## Abstract

Satellite observations and modeling studies show that midlatitude SST fronts influence the marine atmospheric boundary layer (MABL) and atmospheric circulation. Here, the Weather Research and Forecasting (WRF) mesoscale model is used to explore the atmospheric response to a midlatitude SST front in an idealized, dry, two-dimensional configuration, with a background wind oriented in the alongfront direction.

The SST front excites an alongfront wind anomaly in the free atmosphere, with peak intensity just above the MABL. This response is nearly quasigeostrophic, in contrast to the inertia–gravity wave response seen for cross-front background winds. The free-atmosphere response increases with the background wind , in contrast to previously proposed SST frontal MABL models.

The MABL winds are nearly in Ekman balance. However, a cross-front wind develops in the MABL as a result of friction and rotation such that the MABL cross-front Rossby number *ε* ≈ 0.2. The MABL vorticity balance and scaling arguments indicate that advection plays an important role in the MABL dynamics. Surface wind convergence shows poor agreement with MABL depth-integrated convergence, indicating that the MABL mixed-layer assumption may not be appropriate for SST frontal zones with moderate to strong surface winds.

## 1. Introduction

Satellite observations show a robust influence of SST fronts on surface winds (Xie 2004; Chelton et al. 2004; O’Neill et al. 2005; Small et al. 2008; Chelton and Xie 2010; O’Neill 2012; O’Neill et al. 2012) such that surface wind speed and surface stress magnitude increase over warm SST, and decrease over cold SST (Fig. 1). Likewise, the wind stress divergence response is strong when surface winds are oriented in the cross-front direction (perpendicular to SST contours), and the wind stress curl response is strong when surface winds are oriented in the alongfront direction. These wind stress patterns are ubiquitous (Chelton et al. 2004), but here we focus on the midlatitude situation.

The marine atmospheric boundary layer (MABL) dynamics are distinct for the two cases depicted in Fig. 1 (Schneider and Qiu 2015). For the case of “strong” cross-front winds [defined as having *O*(1) cross-front Rossby number *ε* = *U*/*fL*, where *U* is the maximum cross-front component of the MABL wind, *f* is the Coriolis parameter, and *L* is the length scale of the SST front], the MABL momentum budget is dominated by changes in the advection and turbulent stress divergence terms (Samelson et al. 2006; Spall 2007b; Small et al. 2008; Kilpatrick et al. 2014, hereafter KSQ). The air–sea heat flux is vigorous because of the high wind speed and large air–sea temperature difference, but the MABL temperature adjusts to SST over a longer length scale than the turbulent stress divergence term. For winds blowing from cold to warm SST, the MABL response resembles the “vertical mixing mechanism” (Wallace et al. 1989; Hayes et al. 1989), whereby enhanced downward mixing of momentum increases the surface stress and reduces the vertical wind shear in the MABL (KSQ; Samelson et al. 2006; Liu et al. 2013; Byrne et al. 2015). Because of the strong influence of advection, MABL winds do not approximate an Ekman momentum balance (Spall 2007b; KSQ).

For the case of alongfront winds or weak cross-front winds (*ε* ≪ 1), the MABL temperature adjusts to SST over a similar length scale as the Coriolis and turbulent stress divergence terms, and the pressure gradient therefore plays a larger role in the response (Spall 2007b; Small et al. 2008; Shimada and Minobe 2011). As *ε* is reduced, MABL winds approach an Ekman momentum balance (KSQ).

Observational studies and GCMs have shown that SST fronts’ influence extends through the MABL, reaching the full depth of the troposphere (Minobe et al. 2008; Small et al. 2008; Chelton and Xie 2010). The Gulf Stream and Kuroshio Extension SST fronts enhance deep clouds and rainfall (Minobe et al. 2008, 2010; Kuwano-Yoshida et al. 2010; Sasaki et al. 2012) and influence the atmospheric circulation on synoptic and planetary scales (Qiu et al. 2007; Joyce et al. 2009; Taguchi et al. 2009; Frankignoul et al. 2011; Kwon and Joyce 2013; Small et al. 2014; O’Reilly and Czaja 2015; Piazza et al. 2015; Smirnov et al. 2015).

But how is the influence of the SST fronts communicated through the MABL to the free atmosphere? Feliks et al. (2004) proposed an MABL model that includes temperature perturbations induced by the underlying SST front. The resulting MABL pressure gradients induce MABL divergence and convergence (i.e., Ekman pumping). Several studies couple this analytical MABL model to a quasigeostrophic atmosphere and show that the SST front-induced Ekman pumping drives large-scale circulations (Feliks et al. 2004, 2007, 2011). Brachet et al. (2012) found the vertical circulation above the Gulf Stream in a GCM was consistent with the Feliks et al. (2004) MABL model.

In a study closely related to ours, Lambaerts et al. (2013) conducted idealized modeling experiments to evaluate the Feliks et al. (2004) MABL model and a similar MABL model proposed by Minobe et al. (2008). The Lambaerts et al. (2013) “reference” configuration has a vertically sheared background wind that increases from zero at the surface to 1 m s^{−1} at 1200-m height. The Feliks et al. (2004) and Minobe et al. (2008) MABL models were found to be reasonable approximations under these conditions of *weak surface winds*.

Here we evaluate the Feliks et al. (2004) and Minobe et al. (2008) MABL models for stronger background winds than were considered by Lambaerts et al. (2013). As discussed in section 4, a shortcoming of the Feliks et al. (2004) and Minobe et al. (2008) MABL models is that the MABL response to the SST front is insensitive to the background wind’s magnitude and angle of orientation to the SST front. KSQ showed that, for strong cross-front winds, an SST front excites an inertia–gravity wave in the free atmosphere, analogous to a mountain wave in the hydrostatic rotating limit (e.g., Gill 1982). Linear inertia–gravity waves carry no perturbation potential vorticity (Lien and Müller 1992; Salmon 1998) and are therefore not “felt” by the balanced, synoptic-scale flow, so long as the waves do not dissipate. In other words, an SST front does not influence the free atmosphere through Ekman pumping in the case of strong cross-front background winds. KSQ did not explore alongfront background winds but speculated that, for *ε* ≪ 1, the MABL winds would remain close to Ekman balance as air parcels cross the SST front and that the free-atmosphere response would be quasigeostrophic (QG).

Recently, Schneider and Qiu (2015) developed an idealized SST frontal MABL model specifically to study the differing responses for cross-front and alongfront background winds. The Schneider and Qiu (2015) MABL model represents the free atmosphere as a reduced gravity layer and diagnoses the linearized steady-state response; linearization is achieved using a background state with no SST front and an Ekman spiral driven by a prescribed, barotropic, geostrophic wind. MABL temperature is constant with height, and changes in vertical mixing induced by the SST front have no impact on MABL temperature; no shallow internal boundary layer forms over a warm-to-cold SST front (Song et al. 2006; Spall 2007b). The Schneider and Qiu (2015) MABL model reproduces the main features of the surface stress response to SST fronts.

Here we explore the atmospheric response for alongfront background winds using a similar idealized, two-dimensional model configuration as KSQ. Compared to Schneider and Qiu (2015), our model experiments are time-dependent, include a full, continuously stratified atmosphere, and solve the fully nonlinear momentum and heat equations.

We will show that the atmospheric response for alongfront background winds is indeed fundamentally different from the response for cross-front background winds: the free atmosphere is nearly quasigeostrophic; and the MABL winds are nearly in Ekman balance, though advection still plays an important role in the MABL dynamics. As motivation, we note that QuikSCAT surface winds’ (conventional) surface Rossby number Ro = *ζ*/*f* exceeds 0.1 over much of the Gulf Stream for a typical 15-day period (Fig. 2); *ζ* is the vertical component of the relative vorticity.

Section 2 describes the regional atmospheric model configuration. Section 3 describes the free-atmosphere response to the SST front, and section 4 describes the MABL response. Section 5 provides a summary and discussion, including a recommendation for an MABL model that is well suited for SST frontal zones.

## 2. Regional atmospheric model configuration

The atmospheric response to an idealized midlatitude SST front is explored here with the nonhydrostatic Weather Research and Forecasting (WRF) Model, version 3.3.1 (Skamarock et al. 2008). WRF has been used to study ocean-to-atmosphere influence over the Agulhas Return Current (Song et al. 2009; O’Neill et al. 2010) and the Kuroshio (Xu et al. 2010, 2011; Tanimoto et al. 2011).

We use a similar idealized, dry, two-dimensional WRF configuration as that in KSQ (Fig. 3). Our grid dimensions are 3600 km in the zonal direction and 20 km in the vertical. The horizontal grid spacing is 4 km. There are 86 grid points in the vertical, with 44 concentrated in the bottom 2 km. An *f*-plane geometry is used with *f* = 10^{−4} s^{−1} such that the inertial period is 2*πf*^{−1} = 17.5 h. Note that our dry configuration precludes clouds, precipitation, and latent heating, which all contribute to the atmospheric response to SST fronts (Minobe et al. 2008, 2010; Tokinaga et al. 2009).

In contrast to the KSQ configuration, here the meridional (alongfront) background wind is prescribed to different values, while the zonal (cross front) background wind is set to zero in an attempt to filter out the inertia–gravity wave response that characterizes cross-front winds (Glendening and Doyle 1995; KSQ). We define the “background wind” as the prescribed, barotropic, geostrophic wind; this is the same definition as in KSQ but differs from that of Schneider and Qiu (2015), whose background wind includes a frictional component in the MABL. Our background wind is set to 15 m s^{−1} in the base case for comparison to KSQ, but also varies [Table 1; section 3b(5)]. A cross-front wind develops in the MABL because of the influence of friction and rotation such that *ε* ≈ 0.2.

SST is fixed to a cold–warm–cold pattern with SST fronts centered at *x*_{1} = 900 km and *x*_{2} = 2700 km:

with *L*_{T} = 40 such that each SST front is approximately 100 km wide (Fig. 4b, thin line). The SST jump Δ*T* = 3°C. The domain is doubly periodic such that mass is conserved within the domain, in contrast to KSQ, who used a cold–warm SST pattern and open boundary conditions.

The two SST fronts are distinguished by the thermal wind shear they are expected to impart in the MABL (Fig. 3): the background wind has the same orientation as the thermal wind (TW) at the TW SST front, centered at *x* = 900 km; the background wind opposes the thermal wind (OTW) at the OTW SST front, centered at *x* = 2700 km. [Note that the SST fronts associated with the Gulf Stream (Fig. 2) and Kuroshio Extension (e.g., Nonaka and Xie 2003) are primarily TW SST fronts.] Analogous terms used in physical oceanography are “down front” winds and “up front” winds, respectively (Thomas and Lee 2005).

The initial temperature sounding is horizontally homogenous. The buoyancy frequency is *N* = 0.01 s^{−1} in the troposphere (corresponding to a lapse rate of 6.8 K km^{−1}) and *N* = 0.021 s^{−1} in the stratosphere, with the tropopause at 12 km. The potential temperature *θ*_{a} = 16°C at the lowest model level (*z*_{a} ≈ 6 m) such that the sea–air temperature difference *T* − *θ*_{a} is initially positive over warm SST and negative over cold SST.

The surface heat flux is given by

where *c*_{p} is the specific heat of air and **u**_{a} is the wind vector at the lowest model level. The drag coefficient *C*_{Ha} for heat is computed as

where *ψ*_{h} and *ψ*_{m} are stability functions for heat and momentum, respectively (Stull 1988). We define a positive heat flux as upward (i.e., out of the ocean and into the atmosphere).

We use the Nakanishi and Niino (2006) level-3 scheme for subgrid-scale turbulent mixing, which is a Mellor and Yamada (1982)–type scheme. For other details of the boundary layer and surface flux schemes, we refer the reader to KSQ.

The initial condition is (i.e., the geostrophic wind). Each WRF experiment is 15 days long.

## 3. Free atmosphere

### a. Quasigeostrophy in pressure coordinates

KSQ suggested that the atmospheric response to an SST front is analogous to a mountain wave, with the type of wave response determined by the “intrinsic frequency” , where the wavenumber *k* corresponds to the length scale *L* of the SST front. For the cross-front cases considered in KSQ, , corresponding to the hydrostatic rotating wave limit (e.g., Gill 1982).

The mean zonal background wind for the regional atmospheric model experiments considered here so that . Since this satisfies , we anticipate a quasigeostrophic response in the free atmosphere, characterized by an evanescent vertical structure that decays over a Rossby height *fL*/*N* (e.g., Gill 1982), rather than the inertia–gravity wave response seen in KSQ.

Later, we will compare the winds and geopotential heights from the regional atmospheric model experiments to the balanced, quasigeostrophic fields, so here we briefly review quasigeostrophy in pressure coordinates. Following Hoskins et al. (1985, their section 5b), the quasigeostrophic potential vorticity (PV) in pressure coordinates is

where is the vertical unit vector, ∇_{p} is the horizontal gradient operator on pressure *p* surfaces, **u** is the horizontal component of the wind, *θ*_{ref}(*p*) is a reference potential temperature distribution, and *θ*′ is the potential temperature anomaly relative to *θ*_{ref}.

We make the geostrophic approximation . The geostrophic streamfunction *ψ*′ is given by

where *ϕ* is the geopotential and *ϕ*_{ref} is a reference geopotential.

The hydrostatic relation in the pressure coordinate system is ∂*ϕ*/∂*p* = −*Rθ*, where *R*(*p*) is the derivative of the Exner function. The hydrostatic relation may be linearized to obtain

such that the vertical derivative of the geostrophic streamfunction is related to the buoyancy anomaly.

For a reference PV *q*_{ref} = *f*_{0} and static stability , we can substitute Eq. (6) into the rhs of Eq. (4) to obtain

where the Laplacian-like operator is defined by

Equation (7) is the QG balance condition: if the PV anomaly *q* − *q*_{ref} and appropriate boundary conditions are known, then *ψ*′ can be found by inverting the Laplacian-like operator .

### b. Free-atmosphere response in the regional atmospheric model

#### 1) Overview

Here we compute geopotential height anomalies *z*′(*p*) relative to the initial sounding and alongfront wind anomalies relative to the background wind,

Figures 4a and 5a show *z*′(*p*) and *υ*′ at 2 and 4 days, respectively, for the base case ( = 15 m s^{−1}); SST is shown in Figs. 4b and 5b.

A high develops over the TW SST front, and a low develops over the OTW SST front. At 4 days, the 800-hPa level shows a 1-m high and a 6.5-m low (Fig. 5a). The height anomalies have a coherent vertical structure, with extrema near the top of the MABL (dashed line) that decay with height, in contrast to the vertically oscillating inertia–gravity wave response for strong cross-front background winds (KSQ).

As we have anticipated, the TW SST front induces positive *υ*′, and the OTW SST front induces negative *υ*′, consistent with thermal wind shear acting over the MABL, as in the SST frontal Ekman pumping model of Feliks et al. (2004) and previous numerical model simulations (Glendening and Doyle 1995; Feliks et al. 2010). The *υ*′ extrema occur near the top of the MABL but are displaced 70 and 100 km west of the TW and OTW SST fronts, respectively, because of buoyancy advection by the Ekman transport (section 4). The displacement of the *υ*′ extrema increases with height, reaching 150–200 km at the 700-hPa level.

#### 2) Balanced quasigeostrophic winds

We test whether the free-atmosphere response is consistent with quasigeostrophy by comparing the WRF winds to the balanced QG winds. We obtain the QG streamfunction *ψ*′ by inverting Eq. (7) numerically, using Jacobi’s method (Press et al. 1992): the rhs *q* − *q*_{ref} = 0 since no PV sources or sinks exist above the MABL; the top boundary condition is *ψ*′ = 0, and the lower boundary condition is given by Eq. (6), where *θ*′ is taken from the WRF experiment on the pressure surface 30 hPa above the maximum MABL height (bold lines in Figs. 4b, 5b).

The balanced QG heights and winds are shown at 2 (Fig. 4c) and 4 days (Fig. 5c). At both times, the balanced fields agree with the WRF fields, with extrema near the top of the MABL that decay with height. This agreement indicates that the SST front excites a nearly quasigeostrophic free-atmosphere response when forced by alongfront background winds.

The PV anomaly is *q* − *q*_{ref} = 0 for the free atmosphere [Eq. (7)], but there is a PV anomaly associated with the potential temperature anomaly *θ*′ on the lower boundary (Figs. 4b, 5b). As discussed by Hoskins et al. (1985) and originally shown by Bretherton (1966), *θ*′ on the lower boundary is equivalent to a PV anomaly with a delta-function distribution centered just above the lower boundary, with *θ*′ > 0 corresponding to a positive PV anomaly and *θ*′ < 0 corresponding to a negative PV anomaly.

#### 3) Evolution of free-atmosphere winds

The base case with = 15 m s^{−1} generates a free-atmosphere response that is nearly quasigeostrophic (Figs. 4, 5). Here, we run some cases with weaker background winds (Table 1) to test the sensitivity of the free-atmosphere response to .

Figure 6a shows the evolution of the maximum and minimum *υ*′ at 700 hPa for the cases in Table 1. All runs show the magnitude of *υ*′ increases until the end of the experiment, though the rate of increase slows after a few days.

We also see that *the strength of the υ′ response increases with the background wind *. The *υ*′ dependence on is seen more clearly in Fig. 6b, which uses a log–log scale; the slope of the *υ*′ points is between 1 and 2 for both SST fronts, indicating that *υ*′ has a nonlinear, but not quite quadratic, dependence on . Note that Fig. 6b shows *υ*′ at 4 days; if we make the same plot using *υ*′ at 15 days (not shown), the OTW *υ*′ still shows a nonlinear dependence, but the TW *υ*′ shows a linear dependence, perhaps because of the influence of spindown at longer times (Schneider and Qiu 2015).

Whether the *υ*′ dependence on is linear or nonlinear, Fig. 6 identifies a shortcoming in the Feliks et al. (2004) and Minobe et al. (2008) MABL models. In the Feliks et al. (2004) and Minobe et al. (2008) MABL models, the MABL response to the SST front is insensitive to the background wind’s magnitude and angle of orientation to the SST front (section 4a).

Last, Fig. 6 shows an asymmetry between the OTW and TW SST fronts for the cases with stronger background winds (V10 and V15; Table 1). This asymmetry is consistent with the stronger air–sea heat fluxes above the OTW SST front (section 4b).

## 4. Marine atmospheric boundary layer

### a. Linear Ekman/MABL model and application to SST frontal zones

An Ekman momentum balance is a three-way balance between the Coriolis, pressure gradient, and stress divergence terms:

where *ρ*_{0} is a reference air density and ** τ** is the stress. The Ekman transports are obtained from an MABL depth integral of the frictional component of Eq. (10):

where *τ*_{s} = (*τ*_{sx}, *τ*_{sy}) is the surface stress, and we have applied the boundary condition that ** τ** vanishes above the MABL.

The Ekman pumping velocity *w*_{e} is the vertical velocity at the top of the MABL (Beare 2007) and may be derived from the Ekman transports or by taking the curl of Eq. (10), substituting the continuity equation, and vertically integrating from the surface to the top of the MABL:

For a linear Ekman layer, *w*_{e} is diagnosed by the curl of the surface stress.

The traditional view is that the atmospheric boundary layer interacts with the synoptic-scale circulation primarily through Ekman pumping and spindown (Charney and Eliassen 1949; Mahrt and Park 1976; Holton 2004; Beare 2007). Boundary layer convergence forces *w*_{e}, which spins down the synoptic-scale flow through vortex stretching and squashing.

Two MABL models based on Eq. (10) have been proposed for SST frontal zones. In the first MABL model (Feliks et al. 2004), the stress is given by ** τ**/

*ρ*

_{0}=

*K*∂

_{z}

**u**, where

*K*is a constant eddy viscosity and a no-slip surface boundary condition is used. The underlying SST pattern imprints a lateral temperature gradient into the Ekman layer via the air–sea heat flux (Lindzen and Nigam 1987) such that the pressure gradient ∇

*p*is baroclinic. Feliks et al. (2004) derive an Ekman pumping velocity from Eq. (10) that contains two terms: the first term acts to damp the large-scale vorticity, as in traditional spindown; the second term depends on the SST Laplacian and

*h*

^{2}, where

*h*is the (constant) MABL depth. The contribution of the second term to the Ekman pumping acts to drive a quasigeostrophic circulation aloft (Feliks et al. 2004, 2007, 2011).

The second MABL model (Minobe et al. 2008) is a mixed-layer model based on Lindzen and Nigam (1987):

where *α* is a constant Rayleigh friction coefficient. The MABL temperature is in equilibrium with SST; the hydrostatic pressure gradient ∇*p* therefore reflects the underlying SST gradient.

Equation (14) can be manipulated to obtain the following relation between MABL convergence and the pressure Laplacian:

Satellite observations and GCMs show that surface wind convergence in SST frontal zones is correlated with the SLP Laplacian, and this correlation has been taken as evidence for the mixed-layer physics of Eqs. (14) and (15) (Minobe et al. 2008, 2010; Bryan et al. 2010; Kuwano-Yoshida et al. 2010; Shimada and Minobe 2011; Xu and Xu 2015).

The Feliks et al. (2004) and Minobe et al. (2008) MABL models both represent an Ekman momentum balance in which the MABL convergence is determined by hydrostatic pressure adjustments. However, KSQ showed that, for strong cross-front background winds [*ε* = *O*(1)], advection dominates and the MABL winds are unbalanced, rendering Ekman-like models invalid. For the alongfront background winds considered here (*ε* ≪ 1), we will show that an Ekman momentum balance is valid but only to first order in *ε*, and *O*(*ε*) advective terms play an important role in the MABL dynamics.

### b. MABL response in the regional atmospheric model

The rest of this section focuses primarily on the MABL fields for the base case (V15).

#### 1) Overview of the OTW SST front

Snapshots of the WRF Model winds are shown in Fig. 7 for the OTW SST front at 48 h. The cross-front wind *u* is negative (westward) within the MABL (Fig. 7a). The minimum MABL depth-averaged *u* = −2.7 m s^{−1} such that the cross-front Rossby number *ε* ≈ 0.2–0.3. The MABL height (thick black line) deepens over warm SST, analogous to the cold-to-warm case of KSQ (cf. their Fig. 2). The MABL potential temperature (white contours) is well mixed in the vertical; a lateral temperature gradient exists within the MABL from *x* = 2500 to *x* = 2700 km. As noted earlier, cold advection by the Ekman transport causes a misalignment between the MABL temperature gradient and the underlying SST gradient.

The full alongfront wind *υ* (Fig. 7b) is vertically sheared, but the vertical shear is reduced from *x* = 2400 to *x* = 2700 km because the geostrophic (thermal wind) shear opposes the frictional shear. The *υ* minimum is near the top of the MABL.

The vertical wind *w* (Fig. 7c) is strongest within the MABL, where it exceeds 2 mm s^{−1}. Above the MABL, *w* < 0 to the east of *x* = 2500 km, indicating MABL divergence; and *w* > 0 to the west of *x* = 2500 km, indicating MABL convergence.

The evolution of the air–sea heat flux is shown in Fig. 8a. A positive heat flux is maintained over the western edge of the SST front, growing to 20 W m^{−2} at 2 days and eventually exceeding 25 W m^{−2}. Note that the cross-front Ekman transport [Eq. (11)] advects cool air over warm SST, maintaining the air–sea temperature difference and therefore the air–sea heat flux [Eq. (2)].

Figure 9 shows the surface stress components at 2, 4, and 15 days. The alongfront component of the surface stress, *τ*_{sy}/*ρ*_{0}, is several times larger than the cross-front component of the surface stress, *τ*_{sx}/*ρ*_{0}, which is not surprising since the background wind is oriented in the alongfront direction. In contrast to the cross-front wind cases of KSQ, *τ*_{sy}/*ρ*_{0} shows a much larger fluctuation in the SST frontal zone than *τ*_{sx}/*ρ*_{0}.

The maximum alongfront surface stress *τ*_{sy}/*ρ*_{0} is located near the center of the SST front at *x* = 2700 km such that the wind stress curl (WSC) is negative on the eastern flank of the SST front and positive on the western flank of the SST front. The negative WSC signal is sharper than the positive WSC signal, especially at early times.

#### 2) Overview of the TW SST front

For the TW SST front, *u* is again negative (Fig. 10a), with minimum MABL depth-averaged *u* = −2.6 m s^{−1}, but parcels pass from warm to cold SST. The MABL depth shrinks as a shallow internal boundary layer forms over cold SST, analogous to the warm-to-cold case of KSQ (their Fig. 6).

The full alongfront wind *υ* (Fig. 10b) shows that the vertical shear over the MABL is enhanced from *x* = 650 to *x* = 950 km, as the geostrophic (thermal wind) shear and frictional shear are now aligned. The *υ* maximum is near the top of the MABL.

The vertical wind *w* (Fig. 10c) again shows the largest signal within the MABL, where *w* < 3 mm s^{−1}. Above the MABL, *w* > 0 to the east of *x* = 750 km, indicating MABL convergence. Note that the *w* vertical structure seen in Fig. 10c looks different from that of several studies that considered reanalysis and GCM winds in the Gulf Stream region [e.g., Fig. 3 of Minobe et al. (2008); Fig. 12 of Minobe et al. (2010); Fig. 19 of Kuwano-Yoshida et al. (2010)]. These previous studies showed vertical winds that are the same sign in the MABL and the free atmosphere, rather than the “dipole” structure shown in Fig. 10c. These discrepancies could be due to differences in model resolution, boundary layer parameterizations, or other model physics.

The air–sea heat flux for the TW SST front is downward (Fig. 8b). This is in contrast to prominent TW SST fronts in nature, such as the Gulf Stream and Kuroshio Extension, where air–sea heat fluxes are generally upward (e.g., Taguchi et al. 2009; Kelly et al. 2010). However, for the purpose of our idealized experiments, the important thing is that the SST front imprints its lateral structure into the MABL via the air–sea heat flux.

The downward, stabilizing heat flux is responsible for the shallow internal boundary layer that forms over cold SST (Fig. 10). The downward heat flux is weaker than the upward heat flux over the OTW SST front, consistent with the weaker free-atmosphere winds above the TW SST fronts (Fig. 6). We speculate that the atmospheric response to the OTW SST front is stronger because the wind speed *increases* as air parcels cross from cold to warm SST (not shown), amplifying the heat flux via Eq. (2), whereas, for the TW SST front, wind speed *decreases* as air parcels cross from warm to cold SST, reducing the heat flux. The larger heat flux and surface stress for the OTW case lead to a deeper MABL and therefore a larger SST-induced pressure gradient.

Figure 9b shows the surface stress components at 2, 4, and 15 days. Again, *τ*_{sy}/*ρ*_{0} is larger than *τ*_{sx}/*ρ*_{0} and shows a larger fluctuation across the SST frontal zone. The minimum alongfront surface stress *τ*_{sy}/*ρ*_{0} is located near the center of the SST front at *x* = 900 km such that the WSC is positive on the eastern flank of the SST front and negative on the western flank of the SST front.

#### 3) Ekman-balanced winds

Here we diagnose the Ekman-balanced components of the MABL winds. We use the zonal component of Eq. (10) to distinguish the geostrophic component of the wind

from the frictional component of the wind

The total Ekman-balanced meridional wind is *υ*_{g} + *υ*_{F} [Eq. (10)].

We consider MABL depth averages of *υ*_{g} and *υ*_{F} for the OTW SST front (Fig. 11a): is diagnosed from the pressure gradient term [Eq. (16)] in WRF and reaches a minimum near *x* = 2600 km; is diagnosed from the surface stress *τ*_{sx} [Eq. (12)] and shows little variation across the SST frontal zone, consistent with Fig. 9. We compute the depth integrals to *h* = 1050 m, above the MABL. The depth average of the total Ekman-balanced wind *υ*_{g} + *υ*_{F} shows excellent agreement with the depth average of the actual wind *υ*′, reaching a minimum of −2.3 m s^{−1}. Note that the minimum is displaced ~100 km west of the center of the SST front as a result of buoyancy advection by the Ekman transport.

Figure 12a shows the same diagnostics for the TW SST front, but *h* is fixed to 950 m because of the shallower MABL. The Ekman-balanced wind *υ*_{g} + *υ*_{F} again shows excellent agreement with the actual wind *υ*′.

Next we consider the MABL relative vorticity. Taking the horizontal divergence of Eq. (10) and dividing by *f*_{0} yields an expression for the Ekman-balanced relative vorticity:

The geostrophic component *ζ*_{g} is diagnosed by the pressure Laplacian, and the frictional component *ζ*_{F} is diagnosed by the horizontal divergence of the turbulent stress divergence.

Figure 11b shows MABL depth averages of *ζ*_{g} and *ζ*_{F} for the OTW SST front. We diagnose *ζ*_{g} from the pressure gradient term in WRF; we diagnose the depth average of *ζ*_{F} from the surface wind stress divergence. The depth average of *ζ*_{g} shows agreement with the depth average of the actual relative vorticity *ζ*, in contrast to the MABL response for cross-front background winds considered by KSQ (their Fig. 14). Figure 12b shows that the MABL depth-averaged *ζ*_{g} and *ζ* also agree for the TW SST front. The depth average of *ζ*_{F} is negligible for both SST fronts, consistent with the small wind stress divergence (Fig. 9). The *ζ* signals in Figs. 11b and 12b are normalized by *f*_{0}, indicating that Ro = *ζ*/*f*_{0} peaks at about 0.3 above the OTW SST front, consistent with our earlier estimate of *ε*.

Figure 13a shows vertical profiles of the wind anomaly *υ*′(*z*), the geostrophic component *υ*_{g}(*z*), and frictional component *υ*_{F}(*z*) for the OTW case at *x* = 2592 km, the location of minimum depth-averaged *υ*′. The geostrophic component, diagnosed from Eq. (16) using the WRF pressure gradient term, reaches a minimum of −3 m s^{−1} at 800 m and decays with height above the MABL. The frictional component, diagnosed from Eq. (17) using the WRF turbulent stress divergence term, has its largest amplitude near the surface, switches sign mid MABL, and drops to zero above the MABL. The actual wind *υ*′ agrees with the total Ekman-balanced wind *υ*_{g} + *υ*_{F} to within 25% in the MABL, and both decay with height above the MABL.

For contrast, Fig. 13b shows the corresponding wind profiles from the experiment with strong cross-front winds analyzed in KSQ; the wind profiles are taken at the center of their SST front. The actual wind *υ* in KSQ differs from the total Ekman-balanced wind *υ*_{g} + *υ*_{F} by more than 50% in the MABL; as *υ* goes to zero above the MABL, *υ*_{g} + *υ*_{F} decreases to −0.3 m s^{−1}. The poor agreement between *υ* and *υ*_{g} + *υ*_{F} above the MABL is consistent with the unbalanced inertia–gravity wave response (KSQ).

We conclude that the MABL winds are nearly Ekman balanced for the alongfront background winds considered here, in contrast to the unbalanced cross-front case of KSQ. However, the MABL cross-front Rossby number *ε* ≈ 0.2, indicating that advection is still significant.

#### 4) Ekman pumping in the SST frontal zone

Figure 14 shows the evolution of *w* on a *z* surface 100 m above the maximum MABL height in the OTW SST frontal zone. The region above the SST front is characterized by descent, while a western region of ascent develops after 12 h. At 24 h, *w* < 0 to the east of *x* = 2550 km and *w* > 0 to the west of *x* = 2500 km; both signals have a magnitude of 0.5–2 mm s^{−1}. The zero contour drifts westward over the integration, reaching *x* = 2400 km by 96 h, and the *w* signal attenuates to a magnitude of 0–0.5 mm s^{−1}, consistent with spindown (Schneider and Qiu 2015). Note that gravity wave propagation away from the SST front is visible the first few hours, and inertial oscillations are visible in the western half of Fig. 14; the inertial oscillations decay over several inertial periods.

Contours of the linear Ekman pumping velocity *w*_{e} diagnosed from Eq. (13) are overlaid in Fig. 14. Equation (13) diagnoses descent on the eastern edge of the SST front, which is the correct sign but much too strong; Eq. (13) diagnoses ascent on the western edge of the SST front, which is the incorrect sign. The linear Ekman pumping estimate shows better agreement with *w* in the western region of ascent (e.g., from *x* = 2200 to *x* = 2450 km at 48 h).

A similar pattern is seen above the TW SST front (Fig. 15), with ascent on the eastern edge of the SST front and descent on the western edge and trailing region. The zero contour drifts westward over the integration. Inertial oscillations are even more prominent than for the OTW SST front and decay over several inertial periods. The linear Ekman pumping estimate [Eq. (13)] shows better agreement with *w* than in the OTW case but is still too large on the eastern edge of the SST front and the wrong sign over the western edge of the SST front.

The poor agreement between the linear Ekman pumping estimate [Eq. (13)] and *w* in the SST frontal zones suggests that advection may play an important role in the MABL dynamics. We diagnose the importance of advection in the MABL by examining the MABL depth-integrated vorticity equation:

where *δ* = −∂_{x}*u* is the horizontal convergence. We neglect the vorticity tendency ∂_{t}*ζ* and the vertical advection of *ζ* because they are smaller than the other terms. Rearranging so that the planetary vortex stretching term is on the lhs (fstr) and applying the boundary condition that ** τ** vanishes above the MABL,

The rhs of Eq. (20) represents MABL depth integrals of the vorticity advection (Adv), relative vorticity stretching (*ζ*str), and vortex tipping terms (Tip); the last term is the surface wind stress curl (WSC). Note that excluding the terms in the brackets on the rhs of Eq. (20) and substituting the continuity equation on the lhs yields Eq. (13).

We diagnose the terms in Eq. (20) from the WRF fields for the OTW SST front at 48 h (Fig. 16), enough time for the MABL winds to adjust to near Ekman balance. All terms contribute to the MABL depth-integrated *ζ* balance along the SST frontal zone, but the two largest terms overall are the WSC and advection terms. We emphasize that *ζ* advection makes a leading-order contribution to the *ζ* balance and actually determines the sign of the rhs of Eq. (20) over much of the SST frontal zone (2550–2700 km).

Several studies have derived analytical expressions for the Ekman pumping velocity *w*_{e} in weakly nonlinear Ekman layers (e.g., Hart 1996; Bannon 1998). However, these studies used a constant viscosity to make the problem tractable; the Mellor and Yamada (1982)–type turbulent mixing scheme used in our WRF experiments makes the analytical problem more complicated. Therefore we do not attempt to derive an analytical expression for *w*_{e} in SST frontal zones.

#### 5) MABL advection versus the “pressure adjustment mechanism”

We diagnose the magnitude of the MABL depth-integrated *ζ* advection [“Adv” term in Eq. (20)] above the OTW SST front (Fig. 17, triangles). The *ζ* advection increases sharply with , as a result of both the larger Ekman transport and the larger *ζ* that results from a deeper MABL and larger pressure gradients.

We propose that the *ζ* advection scales as follows. The scale of the advecting wind *u* is obtained from the Ekman transport [Eq. (11)], , where *C*_{D} is the drag coefficient; and *ζ* scales as the geostrophic component [Eq. (18)], ∇^{2}*p*/*ρ*_{0}*f*_{0}. For MABL depth *h*, the pressure Laplacian scales as ∇^{2}*p*/*ρ*_{0} ~ *gh*∇^{2}*T*/*θ*_{0}, where *T* is SST and *θ*_{0} is a reference potential temperature. Thus, the advection term in Eq. (20) scales as

so the *ζ* advection should grow at least with . If one assumes that *h* scales with (e.g., Pollard et al. 1973), the *ζ* advection grows with . Figure 17 suggests that the *ζ* advection does grow at least quadratically with , consistent with Eq. (21).

To allow a direct comparison between *ζ* advection and the pressure adjustment mechanism (PAM) of Minobe et al. (2008), we vertically integrate Eq. (15) and multiply by *f*_{0} to obtain

We estimate the PAM strength from the WRF experiments by diagnosing the rhs of Eq. (22) in the OTW SST frontal zone (Fig. 17, circles). We estimate the friction coefficient as , where *C*_{D} = 0.001 and *h* is diagnosed as the mean MABL height in the SST frontal zone. The PAM strength also increases sharply with , apparently because the MABL depth increases with , resulting in stronger pressure gradients.

We propose that the rhs of Eq. (22) scales as

where we have utilized the fact that *α* ≈ 0.2*f*_{0} is roughly constant for the range of background winds explored here. According to Eq. (23), the PAM is sensitive to through the dependence on *h*^{2}. The ratio of the *ζ* advection [Eq. (21)] to the PAM [Eq. (23)] is

indicating that the relative importance of advection increases for stronger background winds and for narrower SST fronts. Since , Eq. (24) can be simplified to Adv/PAM ≈ 5*ε*, where *ε* = *u*/*f*_{0}*L* is the cross-front Rossby number. For weak background winds (e.g., Lambaerts et al. 2013), the ratio Adv/PAM ≪ 1, and the Minobe et al. (2008) pressure adjustment mechanism is a reasonable approximation for the MABL convergence. However, Adv/PAM is *O*(1) for the range of explored in our WRF experiments.

Figure 17 and the above scaling analysis suggest that, for moderate to strong alongfront background winds , advection makes a leading-order contribution to the MABL *ζ* balance and that linear MABL models (Feliks et al. 2004; Minobe et al. 2008) may not be appropriate. There is a substantial meteorological literature on advective effects in the Ekman layer (Mahrt 1975; Bannon 1998; Ishida and Iwayama 2006), but to our knowledge no one has considered the specific case of SST frontal zones.

#### 6) Mixed-layer model

Last, we evaluate the Minobe et al. (2008) mixed-layer assumption [Eqs. (14) and (15)]. We compare the lowest model level convergence *δ*_{a} = −∂_{x}*u*_{a} to the MABL depth-averaged convergence , where *h* = 1050 m, chosen to be above the MABL.

The surface convergence *δ*_{a} shows some agreement with the SLP Laplacian (Fig. 18), as predicted by Eq. (15). However, *δ*_{h} is generally the *opposite sign* of *δ*_{a} in the SST frontal zones, indicating the mixed-layer assumption is not justified for our V15 experiment. This calls into question previous studies that take a correspondence between *δ*_{a} and SLP Laplacian as evidence for the mixed layer of physics of Eq. (15) (Minobe et al. 2008, 2010; Bryan et al. 2010; Kuwano-Yoshida et al. 2010; Shimada and Minobe 2011; Xu and Xu 2015). We note that the variable MABL depth appears to be important for determining the sign of the MABL depth-integrated convergence: for example, in the TW case (Figs. 10a,c) the lower MABL winds are divergent, but the upper MABL winds that exit the collapsing MABL are strongly convergent, resulting in MABL depth-integrated convergence and ascent.

Our approach is to diagnose the MABL depth-integrated convergence *δ*_{h}, which determines *w*_{e} and thus the free-atmosphere response (Figs. 4, 5). Two recent studies use a different approach, focusing on the near-surface portion of the MABL, from the surface to 100 m (Takatama et al. 2012, 2015). They use a realistic GCM configuration, albeit with coarser grid resolution (0.5° and 0.25°) and SST resolution (0.5°) compared to our WRF experiments. Takatama et al. (2012) and Takatama et al. (2015) find that MABL pressure adjustments explain most of their near-surface wind convergence and conclude that the Minobe et al. (2008) MABL dynamics are dominant.

Further modeling studies and observational analysis may be necessary to resolve the discrepancies between these studies.

## 5. Summary and discussion

The atmospheric response to an SST front is explored here for background winds oriented in the alongfront direction. Idealized, dry experiments with the WRF regional atmospheric model show that the free-atmosphere and MABL responses are fundamentally different from the responses for cross-front background winds considered in KSQ. The free-atmosphere response decays in the vertical over a Rossby height (Figs. 4a, 5a), in contrast to the vertically oscillating inertia–gravity wave response for cross-front background winds (KSQ). For = 15 m s^{−1}, the wind anomaly *υ*′ has a magnitude of ~1 m s^{−1} at the 700-hPa level (Fig. 6).

The free-atmosphere response is consistent with the mountain wave analogy of KSQ, whereby the intrinsic frequency determines the type of wave response in the free atmosphere; the wavenumber *k* corresponds to the length scale *L* of the SST front. In KSQ, , so the free-atmosphere response corresponded to the hydrostatic rotating wave limit; in this study, since , corresponding to the quasigeostrophic limit () (e.g., Gill 1982). Accordingly, we find a close agreement between the balanced quasigeostrophic height and wind anomalies and the actual WRF height and wind anomalies (Figs. 4c, 5c).

In the MABL, cross-front winds develop, primarily as a result of friction and rotation. However, the cross-front Rossby number is small enough (*ε* ≈ 0.2) that the MABL winds are nearly in Ekman balance, in contrast to the unbalanced MABL winds [*ε* = *O*(1)] seen for the cross-front background winds considered in previous studies (Song et al. 2006; Spall 2007b,c).

Thus, the MABL response might appear, at first glance, to resemble the linear Ekman-like MABL models that have been proposed for SST frontal zones by Feliks et al. (2004) and Minobe et al. (2008) (section 4a). However, the atmospheric response in our WRF experiments differs in important ways from these previous SST frontal MABL models:

*The free-atmosphere response**υ*′*increases with the background wind**(Fig. 6).*In the MABL models of Feliks et al. (2004) and Minobe et al. (2008), the MABL response to the SST front is insensitive to the background wind’s magnitude and angle of orientation to the SST front; therefore, Fig. 6, which shows*υ*′ has a nonlinear dependence on after 4 days of integration, identifies a shortcoming of the Feliks et al. (2004) and Minobe et al. (2008) MABL models.*Advection plays an important role in the MABL dynamics when**ε*≈ 0.2*or larger.*One might wonder whether an “extended” Feliks et al. (2004) MABL model that allows the MABL height to increase (through larger*K*) with while retaining a linear momentum balance [Eq. (10)], can explain the*υ*′ behavior in Fig. 6. This motivates our MABL diagnostics in section 4b, which indicate the following:Advection makes a leading-order contribution to the MABL

*ζ*balance for the V15 case (Fig. 16).Scaling arguments and Fig. 17 indicate advection and pressure adjustments make comparable contributions to the MABL depth-integrated convergence when

*ε*≈ 0.2 or larger. The Feliks et al. (2004) and Minobe et al. (2008) MABL models are sometimes referred to as the “pressure adjustment mechanism” because the MABL convergence is determined solely by the pressure Laplacian [Eq. (15)]. However, the pressure adjustment mechanism is physically linked to*ζ*advection because of the dependence of the SST front-induced pressure gradient on the MABL depth, which in turn is linked to the surface stress (deeper MABL for stronger surface stress). The alongfront component of the surface stress is associated with an Ekman transport [Eq. (11)] that advects the Ekman-balanced*ζ*[Eq. (18)] across the SST front, altering the MABL dynamics from that of a linear Ekman layer.The linear Ekman pumping formula [Eq. (13)], which diagnoses

*w*_{e}in linear Ekman models (Feliks et al. 2004; Minobe et al. 2008), fails to diagnose*w*_{e}in the SST frontal zone for the V15 case (Figs. 14, 15). [Note that this implies wind stress curl perturbations observed by satellite in SST frontal zones (Chelton et al. 2001, 2004; Chelton and Xie 2010; O’Neill et al. 2003, 2005) should not be related to*w*_{e}via Eq. (13).]^{1}Taken together, our diagnostics indicate the important role of MABL advection when

*ε*≈ 0.2 or larger. This conclusion is actually consistent with some of the discussion by Feliks et al. (2004), who acknowledge that nonlinear MABL effects might be important for SST fronts of the strength considered here.

*The surface wind convergence is not the same sign as the MABL depth-averaged convergence (Fig. 18),*suggesting that the mixed-layer assumption commonly made for SST frontal zones (Bryan et al. 2010; Shimada and Minobe 2011; Xu and Xu 2015) may be inappropriate. We note that several studies that considered reanalysis and GCM winds over the Gulf Stream region appear to show surface wind convergence that is the same sign as the MABL depth-averaged convergence (Minobe et al. 2008, 2010; Kuwano-Yoshida et al. 2010); however, our results indicate there is no a priori reason to assume this is always the case. The variable MABL depth appears to be an important factor in this discrepancy between the surface wind convergence and the MABL depth-averaged convergence (Figs. 7, 10).

Our results call for an SST frontal MABL model that includes momentum and buoyancy advection and variable MABL depth. One such model is the semigeotriptic (SGT) model (Beare and Cullen 2010, 2012, 2013), which extends the semigeostrophic equations (Hoskins 1975) to include a boundary layer such that the zeroth-order momentum balance is an Ekman (geotriptic) balance rather than the geostrophic balance. The Ekman balance condition filters out unbalanced inertial oscillations (Beare and Cullen 2010) and inertia–gravity waves (Beare and Cullen 2012), which dominate the free-atmosphere response when background winds are oriented across an SST front (KSQ).

The SGT model is therefore well suited for studying how the SST frontal MABL interacts with the synoptic-scale circulation. We have focused on the Ekman-balanced component of the MABL response because this component presumably has the strongest interaction with the balanced, synoptic-scale circulation, but this could be tested with the SGT model in more geophysically relevant situations than have been explored here (e.g., a developing baroclinic wave). Resulting insights could help improve the representation of the SST frontal MABL, and its interaction with the dynamics, in weather forecasting and climate models. This is urgent because air–sea interaction around midlatitude SST fronts appears to be important on climate time scales (Qiu et al. 2007; Minobe et al. 2008; Masunaga et al. 2016; Nakamura et al. 2008; Chelton and Xie 2010; O’Reilly and Czaja 2015), with implications for predictability (Hoskins 2013; Qiu et al. 2014).

## Acknowledgments

We thank three anonymous reviewers for their helpful comments. We acknowledge support from the National Science Foundation through Grants OCE-0550233, OCE-0647994, and OCE-0926594; from the U.S. Department of Energy, Office of Science, through Advanced Scientific Computing Research Award DE-SC0006766; and from NASA through Grants NNX10AO90G and NNX14AL83G. We thank Drs. Bob Beare, Shang-Ping Xie, Eric Firing, Roger Lukas, Peter Müller, and Gary Barnes for valuable discussions. QuikSCAT data were produced by Remote Sensing Systems (RSS; Ricciardulli et al. 2011), with thanks to the NASA Ocean Vector Winds Science Team for funding and support. AMSR data are produced by RSS (Wentz et al. 2014) and sponsored by the NASA Earth Science MEaSUREs DISCOVER Project and the NASA AMSR-E Science Team. RSS data are available at the RSS website (www.remss.com).

## REFERENCES

## Footnotes

International Pacific Research Center Publication Number 1195 and School of Ocean and Earth Science and Technology Publication Number 9635.

This article is included in the Climate Implications of Frontal Scale Air–Sea Interaction Special Collection.

^{1}

We have only considered the atmospheric Ekman pumping; the wind stress curl may very well diagnose the oceanic Ekman pumping (Vecchi et al. 2004; Spall 2007a; Chelton et al. 2007; Hogg et al. 2009).