2. Physics of resonance in a slab mixed layer model
a. Unforced and inviscid initial-value problem
b. Resonance: The upper ocean as a field of harmonic oscillators
c. Energetics and the role of lateral shear production
1) Transient spinup/spindown
2) Forced-dissipative equilibrium
3. Numerical simulations with spatial variations in vorticity
a. Low Rossby regime
b. High Rossby regime
4. Conclusions and discussion
Scaling the Slab Mixed Layer Equations
Analytic Solution to Slab Mixed Layer Model
Response of the Slab Model to Variations in the Eccentricity of the Wind Forcing
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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 is constant. The surface boundary layer is forced by a spatially uniform but time-varying wind stress vector (large orange arrow) that rotates at a constant angular frequency ωw. Throughout this paper, positive ωw is defined to indicate rotation in the same sense as free boundary layer inertial oscillations (that is, clockwise in the Northern Hemisphere, as shown here), opposite to the rotation direction of the frame of reference Ω = f/2. At t = 0, the winds point in the direction of the principal axis denoted by τa (red line) and oriented at an angle θw relative to the geostrophic flow. An angle θw = 0 implies τa and the initial winds point in the direction of the geostrophic flow, whereas an angle θw = π/2 indicates the initial winds and τa point in the −y direction. Note that our definition of the wind stress, (15)–(16), allows for either τa ≥ τb or τa ≤ τb.
(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 . The perturbation kinetic energy densities, , vary as a function of time/wave phase as shown in (c) and (d). The oscillations periodically exchange energy with the mean flow via the geostrophic lateral shear production LSP. The blue (red) shaded regions denote negative (positive) LSP.
(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)].
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.
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.
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.
(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.
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.
The θw ensemble-averaged lateral shear production as a function of (a) the Rossby number of the background flow Rog and (b) . Both panels show normalized by the ensemble-averaged wind work , where the average is taken over a large ensemble of 10-day transient spinup/spindown integrations, as in Fig. 8, but with a uniform distribution of wind orientation angle θw ∈ [−π, π) and resonant forcing for the first 24 h. As in Fig. 8, but HML = 25 m, r = 5.79 × 10−6 s−1, and f = 10−4 s−1.
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.
(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.
(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.
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).
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 , where s is a dummy time integration variable, pa is the perturbation pressure, wa is the vertical velocity, and the integral is evaluated at z = −HML = −40m.
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) and (b) corresponding to Rossby numbers 1 and −0.5, respectively. The parameters are the same 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 denotes the resonant frequency ωw = F.
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).
Resonant Generation and Energetics of Wind-Forced Near-Inertial Motions in a Geostrophic Flow
Daniel B. Whitt
Daniel B. WhittEnvironmental Earth System Science Department, Stanford University, Stanford, California
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Leif N. Thomas
Leif N. ThomasEnvironmental Earth System Science Department, Stanford University, Stanford, California
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LatMix: Studies of Submesoscale Stirring and Mixing
https://doi.org/10.1175/JPO-D-14-0168.1
A slab mixed layer model and two-dimensional numerical simulations are used to study the generation and energetics of near-inertial oscillations in a unidirectional, laterally sheared geostrophic current forced by oscillatory winds. The vertical vorticity of the current ζg modifies the effective Coriolis frequency , which is equivalent to the local resonant forcing frequency. In addition, the resonant oscillatory velocity response is elliptical, not circular, because the oscillation periodically exchanges energy with the geostrophic flow via shear production. With damping, this energy exchange becomes permanent, but its magnitude and sign depend strongly on the angle of the oscillatory wind vector relative to the geostrophic flow. However, for a current forced by an isotropic distribution of wind directions, the response averaged over all wind angles results in a net extraction of energy from the geostrophic flow that scales as the wind work on the inertial motions times (ζg/f)2 for ζg ≪ f. For ζg ~ f, this sink of geostrophic kinetic energy preferentially damps flows with anticyclonic vorticity and thus could contribute toward shaping the positively skewed vorticity distribution observed in the upper ocean.
Corresponding author address: Daniel Whitt, Stanford University, 473 Via Ortega, Stanford, CA 94305. E-mail: dwhitt@stanford.edu
This article is included in the LatMix: Studies of Submesoscale Stirring and Mixing Special Collection.
Keywords: Circulation/ Dynamics; Fronts; Inertia-gravity waves; Internal waves; Wave properties; Atm/Ocean Structure/ Phenomena; Jets; Oceanic mixed layer
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 = ≈ f is not substantially modified. Thus, most previous studies on the generation of inertial oscillations in the ocean boundary layer have neglected relative vorticity effects (e.g., Pollard 1970; D'Asaro 1985; Crawford and Large 1996; Skyllingstad et al. 2000; Alford 2001; Watanabe and Hibiya 2002; Alford 2003a; Mickett et al. 2010), although not all (e.g., Weller 1982; Klein and Hua 1988; Klein et al. 2004a; Zhai et al. 2005; Danioux et al. 2008). In Klein et al. (2004a) in particular, numerical simulations of wind-forced inertial oscillations in a mesoscale eddy field suggested that the variation in F may not affect the global flux of energy from the winds to inertial oscillations, even if it may have important effects in certain regions of the flow. However, Klein et al. (2004a) only considered low Rossby number geostrophic flows, and therefore the results may not generalize to situations where the Rossby number is of order one.
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).
An analytically tractable way to study the generation of inertial oscillations by winds in the ocean boundary layer is with a slab mixed layer model (e.g., Pollard and Millard 1970; D'Asaro 1985; Alford 2003a; Mickett et al. 2010). Here, we use a version of the model of Pollard and Millard (1970), modified to account for the presence of a rectilinear, barotropic, geostrophic flow ug(y) that is aligned with the x direction (see Fig. 1):
where (UML, VML) are the velocities in the down x and cross y stream directions, respectively; F2 = f(f − ∂ug/∂y) = f2(1 + Rog) is the square of the effective Coriolis frequency; r is a damping coefficient; τ = (τx, τy) is the wind stress vector in terms of down and cross-stream components, respectively; ρ0 is the density of seawater; and HML is the mixed layer depth (e.g., Weller 1982; Klein and Hua 1988). We will always assume that the flow is inertially stable, that is, F > 0 (e.g., Hoskins 1974). As discussed in D'Asaro (1985), the solutions of system (1)–(2) are not especially sensitive to the magnitude of the damping parameter, and one may choose a value in the range 1/r = 1–10 days without qualitatively changing the results [see Park et al. (2009) for a discussion of global observational estimates of r from satellite drifters, which generally range from 1 to 5 days]. To develop specific examples, we will often assume a damping parameter 1/r = 2 days, a Coriolis frequency f = 10−4 s−1 that is typical of the midlatitude Northern Hemisphere, a mixed layer depth of HML = 25 m, and a reference density of ρ0 = 1025 kg m−3 typical of midlatitude upper-ocean seawater. The scaling assumptions required to derive these simplified equations from the Boussinesq equations, and the analytic solution are presented in appendixes A and B. Additional insightful discussions into these equations can be found in Pollard (1970), Pollard and Millard (1970), Stern (1975), Weller (1982), and Klein and Hua (1988).
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.
A linear transformation,
transforms (1)–(2) to the form of the classic (i.e., no background flow) slab mixed layer model (e.g., Pollard and Millard 1970):
These equations can be reduced to a single equation in complex form that is readily solved for a given F and an arbitrary wind stress time series by taking the Fourier transform and solving the equation in the frequency domain (e.g., D'Asaro 1985; Alford 2003a). However, to elucidate the new physics introduced by the lateral shear, the discussion will remain largely in terms of the untransformed variables and only occasionally refer to the transformed system [(3)] when it helps the interpretation of a result.
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.
Before proceeding with a discussion of the forced problem, we will briefly discuss the unforced and inviscid solutions to (1)–(2) governed by
Solutions to this system are harmonic with frequency F, that is, and , where ϕV − ϕU = π/2. Velocity vectors and fluid parcels rotate clockwise (counterclockwise) in the Northern (Southern) Hemisphere. The amplitudes in the x and y direction, and , respectively, depend on the initial condition, but are always related to each other via the ratio . As F diverges from f, the rotating inertial velocity becomes more eccentric, which implies that the velocity response is composed of both clockwise and counterclockwise rotary components in the untransformed variables, but is always composed of a single clockwise or counterclockwise component (depending on the sign of f) in the transformed system (3) (e.g., Mooers 1970; Gonella 1972).
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 . Both example oscillations have the same initial kinetic energy of EML(0) = ½[UML(0)2 + VML(0)2] = 0.5 m2 s−2, but one oscillation starts with its velocity vector parallel to the geostrophic flow, whereas the other starts with its velocity vector perpendicular to the geostrophic flow. However, the wave period–averaged kinetic energy 〈EML〉T, where and T is the wave period, varies by a factor of 4 between the two cases because 〈EML〉T depends strongly on the phase angle ϕ0 and magnitude of the initial perturbation velocity vector (Figs. 2c,d),
The variation in the wave kinetic energy as a function of time,
where s is a dummy time integration variable, is because of a reversible exchange of energy with the mean flow during each wave period that is expressed in terms of a time-integrated lateral shear production LSP in (8). LSP can take both signs during a wave period depending on the sign of Rog and the direction of the initial perturbation velocity vector relative to the geostrophic flow (Figs. 2c,d). For Rog < 0, initial perturbation velocity vectors parallel (perpendicular) to the geostrophic flow result in positive (negative) values for LSP and conversely for Rog > 0. However, because LSP depends on EML, the initial velocity vectors parallel (perpendicular) to the geostrophic flow result in larger (smaller) magnitudes of |LSP| for Rog < 0 and conversely for Rog > 0.
Thus, for a given Rog ≠ 0, a given initial perturbation kinetic energy, and a range of initial velocity directions (i.e., a range of phase angles ϕ0), the positive LSP is always greater in magnitude than the negative LSP (Figs. 2c,d). In fact, when averaged over an ensemble of oscillations with an isotropic distribution of initial phase angles ϕ0 ∈ [0, 2π), but constant initial energy EML(0), LSP is greater than zero for all Rog ≠ 0, that is,
This result in (9) can be appreciated by noting that the azimuthally averaged energies of the two ellipses shown in Fig. 2 are 1.25 and 0.31 m2 s−2, which average to 0.78 m2 s−2, a value significantly greater than EML(0) = 0.5 m2 s−2. Likewise, an average over an ensemble of hodographs derived from an isotropic distribution of initial velocity phase angles ϕ0 yields an ensemble- and phase-averaged kinetic energy that is also a factor of ~1.5 times larger than EML(0) (Fig. 3).
To understand (9) in terms of the physical parameters of this problem, consider a scaling argument for how depends on Rog and EML(0). First, observe in Fig. 2c that the lateral shear production scales as the difference between the maximum and minimum kinetic energy during a wave period,
which scales as the wave period–averaged kinetic energy times a dimensionless parameter
a result that follows from the polarization relation .
Now, as illustrated in Fig. 2c, LSP can either add or subtract from the initial kinetic energy EML(0). It follows, after substituting and into (10), that
This quantity can be interpreted as a scaling for the average LSP between the two cases ϕ0 = 0 and π/2, shown in Fig. 2c. It is therefore also a representative scaling for . Note that the dimensionless parameter in (12) is positive for centrifugally stable flows, that is, for Rog > −1. This suggests that the time-averaged perturbation kinetic energy, averaged over an isotropic distribution of initial perturbation angles ϕ0 with constant initial kinetic energy EML(0), will be larger than the initial perturbation energy for Rog > −1 and Rog ≠ 0. That is,
as shown in Fig. 3. Moreover, the difference between and EML(0) will be largest in anticyclonic geostrophic flows, with Rog → −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.
The governing system (1)–(2) can be reduced to the classic form of a driven damped harmonic oscillator,
Because (14) is a linear ordinary differential equation, it is possible to examine the response of (14) to each frequency independently. Without loss of generality, we can decompose τ = (τx, τy) into sums of harmonic functions with angular frequencies ωw, which may be inserted into the right-hand side of (14) to solve for VML. To elucidate the fundamental physical properties of the system, we restrict attention to wind forcing that has a single frequency and an elliptic orbit as a function of time oriented at an angle θw relative to the geostrophic jet. We define θw = 0 to correspond to a wind-forcing ellipse with one principal axis oriented parallel to the mean flow and with initial winds pointing in the direction of the jet (positive x). The parameters are defined more generally via the parametric formulas for the wind stress as a function of time t, forcing frequency ωw, and wind orientation angle θw as follows:
where τa and τb are the amplitudes of the wind stress in the direction of the principal axes of the ellipse traced out by the wind vector as shown schematically in Fig. 1.1 For simplicity, the initial wind direction is always oriented along the axis denoted a in Fig. 1. Let τM = max(τa, τb) denote the longer major axis and τm = min(τa, τb) denote the shorter minor axis. Then, the forcing period–averaged wind amplitude is given by
where T = 2π/ωw and the eccentricity of the wind ellipse is given by . Equation (17) is in the form of an elliptic integral (e.g., Abramovitz and Stegun 1965).
The general solution to (14) (see appendix B) has two components: a transient or homogeneous component , which depends on the initial conditions and decays exponentially (except when r = 0), and a forced or particular component , with an amplitude proportional to the forcing. The general solution for the streamwise mixed layer velocity UML (also in appendix B) has a slightly different form because of the asymmetry in the equations introduced by the lateral shear of the background flow. Analytic solutions for wind stress time series that exhibit piecewise constant (in time) frequencies may be obtained by progressively solving piece by piece, using the data from the last time in the previous segment as the initial condition for the next segment. More generally, when the time series cannot be decomposed into regions with piecewise constant frequency, the response may be obtained by integrating the governing equations numerically. We use the piecewise integration approach exclusively, since we always deal with forcing time series that have piecewise constant frequencies.
The response of the system may be interpreted in terms of resonance physics that is encapsulated in the forced part of the solution to (1)–(2) (see appendix B) and therefore may be illustrated with the linear response function. In this general context, where both the response and forcing may be elliptical, we define the nondimensional response function as
where 〈·〉T denotes the average over one forcing period (that is T = 2π/ωw) and is the vector form of the forced solution, the components of which are given by (B12) and (B1).
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 . This suggests that, in a geostrophic flow with lateral variations in vorticity (and hence lateral variations in F), each frequency component of the wind should produce spatially localized regions of strong forcing where F ≈ ωw. Therefore, moderate changes in the frequency of monochromatic forcing, for example, from ωw ≈ 1.25f to ωw ≈ 0.75f, will result in qualitatively different forced responses. This observation is consistent with the idea that trapped subinertial waves, for example, those observed by Kunze (1986), Kunze et al. (1995), and Joyce et al. (2013), must be locally generated in regions of anticyclonic vorticity by the subinertial frequency content in the winds. This phenomenon will be illustrated with two-dimensional numerical solutions of the primitive equations in section 3.
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 and given in appendix B. Moreover, the result that AU/AV ≈ F/f, where ωw ≈ F, is essentially independent of the eccentricity and orientation of the wind forcing.
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.
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.
An equation governing changes in the kinetic energy of slab mixed layer oscillations is obtained by multiplying (1) by UML, multiplying (2) by VML, adding the results, and integrating with respect to time:
where s is a dummy time integration variable, the mixed layer kinetic energy per unit mass is EML = 1/2[UML(t)2 + VML(t)2], the wind work is labeled WORK, the radiative/frictional damping is parameterized with a constant damping parameter r and labeled DAMP, and the remaining term represents the lateral shear production labeled LSP. LSP is nonzero when there is a time-integrated Reynolds stress and nonzero lateral shear ∂ug/∂y. As mentioned in section 2a, the time-integrated Reynolds stress in the unforced inviscid problem is zero when t is a multiple of the effective inertial period 2π/F because UML and VML are in quadrature (see Fig. 2). Thus, the irreversible time-integrated energy exchange between the geostrophic flow and the perturbation is zero in the unforced and inviscid case as mentioned in section 2a (e.g., Mooers 1975; Whitt and Thomas 2013).
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.
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).
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).
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 (i.e., the θw average) for a suite of 10-day integrations forced by winds with the same amplitude but different initial orientations θw ∈ [−π, π) yields positive for all F/f, and is larger the higher |Rog| (Fig. 9a). Moreover, the ensemble-averaged is asymmetric in Rog; negative Rossby numbers yield larger ensemble-averaged than positive Rossby numbers of the same magnitude.
These results can be understood in terms of the unforced, inviscid, initial-value problem discussed in section 2a. Like resonantly forced oscillations, inviscid oscillations are elliptical with (see Figs. 3, 5) because the oscillations exchange kinetic energy with the mean flow via lateral shear production. Yet, the ensemble average over a series of inviscid oscillations with different phase angles ϕ0 for the initial velocity vector but the same initial kinetic energy EML(0) yields a nonnegative lateral shear production, , for all Rog [see (9) and Fig. 3]. Assuming that the wind forcing angle θw is analogous to the initial angle of the velocity vector ϕ0 and the wind work is analogous to the initial energy of the perturbation, the following scaling holds [see (12)]:
As shown in Fig. 9b, this scaling accurately represents the true value of , even when Rog ≈ 0.5. Thus, for Rog ≪ 1, , which is an even function of Rog. On the other hand, for Rog ~ 1, is asymmetric about Rog = 0 with much larger magnitudes for negative Rog.
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 for Rog ≪ 1. Hence, this phenomenon becomes nonnegligible only when Rog ≳ 0.1, as in western boundary currents and Southern Ocean frontal zones. Moreover, forced/dissipating inertial oscillations will tend to damp geostrophic flows more strongly in regions of anticyclonic vorticity than cyclonic vorticity, possibly enhancing the cyclone/anticyclone asymmetry observed in strong frontal jets like the Gulf Stream (e.g., Rossby and Zhang 2001) or other regions where strong relative vorticity coincides with strong wind work on inertial motions (e.g., Rudnick 2001; Shcherbina et al. 2013). This energy exchange would also intensify the horizontal anisotropy in the vertical viscosity in the mechanism of Klein et al. (2003), in which greater near-inertial energy density in anticyclonic (compared to cyclonic) geostrophic relative vorticity leads to more intense turbulent mixing and greater damping of anticyclonic geostrophic kinetic energy via lateral shear dispersion driven by the inertial oscillations. We will return to a discussion of the broader implications of these results in section 4.
The energetics of inertial motions in a forced-dissipative equilibrium is governed by the following equation:
where the average is taken over one full forcing period 2π/ωw, and the e prefix denotes equilibrium energetics. In this case, the combination of forcing and damping introduces a phase shift so that UML and VML are no longer π/2 out of phase. Therefore, although the amplitude in each component does not change with time, there is a phase-averaged flow of energy between the perturbation, mean flow, and damping. Figure 10 shows the phase-averaged terms in the energy budget (eLSP and eWORK = eWORKx + eWORKy and eDAMP) in this case, where the winds are rectilinear (τb = 0) and resonant (ωw = F), but the orientation angle θw varies.
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.
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?
In all eight numerical simulations in this section, the initial condition is an idealized two-dimensional current that varies horizontally in y but not in x or z. The background flow is sinusoidal in y and barotropic. In the first set of four simulations, Rog ≲ 0.3, whereas in the second set of four, Rog ≲ 0.8. Therefore, we refer to the two sets of simulations as the low Rossby regime and the high Rossby regime.
We will consider the evolution of weak perturbations from an initial condition that is in geostrophic and hydrostatic balance:
The pressure pg depends on an appropriately defined free surface and the initial buoyancy structure bg = −gρg/ρ0. The Coriolis frequency f = 10−4 s−1 is a representative midlatitude Coriolis frequency, g = 9.81 m s−2 is the acceleration due to gravity, and ρ0 = 1027 kg m−3 is the reference density. The domain has a flat bottom and periodic boundaries in the cross-stream y direction.
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).
The same metrics used during the discussion of the transient energetics in the previous section will be applied here. In particular, we consider the time-integrated wind work,
and lateral shear production,
In the numerical simulations, the mixed layer depth HML and velocity UML need to be appropriately defined from the output model variables. The decomposition between streamwise geostrophic ug and ageostrophic ua velocity is achieved by assuming that the perturbations to the initially balanced flow are largely ageostrophic and have a relatively small integrated effect on the geostrophic flow over the time period of interest. This is based on the fact that the ageostrophic flow is weak compared to the geostrophic flow in these simulations.3 For example, the perturbation velocities peak at ~10 cm s−1, whereas the geostrophic velocities peak at 50 cm s−1 in the low Rossby regime and 1.5 m s−1 in the high Rossby regime. The perturbation is at least 25 times less energetic than the mean flow. Therefore, we can approximate ug(t) = u(t = 0) and ua(t) = u(t) − ug over short integrations like those we consider here. The cross-stream ageostrophic flow is simply υ, where u(t) and υ(t) are the x and y components of the model velocity at time t. The mixed layer depth HML = 40 m is fixed and is chosen to be representative of the typical boundary layer depth, which is an output of the K-profile parameterization scheme (Large et al. 1994). A constant value for HML is not a poor choice in this case (although it may be in general) because the wind amplitude τ = 0.06 N m−2 is similar in all the simulations presented in this section and the deepening of the boundary layer by mixing is slight. Then the mixed layer velocity UML is computed as the average ageostrophic velocity over the boundary layer:
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.
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.
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).
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: . Thus, as discussed above, LSP is asymmetric about Rog = 0 when |Rog| ~ 1 [see (10)–(11) and Fig. 9]. Therefore, the subinertial winds (ωw = 0.5f) yield a more eccentric velocity ellipse and much stronger overall response than the superinertial winds (ωw = 1.32f), despite the fact that they are both resonant at |Rog| = 0.75.
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 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.
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 (e.g., Figs. 4 and C1). (ii) The inertial response depends on the eccentricity eτ and orientation angle θw of the winds, especially for geostrophic flows with high Rossby numbers Rog (see Figs. 1, 6, 7, C2). (iii) The anisotropy of the unforced, undamped inertial oscillations, which manifest as an elliptic velocity hodograph (e.g., Figs. 2, 3), is because of a reversible exchange of energy with the geostrophic flow via lateral shear production LSP. (iv) With the addition of damping and forcing, this energy exchange can become permanent. For O(1) Rog, LSP can contribute to the energy budget of inertial motions at a magnitude equal or greater than the wind work WORK (e.g., Figs. 7, 8, 12). (v) Although LSP depends strongly on the orientation angle θw (e.g., Fig. 8), averaged over all angles the is positive for any nonzero Rog (e.g., Fig. 9) and scales approximately as at small Rossby number. This suggests that this mechanism represents a net sink of mean/geostrophic kinetic energy under conditions of isotropic wind forcing. (vi) Finally, the wind power input at resonance is relatively insensitive to the effective Coriolis frequency F (Figs. 8c, 10c). Therefore, for a given wind stress magnitude at resonance, the variations in inertial energy density EML are largely attributable to LSP, which depends on wind orientation angle θw and the Rossby number of the geostrophic flow (Figs. 8a, 10a).
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, , that the nonlinear perturbation advection terms, for example, UML · ∇UML, may be neglected and (ii) the aspect ratio of the inertial motions is so small that the Burger number (a tilde above a variable denotes a characteristic scale). Thus, the pressure, buoyancy, and vertical velocity perturbations associated with the near-inertial motions may be neglected (see appendix A). In the midlatitudes, these scaling assumptions are not very restrictive. For example, the winds, and hence the oceanic response, typically have fairly large horizontal length scales, that is, m, and the horizontal mixed layer velocities are usually moderate, that is, m s−1 (Chaigneau et al. 2008), yielding RoML ~ 10−3–10−1. While the mixed layer depth is usually fairly shallow, that is, HML ~ 10–100 m (de Boyer Montegut et al. 2004), even with a relatively strong stratification s−1 and . However, smaller-scale atmospheric variability (e.g., Chelton et al. 2004; Small et al. 2008) or strong low-latitude storms (e.g., hurricanes) may generate inertial motions that could violate the low RoML or low BuML assumptions.
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).
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.
The slab mixed layer system [(1)–(2)] can be derived from the continuously stratified Boussinesq equations on the traditional f plane as follows: First, decompose the flow into a mean part ug and a perturbation that represents the inertial oscillations, for example, ua. Then assume that the mean flow is aligned with the x direction and in hydrostatic and geostrophic balance. Third, assume that the perturbations are so horizontally anisotropic that all derivatives in the x direction are negligible.5 The governing equations for the ageostrophic motions then become
where (X, Y) are sources and sinks of perturbation momentum. We use the following nondimensionalization for the perturbation evolution equations:
where the tilde denotes a characteristic dimensional scale for the variable in question, the subscript a denotes the ageostrophic perturbation, and the subscript g denotes the mean geostrophic flow. Thus, is the vertical scale of the perturbation in meters, is the horizontal scale of the perturbation in meters, is the horizontal velocity of the perturbation in meters per second, is the vertical velocity of the perturbation in meters per second, is the time scale of the perturbation in seconds, is the turbulent eddy viscosity in meters squared per second, is the buoyancy frequency of the geostrophic flow in inverse seconds, is the vertical scale of the geostrophic flow in meters, is the horizontal scale of the geostrophic flow in meters, and is the velocity of the geostrophic flow in meters per second. We emphasize that the geostrophic length scales need not necessarily match the ageostrophic length scales.
With these choices, the nondimensional equations are
is the aspect ratio of the perturbation. When α ≪ 1, the flow is approximately hydrostatic. The Rossby number of the perturbation,
is the ratio of the rotational and advective time scales. The Burger number of the perturbation,
compares the effects of rotation with those of stratification. The Ekman number,
is the ratio of viscous and Coriolis forces. The scaling in terms of the surface stress applies in the boundary layer.6 With regard to the geostrophic flow, we assume that the Froude number , that is, the geostrophic flow, is nearly barotropic. On the other hand, the geostrophic Rossby number may be order one.
To derive the slab mixed layer model, we assume that the aspect ratio α ≪ 1 so that the hydrostatic approximation may be employed. Moreover, we assume that
Therefore, all nonlinear advection terms may be neglected (e.g., u′ · ∇u′). Then we assume that the Burger number of the perturbation is small; that is,
where is the characteristic buoyancy frequency of the upper thermocline (see Pollard 1970).
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; can be O(1) in this case. Although we have assumed that the geostrophic flow is steady, we could have arrived at the same (1)–(2) by assuming that the time rate of change of the geostrophic flow is slow compared to the inertial period 2π/f, an assumption related to the geostrophic momentum approximation of Hoskins (1975).
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, will vary as a function of time if F is nonconstant, even in the absence of forcing. In particular, geostrophic vorticity gradients will tend to cascade existing near-inertial energy to high horizontal wavenumbers over time, leading to a violation of the scaling assumptions (A16)–(A17) (e.g., van Meurs 1998; Klein et al. 2004b). Typically, the winds are of a sufficiently large scale and sufficiently low amplitude that if F is sufficiently close to f (i.e., the Rossby number of the geostrophic flow is small), then the initial aspect ratio of the inertial motions will be small enough for the scaling assumptions (A16)–(A17) to be satisfied. For example, in the midlatitudes, typical scales might be m s−1 (e.g., Chaigneau et al. 2008), f ~10−4 s−1, m, m, and s−1 (e.g., D'Asaro 1985). Therefore, Roa ~ 0.01 and Bua ~ 10−4, and the scaling assumptions should be satisfied.
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.
To derive the solutions of the slab mixed layer Eqs. (1)–(2), we assume a wind stress [(15)–(16)] that rotates at a constant angular frequency with an elliptic trace (see Fig. 1), a fixed mixed layer depth HML, a fixed radiative decay parameter r, and a fixed effective Coriolis parameter . Given these assumptions, the general solution of the slab mixed layer model, in the form of (14), can easily be found. It can be written as a sum of two components: the particular and homogeneous solutions. The particular or "forced" solution is given by
in which ,
The homogeneous or transient solution is obtained by setting the right side of (14) to zero and solving the damped oscillator equation:
and we have used the initial conditions [UML(0), VML(0)] = (U0, V0) and ∂VML/∂t(0) = −rV0 − fU0 + τy(0)/ρ0HML. The general solution is . In the presence of a rotary wind with constant magnitude and no relative vorticity, AV may be divided by |τ|/ρ0HML to obtain the magnitude of the response function given by D'Asaro [1985, his (10)], as discussed in appendix C.
We can similarly compute the analytic form for the streamwise velocity UML, which is governed by
The particular or forced solution
and where
Here, we have defined
The transient solution is obtained by setting the right side of (B11) to zero and solving the damped oscillator equation:
where we have used the initial conditions [UML(0), VML(0)] = (U0, V0) and . The general solution is . The algebra in this appendix has been checked by comparing the general analytic solutions with numerical integrations of the original governing equations (1)–(2).
We now need to compute the phase-averaged forced velocity response . Without loss of generality, we define (note that |Δϕ| → π/2 as r → 0). The implicit quadratic equation for the forced velocity response is given by
When Δϕ = 0, the system is degenerate—the ellipse reduces to a line—and when |Δϕ| = π/2, the system is in canonical elliptical form. In between these two angles, one can extract the lengths of the principal axes from the eigenvalues of the quadratic form,
The lengths of the principal axes of the velocity response ellipse are given by , where we relabel so that LM denotes the length of the larger, major axis and Lm denotes the smaller, minor axis. The slope of the major axis can be written as
Similar to the wind stress (17), the period-averaged, forced, elliptical, two-component velocity response is given by
which is in the form of an elliptic integral of the second kind, similar to the wind stress mentioned in section 2. The eccentricity of the velocity response ellipse is given by and the ratio . With (17) and (B23), the value of the mean response function (18) may be computed for arbitrary elliptical forcing and response.
Finally, to clarify the dynamics of secular growth, consider the limit of resonant forcing with infinitesimally small damping, that is, |F − ωw| ~ r → 0. In this limit, the solutions exhibit secular growth governed by the following formulas:
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 principal axes of the forcing ellipse are labeled τa and τb, but the eccentricity of the ellipse is defined to be , where τm denotes the shorter minor axis and τM denotes the longer major axis. Here, variations in eccentricity will be considered in terms of the parameter τb/τa, which has a more immediately obvious physical interpretation (see Fig. 1). For this section, θw = 0 so τb/τa = τy/τx, and we begin the discussion considering circular winds, that is, τx = τy.
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.
In the transformed variables [(3)], however, winds with τy/τx = f/F are circularly polarized for all Rog. Therefore, for arbitrary ωw, A [(18)] takes the form of the classic (no background flow) slab mixed layer response function (e.g., D'Asaro 1985):
In addition, (C1) becomes as r → 0 and |ωw − F| remains finite or
as ωw → F and r remains finite. The region of parameter space where τy/τx = f/F is denoted by a dashed blue line in Fig. C2. Along this line, (C2) is constant for resonant forcing ωw = F (Fig. C2a), as in the classic Rog = 0 case when ωw = f. The collapse of parameter space in (C1)–(C2) is a consequence of the fact that this particular forcing |τb|/|τa| = |τy|/|τx| = f/F is circular in the transformed variables in (3) and therefore composed of only one rotary component in those variables. Therefore, this region of parameter space is mathematically analogous to the classic resonant forcing problem with circular winds and no background flow (e.g., D'Asaro 1985).
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 for all τy/τx = τb/τa (Fig. 5). Thus, one would expect the maximal A [(18)] for a given 〈|τ|〉T when τy/τx = f/F because this configuration would maximize the wind work over a forcing period. However, the maximum in A occurs for elliptical forcing with stronger winds in the streamwise direction (τx > τy) for F < f and stronger winds in the cross-stream direction (τx < τy) for F > f, that is, when the major axes of the wind and inertial current ellipses are perpendicular (along the red dashed line in Fig. C2a). This configuration would not maximize the wind work, suggesting that there is an additional source of energy that is tapped for the eccentricities that maximize A. This issue is discussed in terms of the energetics of system (1)–(2) in section 2c.
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.
Hoskins, B.