1. Introduction
Midlatitude storms efficiently inject energy into boundary layer inertial oscillations (e.g., Pollard 1970; Pollard and Millard 1970; D’Asaro 1985; D’Asaro et al. 1995; Alford 2003b), and therefore boundary layer near-inertial energy density exhibits spatial and temporal patterns similar to atmospheric storm tracks (e.g., Chaigneau et al. 2008; Elipot et al. 2010). As storm tracks overlie western boundary current regions, which contain energetic geostrophic flows and strong fronts (e.g., D’Asaro et al. 2011; Whitt et al. 2014, manuscript submitted to J. Geophys. Res.), there is ample opportunity for interactions between inertial oscillations and strong geostrophic vorticity (e.g., Mooers 1975; Kunze 1985; Young and Ben-Jelloul 1997; Whitt and Thomas 2013). These wave–mean flow interactions may result in regionally elevated internal wave energy and enhanced turbulent mixing in the boundary layer and upper thermocline of the western boundary current extension regions (e.g., Silverthorne and Toole 2009; Nagai et al. 2009; Inoue et al. 2010; Whalen et al. 2012). Therefore, these interactions could modulate global tracer transport pathways, which may depend on diapycnal mixing in western boundary currents (e.g., Pelegrí and Csanady 1991; Jenkins and Doney 2003), and contribute to the dissipation of the kinetic energy in the oceanic general circulation via energy loss from balanced to unbalanced flows (e.g., Müller 1976; Kunze and Müller 1989; Müller et al. 2005; Polzin 2010; Thomas 2012; Vanneste 2013; Alford et al. 2013).
Here, we investigate the wind-driven generation of inertial oscillations in laterally sheared rectilinear geostrophic flows. The focus in this paper is largely on the process by which the background flow modifies the local flux of near-inertial energy from the winds to the boundary layer, although energy exchanges between the oscillations and the mean flow are also discussed in this context. To a large degree, mesoscale oceanic flows have a low Rossby number Rog = ζg/f ≪ 1. Hence, the geostrophic relative vorticity is small compared to the planetary vorticity ζg ≪ f, and the effective Coriolis frequency F = 
The work is further motivated by recent observations collected as part of the U.S. Climate Variability and Predictability (CLIVAR) Mode Water Dynamics Experiment (www.climode.org) and the Scalable Lateral Mixing and Coherent Turbulence (LatMix) initiative, which show that the winter Gulf Stream exhibits ubiquitous, banded, near-inertial shear and enhanced turbulence parallel to sloping isopycnals (e.g., Marshall et al. 2009; Inoue et al. 2010; Whitt and Thomas 2013; Whitt et al. 2014, manuscript submitted to J. Geophys. Res.; D. B. Whitt et al. 2014, unpublished manuscript). The key observation relevant to this work is that strong fronts like the Gulf Stream are associated with O(1) Rossby numbers Rog = ζg/f, which significantly modify the effective Coriolis frequency in the surface boundary layer across the front. We would like to understand how the generation of the near-inertial waves observed in the Gulf Stream is affected by the strong background vorticity.
Like many previous investigators (e.g., Pollard and Millard 1970; D’Asaro 1985; Alford 2003a; Mickett et al. 2010), we approach the generation problem using analytic solutions of a slab mixed layer model, but modify the equations to include the effects of background relative vorticity (e.g., Weller 1982; Klein and Hua 1988), as discussed in section 2. We then elucidate the physics encompassed in these equations and test the solutions by comparing them with idealized high-resolution two-dimensional numerical simulations that solve the full primitive equations in section 3.
The computational work extends earlier numerical studies of near-inertial waves in fronts (e.g., Rubenstein and Roberts 1986; Wang 1991; Klein and Treguier 1995; Federiuk and Allen 1996) that are focused primarily on the radiation of near-inertial energy from the mixed layer, the dynamics of the waves in the ocean interior, and the general redistribution of near-inertial energy by the background flow after generation. In contrast, the focus here is on the generation process in the boundary layer. In particular, we study how the physics of resonance (e.g., D’Asaro 1985; Crawford and Large 1996; Skyllingstad et al. 2000; Mickett et al. 2010) is modified by the presence of a laterally sheared geostrophic flow, in both low Rossby number (section 3a) and order-one Rossby number regimes (section 3b).
2. Physics of resonance in a slab mixed layer model
A schematic illustrating the idealized problem setup used in this paper. The geostrophic flow ug (blue arrows) is vertically uniform over the depth of the boundary layer HML and points in the x direction. Moreover, the effective Coriolis frequency 
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
The modified slab mixed layer system (1)–(2) are similar to those of Weller [1982, see his (7)] and Klein and Hua [1988, see their (5)]. However, we have neglected entrainment and the subsequent variation of the mixed layer depth and restrict attention to mean flows that only vary in one dimension (y) for simplicity.
In what remains of this section, we will first consider undamped homogeneous solutions to (1)–(2) in order to introduce some of the physical concepts without the added complications associated with damping and forcing. Then, in section 2b, we will consider (1)–(2) as an underdamped harmonic oscillator susceptible to resonance and use the linear response function to interpret the physics of the forced-dissipative equilibrium solutions. Finally, we will use the energy equation associated with (1)–(2) to discuss the full solution, which may be represented as a sum of both homogeneous and forced components.
a. Unforced and inviscid initial-value problem
Figures 2a and 2b show the components of the velocity vector as a function of time for two oscillations with different initial conditions but the same Rog = −∂ug/∂y/f = −0.75 and the same 
(a),(b) Velocity and (c),(d) kinetic energy for two example inviscid inertial oscillations governed by (6)–(8). In both examples, Rog = −(∂ug/∂y)/f = −0.75. Thus, as shown in (b), the velocity hodographs are elliptic with ratio 
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
(a) An ensemble of near-inertial velocity hodographs and the (b) corresponding energy densities EML as a function of time and initial phase angle ϕ0. Greater radii indicate greater EML. As in Fig. 2, the background flow is characterized by Rog = −0.75 and the initial speed of each oscillation is the same 1 m s−1 (dashed black circle) but the initial perturbation velocity vector has a different angle in each case. Therefore, for each hodograph the energy averaged over a wave period differs substantially, and when additionally averaged over all angles ϕ0, it exceeds the initial energy of 0.5 m2 s−2 [dashed black circle in (b)].
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
In any case, because UML and VML vary in quadrature, there is never any irreversible energy exchange with the mean flow in system (6)–(7). Since (6)–(7) lack damping and forcing, EML(t) must equal EML(0) at some point in each wave period. Therefore, we will consider the implications of (9), (12), and (13) in the context of forced and damped oscillations in section 2c. However, before getting to the solutions to the full forced/damped problem, we will consider the properties of the forced-dissipative equilibrium solutions to (1)–(2) using an appropriate linear response function in section 2b.
b. Resonance: The upper ocean as a field of harmonic oscillators
The general solution to (14) 
The question of interest to this study is what wind forcing τ = (τx, τy), of the form given in (15)–(16), maximizes A [(18)]? We will explore this question by considering the amplitude of the response as a function of both the effective Coriolis frequency F and the different forcing parameters in (15)–(16), that is, the angular frequency ωw, the eccentricity eτ, and the orientation angle θw of the winds.
Three key results emerge from studying the parameter space of system (1)–(2). First, varying the forcing frequency ωw and effective Coriolis frequency F highlights the resonance condition ωw = F. Second, the velocity of the forced oscillation is elliptically polarized when F ≠ f, as in the unforced homogeneous problem. Third, varying the angle θw and eccentricity eτ of the winds shows that the winds that do maximal work do not necessarily yield the maximal response. We will now briefly elaborate on each of these results.
The most obvious result is that the resonant frequency is approximately equal to the effective Coriolis frequency F, not the planetary Coriolis frequency f, unlike what is typically assumed.2 This is illustrated for θw = 0 and τa = τb (i.e., a circularly polarized wind stress) in Fig. 4, which shows that the largest values of A occur where 
The linear response function A [(18)] of the slab mixed layer system (1)–(2), as a function of the effective Coriolis parameter F and the angular frequency of the wind ωw. In this case, HML = 25 m, r = 5.79 × 10−6 s−1, f = 10−4 s−1, θw = 0, and the winds are circularly polarized, that is, τa = τb. The dashed red line denotes the approximate resonance condition ωw = F.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
The second key result is that, like the free solutions discussed above, the forced solutions tend to exhibit elliptically polarized velocity when F ≠ f. In fact, when r is small and ωw ≈ F, then AU/AV ≈ F/f (see Fig. 5), where AU and AV are the amplitudes of 
A measure of the eccentricity of the velocity response AU/AV for the same part of parameter space shown in Fig. 4. The results are normalized by F/f because as r → 0, AU/AV → F/f when ωw = F. The parameters are the same as in Fig. 4, that is, θw = 0, HML = 25 m, r = 5.79 × 10−6 s−1, f = 10−4 s−1, and the winds are circularly polarized τa = τb. The dashed red line denotes the approximate resonance condition ωw = F.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
Finally, because the velocity vector of the response traces out an ellipse, one might expect that elliptically polarized winds would yield the maximal response for a given forcing period–averaged wind stress magnitude. Indeed, this does turn out to be the case, but the winds that do the maximal work do not necessarily yield the maximal response A. Consider a situation where the winds are oscillating back and forth at frequency ωw along a single axis oriented at an angle θw relative to the mean flow (see Fig. 1). That is, τb = 0, τa ≠ 0, and θw is varied between 0 and π/2 to illustrate how the orientation of winds relative to the mean flow affects the amplitude of the velocity response. At resonance, that is, ωw = F, the maximum-amplitude response occurs when θw = 0 (the winds are parallel to the mean flow) for F < f and when θw = π/2 (the winds are perpendicular to the mean flow) for F > f, as shown in Fig. 6.
The response A [(18)] of the slab mixed layer system (1)–(2), as a function of the effective Coriolis parameter F and the orientation angle θw of a rectilinear wind (τb = 0) oscillating at the resonant frequency ωw = F. The parameters are the same as in Fig. 4, HML = 25 m, r = 5.79 × 10−6 s−1, and f = 10−4 s−1.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
In contrast, the major axis of the ellipse traced out by the inertial velocity response is aligned with the x axis (AU/AV > 1) for F > f and aligned with the y axis (AU/AV < 1) for F < f. Thus, the maximal response (18) for rectilinear winds oscillating along a single axis (τb = 0) occurs when the major axis of the inertial current ellipse is perpendicular to the axis along which the wind vector oscillates.
This finding is unintuitive, but it is consistent with the results derived in the inviscid framework in section 2a (see Figs. 2, 3). Based on section 2a, we may surmise that the mean flow also represents an important source of energy for the waves at high geostrophic Rossby number. Therefore, the last result of this section should be interpreted in terms of the energetics of the forced oscillations, which will be discussed in section 2c. Further discussion of the implications of different wind eccentricities eτ on the response function (18), including an interpretation of the amplification at ωw ≈ −F in Fig. 4, can be found in appendix C.
c. Energetics and the role of lateral shear production
However, the forcing and damping terms in (1)–(2) open the door for irreversible time-integrated energy exchange between the geostrophic and ageostrophic flows via LSP. This irreversible LSP occurs for two reasons: (i) damping and forcing together break the quadrature between UML and VML. For example, in a forced-dissipative equilibrium, as discussed in section 2b, the ellipse traced out by the velocity vector is tilted relative to the geostrophic flow; hence, there is a correlation between UML and VML. (ii) Even if UML and VML oscillate in quadrature, as in an unforced decaying oscillation, the damping term can induce a wave period–integrated Reynolds stress by itself. Either way, via damping and/or forcing, an inertial oscillation can induce an irreversible time-integrated exchange of energy with the geostrophic flow. Moreover, in some cases the time-integrated shear production term may be of the same magnitude as the wind work (or greater), as we will demonstrate below.
To understand this energy exchange, we will consider inertial oscillations in two scenarios: (i) a transient spinup and spindown and (ii) a forced-dissipative equilibrium. The latter scenario corresponds to the forced linear response discussed in section 2b.
1) Transient spinup/spindown
Transient wind forcing over a steady geostrophic flow leads to the spinup of an ageostrophic perturbation, which subsequently decays after the forcing subsides. This process can lead to an irreversible exchange of energy between the mean flow and the perturbation via LSP, during both the spinup and the spindown periods. The energy exchange is highlighted in two example spinup/spindown problems with Rog = −0.75. In both cases, the wind forcing is oscillatory, resonant (ωw = F), and rectilinear (τb = 0 and τa = 0.06 N m−2). However, the winds oscillate at two different angles relative to the front: θw = 0 in Fig. 7a and θw = π/2 in Fig. 7b. As a result, an oscillatory ageostrophic velocity quickly develops in both cases. Yet, as shown in Fig. 7c, the winds that oscillate parallel to the geostrophic flow (Fig. 7a) induce a much larger perturbation velocity than the winds that oscillate perpendicular to the flow (Fig. 7b). After 24 h, the forcing is turned off and the perturbation’s decay is due to the damping term DAMP in (19).
(a),(b) The four components of the energy budget (19) integrated over 5 days in two example transient spinup/spindown problems. The time-integrated LSP, WORK, and DAMP add to form EML. (c) The velocity hodographs associated with each case; (a) is red and (b) is blue. In both cases, the forcing is resonant (ωw = F) but only active for the first 24 h, after which the solution follows a homogeneous decay. LSP is positive and larger than WORK in (a), where θw = 0, but negative and weaker than WORK in (b), where θw = π/2. In both cases, Rog = −0.75, HML = 25 m, r = 5.79 × 10−6 s−1, f = 10−4 s−1, τa = 0.06 N m−2, and τb = 0.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
Note that the integrated wind work is the same in these two cases (Figs. 7a,b). That is because the governing oscillator equations for the forced components are identical [cf. (14) with (B11) when τx = 0 in (14) while τy = 0 in (B11)]. However, because LSP is of the same order as WORK, the energy densities EML in the two cases θw = 0 and π/2 are markedly different. Moreover, because of the damping, LSP results in a time-integrated exchange of energy from the mean flow to the perturbation.
Note, however, that resonant forcing (ωw = F) without explicit damping cannot result in any permanent exchange of energy from the mean flow to the perturbation in (19) for any τb/τa or θw. That is because in the resonantly forced but undamped case (which is associated with secular growth in the phase-averaged energy density 〈EML〉T; see appendix B), the time-integrated LSP always passes through zero during each forcing period, although it experiences wider and wider deviations from zero in each successive wave period. With the addition of damping, these energy exchanges can then become permanent. On the other hand, permanent exchange can occur without explicit damping for nonresonant forcing frequencies (ωw ≠ F), where the forcing can both add and subtract energy from the perturbation and the energy density is not monotonically growing. With that said, motions at nonresonant frequencies have relatively low amplitudes (see Fig. 4), and therefore the total energy exchange will likely be small compared to that associated with resonantly forced but weakly damped motions.
As discussed above, forcing at resonant frequencies ωw = F can yield substantially different mixed layer energy densities EML, depending on the orientation angle of the wind θw. These differences in EML are caused by variations in the lateral geostrophic shear production LSP and not the wind work. Figure 8 presents the three components on the right-hand side of the energy equation [(19)], LSP, WORK, and DAMP, as a function of θw ∈ [0, π/2] and F/f ∈ [0.5, 2] after a series of 10-day integrations. As in Fig. 7, the perturbation is spun up from rest over 24 h by resonant rectilinear winds with ωw = F, τb = 0, and τa = 0.06 N m−2. Subsequently, the forcing is turned off and the perturbation decays because of the damping term in (19). In all cases, the perturbation begins and ends with EML ≈ 0. However, the total dissipation is not the same as the work done by the wind forcing (WORK), which is effectively constant in the parameter space shown here (Fig. 8c). The differences in dissipation (DAMP) as a function of θw (Fig. 8b) arise because the inertial oscillations exchange kinetic energy with the geostrophic flow via LSP (Fig. 8a).
The terms in the energy budget (19) integrated over 10 days and plotted after the tenth day as a function of the effective Coriolis parameter F and the angle of the rectilinear forcing θw. The temporal variation of the amplitude of the wind stress is the same as the example in Fig. 7. The forcing is resonant (ωw = F) but only active for the first 24 h, after which the solutions follow an unforced homogeneous decay. (a) LSP and (b) DAMP are normalized by (c) WORK. All three are order-one contributors to the energy of the perturbation in some parts of parameter space. Integrating over a unit wave period (2π/F instead of 24 h) yields very similar patterns in (a) and (b), but the total work in (c) varies with the integration time. For this calculation, HML = 25 m, r = 5.79 × 10−6 s−1, f = 10−4 s−1, τa = 0.06 N m−2, and τb = 0.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
In a sense analogous to the unforced inviscid initial-value problem discussed in section 2a (see Figs. 2, 3), winds aligned parallel to the mean flow (θw ≈ 0) yield positive LSP and perturbations that extract energy from a geostrophic flow with anticyclonic vorticity (F < f), whereas winds aligned perpendicular to the mean flow (θw ≈ ±π/2) yield positive LSP and perturbations that extract energy from geostrophic flows with cyclonic vorticity (F > f). Conversely, winds aligned with the mean flow (θw ≈ 0) yield negative LSP and perturbations that inject energy into geostrophic flows with cyclonic vorticity (F > f), whereas winds aligned perpendicular to the mean flow (θw ≈ ±π/2) yield negative LSP and perturbations that inject energy into geostrophic flows with anticyclonic vorticity (F < f).
At every Rog ≠ 0, the lateral shear production is a strong function of the orientation angle of the winds (Fig. 8a). Yet, realistic winds are not oriented at only one angle. Rather the winds are composed of a sum of frequencies and orientation angles. To address the question of how a more realistic wind forcing might affect the geostrophic flow, we consider an ensemble of integrations with different forcing orientation angles θw. We continue to consider only resonant frequencies, however, because the response and energy exchange is much larger near resonance. Thus, resonant frequencies should dominate the overall energy budget.
The ensemble-averaged 
The θw ensemble-averaged lateral shear production 
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
The result (20), shown in Fig. 9, suggests that an isotropic distribution of wind directions will generate inertial motions that act as a net sink of geostrophic kinetic energy over long time periods with a rate that scales with the local wind work on inertial motions. However, this energy sink also scales as 
2) Forced-dissipative equilibrium
The terms in the energy budget (21) of near-inertial motions in a forced-dissipative equilibrium when the forcing frequency is resonant ωw = F. (a) Lateral geostrophic shear production (eLSP), (b) damping (eDAMP), (c) wind work (eWORK), and (d) the forcing period–averaged energy 〈EML(t → ∞)〉, all normalized by eWORK. The forcing period–averaged rates, eLSP, eDAMP and eWORK, are multiplied by 2π/f to put them in units of energy rather than power. As in the transient case (Fig. 8) all three components may be order-one contributors to the energy of the perturbation. As in Fig. 8, but HML = 25 m, r = 5.79 × 10−6 s−1, f = 10−4 s−1, τb = 0, and τa = 0.06 N m−2.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
Figure 10 gives some insight into the unintuitive result that emerged in section 2b (see Figs. 6, C2); that is, the amplitude of the resonant (ωw = F) forced response is maximal when θw = 0 for F < f and θw = π/2 for F > f (and along the red dashed line in Fig. C2 in the case of varying τb/τa; see appendix C). These are the same regions of parameter space associated with substantial energy extraction from the geostrophic flow and the largest eLSP/eWORK ratios. This finding explains why the largest resonant response occurs for winds that do not yield the maximum wind work, an unintuitive result of section 2b. It is because the largest response is associated with winds that both trigger a strong extraction of kinetic energy from the mean flow via eLSP and do substantial work on the inertial motions.
3. Numerical simulations with spatial variations in vorticity
In this section, two sets of numerical simulations with the spatially variable geostrophic Rossby number Rog will be used to test some of the predictions from the slab mixed layer model. The following questions will be considered:
The slab mixed layer model has no explicit lateral spatial variability (it varies only in time). Can the slab model represent local boundary layer inertial oscillation physics accurately when there are horizontal gradients in the slab model parameters such as the geostrophic relative vorticity ζg and therefore F?
The radiative decay parameter in the slab model r is designed to represent both viscous and inviscid physics, that is, the radiation of wave energy from the boundary layer to the interior. Are the results derived from the slab model relevant when there is an ocean below the boundary layer where near-inertial energy may radiate?
Each simulation is forced for the first 24 h with a spatially uniform wind stress of constant angular frequency ωw (defined positive clockwise as in section 2; see Fig. 1). The simulations are conducted using the Regional Ocean Modeling System (ROMS) (e.g., Shchepetkin and McWilliams 2005), a fully nonlinear three-dimensional hydrostatic primitive equation numerical ocean model that is run for this application in a two-dimensional mode where solutions are spatially uniform in x, but vary in y, z, and time. The computational grid has a uniform 300 m by 4 m resolution over a 120-km-wide by 1-km-deep domain. Momentum and tracer advection are computed with a third-order upwind scheme in the horizontal and a fourth-order centered scheme in the vertical. Diffusive terms are computed using a constant horizontal Laplacian eddy viscosity of 1.0 m2 s−1 and vertical diffusivity determined by the K-profile parameterization (KPP) mixing scheme of Large et al. (1994).
a. Low Rossby regime
Here, we present simulations of a sinusoidal barotropic background flow with relatively low Rossby number. The background velocity field is given by ug(y) = U0 sin(2πy/Ly), where U0 = 0.5 m s−1, Ly = 120 km, and y ranges from −Ly/2 to +Ly/2. The perturbation dynamics should be analogous to those in mesoscale quasigeostrophic flows, which are generally characterized by O(0.1) or lower Rossby numbers and high Richardson numbers (e.g., Charney and Stern 1962). Here, the maximum Rossby number is 0.26, and the Richardson number is infinite. To simplify matters, the background buoyancy frequency N = 4.5 × 10−3 s−1 is constant, except in the top 32 m, where the flow is initially unstratified.
Four simulations were run with the same background flow and forced by a spatially uniform, circularly polarized wind stress with a magnitude τ = 0.06 N m−2 over a forcing time t = 24 h (which also equals the total simulation time), but with different angular frequencies ωw = 0, 0.7f, f, and 1.2f [positive clockwise, as defined in (15)–(16) in section 2; see Fig. 1]. In this section, the wind always starts pointed in the negative y direction (θw = π/2). For the run with ωw = 0, the wind remains pointed in that direction for 24 h. The three rotating wind cases are resonant when the Rossby number of the geostrophic flow is −0.5, 0, and 0.5, respectively. However, in this mesoscale background flow, the superinertial and subinertial winds are not resonant anywhere in the domain because |Rog| < 0.5 everywhere. Nevertheless, the superinertial winds do about the same amount of work in the cyclonic part of the flow (ζg > 0) as the inertial winds, whereas the subinertial winds do about the same amount of work as the inertial winds in the anticyclonic part of the flow (ζg < 0) (see Figs. 11a,b). However, superinertial winds perform less work in the anticyclonic side and subinertial winds perform less work in the cyclonic side because the forcing frequency is farther from the resonance than the inertial winds in those regions of space. Therefore, both the sub- and superinertial winds do substantially less work overall than the inertial winds.
(a),(b) The 24-h-integrated wind work WORK and (c),(d) lateral shear production LSP in a sinusoidal barotropic geostrophic jet with amplitude 0.5 m s−1. In each of four simulations, the wind stress rotates clockwise at a fixed angular frequency ωw and amplitude τ = 0.06 N m−2. (a) and (c) show the result from a slab mixed layer model computed with HML = 40 m, r = 5.79 × 10−6 s−1, and f = 10−4 s−1. (b) and (d) show the result computed from a two-dimensional numerical simulation that solves the full primitive equations. The forcing frequency in each case is labeled with an arrow.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
Based on the scaling discussion in appendix A, it is not surprising to find that the simulation results compare fairly well with results from the slab mixed layer model over a 24-h forcing period [cf. WORK (Fig. 11b) and LSP (Fig. 11d) from the numerical model with WORK (Fig. 11a) and LSP (Fig. 11c) from the slab mixed layer model in Fig. 11]. The dominant physics in both the transient slab model (Fig. 11a) and the numerical simulations (Fig. 11b) is mechanical resonance. In these transient simulations with spatially variable F, the resonance physics manifests as a spatial variation in wind work and perturbation energy density, consistent with the interpretation of the linear response function (e.g., Fig. 4). Moreover, for circular winds and moderate geostrophic Rossby numbers, WORK tends to dominate LSP, although LSP still amounts to a 10% correction to the wave energy budget, as shown in Figs. 11c and 11d.
b. High Rossby regime
In this section, we compare four transiently forced numerical simulations of a sinusoidal barotropic jet with O(1) Rossby number characteristic of semigeostrophic flow (e.g., Hoskins 1975). The background velocity field is given by ug(y) = U0 sin(2πy/Ly) m s−1, where U0 = 1.5 m s−1, Ly = 120 km, and y ranges from −Ly/2 to +Ly/2. Our goal is to test the dependence of the resonant response on the orientation angle of the winds θw predicted by the theory (e.g., Fig. 7). In this section, the barotropic jet is forced for the first 24 h with a spatially uniform, rectilinear wind that oscillates at a frequency ωw = 0.5f or ωw = 1.32f, which yield a resonant response for Rog = ±0.75 such that resonance occurs at y ≈ 0 for Rog = +0.75 and y ≈ ±Ly/2 for Rog = −0.75. After 24 h, the system undergoes a homogeneous decay (with no forcing) for four more days so that the total integration time is t = 5 days. As in Fig. 7, τa = 0.06 N m−2 and the winds oscillate parallel (θw = 0) or perpendicular (θw = π/2) to the geostrophic flow.
We again find that the predictions of the slab mixed layer model and the physics of mechanical resonance are consistent with the results of the numerical simulation (see Fig. 12). To effectively generate inertial oscillations, the wind forcing must be near the local effective Coriolis parameter, as discussed in section 2. Therefore, for a forcing frequency of ωw = 0.5f (ωw = 1.32f), the large wind work and large-amplitude responses are restricted to the regions of the flow with a large and negative (positive) Rossby number (see Figs. 12a,b).
(a),(b) The 5-day-integrated wind work WORK and (c),(d) lateral shear production LSP in a transient spinup/spindown of inertial motions in a sinusoidal barotropic geostrophic jet with amplitude 1.5 m s−1. (a) and (c) are computed with the slab model, whereas (b) and (d) are computed with two-dimensional numerical model. Here, Rog ranges between ±0.8; the wind forcing, which runs for the first 24 h, is rectilinear (τb = 0), oscillates at a fixed frequency (ωw = 0.5f or 1.32f), has orientation angles θw = 0 or π/2 as labeled, and amplitude τa = 0.06 N m−2. In the slab model, HML = 40 m, r = 5.79 × 10−6 s−1, and f = 10−4 s−1.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
Furthermore, as in Fig. 7, there is a substantial asymmetry in the amplitude of the response when the winds are aligned parallel and perpendicular to the mean flow that is particularly evident when ωw = 0.5f. The asymmetry arises because there is a large integrated lateral shear production LSP that adds to the energy of the perturbation when the winds are parallel to the mean flow in resonantly forced regions of anticyclonic vorticity. In fact, after 5 days LSP contributes about twice as much as the wind work to the wave energy budget, consistent with the theoretical prediction (cf. Figs. 12c and 12d). On the other hand, LSP is negative when the winds are aligned perpendicular to the mean flow, but the magnitude is much smaller, as discussed in section 2.
The simulations also reveal a marked asymmetry in the strength of the ageostrophic response as a function of forcing frequency (see Fig. 13). This is because the strength of the resonant response where ωw ≈ F depends on the strength of LSP, which scales as the difference between the maximum and minimum kinetic energy during a wave period: 
The cross-stream velocity υ, which is all ageostrophic, is plotted in color after 5 days for the four numerical simulations in the high Rossby regime shown in Figs. 12 and 14. The potential density anomaly is contoured in kg m−3. The simulations are forced over the first 24 h with a rectilinear oscillating wind with frequency (a),(b) ωw = 0.5f or (c),(d) ωw = 1.32f and forcing orientation angle θw = 0 (parallel to the x axis) in (a) and (c) or θw = π/2 (parallel to the y axis) in (b) and (d). The wind stress has an amplitude |τ| = 0.06 N m−2. The forcing is resonant ωw = F in the center of the domain where ζg ≈ +0.75f in (c) and (d) or edges where ζg ≈ −0.75f in (a) and (b).
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
Although the assumptions of the slab mixed layer model are violated to some degree in all the two-dimensional nonlinear simulations with spatially varying F, the patterns in WORK, LSP, and other variables (not shown) as a function of space and parameters are qualitatively and quantitatively similar in both models, as shown in Figs. 11 and 12. The similarity between the slab model and ROMS simulation results suggests that the scaling assumptions used in the slab model (see appendix A) may be justified even in geostrophic flows with O(1) Rossby numbers. Thus, the insights derived from the slab mixed layer model provide a useful first-order description of the generation of near-inertial waves by winds in geostrophic flows with strong vertical vorticity, like western boundary currents.
This is despite the fact that much of the “damping” of near-inertial energy in the boundary layer of the numerical simulations is associated with wave radiation from the boundary layer to the interior, an inviscid process. By the end of the 5-day simulation shown in Fig. 13a, more than three quarters of the wave kinetic energy is in the interior (below 40 m). As revealed by a diagnostic calculation of the wave energy flux from the simulations (e.g., Fig. 14), more than twice as much wave kinetic energy has radiated from the surface to the interior when ωw = 0.5f and θw = 0 than when ωw = 0.5f and θw = π/2 (cf. Figs. 13a and 14a to Figs. 13b and 14b). The enhancement of interior wave energy when θw = 0 is consistent with the substantially enhanced boundary layer wave energy extracted from the mean flow via LSP when θw = 0 (e.g., Figs. 7 and 12).
The time-integrated downward wave energy flux evaluated just below the boundary layer and plotted as a function of time and cross-stream position in the four ROMS simulations in the high Rossby regime. We define the wave energy flux as the time integral of the perturbation pressure work 
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
The favorable comparisons between the slab model and ROMS simulations shown in Figs. 11 and 12 suggest that the slab model is capable of representing the boundary layer dynamics of near-inertial motions even when the damping is associated with a substantially inviscid downward wave radiation process. In fact, the spatiotemporal maps of downward wave energy flux in Fig. 14 demonstrate that inviscid wave radiation (not dissipation) is the dominant physical mechanism leading to the damping of the near-inertial motions in the boundary layer of the ROMS simulations after the wind stress is turned off at 24 h. Thus, the results discussed in this paper are not artifacts of the form of the slab model damping parameterization.
4. Conclusions and discussion
The results derived in this paper are most relevant in regions of the ocean with strong and episodic wind forcing and energetic geostrophic flows with horizontal length scales that are small relative to the wind forcing. This situation is representative of midlatitude western boundary currents and Southern Ocean fronts. These currents underlie atmospheric storm tracks and are associated with substantial wind work on near-inertial motions and large near-inertial energy densities near the ocean surface (e.g., D’Asaro 1985; Alford 2003b; Chaigneau et al. 2008; Silverthorne and Toole 2009).
In this context, the key results are as follows: (i) the amplitude of the local inertial response depends most strongly on the amplitude |τ(ωw)| of the wind stress at frequencies that are close to the local effective Coriolis frequency 
As in previous analyses with slab mixed layer models (e.g., Pollard and Millard 1970; Alford 2003a; Mickett et al. 2010), the results derived here depend on two scaling assumptions: (i) the Rossby number of the inertial response is so small, 
Having said that, as discussed in appendix A, geostrophic flows with large and variable effective Coriolis parameter tend to be associated with inertial oscillations that have smaller horizontal length scales. Hence, the inertial motions in these flows are more likely to violate the assumptions of the slab mixed layer model, which requires the motion to have a very low aspect ratio. There are two primary reasons for the association between high geostrophic Rossby numbers and smaller horizontal near-inertial wavelengths: (i) the resonant frequency for near-inertial motions, the effective Coriolis frequency F, varies in physical space (e.g., across a geostrophic front). Therefore, if the wind forcing is narrowbanded in frequency space, the inertial response to that forcing will vary substantially in physical space. (ii) Gradients in relative vorticity tend to cascade existing near-inertial energy to higher horizontal wavenumber over time (e.g., van Meurs 1998; Klein et al. 2004b). The numerical simulations in the high Rossby number regime exhibit both of these effects (see Figs. 12, 13). Thus, over time the near-inertial motions may eventually violate the low aspect ratio assumption, even if they do not initially.
Although the qualitative results presented in this paper are not especially sensitive to the choice of the radiative damping parameter r in the slab model, small quantitative improvements in the comparison between the slab model and ROMS simulations can be made by modifying r. However, the appropriate local value of r depends on properties of the geostrophic flow, including the horizontal spatial gradients of F, in a manner that is not fully understood (e.g., D’Asaro 1995; Klein and Treguier 1995; van Meurs 1998; Balmforth and Young 1999; Zhai et al. 2005; Danioux et al. 2008). In any case, nonlinear two-dimensional numerical simulations with Rog ~ 1 are consistent with the slab model over a few days as shown in Fig. 12 and thus justify the assumptions used in the modified slab model in this context.
The results derived in this paper have several important implications for the interpretation of inertial motions observed in the vicinity of strong geostrophic vorticity. For example, several investigators have observed energetic inertial motions trapped in anticyclonic relative vorticity (e.g., Kunze and Sanford 1984; Kunze 1986; Kunze et al. 1995; Joyce et al. 2013) and interpreted these observations based on theories that explain how near-inertial energy is redistributed by a geostrophic flow after generation. The analysis presented here (see, e.g., Figs. 4, 12, 13) allows us to reinterpret these observations in the context of resonance physics and infer that these oscillations in regions of anticyclonic vorticity were likely locally generated by winds that contained power near the local effective Coriolis frequency. We hypothesize that these observed waves exhibited elevated energy densities not only because the wave energy was spatially trapped and focused into a small region by the front/vortex waveguide but also because of the specific history of the local winds.
This work also highlights a new source of energy for upper-ocean near-inertial motions—the geostrophic flow. Although a rigorous global estimate of the energy exchange from geostrophic flows to near-inertial motions via LSP is beyond the scope of this paper, we discuss some back-of-the-envelope estimates to link this work to the broader questions about the oceanic kinetic energy budget (e.g., Ferrari and Wunsch 2009).
A scaling for the 〈LSP〉 could be calculated from (20) if the joint distributions of the wind work on near-inertial motions and the Rossby number of the underlying geostrophic flow were available. While we do not have the observations to do this, a crude estimate of (20) could be obtained from observations of relative vorticity in the boundary layer by assuming that the wind work on near-inertial motions is uniformly distributed relative to the distribution of vorticity. Then we can obtain a histogram of 〈LSP〉 from the histogram of the Rossby number using (20) and find its mean.4
The computation is complicated by the fact that accurate calculation of vorticity distributions requires high-resolution velocity observations. We know of two datasets with sufficient resolution to do this that were collected in the subtropical gyres of the North Pacific (Rudnick 2001) and North Atlantic (Shcherbina et al. 2013). The observations cover moderate and high Rossby number regimes, with rms Rossby numbers of 0.18 (Rudnick 2001) and 0.94 (Shcherbina et al. 2013). To estimate LSP using these datasets, we fit the observed probability density functions of vorticity to a Pearson family distribution that could then be substituted into (20). Expressed in terms of a fraction of the wind work on near-inertial motions, the resultant energy transfer from the background, presumably dominantly geostrophic, flow is 〈LSP〉/〈WORK〉 ~1% and 30% for the moderate and high Rossby number regimes, respectively.
Given these percentages, it seems unlikely that this lateral shear production mechanism represents a global source of energy for the internal wave field at the same order of magnitude as the 0.1–1 TW direct wind work on near-inertial motions (e.g., Alford 2001; Watanabe and Hibiya 2002; Alford 2003a; Jiang et al. 2005; Furuichi et al. 2008; Rimac et al. 2013), tidal conversion over rough topography (0.7–1.3 TW) (e.g., Egbert and Ray 2000, 2001; Nycander 2005; Garrett and Kunze 2007), or lee-wave conversion from geostrophic flows (0.2−0.8 TW) (e.g., Nikurashin and Ferrari 2011; Wright et al. 2014). However, it may be comparable to spontaneous loss of balance, which has not been precisely quantified on a global scale to our knowledge, but also depends strongly on the Rossby number (e.g., Müller et al. 2005; Danioux et al. 2012; Vanneste 2013; Alford et al. 2013; Shakespeare and Taylor 2013).
Acknowledgments
The authors thank Matthew Alford and an anonymous reviewer for suggestions that improved the manuscript. Discussions with John Mickett were also useful at an early stage. This work was supported by the Office of Naval Research Grant N00014-09-1-0202 and the National Science Foundation Grant OCE-1260312 as well as the William Whiteford Fellowship at Stanford University.
APPENDIX A
Scaling the Slab Mixed Layer Equations
One could proceed with a higher-order asymptotic analysis (e.g., Young and Ben-Jelloul 1997; Reznik et al. 2001; Zeitlin et al. 2003), but we keep only those terms that are O(1). Thus, we completely neglect the pressure gradient and buoyancy perturbations as well as vertical advection. Assuming that turbulent viscous terms may be parameterized by a Raleigh damping yields (1)–(2). Pollard and Millard (1970) suggest an inviscid interpretation of the damping terms in (1)–(2), noting that these terms could be used to parameterize vertical energy fluxes, which may dominate dissipation, particularly as (A17) gets closer to one. A judicious selection of the damping parameter r may therefore allow (1)–(2) to perform reasonably well even when (A17) becomes nonnegligible. Alternative forms of damping to parameterize wave radiation have also been proposed, including a lateral viscosity (Gill 1984) and an imaginary viscosity (Balmforth and Young 1999).
Note that we have not assumed that the Rossby number of the geostrophic flow is small; 
However, a geostrophic flow with a large Rossby number will tend to push the perturbations toward parts of parameter space that violate the assumption (A17). That is because the background flow leaves an imprint on the perturbation in a variety of ways. For example, a spatially uniform wind stress with an angular frequency will result in responses that have a spatially variable amplitude that depends on the vorticity of the underlying geostrophic flow, which modifies the local natural resonant frequency of inertial oscillations (e.g., Fig. 4). Moreover, 
Here, the analysis is conducted as if the scaling assumptions hold at all times of interest. This is accomplished by letting the winds be spatially uniform and letting F be a constant (i.e., ∂2ug/∂y2 = 0) in the discussion of the slab model [(1)–(2)] (section 2) and by considering only fairly short numerical simulations with integration times of a few days or less and spatially varying F in section 3. Moreover, as discussed above, a judicious choice for the radiative damping parameter r in (1)–(2) may improve the results by modeling the downward wave energy flux that occurs as Bua → 1.
APPENDIX B
Analytic Solution to Slab Mixed Layer Model
APPENDIX C
Response of the Slab Model to Variations in the Eccentricity of the Wind Forcing
In this section, three key results emerge from varying the eccentricity eτ of the wind forcing while holding θw = 0 constant. The first is that rotating winds with any eccentricity other than τy/τx = f/F force both clockwise and counterclockwise rotary components in the response. The second is that the maximum response A is not necessarily associated with winds that do maximal work on the inertial motions, as discussed in the body of the text. The third is that when the forcing frequency ωw is close to the resonant frequency F, variations in eτ have a small effect on the amplitude of the response compared to variations in ωw. We illustrate these results using two background flows with Rog = −0.5 and 1 forced by an oscillatory wind with varying frequency and eccentricity (Fig. C1).
The response A [(18)] of the slab mixed layer system (1)–(2) as a function of the forcing frequency ωw and the aspect ratio of the wind forcing τb/τa. In this case, only two values of the effective Coriolis frequency are presented: (a) 
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
The principal axes of the forcing ellipse are labeled τa and τb, but the eccentricity of the ellipse is defined to be 
The asymmetry introduced by the laterally sheared geostrophic flow implies that circular winds are eccentric in the transformed variables [(3)]. Therefore, circular winds, although composed of only one clockwise or counterclockwise rotary component in the untransformed physical variables, are composed of both clockwise and counterclockwise components in the transformed system [(3)]. Thus, when F ≠ f, both ωw = ±F can yield an elevated response when the winds are circularly polarized, as shown in Figs. 4 and C1. We illustrate this phenomenon further in Fig. C2 by comparing the amplitudes of the oscillations for ωw = +F and ωw = −F as a function of the Rossby number of the geostrophic flow Rog and τy/τx.
The response A [(18)] of the slab mixed layer system (1)–(2) as a function of the effective Coriolis parameter F and the aspect ratio of the wind forcing τb/τa = τy/τx for wind forcing that is not necessarily circularly polarized. (a) The angular frequency of the wind is defined to be resonant everywhere, that is, ωw = F, whereas (b) the frequency is resonant but the winds rotate counterclockwise, that is, ωw = −F. As in Fig. 4, θw = 0, HML = 25 m, r = 5.79 × 10−6 s−1, and f = 10−4 s−1. The dashed red line in (a) denotes the ratio τb/τa that produces the maximum resonant response at a given frequency, ωw = +F. The dashed blue line denotes the part of parameter space where τy/τx = f/F and A = f/r for ω = +F, whereas A = f/2F for ωw = −F; see (C1)–(C2).
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0168.1
For ωw = −F, winds that rotate with τy/τx = f/F yield the response with minimal amplitude in τy/τx parameter space (marked by the blue dashed line in Fig. C2b). That minimum, A = f/2F for r ≪ F, is achieved where the ellipse traced out by the winds transforms to a circle, that is, a single rotary component, via (3). Because ωw = −F, the forcing rotates counter to a free inertial oscillation, which also traces out a circle in the transformed variables. In contrast, rotating winds with any other τy/τx trace out an ellipse in the transformed variables in (3). Therefore, because any ellipse with nonzero eccentricity can be expressed as a sum of both clockwise and counterclockwise rotary components, winds with τy/τx ≠ f/F can be decomposed into a sum of both clockwise and counterclockwise rotary components in the transformed variables (e.g., Gonella 1972). The clockwise (counterclockwise) component of the transformed wind forcing rotates in the same direction as a free inertial oscillation—and is therefore resonant—in the Northern (Southern) Hemisphere. Therefore, the amplitude of the response A [(18)] to winds with frequency ωw = −F is locally elevated relative to winds with τy/τx = f/F, as shown in Fig. C2b.
Yet, the question remains: what τy/τx induces the maximum resonant response A? When ωw = F and r → 0, the forced velocity response takes on the ratio 
In any case, the amplitude of the response at resonance (when ωw = F) varies relatively weakly as a function of F and τy/τx for 0.5f < F < 2f compared to the variations in A as a function of |ωw − F| near resonance (e.g., Fig. C1). Moreover, the precise magnitudes of A at resonance are not well constrained because they depend strongly on r, as shown in (C2). For example, when r is lowered by several orders of magnitude, the amplitudes of A are much larger; however, the relative differences between A in different parts of parameter space in Fig. C2 do not change qualitatively.
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Joyce,