## 1. Introduction

Satellite-borne observations of the atmospheric response to fronts of sea surface temperature (SST) have revolutionized the understanding of midlatitude air–sea interaction (Xie 2004; Small et al. 2008). While the traditional, large-scale view holds that the ocean primarily responds to forcing by the atmosphere, the ocean mesoscale shows a ubiquitous imprint of SST fronts on the atmospheric boundary layer (Chelton and Xie 2010; Xie 2004). For scales shorter than about 1000 km, wind speeds are proportional to SST perturbations, and wind stress divergence and curl are proportional to the downwind and crosswind gradients of SST, respectively (O’Neill et al. 2003; Chelton and Xie 2010; Song et al. 2009). The associated regression coefficients (i.e., coupling coefficients) vary seasonally and regionally, but the coupling coefficients between divergence and downwind SST gradients are consistently larger than those between wind stress curl and crosswind SST gradients in observations (Chelton and Xie 2010) and in high-resolution numerical models (Seo et al. 2007; Song et al. 2009; Bryan et al. 2010). Kinematically, this results from gradients of the frontally induced surface stress direction that diminish the wind stress curl but enhance the wind stress divergence (O’Neill et al. 2010a). Here, we seek to dynamically explain these observations using a linearized model for the atmospheric boundary layer that includes advection by background Ekman winds, frontally induced air–sea fluxes of heat, and their impact on the momentum budget. In the process, we provide a unified framework for all processes put forth in the context of frontal air–sea interaction, cast frontal air–sea interaction as a classical Rossby adjustment problem, and explore its scale and parameter dependence.

Two mechanisms are invoked to explain the response of boundary layer winds and surface stresses to fronts of SST: adjustments of vertical mixing and of baroclinic pressure gradients. Modulation of air–sea temperature fluxes affects vertical mixing between winds near surface and aloft and accelerate (decelerate) surface winds downstream of a cold-to-warm (warm-to-cold) front (Wallace et al. 1989; Hayes et al. 1989). The increase in vertical mixing and associated deepening of the boundary layer may be limited to the wake of the nonequilibrated difference of boundary layer and sea surface temperatures, or it may extend farther downstream if the boundary layer depth is permanently altered (Samelson et al. 2006). Support for this process comes from observations of higher surface winds (Sweet et al. 1981) and deep boundary layers on the warm side of SST fronts (Businger and Shaw 1984; Wai and Stage 1989).

Air–sea temperature fluxes downstream of an SST front also imprint the oceanic conditions on the hydrostatic, baroclinic pressure in the boundary layer (Lindzen and Nigam 1987). Evidence for this pressure effect comes from the covariations of divergences of surface winds and surface wind stress with the Laplacian of surface temperature and sea level pressure (Shimada and Minobe 2011; Tokinaga et al. 2009; Lambaerts et al. 2013).

The distinct responses of the wind stress divergence and curl emerge from the response of the atmospheric boundary layer momentum, heat, and mass balances to the accelerations induced by vertical mixing and pressure gradient mechanisms. In the absence of horizontal advection, Ekman pumping due to the divergence of the frontally induced Ekman transports displaces the stratification outside of the boundary layer and generates a back pressure in the boundary layer (Lindzen and Nigam 1987; Battisti et al. 1999) that spins down (Greenspan and Howard 1963; Holton 1965a,b; Pedlosky 1967) the wind stress curl.^{1} Atmospheric observations show the back pressure, in that frontal adjustments extend beyond the boundary layer and partially offset the surface pressure gradients induced by mesoscale SST (Hashizume et al. 2002). A series of numerical investigations on the response of the free troposphere to SST fronts relies on this frontally induced Ekman pumping to couple the boundary layer with the atmosphere aloft (Feliks et al. 2004, 2007, 2011). In these studies, the atmospheric boundary layer temperature is assumed to be in equilibrium with the underlying SST, and the stress results from a one-dimensional momentum budget in the boundary layer in response to the frontally induced baroclinic and back pressures. The role of advection that leads to imbalances of boundary layer temperature and SST and that affects the boundary layer momentum budget is not considered.

Recent modeling studies (Kilpatrick 2013; Kilpatrick et al. 2014) that build upon the two-dimensional simulations of frontal air–sea interaction (Wai and Stage 1989; Spall 2007) show the dynamics of frontal interaction to be a strong function of the large-scale wind magnitude and direction relative to the front. For strong cross-frontal winds, the effective time scale of an air column crossing the front is shorter than an inertial period. Resulting boundary layer transport divergences excite vertically propagating gravity waves in the free troposphere (Kilpatrick et al. 2014). In contrast, for alongfront winds, the effective time scale experienced by an air column that crosses the front is longer than an inertial period so that a surface trapped response results consistent with a spindown of the lower troposphere induced by the frontally induced secondary circulation (Kilpatrick 2013).

In this manuscript, we explore the hypothesis that spindown is responsible for the consistently smaller magnitude of the coupling coefficients for the wind stress curl than for the wind stress divergence. To this end, we adopt a reduced-gravity model for the atmospheric boundary layer (Lindzen and Nigam 1987; Battisti et al. 1999) to include advection by a prescribed, uniform, geostrophic wind and forcing by an arbitrary SST distribution (sections 2 and 3). We explore the responses of the model vis-à-vis observed characteristics (section 4), and show the distinct dynamics governing the wind stress divergence and curl and their coupling coefficients (section 5). Parameter sensitivity of the results and a comparison with observations are presented (section 6), followed by conclusions (section 7).

## 2. Reduced-gravity model

We employ a minimal model that includes the frontal physics outlined above and simplifies the vertical structure of the lower atmosphere as an active layer adjacent to the surface separated by a sharp inversion from the resting troposphere aloft (Battisti et al. 1999). The full, nondimensional equations are formulated on an *f* plane using classical scaling and are then linearized about a background Ekman spiral due to a prescribed, geostrophic wind.

The active layer of depth *h* is capped by an inversion with potential temperature jump *g* is Earth’s gravitational acceleration, and *T*. We consider the system in steady state, restricting time scales to longer than the maximum of an inertial period, the spindown time scale (see below), and the thermal adjustment time of the layer.

Equations and variables are nondimensionalized using the Rossby radius *f* the Coriolis frequency, the mean inversion height *H* as the vertical scale, and the gravity wave speed *z* is the vertical distance, so that *s* = 0 is the sea surface, and *s* = 1 is the time-dependent inversion height.

*T*with an adjustment rate

*A*being the vertical exchange coefficient that, in general, is a function of

*s*, and by the reduced gravitythat includes the impacts of the varying temperature in the boundary layer.

*s*= 0, horizontal winds equal ocean currents, which we assume to be zero:so that the surface stress

*E*and near-surface winds evaluated at

*s*= 1, turbulent fluxes vanish:which implies for

*E*on the surface stability

*E*for unstable,

*E*and drag coefficient in Eq. (9) that are responsible for differences in coupling coefficients between SST and wind speed and stress (O’Neill et al. 2012). Their consideration would render coefficients of the frontal responses, introduced below, a function of background wind speed but would otherwise not change results.

## 3. Linearization for small-amplitude mesoscale sea surface temperature variations

*ε*, where

*ε*. The condition of small

*ε*is satisfied if first-order winds are small compared to background fields [i.e.,

### a. Background Ekman spiral

**x**. To order

### b. Sea surface temperature–induced circulation

*ε*of the heat budget Eq. (1)balances horizontal advection by the vertically averaged background winds with the air–sea fluxes and lateral mixing. The frontally induced atmospheric temperatures

*ε*momentum and continuity equations. These extend the classic, forced, shallow water equations (Gill 1982) to frontal air–sea interaction and include horizontal advection by the background winds and vertical displacement by frontally induced updrafts of the background Ekman spiral, the Coriolis acceleration, mixing by the background eddy viscosity, and the back pressure due to gradients of inversion height:where we have used Eq. (15).

**F**consists of the baroclinic pressure gradient (Lindzen and Nigam 1987) and of the vertical mixing effect (Wallace et al. 1989; Hayes et al. 1989; Samelson et al. 2006). The latter vertically redistributes background momentum in response to frontally altered stability

*ε*the advection of the inversion height by the background wind with the divergence of the frontally induced horizontal winds and updrafts. Boundary conditionsimply that boundary layer transport divergences are balanced by inversion height advection by background transports. From Eq. (9), the surface stressconsists of contributions due to background mixing acting on the frontally induced surface winds and the surface stress due to the vertical mixing effect.

### c. Dynamical regimes

*E*up to order

*ε*form a linear system for the dependent variables

**x**independent, and equations are expanded in a Fourier series

**k**, the vertical

*s*dependence is solved and yields the Fourier coefficients, marked by a tilde, as a convolution of the transfer functions

The character of the solution is determined by the length scales of the thermal wake and the Froude number. Comparing in Eq. (19) the air–sea heat flux with the background advection or the lateral mixing term yields the downwind wavenumber component

**F**in Eq. (21) iswhere we have scaled the stability and temperature gradient with the solution of Eq. (19) for a straight SST front. For zero background winds, accelerations due to frontally induced turbulent mixing vanish, and the baroclinic pressure gradient is the only forcing. As the cross-frontal background winds increase, the downstream wake broadens and reduces the frontally induced baroclinic pressure gradient, while vertical mixing–induced accelerations increase (Small et al. 2008). As background winds rotate from cross- to alongfront,

The response of momentum balance and continuity, Eqs. (20) and (22), to the forcing **F** in Eq. (21) represents the final state of a Rossby adjustment and spindown. Its character is well known from the study of mountain waves and the ocean’s response to cyclones (Gill 1982; Suzuki et al. 2011), as well as from the time-dependent but nonadvective linear formulation for the diurnal sea breeze (Rotunno 1983; Niino 1987). It is governed by the Froude number of background advective wind *E*. In the limit of vertically constant advection and vanishing *E*, subsystems for the boundary layer transport and boundary layer shear exist. The wind shear is independent of the barotropic back-pressure gradient and reduces to a parabolic equation for **k**|^{−1}|**k ⋅ u**^{(0)}| > 1 and supports standing inertia–gravity, Poincaré waves in the lee of an SST disturbance (Spall 2007; Kilpatrick et al. 2014). For |**k**|^{−1}|**k ⋅ u**^{(0)}| < 1, the system is elliptic and yields an evanescent, geostrophic response.

A nonzero *E* couples the systems for layer shear and transport by the surface stress and yields, in the limit of vanishing advection, a balance of the forcing **F** with the Coriolis acceleration and vertical mixing—a frontally induced Ekman circulation in the presence of a thermal wind shear (Cronin and Kessler 2009). In the frontal equivalent of the classical spindown (Greenspan and Howard 1963; Holton 1965a,b; Pedlosky 1967) and buoyancy shutdown (MacCready and Rhines 1991; Benthuysen 2010), convergences of Ekman transports reduce the wind stress curl by adjusting the inversion height gradient (i.e., the back pressure).

## 4. Response to an undulating front

The frontally induced system is valid for any small-amplitude SST field, including that associated with ocean mesoscale eddies and fronts. To solve for background Ekman spiral and frontally induced circulation, Eqs. (16), (20), and (22) are vertically discretized using first-order finite differences on an equally spaced grid with 10 *s* levels. Levels for the vertical velocity and Ekman number include the surface and inversion. Horizontal winds are staggered in between so that surface wind stresses are proportional to the winds at the lowest level adjacent to the surface. The horizontal dependence of SST and Eqs. (19), (20), and (22) are Fourier transformed to horizontal wavenumber space **k**; the transfer functions

### a. Model parameters

The adjustment time of the boundary layer temperature

Parameters used in the frontal model. Horizontal and vertical length scales in units of Rossby radius of deformation and mean inversion height, respectively; time scales in inertial periods; and temperature as a fraction of mean inversion strength.

*E*has a midlayer maximum, decays toward the surface and the free troposphere (Fig. 1), and mimics the observed structure (Stull 1988; Hong and Pan 1996):where

*γ*denotes the height

*s*= 0. Frontally induced surface stresses in Eq. (24) therefore result only from adjustments of the vertical wind profile. A Taylor expansion of Eq. (27) in

*δ*then yields the frontally induced eddy coefficientsas a fraction of

### b. Background winds

Background winds (Fig. 2) point to the left of the geostrophic wind, are small close to the surface, and approach geostrophic speeds close to the inversion, similar to the classical Ekman spiral with constant eddy coefficients at the bottom of an infinitely deep atmosphere. The vertically averaged wind has a magnitude of 72% of the geostrophic wind, with components in the direction of the geostrophic wind of 57% and in the direction of the prescribed pressure gradient force of 45%. Together with

The background shear and

### c. Frontally induced circulation

*y*direction, and frontal wavelength

*B*in

*x*(see Table 1). The coordinate system

*x*. The domain is a doubly periodic square with side lengths of 50 Rossby radii. This SST prototype exemplifies all physical regimes—large

*B*yield high-curvature fronts relevant for comparison with observations of Gulf Stream and Southern Ocean rings, and their atmospheric response (Park et al. 2006; Frenger et al. 2013).

We focus on the observed characteristics of wind speed and direction, of wind stress divergence and curl, and of the coupling coefficients, all in relation to SST and its gradients. For observations, frontal impacts are separated from large scales by application of a filter that retains scales smaller than 10° latitude and 20° longitude only (Chelton et al. 2004; Chelton and Xie 2010). In our model, large-scale variability is reduced to a prescribed geostrophic wind *B* of the SST front (Table 1). The downwind and crosswind components of the SST gradient are determined relative to the large-scale winds in observations (O’Neill et al. 2010a) and relative to surface background winds

Impacts of the background winds are explored with *x* = 10 (Fig. 3a) where background winds cross at a near-right angle as “cross front” and segments where the background winds are approximately parallel to the front as “alongfront” [centered at *x* = −10 (Fig. 3a)]. The regions around the cusps of the SST front at *x* = 0 and *x* = ±25 afford a view of the response when winds cross the front at increasing angles and the associated increases of the cross-frontal advection. To distinguish these regimes from the type of forcing, we show the responses to the total and individual components of the forcing **F** in Eq. (21).

#### 1) Wind speed and direction

For **F** only acts at and downstream of the SST front. The in-phase relationship between high SST and wind speed is more pronounced for stronger background winds and shows a downstream oscillatory wake associated with a damped, lee gravity wave for Froude numbers greater than one (Fig. 3d). The transition between the alongfront evanescent response to the cross-front gravity wave is seen at the cusp of the front: as the cross-frontal component of the background wind increases, the downwind wake grows, while the upwind expression diminishes, until the latter vanishes and the former exhibits an oscillation.

As expected from the scaling Eq. (26), the importance of the vertical mixing effect increases with

The direction of frontally induced surface winds (Fig. 4) reflects the strong impact of the baroclinic pressure forcing for the spindown and inertial turning for the gravity wave regime. In the alongfront segment, winds over the front turn counterclockwise, consistent with a sea breeze from the cold to the warm side of the front. In the cross-front segment, winds turn in a clockwise direction (anticyclonic), because of a geostrophic response to the across-front pressure gradient. The directional modulation because of the vertical mixing effect alone (Fig. 4b) leads to a weak anticlockwise response, associated with the development of a back-pressure gradient. For the strong wind case with

#### 2) Wind stress curl and divergence

The model reproduces the observed characteristics of the frontally induced wind stress divergence and curl. In the cross-front segment, the wind stress divergence displays a dipole with large, positive values aligned with a downwind gradient of SST for the slow and fast background wind cases (Figs. 5, 6). Downstream, the wind stress divergence turns negative for

The wind stress curl (Figs. 7, 8) is more complicated than the strong curl in the alongfront segment suggested by qualitative reasoning based on the vertical mixing effect (Businger and Shaw 1984; Chelton et al. 2004). In the cross-front segment, the wind stress curl forms a dipole, with negative values upstream and positive values downstream of the front. In the alongfront segment, the curl vanishes at the SST front and reaches largest negative values on the cold and largest positive values on the warm side of the front toward the cusps of the front, when the air–sea temperature difference is no longer in equilibrium and advection comes into play. Negative wind stress curl is collocated with a positive crosswind SST gradient (Fig. 8), as expected from observations. This relationship stems from the vertical mixing term (Figs. 8b,e), while the pressure gradient produces a wind stress curl that is out of phase with the crosswind SST gradient. Since the vertical mixing term is dominant for the strong background wind case, the total response is collocated with the crosswind SST gradient for

In observations, frontally induced surface wind speed and direction are phase shifted with respect to the angle of SST gradient and background winds, so the wind stress divergence is enhanced, but the wind stress curl is reduced (O’Neill et al. 2010a). The model reproduces this finding. Use of the surface wind component Eqs. (30) and (31) yields the linearized formulation of the surface stress divergences in terms of downwind changes of the speed and crosswind changes of the direction, as well as of the curl in terms of crosswind changes of speed and downwind changes of direction (e.g., O’Neill et al. 2010a). Considering the cusp of the front at

## 5. Coupling coefficients

Coupling coefficients times 100 between wind stress divergence and downwind gradient of SST *γ* are doubled, while the remaining parameters remain at their reference values. Results are shown for the undulating front of Eq. (29), the reference case, a broad and a short wavelength, and strongly undulating fronts.

Since the dominance of

For

For

**F**in Eq. (21) is cancelled by the barotropic back-pressure gradient so that the surface geostrophic winds and surface stress vanish. For a strong baroclinic pressure gradient in

**F**, this balance is modified by the surface stress because of the geostrophic and ageostrophic shears (Cronin and Kessler 2009), but it again adjusts the back pressure to render the total wind stress curl zero. Since the back pressure directly affects only the horizontal transports, the hallmark of spindown is that the curl of the frontally induced wind stress [Eq. (24) with our choice of

Transforming Eqs. (34) and (35) to wavenumber space yields the budgets governing the transfer functions

For

*ϕ*is the angle between

**k**and

## 6. Sensitivity and comparison with observations

The sensitivity of the coupling coefficients to the formulation of the vertical exchange coefficients and to the SST distribution are explored by doubling the background eddy coefficient amplitude *γ* and by considering a broad SST front with

Coupling coefficients show a strong dependence on the background winds, the vertical mixing formulation, and, as expected from the transfer functions, the minimum SST scale (Table 2). The simulated order of nondimensional

Coupling coefficients increase with background winds, including the experiment with forcing by only the vertical mixing effect. This is consistent with an increase of the vertical mixing effect following Eq. (26), the steeper slopes of the transfer functions (Figs. 9, 10), the reduction of

For all but two cases, the coupling coefficient magnitudes for wind stress divergence are larger than for the wind stress curl (Table 2). This implies the ratios

The responses to a smooth front are equivalent with a coarsening of the model resolution and yield a reduction of the coupling coefficients (Table 2). This mimics the simulation at a variety of resolutions and changes induced in the atmospheric reanalysis by the use of a higher-resolution SST (Song et al. 2009; Bryan et al. 2010). A doubling of *γ* of the Ekman number decreases the coupling coefficients and leads to a consistent negative sign of

## 7. Conclusions

Observations of the impacts of SST fronts on the atmospheric boundary layer in the extratropics show ubiquitous covariations of the wind stress divergence with the downwind SST gradient and of a negative wind stress curl with the positive crosswind SST gradient (for SST increasing to the left of the winds). The associated regressions—called coupling coefficients—are positive for the divergence and negative for the curl, and their magnitudes are systematically larger for the divergence than for the curl. We explain this observational finding by distinct dynamics of the wind stress divergence and curl. Wind stress divergence results from either large-scale winds crossing the front or from a thermally direct, cross-frontal circulation. Wind stress curl, expected to be largest when winds are parallel to SST fronts, is reduced through geostrophic spindown and thereby yields weaker coupling coefficients.

To show these dynamics, we introduce a shallow water model for the atmospheric boundary layer, coupled by air–sea heat fluxes to SST and bounded aloft by a strong inversion with zero turbulent and radiative fluxes. The model is forced by a prescribed, large-scale, barotropic, and time-independent geostrophic wind. Model dynamics are obtained by linearization of the circulation induced by weak SST variations about a background state of an Ekman spiral and constant SST. The heat budget assumes vertically constant temperatures and balances advection by background winds and air–sea heat fluxes. For spatial scales smaller than a Rossby radius of deformation, lateral mixing parameterizes the smoothing action of a sea breeze. The resulting frontally induced temperatures are independent of frontally induced winds and force the momentum equations by baroclinic pressure gradients (Lindzen and Nigam 1987) and by the vertical mixing effect (Hayes et al. 1989; Wallace et al. 1989). The importance of the latter relative to the former forcing is given by Eq. (26) and increases with background winds and stress, with the thermal adjustment time of the boundary layer, and with the sensitivity of vertical mixing to the air–sea temperature difference.

Frontally induced winds are governed by the classical Rossby adjustment problem, albeit in the presence of vertical mixing and background advection. For cross-frontal background winds faster than the gravity wave speed (Froude numbers greater than one) the frontally induced circulation is characterized by lee gravity waves (Spall 2007; Kilpatrick et al. 2014). Froude numbers less than one yield the geostrophic regime and a dependence of the length scale relative to

The model reproduces the observed characteristics of frontal air–sea interaction: wind speeds increase (decrease) over warm (cold) waters downwind of a front; and the wind stress divergence is correlated with the downwind gradient of SST, while the regression of the wind stress curl with the crosswind gradient of SST is weaker, and negative because of the dominance of the vertical mixing mechanism. Overall, the coupling coefficients span the range of observations (O’Neill et al. 2010a) and depend on the background winds, the scales of the SST distribution, and the vertical mixing formulation. The increase of the coupling coefficients found as the SST variance includes scales close to the Rossby radius of deformation is consistent with model simulations comparing the atmospheric responses to smooth versus high-resolution SST fields (Song et al. 2009).

While our model includes all physical processes cited in the context of frontal air–sea interaction and captures the observed characteristics, it makes a number of assumptions. The model is in steady state, consistent with the observational focus on time scales long compared to synoptic atmospheric variability (Chelton and Xie 2010; O’Neill et al. 2010a) and a spindown time scale approaching an inertial period as the scales approach the Rossby radius of deformation (Holton 1965a). The vertical structure of the free troposphere is a reduced-gravity layer, a choice in midlatitudes that excludes vertically propagating waves (e.g., Kilpatrick et al. 2014) and interactions of the boundary layer with the evolution of potential vorticity in the free troposphere considered in Feliks et al. (2004, 2007, 2011). This simplified vertical structure, however, is a conceptual prototype to show the similarity of the boundary layer response to the classical Rossby adjustment and to determine the impact on the coupling coefficients, and is best suited to stratocumulus regimes (Schubert et al. 1979b). The model is formulated on a midlatitude *f* plane and cannot explain directly observations of

Given the vigorous atmospheric synoptic variability in the midlatitude regions of large SST gradients of the western boundary currents and Southern Ocean, the accuracy of the linear approximation of our model may be limited, and it remains to be seen if analyses of observations or high-resolution models confirm its relevance. However, the linear dynamics discussed here provide a unified framework that we hope is useful for future studies of frontal air–sea interaction.

## Acknowledgments

This work has benefited from discussions with Drs. Jessica Benthuysen, Thomas Kilpatrick, Emanuele Di Lorenzo, Shoshiro Minobe, Masami Nonaka, Larry O’Neill, Jim Potemra, Yoshi Sasaki, Justin Small, Bunmei Taguchi, Kohei Takatama, Axel Timmermann, Hiroki Tokinaga, and Shang-Ping Xie. Thoughtful comments by Drs. Larry O’Neill, Hisashi Nakamura, Mr. Ryusuke Masunaga, and an anonymous reviewer greatly improved an earlier draft of this manuscript. We gratefully acknowledge support by National Science Foundation (Grants OCE06-47994, OCE13-57157, and OCE09-26594); by the Department of Energy, Office of Science (Grants DOE-SC0006766 and DOE DE-SC0005111); by the National Aeronautics and Space Administration (Grant NNX14AL83G); and by the Japan Agency for Marine-Earth Science and Technology as part of the JAMSTEC–IPRC Joint Investigations (JIJI).

## APPENDIX

### Derivation of the Nondimensional Reduced-Gravity Model

*w*is the vertical component of the velocity,

*f*is the Coriolis frequency,

*ρ*is the density,

*p*is the pressure,

*A*is the vertical exchange coefficient, and

*z*is the vertical coordinate. For a 1.5-layer reduced-gravity framework,

*g*is Earth’s gravitational acceleration.

*z*to sigma levels

*s*=

*z*/

*h*and neglecting the term proportional to

Scales used in the nondimensionalization.

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This effect is called “buoyancy shutdown” in the oceanic bottom boundary layer (MacCready and Rhines 1991) and drastically reduces bottom friction (Benthuysen 2010).