman transport across the eastern front, and cold-to-warm (destabilizing) Ekman transport across the western front. The wind forcing increased linearly from zero to the fixed prescribed value over the first 12 simulation hours. The frontal-zone simulations have instead a uniform, imposed, horizontal temperature gradient, Γ = dt/dx, and an initial geostrophic current, V G, that is in thermal-wind balance with the thermal gradient, so that 142!!! =!" Γ, (1)!"! where f is the Coriolis parameter, g = 9.81 m s -2 is gravitational acceleration, and α = 2 x 1-4 is the approximate expansion coefficient for sea water.. The corresponding horizontal buoyancy gradient is denoted M! =!" dt = gα!" dx. These simulations were initialized with this geostrophically balanced thermal gradient superimposed on a surface mixed-layer thermal structure that varies only in the vertical (Figure 1). The domain size was 2,56 m in both x and y, and 15 m in the vertical with grid spacing of 2.5 m in all directions and periodic lateral boundaries. Rigid upper and lower boundary

8 conditions are applied with a Rayleigh damping sponge layer over the bottom 12 levels. For the frontal-zone simulations, we explore a range of temperature gradients and wind forcing as presented in Table 1. Experiment names indicate the frontal strength (SF, MF, WF, denoting Strong, Medium, Weak Front) and the wind direction (N,S,E,W, with the meteorological convention) and strength (S, M, W). With winds from the north (cases xxnx), the classical steady Ekman transport would tend to advect cold water over warm, leading to static instability or mechanically forced convection, or to related instabilities. Two additional simulations are considered, one equivalent to case SFNM but including Stokes drift and the vortex force, and one with weak surface cooling and no wind forcing (case SFC). The prescribed Stokes drift profile is based on a monochromatic wave field with wave height of 1 m and wavelength of 3 m for this single case. The frontal-zone simulations each covered 36-hour periods, with wind (and wave, in the wave-driven case) forcing increasing linearly from zero to the fixed prescribed value over the first 12 simulation hours. Wind-forced Ekman transport from the cold (or saline) side of a front toward the warm (or fresh) side causes a horizontal buoyancy advection that, when the temperature (or salinity) are vertically uniform, is proportional to the wind stress and the horizontal buoyancy gradient: 168 EkBF =!!!!!! (2) where τ is the wind stress and ρ o is the mean density. If this Ekman flux is confined close to the surface, relative to the depth of the layer of interest, it may be interpreted as an effective surface buoyancy flux. The value of EkBF in (2) can equate to 1 s of W m -2 (e.g. Thomas et al., 13), and corresponds to a surface heat loss when surface winds are in the direction that will cause de-stabilizing Ekman heat transport. Winds in the opposite direction generate a stabilizing

9 effective surface flux that promotes restratification of the ocean boundary layer. A constant EkBF assumes that the upper ocean current structure is in a balanced equilibrium with the surface wind and wave forcing, which is rarely the case over the midlatitude ocean, where winds vary in response to synoptic weather events, generating currents that rotate with an inertial period Results: warm filament The evolution of the warm filament simulation with wind forcing from the north is fundamentally different than that of the unforced warm filament considered by Skyllingstad and Samelson (12). The strength and rapid development of both vertical and lateral mixing generated by the wind-forced processes prevents the formation of the baroclinic waves and eddies that dominated the unforced evolution. This difference can be illustrated (Figure 2) by comparing a warm filament case similar to the example from Skyllingstad and Samelson (12) with a wind-forced experiment. Significant baroclinic eddy transport takes roughly 5 days to develop in the non-forced case, whereas disturbance growth in wind-forced case requires only 12 hours to generate significant turbulence mixing. The structures of the turbulent response in the two frontal zones in this simulation also differ fundamentally. At the western front, where the northerly wind would cause warm-to-cold steady Ekman transport, the wind-driven currents strengthen the stratification and damp the turbulence, leading to a diffuse, uniform frontal gradient. At the eastern front, where the northerly wind would cause cold-to-warm steady Ekman transport, the wind-driven currents weaken the stratification and induce a strong turbulent response. The TKE budget terms (not shown) associated with the wind-forced front in Figure 2 are about 2 orders of magnitude greater

10 than the levels of turbulence that were found to arise from the collapsed baroclinic fronts in Part I (Skyllingstad and Samelson, 12, their Fig. 13). Thus, in the presence of wind forcing, the wind-forced turbulence and Ekman transport of the background front completely dominates the boundary layer development, while the weak, downscale transfer of energy from the baroclinic waves through frontal collapse and K-H transition to turbulence can only be detected in the model in the absence of wind forcing. Of particular interest is the characteristic banded structure in at the eastern front (figure 2b). These structures can be neither Langmuir circulations, because no wave or vortex forcing is present in these simulations, nor can they be symmetric instabilities, because they are oriented at an angle away from the frontal axis. Additional northerly-wind-forced warm-filament simulations indicated that these structures are formed robustly at the eastern front. The constant horizontal gradient, frontal-zone simulations described in the next section were designed specifically to explore and address the dynamics of these coherent roll structures Results: turbulence in frontal zones a) Northerly (de-stabilizing) winds In our first example, Case SFNM (strong front, north wind of moderate strength), the onset of northerly winds generates an inertial oscillation in the average meridional velocity component, v, as shown Figure 3, which produces alternating destabilization and restratification as shown by the potential temperature, depending on the direction of the inertial current relative to the imposed background density gradient. Initially, the wind stress increases the surface current and generates turbulence kinetic energy (TKE) that mixes momentum downward as shown by the changing current structure with depth over the first 6 hours. By hour 8, the near surface flow has rotated to an eastward direction so that warmer water is now being advected

11 over colder water increasing the near-surface stratification. Reduced turbulence and increased stratification limits vertical mixing at this time, trapping wind-forced momentum near the surface. After roughly hour 12, the near surface currents again advect cold water over warm, destabilizing the upper boundary layer, which generates a pulse of turbulence that extends downward to the thermocline at 8 m depth. This pattern repeats over the next 18 hours with turbulence strengthening further by the end of the simulation. For comparison we include plots of the average v component and TKE from cases with no temperature gradient, NFNM, and stronger surface winds, SFNS, in figure 4. Average conditions for case NFNM show a much weaker inertial response with nearly uniform currents and TKE over the 36-hour simulation. The maximum strength of mixing as indicated by average TKE, is roughly half the value achieved when the horizontal temperature gradient is imposed, and the boundary layer temperature remains relatively constant over the simulation period in contrast to a significant cooling in case SFNM. A consistent inertial response is also absent when the wind stress is increased in case SFNS (figure 4b). Here, inertial re-stratification is not sufficient to prevent strong turbulence and the entire boundary layer takes on a more uniform structure with only slight variations tied to the inertial currents. Turbulence in this case is about 3 times stronger than the maximum values in case SFNM, mostly due to the increased momentum flux. We examine the relative contribution of geostrophic and wind-forced shear in the TKE budget later in the paper. Plots of the surface meridional velocity presented in Figure 5 show how the horizontal temperature gradient in case SFNM produces strong, coherent circulations that form near the surface and grow in horizontal scale with time. At the beginning of the first significant mixing event around hour 12, structures form with horizontal spacing similar to the boundary layer

12 depth. By hour 18, these circulations are well defined with cross scale length of ~5 m and an orientation between -25 degrees to the right of the wind direction. Similar coherent structures (not shown) are produced in case SFNS, but with stronger velocity variations in response to the larger wind stress. Circulations in case NFNM (no horizontal temperature gradient) are also oriented to the right of the wind (figure 5b and 5d), but have a reduced separation of about 1 m and weaker intensity. Circulations do not evolve significantly in this case as shown by the small difference between hours 12 and 18. b) Dynamics of coherent roll structures The orientation angle of the roll structures in case SFNM departs from the frontal axis by roughly 25 degrees, similar to the departure angle for the roll structures observed on the destabilizing-wind side of the warm-filament simulation (Fig. 2). This departure indicates that the disturbances cannot be explained by appealing to symmetric instability, which by definition can only produce structures that are aligned along the frontal axis. They also cannot be Langmuir circulations, because no wave-vortex forcing was applied in case SFNM. A dynamical explanation for these structures must therefore be sought elsewhere. To diagnose the relevant dynamics, a linear instability calculation was carried out that incorporates both the Ekman shear and the geostrophic shear, as well as the horizontal frontal gradient and the background stratification. The linear instability theory essentially combines the various instability elements considered separately by Lilly (1966), Stone (1966, 197), and Asai and Nakasuji (1973), using an adjusted surface flux boundary condition to allow the consistent definition of a steady state. The stability problem for linear disturbances to this generalized background flow is then solved numerically, using finite-difference approximations and matrix

13 eigenvalue methods. A complete discussion of the linear theory, with more extensive comparisons to the LES results, is given by Duncombe et al. (manuscript in preparation). The results of this calculation indicate that the roll structures should be understood as, primarily, a form of Ekman instability (Lilly, 1966) that is modified by the horizontal and vertical buoyancy gradients. The angle of departure from the frontal axis and the horizontal scale separation for coherent structures in case SFNM correspond to those predicted by the linear theory for basic state profiles that approximate the LES mean fields (Fig. 6). The linear instability analysis found a localized region of instability for perturbations with wavenumbers oriented to the right of the wind direction. The most rapidly growing mode occurs at an angle of 26 degrees and at the scaled wavenumber.55δ E, where δ E is the Ekman depth. These results, which are based on horizontal mean LES profiles extracted at hour 16, are consistent with the wavelength and angle of the coherent structures in the LES model at hour 18. The linear-theory and LES wavelengths are 33 m and 257 m, respectively, and the angles are 26 and 24 degrees to the right of the wind. Together, the departure of the roll-structure angle from the frontal axis and the independent results for the modified Ekman instability persuasively demonstrate that symmetric instability is not the primary mechanism leading to these disturbances. Further insight can be obtained by examining the average potential vorticity, 283 q = fk + ω b (3) and Richardson number defined as Ri =!!!!"!", (4)

14 where ω = V is the relative vorticity and V is the three-dimensional velocity to better understand the flow behavior. Potential vorticity for each scalar grid point was calculated every model time step and averaged horizontally and over a 3 second time period. Symmetric instability requires average q < and Ri between.25 and 1. over a sufficient period of time to allow instability growth (Stone 1966). Shaded plots of Ri are shown in figure 7 along with contours of Ri =.25 and 1. (green and black) and q = (red). For almost all of the simulation, the value of q is negative in the upper water column, which would support growth of symmetric instability. However at hour 18, Ri is below.25 over most of the water depth indicating conditions that are unstable to shear instabilities such as Ekman instability mentioned above. Conditions at hour 12 when Ri is between.25 and 1. would indicate growth of symmetric instability between depths of ~25-7 m, however plots of the velocity field at these depths do not show circulations aligned with the temperature gradient or growing disturbances. We believe that strong episodic mixing every ~12 hours associated with the inertial current shear disrupts the slower growing symmetric instability. The time period between these mixing events is not long enough for symmetric instability to develop for the prescribed conditions. Surface gravity waves influence upper ocean mixing by generating a Lagrangian transport (Stokes drift) and Langmuir circulation via the Stokes term defined as ω + f V!, where V s is the wave Stokes drift velocity vector. These two factors cause changes in the inertial velocity response and strength of vertical mixing, for example the Stokes drift term accelerates the formation of strong mixing at the beginning of the simulation, and generates an overall increase in turbulence strength and vertical transport of momentum. When Langmuir forcing is added to Case SFNM, the resulting velocity fields show distinctive structures associated with Langmuir circulation that are superposed on the larger-scale Ekman-instability structures. In

15 general, the simulated effects of surface waves are relatively minor and do not cause a major change in the boundary layer behavior. For the remainder of this paper, we do not consider the effects of Stokes drift on the boundary layer mixing. c) Southerly winds warm-to-cold Ekman transport Cases presented so far have considered only northerly winds where the Ekman heat advection tends to destabilize the stratification, resulting in strong mixing, the amplitude of which depends on the strength of the background temperature gradient. For southerly winds, we expect a reduction in turbulence because the flux of momentum reduces the geostrophic shear and the Ekman transport vector is on average acting to restratify the upper boundary layer. Plots of the average currents and TKE for Case SFSM in figure 8 demonstrate how these two factors affect the boundary layer structure. Mixing in this case is much weaker than any of the destabilizing wind examples, with TKE about 1 times smaller in comparison with case SFNM. Consequently, the boundary layer is stratified throughout the simulation with only small variations in response to a weak inertial current oscillation. Interestingly, this case presents conditions that are conducive to the formation of symmetric instability near the bottom of the boundary layer, namely, a region of reduced stratification with geostrophic shear between 3 and 8 m depth. Plots of average Ri and q show this more clearly, with Ri between.25 and 1. overlapping negative q values (Figure 9). Decoupling of the symmetric instability layer from the surface, as shown by the higher Ri between and 3 m depth, is key in this case and allows instabilities to grow without disruption from surface forced turbulence. Plots of the v component from 5 m depth in Figure 1 shows coherent structures that are aligned with the geostrophic shear, consistent with symmetric instability.

16 d) Surface cooling, no wind forcing In the simulations presented so far, only the decoupled boundary layer generated in the southerly wind case produced conditions that would support the generation of symmetric instability. We next consider a case (SFC) without wind forcing, but with weak surface cooling promoting negative potential vorticity without a strong inertial current response, similar to Taylor and Ferrari (1). Horizontally averaged plots for this case shown in figure 11 have, somewhat surprisingly, a significant inertial response that is generated by mixing of the geostrophic flow. As in the wind forced case, buoyant convection generates vertical momentum flux, disrupting the geostrophic balance and forcing accelerations in the u component velocity by the imposed pressure gradient. Temperature advection by the inertial current again modulates the boundary layer stratification, setting up periods of enhanced TKE associated with coherent, organized circulations. In this case, however, much of the boundary layer has Ri between.25 and 1., which would promote the formation of symmetric instability. Indeed, plots of the v velocity component at hour 23 (figure 12) indicate roll instabilities aligned with the temperature gradient consistent with symmetric instability. We note that the TKE associated with these disturbances is about an order of magnitude weaker than the Ekman instabilities generated in case SFNM with moderate wind stress, suggesting that symmetric instability alone is not responsible for explosive mixing events in boundary layer fronts Turbulence kinetic energy a) TKE budget Our simulations show that stronger fronts lead to more intense turbulence, but only during specific times in the inertial cycle and when winds are from preferred directions relative

17 to the front. Integrating the total TKE over the simulation period (figure 13) allows us to quantify how much the total change in turbulence is tied to the strength of horizontal temperature gradient. In general, the stronger the gradient, the greater the total integrated TKE and subsequent mixing at the bottom of the boundary layer, if the wind is in the de-stabilizing direction. For different wind directions having no de-stabilizing component (figure 13b), results suggest that the background gradient does not generate increased turbulence, and may in fact decrease total mixing in comparison with no gradient conditions (case NFNM). We can gain more insight on the dominant processes in the mixed layer by analyzing the TKE budget equation, 364!"#$!" = g!!! u!!! w!!!"#$ u!!!"! w!!!"#!!"!"!!! wtke + K!"#$!!!" + ε (5) where u i are the horizontal velocity components with i=1,2 denoting x and y directions, V ageo and V geo are the horizontally averaged ageostrophic and geostrophic velocity vectors, K m is the subgrid scale eddy viscosity, the overbar represents horizontal averaging, and primes indicate deviations from the mean. Terms in (5) are defined in order as the storage of TKE, buoyancy production, ageostrophic shear production (SP ageo ), geostrophic shear production, vertical transport, and dissipation. Plots of the horizontal and vertical averaged TKE budget terms are presented in figure 14 for a range of temperature gradient and wind stress values. In general, the TKE budget is dominated by shear production from wind stress and/or geostrophic shear. The effects of the inertial current are significant when the geostrophic term is large and winds are moderate as in case SFNM. In all cases, total shear production is nearly balanced by dissipation, which converts kinetic energy into heat. Buoyancy is the third player that can upset this balance leading to a decrease in the overall turbulence energy. For example, in case SFNM, negative

18 buoyancy production around hours 7 and 17 lead to a slight decrease in TKE (see figure 3). The negative buoyancy production is partially in response to mean horizontal advection of the background temperature gradient by the inertial current as discussed previously. In case SFC (figure 14e), nearly all of the TKE is produced by the geostrophic shear term as predicted for traditional symmetric instability (Thomas et al 16). Buoyancy flux is also positive in this case corresponding to the weak inertial current advection that is a maximum near the surface where mixing of momentum from surface convective forcing has generated an inertial response. We note that this case is very similar in appearance to the stability analysis results presented in Thomas et al. (16) with negative average ageostrophic shear production throughout the simulation. Here, however, there is inertial shear that generates positive buoyancy flux. In contrast, the cases with surface wind stress (figure 14a-d) have ageostrophic shear production that is often similar in strength to the geostrophic shear production. Formation of significant coherent structures are mostly restricted to times when the geostrophic shear is relatively large in these experiments, suggesting that the mechanisms responsible for symmetric instability are also influencing the growth of Ekman instabilities. The vertical distribution of TKE budget terms demonstrates more clearly the interplay of ageostrophic and geostrophic shear production over time for case SFNM (Figure 15). At hour 7, TKE growth is dominated by geostrophic shear production with ageostrophic shear production showing a transfer of energy from turbulence into the mean background velocity field. This exchange is more easily understood if we consider the role of the inertial current acting on the geostrophic flow (figure 15b). The inertial current is a maximum at the surface in response to wind stress forcing and is the dominant surface TKE production term at all times. When the inertial current has rotated so that it opposes the geostrophic flow at hour 7, vertical mixing of

19 the total velocity acts to decrease the overall shear as shown by the horizontally averaged meridional velocity. The average shear is reduced compared with the geostrophic velocity shear, consequently the ageostrophic term acts to limit the TKE production by the geostrophic shear. We again note that the buoyancy production is not a significant direct source of TKE in the budget as shown by figures A similar plot of the TKE budget terms for case SFSM with southerly, stablizing winds and case SFC is presented in figure 16. As noted above, we believe traditional symmetric instability is active in both of these examples. Geostrophic shear production is the dominant source of TKE in both cases consistent with symmetric instability as the primary instability process. We also note that the buoyancy production contributes to the disturbance growth, with balance maintained by dissipation, transport, and ageostrophic shear production. These profile plots differ from the Ekman layer instability scenario where both buoyancy and vertical transport are negligible for most of the profile. Nevertheless, there does not appear to be a unique TKE budget analysis for differentiating symmetric instability from other instabilities that depend on the geostrophic shear. Likewise, potential vorticity is negative in all of our simulations that have instabilities. A more reliable predictor is provided by the Richardson number, which is between.25 and 1. for symmetric instability and less than or equal to.25 for the surface based Ekman instability. b) Simple TKE budget model For many mixed layer parameterizations, for example the KPP scheme (Large et al., 1994), estimation of eddy viscosity depends on assumptions about the scale of TKE production terms. For example, the commonly used Obukhov length, L, provides a measure of stratification versus shear allowing universal functions of z/l. Here we perform a scale analysis of the TKE

20 budget equation and define a new length scale analogous to L, but relating ageostrophic and geostrophic shear production. Our intent is to demonstrate that the relative importance of geostrophic shear versus wind-forced shear can be estimated through a simple parameter that is dependent on horizontal temperature (or salinity) gradient and the de-stabilizing wind component velocity. Traditional scaling of the TKE budget for the surface layer begins with the definition of a reference velocity (or friction velocity), u *, defined using the surface wind stress as, τ = ρu! where ρ is the fluid density and τ is the surface wind stress (see Kaimal and Finnigan 1994). For a wind-forced, neutral boundary layer, average shear near the surface can be scaled using!!!" =!!" where κ is the von Karman constant (~.4), and z is the depth. Turbulence kinetic energy dissipation for an unstratified boundary layer is roughly equal to shear production, or (6) (7) ε = u w u = u 2 u z κz = u 3 κz. (8) This scaling for dissipation or shear production can be used to compare various terms in the TKE budget equation for flows near the surface. Here, we apply a similar scaling for the boundary layer following methods presented in Moeng and Sullivan (1994) and Belcher et al. (12). For boundary layer mixing in frontal zones where the buoyancy term is relatively small, we would like to know which is more important; TKE production from wind and/or wave mixing through the ageostrophic shear production terms (for now we ignore wave effects through V s ), or TKE growth from geostrophic shear production. Assuming steady state, we can express the

21 TKE budget as a balance between shear production from ageostrophic shear, SP ageo, geostrophic shear, SP geo, and dissipation, or ε = SP!"#$ + SP!"#, (9) where parameters are assumed to be averaged over time. We ignore the buoyancy term, which is typically small in our simulations. We can replace the shear production terms in this equation using surface layer scaling for turbulence, for example, as used in equation (8), ε z = A u 3 + u κz 2 V geo z 1!! (1) 459 where we have scaled u!! w with u * 2 and applied a simple model for the linear variation of budget terms over the boundary layer depth, h, following Moeng and Sullivan (1994). We include an empirical constant, A =.5, which accounts for the reduction in shear and subsequent scaled dissipation rate because of the rotating inertial current, and differences between the surface layer and mixed layer scaling. We can also express (1) in terms of a characteristic length scale, 464 L!" =!!!"#!"!", (11) analogous to the Obukhov length, but with buoyancy production replaced with shear production by geostrophic shear. Equation (1) can be rearranged to yield, ε z =!!!".5 +!!!" 1!!. (12) When z = L sp, shear production from geostrophic shear is roughly equal to shear production from wind stress acting on the ocean surface, consequently, large L sp implies either strong wind forcing or a weak horizontal temperature gradient.

22 Plots of the average dissipation rate from the LES model are presented in Figure 17 for both weak and moderate wind forcing cases along with predicted profiles from Eq. (1). Overall, results from the simple model are in good agreement with the LES and indicate how the geostrophic shear contributes to increased turbulence, especially near the base of the boundary layer where geostrophic shear dominates. For moderate winds, even the weakest gradient of.2 o C/1 km generates a significant increase (factor of 2-3) in turbulence dissipation rate over the lower half of the boundary layer Conclusions The wind-forced warm-filament simulation demonstrates that along-front wind forcing can dramatically alter the evolution of shallow baroclinic (mixed-layer) instabilities, relative to unforced conditions. The strength and rapid development of both vertical and lateral mixing generated by the wind-forced processes was seen to prevent the formation of the baroclinic waves and eddies that dominated the unforced evolution. In addition, coherent roll structures that qualitatively resembled some recent oceanographic observations were found to form in one of the frontal zones, but not the other. The primary goal of this study was to examine how the interaction between surface windforced shear and geostrophic shear generates the coherent roll structures. We found that the coherent-structure dynamics were more in line with shear instabilities of the Ekman layer originally described in Lilly (1966) and discussed in detail in Duncombe et al. (manuscript in preparation) than, for example, a symmetric instability such as that described by Stone (1966). These shear instabilities, which we refer to as Ekman instabilities, were found to depend strongly on the phase of inertial currents, releasing energy stored in near surface currents generated by stratification associated with advection of the background temperature gradient. Analysis of the

23 TKE budget indicate that geostrophic shear is a key component in providing energy for instability growth, indicating that while these structures don t appear to result from symmetric instability, they do gain energy from the geostrophic flow and would act to decrease baroclinicity if the horizontal gradient were not held constant. As pointed out in Stone (1966), one cannot rule out the possibility that instabilities are possible that are neither of the baroclinic type or purely symmetric under non-geostrophic conditions. In two cases, coherent structures formed that were aligned with the temperature gradient consistent with symmetric instability. Winds in one case were opposed to the surface geostrophic flow, forcing restratification through Ekman transport. Consequently, the boundary layer developed into two regions; one near the surface dominated by wind-forced shear, and second deeper layer separated by a region of stable stratification. This lower layer had both negative potential vorticity and Richardson number between.25 and 1., suggesting that the circulations forced at this level formed from symmetric instability. A second case with possible symmetric instability did not have wind forcing, but only weak surface cooling. Again, the simulation exhibited Richardson number between.25 and 1., and negative potential vorticity. Our analysis of the TKE budget suggests that there is not a unique signature for symmetric instability in cases where ageostrophic shear has scales similar to geostrophic shear. Both the weak symmetric instability case with winds that drive stabilizing Ekman transport and the strong Ekman instability case with winds that drive destabilizing Ekman transport have conditions where geostrophic shear is the dominant TKE production term. The only discerning difference between the two cases is the Richardson number, which is below or near.25 in the Ekman instability case. Nevertheless, most of the TKE is produced by geostrophic shear indicating that baroclinicity is a dominant factor in generating boundary layer mixing.

24 An important question concerns when and where we would expect strong boundary layer mixing in the ocean because of frontal instabilities as modeled here. Using a range of wind forcing and horizontal temperature gradients, we examined how important wind forced shear is relative to geostrophic shear and related this to wind speed and horizontal temperature gradient. Enhanced mixing is primarily associated with winds aligned to cause destabilizing Ekman transport; other wind directions examined produced turbulence intensity similar to cases without a horizontal temperature gradient. We find that for these Ekman-destabilizing winds, the importance of geostrophic shear production can be accurately estimated using a simple model based on wind speed and horizontal temperature gradient. Many existing boundary layer parameterizations that resolve upper ocean shear should account for this enhancement because they include the effects of geostrophic shear simulated by the large-scale circulation. Methods that base upper-ocean mixing on surface heat and momentum fluxes alone will not accurately depict the effects of fronts and geostrophic shear. More research is need to determine if simply adding geostrophic shear in typical parameterizations is sufficient (e.g. Mellor and Yamada 1982), or if the effects of coherent structures as modeled here requires special treatment, for example, through a non-local term in the transport equations (e.g. Smyth et al. 2). Both field measurements (e.g. Thomas et al. 16) and our numerical experiments indicate that mixed layer fronts can have a significant effect on the character of turbulence and the overall strength of mixing. What is not yet understood is how often frontal regions dominate the dynamics of boundary layer turbulence. Strong horizontal density gradients, similar to the examples studied here, are found in limited areas of the world ocean. Primary areas of strong gradients are located near the western boundary currents (Gulf stream and Kurashio) and in the

25 southern ocean. These regions are likely to have enhanced boundary layer mixing when winds are aligned along front in the direction corresponding to destabilizing (cold to warm) Ekman transport Acknowledgments. This research was funded by the National Science Foundation Grant OCE We would like to acknowledge high-performance computing support from Yellowstone (ark:/8565/d7wd3xhc) provided by NCAR's Computational and Information Systems Laboratory, sponsored by the National Science Foundation

26 References Asai, T., and I. Nakasuji, 1973: On the stability of Ekman boundary layer flow with thermally unstable stratification. J. Meteor. Soc. Japan, 51, Belcher, S. E., and Coauthors, 12: A global perspective on Langmuir turbulence in the ocean surface boundary layer. Geophys. Res. Lett., 39, L1865, doi:1.129/12gl Capet, X., J. C. McWilliams, M. J. Molemaker, and A. F. Shchepetkin, 8: Mesoscale to submesoscale transition in the California Current System. Part I: Flow structure, eddy flux and observational tests. J. Phys. Oceanogr., 38, Craik, A.D. D., and S. Leibovich, 1976: A rational model for Langmuir circulations. J. Fluid Mech., 73, D Asaro, E., C. Lee, L. Rainville, R. Harcourt, and L. Thomas, 11: Enhanced turbulence and energy dissipation at ocean fronts. Science, 332, Deardorff, J. W., 198: Stratocumulus-capped mixed layers derived from a 3-dimensional model. Bound.-Layer Meteor., 18, , doi:1.17/bf Ducros, F., P. Comte, and M. Lesieur: 1996: Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate, J. Fluid Mech., 326, Foo, E.-C., 1981: A Two-Dimensional Diabatic Isopycnal Model Simulating the Coastal Upwelling Front. J. Phys. Oceanogr., 11, Hamlington, P. E., L. P. Van Roekel, B. Fox-Kemper, K. Julien, G. P. Chini, 14: Langmuirsubmesoscale interactions: Descriptive analysis of multiscale frontal spindown simulations. J. Phys. Oceanogr., 44,

27 Kaimal, J. C., and J. J. Finnegan, 1994: Atmospheric Boundary Layer Flows, Their Structure and Measurement. Oxford University Press. 289 pp. Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization. Rev. Geophys., 32, , doi:1.129/94rg1872. Lilly, D. K., 1966: On the instability of Ekman boundary flow. J. Atmos. Sci., 23, McWilliams, J. C., E. Huckle, J.-H. Liang, and P. P. Sullivan, 12: The wavy Ekman layer: Langmuir circulations, breaking waves, and Reynolds stress. J. Phys. Oceanogr., 42, Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. Space Phys.,, Moeng, C.-H., and P. P. Sullivan, 1994: A comparison of shear- and buoyancy-driven planetary boundary layer flows. J. Atmos. Sci., 51, Noh, Y., H. Ok, E. Lee, T. Toyoda, N. Hirose, 16: Parameterization of Langmuir circulation in the ocean mixed layer model using LES and its application to the OGCM. J. Phys. Oceanogr., 46, Samelson, R. M., and R. A. de Szoeke, 1988: Semigeostrophic wind-driven thermocline upwelling at a coastal boundary. J. Phys. Oceanogr., 18, Samelson, R. M., and C. A. Paulson, 1988: Towed thermistor chain observations of fronts in the subtropical North Pacific. J. Geophys. Res., 93(C3), Samelson, R. M., 1993: Linear instability of a mixed-layer front. J. Geophys. Res., 98(C6), 1,195-1,4.

28 Samelson, R. M., and D. C. Chapman, 1995: Evolution of the instability of a mixed-layer front. J. Geophys. Res., 1(C4), Skyllingstad, E.D., W.D. Smyth, and G.B. Crawford, : Resonant wind-driven mixing in the ocean boundary layer. J. Phys. Oceanogr., 3, Skyllingstad, E. D., and R. M. Samelson, 12: Baroclinic frontal instabilities and turbulent mixing in the surface boundary layer. Part I: Unforced simulations. J. Phys. Oceanogr., 42, Smyth, W. D., E. D. Skyllingstad, G. Crawford, and H. Wijesekera, 2: Nonlocal fluxes and Stokes drift effects in the K-profile parameterization. Ocean Dyn., 52, Stone, P. H., 1966: On non-geostrophic baroclinic stability. J. Atmos. Sci., 23, Sundermeyer, M. A., E. Skyllingstad, J.R. Ledwell, B. Concannon, E. A. Terray, D. Birch, S. D. Pierce, and B. Cervantes, 14: Observations and numerical simulations of large-eddy circulation in the ocean surface mixed layer, Geophys. Res. Lett., 41, doi:1.12/14gl61637, Taylor, J. R., and R. Ferrari, 1: Buoyancy and wind-driven convection at mixed layer density fronts. J. Phys. Oceanogr., 4, Thomas, L. N., and C. M. Lee, 5: Intensification of ocean fronts by down-front winds. J. Phys. Oceangr., 35, Thomas, L. N., J. R. Taylor, R. Ferrari, T. M. Joyce, 13: Symmetric instability in the Gulf Stream. Deep-Sea Res. II, 91, Thomas, L. N., J. R. Taylor, E. A. D Asaro, C. M. Lee, J. M. Klymak, A. Shcherbina, 16: Symmetric instability, inertial oscillations, and turbulence at the Gulf Stream front. J. Phys. Oceanogr., 46,

29 Thorpe, A. J., and R. Rotunno, 1989: Nonlinear aspects of symmetric instability. J. Atmos. Sci., 46,

30 Table 1. Simulation parameters. Case SFC also has surface cooling of - W m -2. Case dt/dx ( o C/1 km) τ x (N m -2 ) τ y (N m -2 ) Μ (s -2 x 1-7) 2 NFNM SFNS SFNM SFNW MFNS MFNM MFNW WFNM WFNW SFSW SFWW SFEW SFC

31 Figure Captions List Figure 1. Initial (a) potential temperature and (b) u (blue) and v (red) velocity component profiles Figure 2. Surface potential temperature for a warm filament simulation initialized in geostrophic balance with (a) no surface forcing after 5 days and with (b) -.1 N m -2 wind forcing after 16 hours Figure 3. Horizontally averaged (a) v component (m s -1 ) and (b) TKE (m 2 s -3 ) for case SFNM. Contours of horizontally averaged potential temperature ( o C) are shown in both plots Figure 4. Same as figure 3 for (a) case NFNM and (b) case SFNS Figure 5. Horizontal cross sections of v component velocity (m s -1 ) at hour 12 for (a) case SFNM and (b) case NFNM, and hour 18 from (c) case SFNM and (d) case NFNM Figure 6. Growth rates for Ekman instability as predicted by linear analysis presented in Duncombe (in preparation). The wavenumber is scaled using the Ekman depth, δ E Figure 7. Horizontally averaged Richardson number, Ri, along with contours of Ri =.25 (green) and Ri = 1. (black). Also show is a contour of potential vorticity, q =., in red, separating negative values above the contour from positive values in the lower section of the domain.

32 Figure 8. Same as figure 3, but for case SFSM with warm-to-cold Ekman transport Figure 9. Same as figure 7, but for case SFSM Figure 1. Meridional velocity from a depth of 5 m for case SFSM with upfront winds Figure 11. Horizontally average (a) v velocity component (m s -1 ) and temperature ( o C contoured), (b) TKE and temperature, and (c) Richardson number for case SFC. Also overlaid on (c) are contours of Ri =.25 (green), Ri = 1. (black), and potential vorticity, q =., in red, separating negative values above the contour from positive values in the lower section of the domain Figure 12. Surface meridional velocity (m s -1 ) at hour 23 for case SFC Figure 13. TKE averaged in the horizontal and over the simulation period for cases with (a) different temperature gradient values (cases SFNM, SFSM, MFNM, WFNM, and NFNM), and (b) wind directions (cases SFNM, SFEM, SFWM, SFSM). Wind stress in all cases is.1 N m Figure 14. Vertically averaged ageostrophic shear production (black), geostrophic shear production (red), buoyancy production (blue) and dissipation (black dash) for case (a) SFNM, (b) NFNM, (c) SFNS, (d) WFNM, and (e) SFC. 672

33 Figure 15. (a) Horizontally averaged TKE budget terms and (b) meridional velocity (blue) and geostrophic velocity (red) from case SFNM at hour 7. Terms plotted in (a) are ageostrophic shear production (black), geostrophic shear production (red), buoyancy production (blue), vertical transport (magenta), and dissipation (dashed) Figure 16. Horizontally averaged TKE budget terms from (a) case SFSM with upfront winds, and (b) case SFC with surface cooling, but no wind forcing. Terms are defined as in figure 15a Figure 17. Plots of the dissipation rate, ε, from the LES model averaged horizontally and between the hours of 15 and 36 for (a) moderate and (b) weak winds, and as predicted by Eq. (1) for (c) moderate and (d) weak winds. 685

34 (a) (b) 5 5 Depth(m) Depth(m) Potential Temperature ( o C) Velocity (m s 1 ) Figure 1. Initial (a) potential temperature and (b) u (blue) and v (red) velocity component profiles.

35 (a) Meridional (m) (b) Meridional (m) Zonal (m) Zonal (m) Figure 2. Surface potential temperature for a warm filament simulation initialized in geostrophic balance with (a) no surface forcing after 5 days and with (b) -.1 N m -2 wind forcing after 16 hours.

36 (a) Depth (m) (b) Depth (m) Time (hr) Time (hr) Figure 3. Horizontally averaged (a) v component (m s -1 ) and (b) TKE (m 2 s -3 ) for case SFNM. Contours of horizontally averaged potential temperature ( o C) are shown in both plots x

37 (a) Depth (m) Time (hr) x Depth (m) (b) Depth (m) Depth (m) Time (hr) Time (hr) Time (hr) Figure 4. Same as figure 3 for (a) case NFNM and (b) case SFNS x

38 (a) (b) Meridional (m) Meridional (m) Zonal (m) 25.3 (c) Zonal (m) 25 (d) Meridional (m) Meridional (m) Zonal (m) Zonal (m) 25 Figure 5. Horizontal cross sections of v component velocity (m s-1) at hour 12 for (a) case SFNM and (b) case NFNM, and hour 18 from (c) case SFNM and (d) case NFNM.

39 39.4 Instability Growth Rate, σ i (n) x Scaled wavenumber (α / δ E ) Orientation angle, φ (degree) Figure 6. Growth rates for Ekman instability as predicted by linear analysis presented in Duncombe (in preparation). The wavenumber is scaled using the Ekman depth, δ E.

40 4 1.8 Depth (m) Time (hr) Figure 7. Horizontally averaged Richardson number, Ri, along with contours of Ri =.25 (green) and Ri = 1. (black). Also show is a contour of potential vorticity, q =., in red, separating negative values above the contour from positive values in the lower section of the domain.

41 (a) Depth (m) (b) Depth (m) Time (hr) Figure 8. Same as figure 3, but for case SFSM with warm-to-cold Ekman transport Time (hr) x

42 Depth (m) Time (hr) Figure 9. Same as figure 7, but for case SFSM.

43 Meridional (m) Zonal (m) Figure 1. Meridional velocity from a depth of 5 m for case SFSM with upfront winds..7

44 (a) Depth (m) Time (hr) (b) x Depth (m) Time (hr) (c) Depth (m) Time (hr) Figure 11. Horizontally average (a) v velocity component (m s -1 ) and temperature ( o C contoured), (b) TKE and temperature, and (c) Richardson number for case SFC. Also overlaid on (c) are contours of Ri =.25 (green), Ri = 1. (black), and potential vorticity, q =., in red, separating negative values above the contour from positive values in the lower section of the domain.

45 Meridional (m) Zonal (m) Figure 12. Surface meridional velocity (m s -1 ) at hour 23 for case SFC..14