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    Energy time series for four simulations with different wave and vortex forcings. (a),(b) τ = 0.125 IP, fw = 0 m s−1; (c),(d) τ = 0.0125 IP, fw = 0 m s−1; (e),(f) τ = 0.125 IP, fw = 1 × 10−4 m s−1; and (g),(h) τ = 0.0125 IP, fw = 1 × 10−4 m s−1. (left) Total energy (Etot, solid), KE (dashed), and APE (dotted–dashed). (right) Total (Etot, solid), vortex (EV, dashed), wave (EW, dotted–dashed), and shear mode (ESM, solid gray) energy and a linear fit to EV (bold dashed).

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    Evolution of PV, velocity, and ρ′ for frequent vortex forcing (τ = 0.0125 IP) and no wave forcing, showing the gradual buildup of energy at large scales. Columns are for times (left) early, (middle) middle, and (right) late in the simulation. Horizontal and vertical slices through the domain are given in the upper and lower three rows, respectively. (b)Velocity is , (e) velocity is υ.

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    Isotropic (a),(b) horizontal and (c),(d) vertical spectra of (left) wave and (right) vortex energy density following model spinup (>100 IP) for simulations with frequent vortex generation (τ = 0.0125 IP) and no wave forcing showing buildup of vortex energy at the largest horizontal scales (smallest horizontal wavenumber). Colors progressing from blue to red in each subpanel are spectra from early to late times after model spinup. Reference slopes are k−5/3, k−3, and k−5. Reference spectra computed at 110 IP from simulations with τ = 0.125 IP and no wave forcing are also shown (solid black spectra). Vortex forcing for all base-case runs is at nondimensional wavenumber 40 in the horizontal and 20 in the vertical; here only the first 30 nondimensional wavenumbers are shown.

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    Evolution of PV, velocity, and ρ′ for infrequent vortex forcing (τ = 0.125 IP) plus wave forcing (fw = 1 × 10−4 m s−1), as in Fig. 2.

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    Energy spectra following model spinup (>100 IP) for infrequent vortex forcing (τ = 0.125 IP) plus wave forcing (fw = 1 × 10−4 m s−1), as in Fig. 3. Reference spectra computed at 110 IP from simulations with infrequent vortex (τ = 0.125.IP) and no wave forcing (solid black spectra), and frequent vortex (τ = 0.0125.IP) and no wave forcing (dashed black spectra) are also shown.

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    Wave forcing vs realized wave and vortex energy densities at the forced wavenumber for two sets of simulations, showing an increase of both wave and vortex energies with increased wave forcing. Open circles are wave energies EW for simulations where only a wave was forced. Squares are wave energies EW and diamonds vortex energies EV for simulations with wave plus infrequent vortex (τ = 0.125 IP) forcing. Error bars are based on standard deviations. The onset of dipole splitting (D.S.) and upscale energy transfer (U.T.) are marked with arrows.

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    Energy spectra following model spinup (>100 IP) for frequent vortex forcing (τ = 0.0125 IP) plus wave forcing (fw = 1 × 10−4 m s−1), as in Fig. 3. Reference spectra computed at 110 IP from simulations with infrequent vortex (τ = 0.125 IP) and no wave forcing (solid black spectra), and frequent vortex (τ = 0.0125 IP) and no wave forcing (dashed black spectra) are also shown.

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    Frequency spectra of the three linear normal modes, B0, B+, and B, for the case of infrequent vortex forcing (τ = 0.125 IP) and wave forcing (fw = 1 × 10−4 m s−1) showing the separation of subinertial motions, dominantly into the B0 mode, from motions with frequencies f < ω < N, dominantly into the B+ and B modes. Inertial f and buoyancy N frequencies are also marked on the plot, along with a reference ω−2 spectral slope, as expected for internal waves and quasigeostrophic turbulence.

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    Horizontal wavenumber bispectra of the three normal modes B0, B+, and B, defined by Eqs. (12)(13), showing the dominance of vortical mode nonlinear interactions for wavenumbers less than the forced wave. Axes represent horizontal wavenumbers, k1 and k2, with color indicating the degree of nonlinear coupling via triad interactions at wavenumber k3 = k1 + k2. Note the different color scales among the panels.

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    Time series for four simulations with reduced vortex amplitude (ΔN2/N2 = 0.1), as in Fig. 1.

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Upscale Energy Transfer by the Vortical Mode and Internal Waves

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  • 1 University of Massachusetts, Dartmouth, Massachusetts
  • | 2 NorthWest Research Associates, Redmond, Washington
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Abstract

Diapycnal mixing in the ocean is sporadic yet ubiquitous, leading to patches of mixing on a variety of scales. The adjustment of such mixed patches can lead to the formation of vortices and other small-scale geostrophic motions, which are thought to enhance lateral diffusivity. If vortices are densely populated, they can interact and merge, and upscale energy transfer can occur. Vortex interaction can also be modified by internal waves, thus impacting upscale transfer. Numerical experiments were used to study the effect of a large-scale near-inertial internal wave on a field of submesoscale vortices. While one might expect a vertical shear to limit the vertical scale of merging vortices, it was found that internal wave shear did not disrupt upscale energy transfer. Rather, under certain conditions, it enhanced upscale transfer by enhancing vortex–vortex interaction. If vortices were so densely populated that they interacted even in the absence of a wave, adding a forced large-scale wave enhanced the existing upscale transfer. Results further suggest that continuous forcing by the main driving mechanism (either vortices or internal waves) is necessary to maintain such upscale transfer. These findings could help to improve understanding of the direction of energy transfer in submesoscale oceanic processes.

Corresponding author address: Miles A. Sundermeyer, School for Marine Science and Technology, UMass Dartmouth, 706 S Rodney French Blvd., New Bedford, MA 02744. E-mail: msundermeyer@umassd.edu

This article is included in the LatMix: Studies of Submesoscale Stirring and Mixing Special Collection.

Abstract

Diapycnal mixing in the ocean is sporadic yet ubiquitous, leading to patches of mixing on a variety of scales. The adjustment of such mixed patches can lead to the formation of vortices and other small-scale geostrophic motions, which are thought to enhance lateral diffusivity. If vortices are densely populated, they can interact and merge, and upscale energy transfer can occur. Vortex interaction can also be modified by internal waves, thus impacting upscale transfer. Numerical experiments were used to study the effect of a large-scale near-inertial internal wave on a field of submesoscale vortices. While one might expect a vertical shear to limit the vertical scale of merging vortices, it was found that internal wave shear did not disrupt upscale energy transfer. Rather, under certain conditions, it enhanced upscale transfer by enhancing vortex–vortex interaction. If vortices were so densely populated that they interacted even in the absence of a wave, adding a forced large-scale wave enhanced the existing upscale transfer. Results further suggest that continuous forcing by the main driving mechanism (either vortices or internal waves) is necessary to maintain such upscale transfer. These findings could help to improve understanding of the direction of energy transfer in submesoscale oceanic processes.

Corresponding author address: Miles A. Sundermeyer, School for Marine Science and Technology, UMass Dartmouth, 706 S Rodney French Blvd., New Bedford, MA 02744. E-mail: msundermeyer@umassd.edu

This article is included in the LatMix: Studies of Submesoscale Stirring and Mixing Special Collection.

1. Introduction

Submesoscale vortices and internal waves are closely linked by their generation mechanism, location, and scale. Internal wave (IW) wave breaking causes diapycnal mixing, resulting in patches of well-mixed fluid. Such IW wave-breaking events and mixed patches are ubiquitous, but sporadic in time and space [e.g., Stellwagen Bank (Haury et al. 1979), the California Current (Gregg et al. 1986), off the California coast (Alford and Pinkel 2000), the North Atlantic (Polzin et al. 2003), the New England shelf (Sundermeyer et al. 2005), and the Sargasso Sea (Goodman 2012)]. Typical scales of mixed patches are comparable to internal wave scales, namely, 1–15 m vertically and 100–1000 m horizontally (Haury et al. 1979; Sundermeyer et al. 2005; Goodman 2012). Once generated, mixed patches can evolve into vortices through geostrophic adjustment (McWilliams 1988; Lelong and Sundermeyer 2005; Sundermeyer et al. 2005). During adjustment, fluid within a mixed patch moves first radially outward owing to the pressure gradient, then azimuthally owing to Coriolis acceleration, resulting in an anticyclone. Meanwhile, above and below the anticyclone, fluid moves radially inward, resulting in two weaker cyclones (Lelong and Sundermeyer 2005; Stuart et al. 2011). Such a compound structure of anticyclone and cyclones is known as an “S vortex” (Morel and McWilliams 1997). Submesoscale vortices such as S vortices are important because they contribute to enhanced lateral stirring and mixing (Sundermeyer 1998). Lateral mixing in turn affects important environmental and ecological factors such as temperature and chlorophyll distributions, plankton dynamics, and pollutant dispersal (Mahadevan and Campbell 2002; Martin 2003; Hazell and England 2003).

Under certain conditions, a field of submesoscale S vortices can undergo upscale energy transfer, resulting in an inverse cascade. Such fields were studied numerically by Sundermeyer and Lelong (2005). When mixed patches were generated infrequently in time and space, they found that the flow was weakly nonlinear in the sense that resulting vortices diffused away without interacting, and mean kinetic energy (KE) equilibrated to a statistically steady state. However, if mixed patches were generated so frequently that new patches were formed before existing vortices could dissipate, the flow became strongly nonlinear and KE did not equilibrate. In this regime, nonlinear interactions caused vortices to merge, moving KE from smaller to larger horizontal scales. Such upscale transfer is characteristic of an inverse energy cascade and is often associated with two-dimensional, rotating stratified, or quasigeostrophic turbulence (e.g., Rhines 1975; Stammer 1997; Rivera and Wu 2000; Smith and Waleffe 2002; Waite and Bartello 2004; Read 2005; Scott and Wang 2005; Waite and Bartello 2006a,b; Vallis 2006).

When internal waves and S vortices coexist, energy may also transfer between them through triad interactions. In this context, the vortex field can be represented by the vortical mode (Mueller 1988). Vortical mode triad interactions by themselves typically transfer energy upscale (Lelong and Riley 1991). Conversely, internal wave triads transfer energy downscale (Garrett and Munk 1979; Lelong and Riley 1991). When the vortical mode and internal waves are combined, the vortical mode can facilitate downscale energy transfer in the internal wave field, even though there is no energy transfer between vortical mode and internal waves in this scenario. Meanwhile, near-inertial waves may inhibit upscale energy transfer in the vortical mode, making it unclear a priori whether the net transfer will be upscale or downscale (Lelong and Riley 1991). The direction of net energy transfer has been shown to depend on the background rotation rate by Waite and Bartello (2006b). Specifically, they found the transition from forward cascade-dominated flow (associated with stratified turbulence) to inverse cascade-dominated flow (associated with quasigeostrophic turbulence) occurs when the volume-averaged vorticity-based Rossby number Ro = (∂υ/∂x − ∂u/∂y)/f < 3.

Details of the energy transfer may also be affected by the stability of individual vortices, which in turn can be influenced by large-scale shear and strain fields, the potential vorticity (PV) distribution of the vortex, large-scale internal waves, and/or the vortex Burger number [Bu = (h2ΔN2)/(f2L2), where h and L are the mixed patch half height and radius, f is Coriolis frequency, and ΔN2 is the difference between the mixed patch and background stratification]. For example, Brickman and Ruddick (1990) showed that an anticyclonic lens thinned and oriented itself along the major axis of horizontal strain once shear exceeded a critical threshold. Similarly, Mariotti et al. (1994) found that a vortex split into dipoles when horizontal shear was strong compared to vortex shear. Vandermeirsh et al. (2002) found that a vertical shear could inhibit vertical separation of hetons with a radius larger than 0.5 Rossby radii of deformation, RD = ΔNh/f (or equivalently, ). An S vortex is unstable by itself, owing to vertical variations of its meridional PV (Flierl 1988; Morel and McWilliams 1997) as well as its horizontal PV (Kloosterziel and Carnevale 1999; Beckers et al. 2003). Furthermore, when perturbed by a large-scale internal wave, an S vortex will break into dipoles, with time to breakup decreasing with increasing internal wave strength (Brunner-Suzuki et al. 2012). Resulting dipoles have smaller horizontal diameter than the original S vortex, indicating a downscale energy transfer, counter to the canonical vortex-induced upscale transfer. Last, the stability of individual vortices is also affected by their Burger number (e.g., Griffiths and Linden 1981; Helfrich and Send 1988; Hopfinger and van Heijst 1993; Flor and van Heijst 1996; Lelong and Sundermeyer 2005).

In a forward cascade, energy moves from larger to smaller scales, where it is ultimately dissipated by molecular viscosity; this is typical for three-dimensional turbulence. Conversely, in an inverse cascade, energy continues to increase at the largest scales unless and until some external forcing or dissipation removes it. Mechanisms that may limit upscale energy transfer exist at different scales and include, at smaller scales, submesoscale surface frontogenesis (D’Asaro et al. 2011; Capet et al. 2008a,b); at somewhat larger scales, topography, bottom drag, suppression by wind stress, and interactions with internal waves (Ferrari and Wunsch 2009); and at the largest scales (e.g., mesoscale geostrophic turbulence) zonation associated with the Rhines scale (Rhines 1979; Vallis 2006).

In this paper, we study interactions between a field of vortical modes and a near-inertial internal wave, examining numerically the effect of such a wave on upscale versus downscale energy transfer in the vortical mode field. As in Sundermeyer and Lelong (2005), we model our internal waves and submesoscale mixed patches after the Coastal Mixing and Optics Experiment (CMO) conducted over the New England shelf. There, near-inertial wave shears reached values on the order of 6.5 × 10−3 s−1 following storm events (Sundermeyer 1998; MacKinnon and Gregg 2005). Meanwhile, the aspect ratio of mixed patches (height h to length L; see also Table 1) was estimated to be of order ≈N/f. Despite the potential importance of Bu noted above, for simplicity, the primary simulations reported here will be for fixed Bu, with only a few simulations with altered Bu included.

Table 1.

Base-case parameters for simulation with infrequent vortex plus wave forcing. Oceanic values were chosen based on observations during CMO. Note that IP refers to inertial periods.

Table 1.

The outline of this paper is as follows: Section 2 describes the numerical methodology and normal mode decomposition of model output into geostrophic (vortex) and ageostrophic (wave) components (see the appendix for further details). Section 3 describes our results on the influence of vortices and internal waves on upscale energy transfer. Section 4 provides a discussion of our major findings, while section 5 summarizes and concludes.

2. Methods and analysis

a. Model equations, setup, and N/f scaling

Numerical simulations are conducted using a modified version of the model by Winters et al. (2004), which solves the nonlinear, three-dimensional, Boussinesq equations on an f plane with Newtonian and hyperdiffusivity and viscosity. The momentum, perturbation density, and continuity equations are
e1
e2
e3
where all variables have their traditional meanings. Here, total density consists of a reference density ρ0, a linear background stratification , and a perturbation density ρ′(x, y, z), while P is the perturbation pressure. The term F is a forcing term used to drive the large-scale internal wave. Time stepping uses a third-order Adams–Bashforth scheme. Linear terms are computed in spectral space, and nonlinear terms are computed in physical space. Boundary conditions are triply periodic. Hyperdiffusive and viscous terms are normalized by the maximum nondimensional wavenumbers and , which depend on grid resolution. This normalization prevents buildup of small-scale energy caused by finite resolution without dynamically affecting scales of interest. To further maintain numerical accuracy while minimizing aliasing and retaining more wavenumbers, one-ninth of wavenumbers are truncated following Patterson and Orszag (1971). The behavior of numerically simulated homogeneous turbulence has been shown to be insensitive to the method of energy extraction at small dissipative scales, hence validating usage of hyperviscosity (e.g., Borue and Orszag 1995b,a). To verify that it does not affect the main dynamics, a simulation with ν6 reduced by a factor of 10−19 is compared to a base simulation. While details differ somewhat (e.g., the total energy is 3% smaller after 800 inertial periods; not shown), the main results are the same: the simulation is dominated by upscale energy transfer. This suggests that the exact manner of energy extraction does not affect our major results.

Model domain size is chosen to accommodate multiple submesoscale mixed patches, while also representing the flat aspect ratio of individual vortices and near-inertial waves (see Table 1 for a list of base parameters). Grid sizes range from n = 64 to 256 grid points in all directions, with larger domains used to resolve the inertial range at wavenumbers less than the wave forcing and smaller ones used to explore various parameter sensitivities, including forced wave amplitudes and mixed patch intensities. Both large and small domains resolve relevant physical processes, including geostrophic adjustment of individual mixed patches, dipole splitting, internal waves, their interactions, and any upscale energy transfer toward the lowest wavenumbers. The model time step, Δt = 15 s, is set to accommodate high-frequency internal waves generated during geostrophic adjustment (Arneborg 2002; Gal-Chen 2002). These high-frequency waves act on shorter time scales than the forced near-inertial internal wave. Vortices evolve on even longer time scales. To accommodate the latter, all simulations are run for at least 800 inertial periods.

To make spanning the requisite length and time scales in the model more computationally tractable, simulations are performed using a reduced N/f ratio (Lelong and Dunkerton 1998a; Sundermeyer and Lelong 2005; Lelong and Sundermeyer 2005). Here f is artificially increased 10-fold compared to typical midlatitude values (see Table 1 for a comparison of model and typical oceanic values), while keeping N constant. To ensure that reduced N/f simulations are dynamically similar to their physical counterparts, key nondimensional parameters including Ro (typically = 0.15 for individual vortices) and Bu (=0.25) are preserved. That is, horizontal domain size and vortex size are also decreased 10-fold, while keeping vertical domain size fixed. Furthermore, to maintain the Ekman number Ek = ν/(h2f), model diffusivity and viscosity are increased by the same factor of 10. Throughout this manuscript we shall refer to realistic midlatitude oceanic values when discussing relevant dynamical parameters. Table 1 contains both realistic and reduced parameter values.

b. Initial conditions and forcing

The model is spun up from a state of rest by generating Gaussian-shaped mixed patches at regular time intervals, but random locations throughout the domain, similar to Sundermeyer and Lelong (2005). Mixed patches are generated by temporarily and locally imposing a vertical diffusivity at random locations (x0, y0, z0) of the form
e4
where ΔN2/N2 is the ratio of mixed patch and background stratification, such that for an initially linear density profile
e5
with parameters as defined in Table 1. More generally, κV mixes the total density profile at the location of the mixed patch. The velocity field at the same location is also mixed using a similar form for viscosity. For an initially linear density profile, the resulting mixed patch is approximately Gaussian, such that
e6
where is obtained from the finite difference diffusion equation:
e7
Once the background density is no longer linear, the mixed patch density profile will also deviate from . However, such deviations do not significantly affect the subsequent adjustment (Lelong and Sundermeyer 2005). Additional structural details of mixed patches can be inferred from our figures (see section 3), but these are not our main focus. The interested reader is referred to Lelong and Sundermeyer (2005) and Brunner-Suzuki et al. (2012).

In our primary simulations (Table 2, runs A–E), mixed patches are assumed fully mixed at their centers (i.e., ΔN2/N2 = 1), yielding a time- and volume-averaged energy dissipation rate of approximately 8 × 10−8 W kg−1. The patches then adjust geostrophically on time scales of approximately five inertial periods (IP) after which the resulting S vortices are gradually damped by background viscosity. As long as individual S vortices carry available potential energy (APE), they continually readjust, retaining kinetic energy until they are of the order of the vertical diffusive time scale, Tdiff = h2/κ2 ≈ 100 IP, when they finally diffuse away. This diffusive time scale thus also determines when the flow reaches a statistically steady state. Simulations with reduced mixing (ΔN2/N2 = 0.1; see Table 2, run F) are also included. Reduced ΔN2/N2 reduces RD and, hence, Bu. Bu ≪ 1 implies a more geostrophic regime where mixed patches convert less potential energy (PE) to KE during geostrophic adjustment, resulting in weaker vortices (Lelong and Sundermeyer 2005).

Table 2.

Model parameters (rescale oceanic values) varied from base case (Table 1) for different suites of simulations.

Table 2.

Superimposed on the mixed patches is a forced monochromatic near-inertial plane wave with frequency ω2 = (N2k2 + f2m2)/|k|2 = (1.1 f)2 and wave vector k = (k, 0, m). Note that the wave frequency is unaffected by the reduction in N/f since both f and k are increased by the same factor. The wave is forced at every time step via the forcing term in Eq. (1), that is,
e8
with k ranging from 1 to 8 × 2π/Lx and m ranging from 1 to 4 × 2π/Lz, depending on the simulation. Coupling of the momentum equations through Coriolis acceleration and the continuity equation gives rise to a wave propagating in the xz plane. The wave amplitude reaches an equilibrium when forcing by Fu is balanced by viscous losses. Wave amplitudes are set by varying the forcing factor fw. The maximum wave amplitude in any of our simulations is 12% of the overturning amplitude, that is, the amplitude at which the wave velocity exceeds its phase speed. By keeping wave amplitudes small, convective and/or shear instability are avoided. Low amplitudes are justified since most oceanic internal waves have small steepness (Staquet 2004). After storm events, oceanic near-inertial waves can have vertical shears of 6.5 × 10−3 s−1 (Sundermeyer 1998; MacKinnon and Gregg 2005); this is twice as large as our largest vertical shear of 3.6 × 10−3 s−1. When wave forcing is further increased to fw = 10−2 m s−1, we observe wave breaking via shear instability. Incidentally, the onset of shear instability before convective instability is consistent with results of Lelong and Dunkerton (1998a,b), who find that plane near-inertial waves experience the fastest shear instability growth rates. With this in mind, our strongest wave-forced simulations employ the largest wave amplitude possible while maintaining a stable plane wave throughout the simulation.

c. Normal-mode decomposition

To distinguish between vortical mode and internal wave energy in our simulations, we employ a linear, normal-mode decomposition based on PV, following Bartello (1995). A more detailed description of the normal-mode decomposition is provided in the appendix. In short, we can define, as a function of wavenumber k, geostrophic E0, and ageostrophic E± energies, plus a shear mode that represents pure inertial oscillations, as
e9
e10
e11
where , , and represent the three normal mode amplitudes at wave vector k. Resulting wave and vortical mode energies are used to track the spectral evolution of energy over time. Here, isotropic horizontal spectra are computed by averaging over all vertical wavenumbers m and sorting by horizontal wavenumber kH = (k2 + l2)1/2, while vertical spectra are averaged over all horizontal wavenumbers. Summing the wave and vortical mode energies over all wavenumbers, k, l, m, yields time series of EW and EV, respectively. These time series help identify upscale energy transfer more easily, as it is typically accompanied by unbounded growth when energy accumulates at the lowest wavenumbers.

The normal mode decomposition also enables us to determine the actual, realized, forced wave amplitude in our model domain, as this cannot be directly prescribed with our method of wave forcing. When the wave is forced without any vortex forcing, the realized wave amplitude can readily be determined from equilibrium energy levels. However, estimating the wave amplitude in the presence of vortex forcing is more difficult (a detailed explanation is presented in section 3). To facilitate such estimates, EW is further decomposed into horizontal kinetic KEW and available potential energy APEW (Polzin et al. 2003, see also the appendix). In our wave plus vortex-forced cases, KEW is used to quantify the internal wave shear and strain.

Noting that the normal mode decomposition given by Eqs. (9)(11) is strictly linear, we test the validity of this assumption in the present simulations. Previous studies have shown that, in practice, the linear PV decomposition is still a good approximation even in apparently nonlinear situations such as ours (e.g., Bartello 1995; Riley and Lelong 2000; Waite and Bartello 2006b). Specifically, Waite and Bartello (2006a,b) show that full PV is well represented by its linear component when the vertical Froude number Frz = U/NH ≪ 1. Examining one of our base-case wave plus vortex-forced runs, we compute both micro and macro vertical Froude numbers, as described by Waite and Bartello (2006b). The first of these, , uses horizontal vorticity, ζx, and ζy, to infer U/H; while the second, , is computed using the domain-averaged rms velocity as U and an energy-weighted vertical wavelength as H [e.g., see Eq. (3.8a) of Waite and Bartello 2006b]. Here, we find and , that is, both ≪ 1. As a cross check, we also compute the vertical Froude number of the forced wave itself, , where U is the maximum wave velocity difference, and H is the vertical wavelength. Again, we find , that is, all confirming the linear PV approximation, PV ≈ (∂υ/∂x − ∂u/∂y) − (f/N2)(g/ρ0)∂ρ′/∂z.

In addition to confirming the linearity of PV, we further verify that the linear decomposition given by Eqs. (9)(11) accurately separates subinertial motions, represented by the PV mode, from motions with frequencies f < ω < N, represented by the wave modes. Frequency spectra are computed for each of the linear modes for three of our base-case simulations. Here, full-resolution model fields are saved twice per buoyancy period for a total of 10 inertial periods, the normal mode decomposition applied and volume-averaged frequency spectra of the normal modes B+, B, and B0 computed. Results (discussed in section 3) confirm that the linear normal-mode decomposition reasonably separates zero-frequency (allowing for Doppler shifting) PV modes from superinertial internal wave modes in our simulations.

d. Some additional checks

We test the sensitivity of our results to how well the inertial range of upscale energy transfer is resolved at wavenumbers below the forced wavenumber. In one set of simulations, the wave is forced at the lowest wavenumber, leaving no wavenumbers below this for upscale energy transfer (Table 2, runs B). These simulations are compared to our base-case simulations (Table 2, runs A) in which the forced wave has a nondimensional horizontal wavenumber k = 8, thus providing nearly a decade of inertial range for upscale energy transfer between the forced wavenumber and the gravest mode. In all cases, there is a pronounced increase in vortical mode energy at the gravest mode, consistent with net upscale energy transfer over the course of the runs. In cases with higher wavenumber forcing, and hence greater inertial range, this increase at the gravest mode is consistently accompanied by an increase in vortex energy at all wavenumbers below the forced wave, but with no anomalous gain in vortex energy at the forced wavenumber (see section 3 for more detail). This supports our assertion that, even in simulations with limited upscale inertial range, the increase in vortical mode energy at wavenumbers at and below the wave forcing is due to upscale energy transfer in the vortex field and not direct transfer of energy from the wave forcing.

Sensitivity of our results to grid resolution is also tested using two paired sets of simulations in which the domain size is held fixed, but the grid resolution is varied from (nx = ny = nz = 64) to (nx = ny = 256, nz = 128) (see Table 2, runs B and C). Results are similar between the pairs of runs—after model spinup, energy in the higher resolution runs is within 6% of runs with 643 grid points. Differences can be explained by better-resolved spectral roll-off characteristics for higher resolutions, which results in increased energy at the highest wavenumbers. Meanwhile, upscale energy transfer toward the lowest wavenumbers still dominates the energy budget, confirming that 643 grid points are sufficient for our purposes. This notwithstanding, for reasons described in section 2a, and unless otherwise noted, we use (nx = ny = 256, nz = 128) grid points in our base-case simulations (Table 2, runs A), reverting to 643 only for runs examining further parameter dependence.

Finally, our assertion that increases in vortex energy at wavenumbers at and below the wave forcing is due to upscale energy transfer in the vortical mode is further tested by computing bispectra of the three normal modes B0, B+, and B. Here, bispectra are computed as the discrete Fourier transform of the third-order cross-cumulant of the different normal-mode combinations:
e12
with ; B[j, r, s] representing some combination of the three normal modes B0, B+ and/or B; ξ1 and ξ2 representing different spatial lags relative to n; and E{⋅⋅⋅} denoting the expected value. As with power spectra, bispectra can equivalently be computed directly from the Fourier transform of the data series, in this case as
e13
where a caret denotes the Fourier transform of a given mode, and an asterisk denotes the complex conjugate (e.g., Rao and Gabr 1984). As discussed by McComas and Briscoe (1980), the wavenumber bispectrum can be interpreted as a measure of the rate of energy transfer via nonlinear interaction between a given mode, Bj, at wavenumber k = (p + q) and two other modes, Br and Bs, at wavenumbers p and q, respectively. In the case of waves and the vortical mode, bispectra provide a measure of the degree of nonlinear coupling via triad interactions between and among the different normal modes. Equation (13) thus bears close resemblance to the vortical and wave spectral energy transfer functions discussed by Waite and Bartello (2004, 2006a) [see also Eq. (A1)], except that the bispectrum does not reveal the sign of the energy transfer. Nevertheless, this suffices for the present purposes, as we are mainly interested in discerning whether accumulation of low-wavenumber energy in the vortical mode is the result of vortical mode triad interactions or directly involves the forced wave.

3. Results

a. Vortex-induced upscale energy transfer

1) Base case—Infrequent vortex forcing, no wave forcing

Before exploring cases when both wave and vortex forcing are present, it is useful to first understand a field of S vortices by itself, without background wave forcing. Consider the case where mixed patches are generated infrequently (τ = 0.125 IP), such that they decay before they interact (e.g., Sundermeyer and Lelong 2005; see Table 2, run A1). After 100 IP, energies reach an approximately steady state (Figs. 1a,b). Examining time series of the various energy components, either as KE and PE or as EW, EV, and ESM, we find APE represents 60% and EV represents 88% of the total energy (Figs. 1a,b). This is similar to findings for single vortex simulations by Brunner-Suzuki et al. (2012). Note that, although energies are nearly steady after model spinup, KE and EV continue to increase slightly throughout the simulation (Fig. 1b). This small growth can be explained by the fact that in a finite domain, every Gaussian-shaped mixed patch has a small barotropic component. In low aspect ratio domains, energies in the barotropic mode are diffused much slower than higher baroclinic modes, leading to a slow buildup of barotropic energy (Jacobs 2012). This notwithstanding, we find spectral shapes of EW and EV vary little once energies reach a steady state (not shown). Last, we note that EW ≠ 0 in our vortex-only forced simulations owing to high-frequency waves generated during mixed patch adjustment (Fig. 1b).

Fig. 1.
Fig. 1.

Energy time series for four simulations with different wave and vortex forcings. (a),(b) τ = 0.125 IP, fw = 0 m s−1; (c),(d) τ = 0.0125 IP, fw = 0 m s−1; (e),(f) τ = 0.125 IP, fw = 1 × 10−4 m s−1; and (g),(h) τ = 0.0125 IP, fw = 1 × 10−4 m s−1. (left) Total energy (Etot, solid), KE (dashed), and APE (dotted–dashed). (right) Total (Etot, solid), vortex (EV, dashed), wave (EW, dotted–dashed), and shear mode (ESM, solid gray) energy and a linear fit to EV (bold dashed).

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

2) Frequent vortex forcing, no wave forcing

Expanding on the base case, we next examine the case where vortices are generated more frequently (τ = 0.0125 IP), thereby enhancing vortex interaction and spurring upscale energy transfer (Table 2, run A2). Such upscale transfer is evidenced by a significant and continuous growth of total energy even after model spinup (Fig. 1c), as energy is carried away from dissipative scales. Here, total energy gain is dominated by EV (Fig. 1d), which grows at a rate of 2.3 × 10−6 J m−3 IP−1. Meanwhile, EW remains approximately steady, while ESM increases slightly over the 1000 IP simulation, but always remains less than EW. Similarly, APE remains nearly constant, while KE increases. Associated with the increase in KE is a transition to larger horizontal scales over the course of the simulation, as revealed by snapshots of PV, velocity, and ρ′ (Fig. 2). Even after model spinup, individual S vortices can be seen in vertical slices of PV, while plan view slices show cross sections of both anticyclones and cyclones. By 800 IP, the horizontal velocity is dominated by larger structures, while vertical slices of PV continue to reveal vorticity at smaller scales (Fig. 2d).

Fig. 2.
Fig. 2.

Evolution of PV, velocity, and ρ′ for frequent vortex forcing (τ = 0.0125 IP) and no wave forcing, showing the gradual buildup of energy at large scales. Columns are for times (left) early, (middle) middle, and (right) late in the simulation. Horizontal and vertical slices through the domain are given in the upper and lower three rows, respectively. (b)Velocity is , (e) velocity is υ.

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

Comparing wavenumber spectra of EW and EV from the τ = 0.0125 IP case to reference spectra from the τ = 0.125 IP base case, we find that low-wavenumber EV increases with time, while EW remains relatively steady across the spectrum (Fig. 3). More specifically, EV approaches a slope of over approximately a decade of wavenumbers before rolling off at large kH. Meanwhile, vertical spectra of EV are relatively flat for low wavenumbers, except for a rise at the gravest modes, before also rolling off for large m. Flat vertical spectral slopes and steepening horizontal EV spectra are typical for quasigeostrophic turbulence undergoing an inverse energy cascade (Stammer 1997; Laval et al. 2003; Waite and Bartello 2006b). Here, the fastest growing mode is kH = 1, m = 0, corresponding to single barotropic dipole, which also explains the rise in EV vertical wavenumber spectra at m = 0. To summarize, for τ = 0.0125 IP, upscale energy transfer is evidenced through a transition to larger scales in horizontal slices of velocity, an increase in total energy driven by EV and KE, and low horizontal wavenumber EV approaching a slope.

Fig. 3.
Fig. 3.

Isotropic (a),(b) horizontal and (c),(d) vertical spectra of (left) wave and (right) vortex energy density following model spinup (>100 IP) for simulations with frequent vortex generation (τ = 0.0125 IP) and no wave forcing showing buildup of vortex energy at the largest horizontal scales (smallest horizontal wavenumber). Colors progressing from blue to red in each subpanel are spectra from early to late times after model spinup. Reference slopes are k−5/3, k−3, and k−5. Reference spectra computed at 110 IP from simulations with τ = 0.125 IP and no wave forcing are also shown (solid black spectra). Vortex forcing for all base-case runs is at nondimensional wavenumber 40 in the horizontal and 20 in the vertical; here only the first 30 nondimensional wavenumbers are shown.

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

b. Wave-induced upscale energy transfer

Infrequent vortex forcing, plus wave forcing

Next we describe a suite of simulations in which wave forcing, superimposed on the above vortex forcing, is increased over several orders of magnitude (Table 2, runs A3, A4, and runs D). Consider first the case of infrequent vortex forcing, τ = 0.125 IP, but additionally forced by a background wave with fw = 1 × 10−4 m s−1. The time evolution of the various components of energy (Figs. 1e,f) shows that, with the addition of the wave, EV increases at a rate of 6.9 × 10−7 J m−3 IP−1 or approximately 40 times faster than the comparable, vortex-forced base case with no wave. Here, EW exceeds EV for most of the simulation, owing entirely to the forced wave. This differs from both infrequently and frequently forced vortex-only cases, which are dominated early on by EV.

Similar to the τ = 0.0125 IP base case with no wave, plan view sections of velocity and PV for the present wave plus vortex-forced case show a progression from small, individual S vortices to a large, barotropic dipole with increasing velocity and PV (Fig. 4). Meanwhile, vertical slices of PV also reveal tilted S vortices (e.g., see PV field at 383 IP; x = 13 km, z = 19 m in Fig. 4), suggesting one possible mechanism for how a large-scale wave affects the stability of S vortices (e.g., Flierl 1988; Brunner-Suzuki et al. 2012). Vertical slices of υ velocity are initially dominated by the forced internal wave (Fig. 4e, 105 IP). However, with continued vortex merging and growth of a barotropic dipole, the dipole velocity eventually becomes comparable to the wave velocity (Fig. 4d, 801 IP). Nevertheless, the wave remains cohesive and does not itself appear to go unstable.

Fig. 4.
Fig. 4.

Evolution of PV, velocity, and ρ′ for infrequent vortex forcing (τ = 0.125 IP) plus wave forcing (fw = 1 × 10−4 m s−1), as in Fig. 2.

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

Considering spectra of the τ = 0.125 IP wave-forced run, we find that EV increases at horizontal wavenumbers less than the forced wave (Fig. 5b), approaching a slope, which is again typical of quasigeostrophic turbulence and similar to the vortex-induced upscale energy transfer case, τ = 0.0125 IP with no wave. At all wavenumbers EV exceeds the reference spectral levels without wave forcing. Meanwhile, EW spectra remain relatively steady following model spinup (Figs. 5a,c). The increase at low wavenumbers of horizontal EV spectra is again indicative of upscale energy transfer. Notable here, however, is that such enhanced upscale energy transfer was not apparent in the similarly forced τ = 0.125 IP base case without wave forcing. Thus, the addition of wave forcing appears to have spurred upscale energy transfer in the vortex field. Henceforth, we refer to this situation as wave-induced upscale energy transfer. Note, by this we do not imply that the wave field itself transfers energy upscale, but rather that it feeds energy to and, hence, facilitates upscale transfer in the vortex field (e.g., Smith and Waleffe 2002).

Fig. 5.
Fig. 5.

Energy spectra following model spinup (>100 IP) for infrequent vortex forcing (τ = 0.125 IP) plus wave forcing (fw = 1 × 10−4 m s−1), as in Fig. 3. Reference spectra computed at 110 IP from simulations with infrequent vortex (τ = 0.125.IP) and no wave forcing (solid black spectra), and frequent vortex (τ = 0.0125.IP) and no wave forcing (dashed black spectra) are also shown.

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

The increase in EV, the transition to larger horizontal scales, and the steepening of horizontal spectral slopes at the smallest wavenumbers are all similar to two-dimensional turbulence (e.g., Vallis 2006) and flows dominated by the vortical mode (Laval et al. 2003; Waite and Bartello 2006b). Unlike the cases studied by Smith and Waleffe (2002), the larger horizontal scales are dominated by vortical modes and not shear modes. This difference is most likely due to the dominance of vortical motions in the mixed-patch forcing scheme. In our simulations, the vorticity-based Ro of a single S vortex ≈0.15, and even though Ro increases once individual S vortices split into dipoles, it never exceeds 1. This is consistent with Waite and Bartello (2006b), who find that Ro < 3 is required for an inverse energy cascade. Further, as noted previously, the vertical Froude number computed in a variety of ways in our simulations is always ≪1. Again, this suggests our simulations are most likely in the quasigeostrophic regime (Riley and Lelong 2000), consistent with the observed upscale energy transfer.

To better understand under what conditions upscale energy transfer occurs, consider the effect of a large-scale internal wave on a single vortex. Brunner-Suzuki et al. (2012) showed that with increasing wave amplitude the splitting of S vortices into dipoles occurs sooner. Moreover, resulting dipoles contain more KE, rotate faster compared to stationary S vortices, and travel farther and faster from their points of origin. Conversely, for weaker waves, viscosity and diffusivity have longer to damp S vortices before they split, so the resulting dipoles are weaker. Dipole splitting in single vortex simulations was only observed if the time required for dipole splitting was less than the viscous time scale. Below this wave strength, the S vortex simply diffuses. In simulations with multiple vortices, we presume the wave has a similar effect. Resulting dipoles will propagate through the domain, potentially interacting and merging with other dipoles and S vortices. Through this wave-induced dipole splitting and subsequent vortex interaction and merging, upscale energy transfer is induced. However, dipole splitting alone is not sufficient to explain the occurrence of upscale energy transfer. Only when the resulting dipoles are sufficiently energetic so as to enhance vortex–vortex interactions is upscale energy transfer observed. It is not entirely clear what mechanism triggers this onset. It does not appear to be solely related to the diffusive time scale. In our simulations, upscale energy transfer occurs once EW is comparable to EV at the forcing wavenumber, provided that both of these energies are sufficiently large.

A summary of wave and vortex energies for a suite of runs with varying wave forcing amplitude is shown in Fig. 6 (see also Table 2, runs D). Noteworthy here is that the realized wave amplitude in our simulations is determined not only by the wave forcing factor fw, but also by the strength of the vortex field. Comparing EW and EV at the forced wavenumber in simulations with and without vortices, the effect of vortices on the wave forcing is clear. In simulations with only a wave, the relation between fw and EW is linear. However, for wave plus vortex-forced runs, EW is always smaller than the comparable wave-only forced runs, and the relationship is no longer linear. Recall here that it is the realized wave amplitude, or EW, that determines the onset of dipole splitting, not simply the magnitude of fw. Three possible mechanisms may explain the reduction of EW compared to wave-only simulations: 1) each new mixed patch also mixes the internal wave velocity and density structure and thus counteracts the wave forcing; 2) adjustment vortices enhance lateral stirring and mixing (e.g., Sundermeyer and Lelong 2005), which damps the wave; and 3) when vortices go unstable, they draw energy from the wave (Brunner-Suzuki et al. 2012). Note that at low wave forcing, where no dipole splitting is expected, points 1 and 2 are most likely to explain the ~90% reduction in EW compared to the wave-only case. Once dipole splitting occurs, point 3 can become important, decreasing the realized wave energy even further, up to 98% in some extreme cases. Once energy begins to be transferred upscale, EW is only reduced by 30%–40% compared to wave-only simulations. There are several possible explanations for this: 1) an increasing projection of the vortex field onto the wave field, 2) the vortex field draws less energy from the wave field, or 3) near-resonant interactions between wave and vortex at the lowest wavenumbers put energy back into the wave field. Determining which scenario is most likely is left for future investigation.

Fig. 6.
Fig. 6.

Wave forcing vs realized wave and vortex energy densities at the forced wavenumber for two sets of simulations, showing an increase of both wave and vortex energies with increased wave forcing. Open circles are wave energies EW for simulations where only a wave was forced. Squares are wave energies EW and diamonds vortex energies EV for simulations with wave plus infrequent vortex (τ = 0.125 IP) forcing. Error bars are based on standard deviations. The onset of dipole splitting (D.S.) and upscale energy transfer (U.T.) are marked with arrows.

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

c. Effect of a wave on vortex-induced upscale energy transfer

To complete our base set of simulations, we last examine the case in which vortex forcing is frequent (τ = 0.0125 IP) and wave forcing strong enough (fw = 1 × 10−4 m s−1) to expect dipole splitting after 16 IP (Table 2, run A4). This is slightly later compared to the base-case simulation with infrequent vortex plus wave forcing, possibly due to the stronger vortex field affecting EW, which is now half the value reached with infrequent vortex forcing. Again, we find that the addition of the wave boosts both KE and EV during model spinup (there is approximately 7 times more total energy after 100 IP), while enhancing the vortex energy growth rate later in the simulation (EV increases 3 times faster than the analogous case with no wave forcing; Figs. 1g,h). Again it appears that the wave enhances vortex–vortex interactions, presumably by causing S vortices to break into dipoles. A boosted upscale energy transfer is also again evident in both velocity and PV, such that they are both quickly dominated by one large dipole (not shown). Energy spectra confirm this, showing enhanced EV at low horizontal wavenumbers compared to simulations without wave forcing (Fig. 7, cf. Fig. 3). Spectral slopes are again approximately for low and intermediate kH, rolling off at high kH. Vertical EV spectra also again remain approximately steady, with enhanced energy levels at the gravest modes. The EW spectra decrease slightly below the forced wavenumber, while remaining relatively steady at high wavenumber over the course of the simulation. A key point both here and in the previous case of infrequently forced vortices is that the forced near-inertial wave does not appear to inhibit upscale energy transfer in the vortex field. Rather, as evidenced here, it enhances the existing upscale energy transfer. Last, we reiterate here that we expressly avoid unstable internal waves and do not expect nor observe overturns in the wave field.

Fig. 7.
Fig. 7.

Energy spectra following model spinup (>100 IP) for frequent vortex forcing (τ = 0.0125 IP) plus wave forcing (fw = 1 × 10−4 m s−1), as in Fig. 3. Reference spectra computed at 110 IP from simulations with infrequent vortex (τ = 0.125 IP) and no wave forcing (solid black spectra), and frequent vortex (τ = 0.0125 IP) and no wave forcing (dashed black spectra) are also shown.

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

d. Verifying the normal-mode decomposition

Having explored both infrequent and frequent vortex-forced simulations, both with and without wave forcing, we now perform some brief consistency checks regarding the frequency content of the normal mode decomposition, and the observed upscale energy transfer in the vortical mode. Regarding the first, we seek to determine the effectiveness of the normal-mode decomposition in separating wave and vortical mode components. Figure 8 shows frequency spectra for the three normal modes, B0, B+, and B, evaluated over more than two decades, from 0.1 f < ω < N, based on our infrequent vortex plus wave-forced simulation. Results show that both PV and wave modes exhibit a ω−2 slope, consistent with theories of quasigeostrophic turbulence and internal waves, respectively (e.g., Arbic et al. 2012; Garrett and Munk 1979). Further, the PV mode B0 contains an order of magnitude more energy than either wave mode at frequencies ω < f, except for some leakage from the forced wave, which itself is near inertial. Meanwhile, a clearly elevated signal corresponding to the internal wave frequency band, f < ω < N, is apparent in the B± modes, including a distinct peak in the B spectrum at the frequency of the downward-propagating forced wave. These results confirm that the linear normal-mode decomposition reasonably separates zero/subinertial (allowing for Doppler shifting) frequency motions from superinertial internal wave motions in our simulations. (Note that the decrease in the wave mode spectra at frequencies > 5 × 10−3 in Fig. 8 is an artifact of subsampling the domain to compute frequency spectra and not a real decrease in wave energy in the model at these frequencies.)

Fig. 8.
Fig. 8.

Frequency spectra of the three linear normal modes, B0, B+, and B, for the case of infrequent vortex forcing (τ = 0.125 IP) and wave forcing (fw = 1 × 10−4 m s−1) showing the separation of subinertial motions, dominantly into the B0 mode, from motions with frequencies f < ω < N, dominantly into the B+ and B modes. Inertial f and buoyancy N frequencies are also marked on the plot, along with a reference ω−2 spectral slope, as expected for internal waves and quasigeostrophic turbulence.

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

Regarding the upscale energy transfer observed in our simulations, we next examine the contributions of nonlinear interactions directly involving the forced wave versus interactions entirely within the vortical mode field. To this end, we compute horizontal wavenumber cross-bispectra of the various combinations of the three normal modes per Eqs. (12)(13). A total of 27 possible combinations and hence cross-bispectra exist between the three modes. However, here we focus on the three autobispectra, [0, 0, 0], [+, +, +], and [−, −, −] (where 0 refers to normal mode B0, + refers to B+, and − refers to B), as these reveal the primary interactions of interest (Fig. 9). Most notably, we find the [0, 0, 0] bispectrum is dominated by nonlinear transfers at low wavenumbers, suggesting that the observed growth in EV energy spectra in Figs. 3, 5, and 7 at low wavenumbers is primarily due to nonlinear interactions within the vortical mode field. Meanwhile [−, −, −] bispectra are dominated by triad interactions at the forced wavenumber and higher, suggesting a minimal direct role at low wavenumbers compared to [0, 0, 0] interactions. Last, [+, +, +] bispectra show a broad distribution of interactions at wavenumbers both above and below the forced wave, but all more than three orders of magnitude weaker than the interactions observed in either of the other modes. Incidentally, cross-bispectra between the forced wave and the vortical mode (not shown) also show significant interactions at the forced wavenumber, consistent with near-resonant interactions between a near-inertial wave and the vortical mode (Lelong and Riley 1991). The main point here, however, is that bispectra of the three normal modes confirm that the observed upscale energy transfer described in the previous sections is, indeed, primarily the result of nonlinear interactions in the vortical mode field.

Fig. 9.
Fig. 9.

Horizontal wavenumber bispectra of the three normal modes B0, B+, and B, defined by Eqs. (12)(13), showing the dominance of vortical mode nonlinear interactions for wavenumbers less than the forced wave. Axes represent horizontal wavenumbers, k1 and k2, with color indicating the degree of nonlinear coupling via triad interactions at wavenumber k3 = k1 + k2. Note the different color scales among the panels.

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

e. Can upscale energy transfer be arrested?

We now explore different scenarios that could influence and possibly arrest upscale energy transfer. First, we consider whether a wave can disrupt the vertical coupling of upscale energy transfer via its vertical shear. That is, although it is the internal wave shear that accelerates the breakup of individual S vortices into dipoles and hence induces the upscale energy transfer to begin with, the vertical scale of the shear may also impose an upper limit on vertical coupling and hence eventually limit upscale energy transfer. The question of which wins was answered by our base simulations with different wave forcing. The associated particle displacement of our largest realized internal wave is small (17.4 m with fw = 10 × 10−4 m s−1) compared to the vortex radius (250 m). Apparently, such small displacements are not strong enough to disrupt the vortex interactions discussed in the previous subsections, and upscale energy transfer occurs despite the relatively strong shear field. If the forced wave amplitude and consequently particle displacements are increased further, the wave becomes unstable and breaks, which is another problem entirely. For the simulations discussed above, we thus conclude that the vertical shear associated with the wave itself does not disrupt upscale energy transfer.

Second, we investigate whether the internal wave’s vertical wavenumber more generally imposes a maximum vertical scale on vertical vortex coupling associated with upscale energy transfer. Associated with the inverse energy cascade of quasigeostrophic turbulence is a growth in horizontal scales from the traditional baroclinic deformation radius (typically ≈50 km) to scales of the order of the barotropic deformation radius (typically ~1000 km). Such large scales can only be achieved through vertical coupling, which increases the effective deformation radius by increasing the vertical scale of the flow. Although in the present simulations we are dealing with a much smaller effective deformation radius, the same principle should apply. Considering our results in this context, that is, that vortex size may be limited by the forced wave height, we note that in simulations presented thus far, the greatest vortex height, that is, that of the single large dipole eventually formed by the upscale energy transfer, is of the order of the domain size. The associated deformation radius for the same domain-filling dipole is half the horizontal domain width. Meanwhile, in our base-case simulations, our forced wave has a vertical wavelength equal to ¼ the vertical domain scale. Hence, if the vertical scale of the wave was limiting the vertical scale of the vortex field, this would have already been apparent in our simulations. We thus conclude that the wave itself does not limit upscale energy transfer.

An alternative test of whether the vertical scale of the wave shear could limit or halt upscale energy transfer is to vary the vertical wavenumber of the forced wave, while keeping Lz constant or vice versa. Here, we force the wave at a fixed percentage of the overturning amplitude compared to our base case (see Table 2, runs e). Incidentally, this also reduces the vertical scale of the wave relative to the vortex height. Examining these simulations (not shown), the conclusion remains that varying the vertical wavenumber of the forced wave relative to the domain size does not affect our previous findings—the wave does not limit upscale energy transfer. If wave forcing is further increased, again the wave breaks owing to the enhanced vertical shear; a scenario we try explicitly to avoid. Given these findings, we conclude that internal wave shear does not halt wave-induced upscale energy transfer nor does it halt a vortex-induced upscale energy transfer.

Third, we examine whether the observed upscale energy transfer can maintain itself in the absence of forcing. We start with simulations with wave forcing (fw = 1 × 10−4 m s−1) and infrequent vortex forcing (equivalently τ = 0.125 IP, adjusting for domain size). Then, when upscale energy transfer is well established (500 IP), wave forcing is turned off. Upscale energy transfer subsides quickly; that is, KE and EV begin to drop within 5 IP. This is consistent with the notion that the upscale energy transfer was induced by the wave. Once the wave is no longer forced, EW decays with an e-folding time scale of about 12 IP, or roughly the vertical viscous time scale. This is also consistent with EW being dominated by KE. Meanwhile, EV decays more slowly since it is dominated by a domain-filling, barotropic dipole whose decay is governed by horizontal diffusivity, which in our case has a time scale of Th = L2/νB ≈ 3.7 × 105 IP (i.e., much longer than any of our simulations). Such a long decay time is consistent with findings of Jacobs (2012), who showed a much slower decay of barotropic energy compared to baroclinic energy in triply periodic models with low aspect ratios. Repeating the above but for the case when vortices are generated frequently (τ = 0.0125 IP), and with wave forcing turned off, total energy drops by 30% over 60 IP. Thereafter, however, it rises, as upscale energy transfer continues at approximately the same rate as without the wave (similar slope for growth of EV), but supplemented by the initial boost obtained from the wave forcing. Last, when vortex forcing is turned off while wave forcing is maintained, upscale energy transfer subsides within 5 to 10 IP, regardless of whether vortex forcing is frequent or infrequent. In this case, EW remains at its established level, while EV slowly decays, according to horizontal diffusivity. However, this time the estimated decay rate is faster than when wave forcing is discontinued since the wave imposes an additional shear and decay term. Based on these results, we conclude that whatever the driving force of upscale energy transfer, either significant vortex forcing or vortex forcing supplemented by wave forcing, this combined forcing must be maintained in order for upscale transfer to persist.

Finally, we explore the effect of a weakened vortex field on upscale energy transfer. Here, the strength of the vortex field is determined by the frequency of mixing events and the strength of individual vortices, given by ΔN2/N2. Thus far, the strength of the vortex field has been set by the vortex generation period, with ΔN2/N2 = 1 [see Eqs. (4)(6)]. We now examine the case of ΔN2/N2 = 0.1 (see Table 2, runs F). The latter is consistent with the idea that mixing in the ocean is almost never complete (e.g., Alford and Pinkel 2000; Sundermeyer et al. 2005). Here, we again contrast four simulations—finding that, without wave forcing, vortex interactions alone are too weak to spur upscale energy transfer. This is true for both τ = 0.125 IP and τ = 0.0125 IP equivalent vortex forcing (adjusting for domain size), where the reduction of ΔN2/N2 to 0.1 reduces both EV and EW 70- to 100-fold compared to our base-case simulations (Figs. 10a–d). However, once wave forcing is added, upscale energy transfer is again observed, with growth rates of EV of 1.3 × 10−8 (τ = 0.125 IP) and 1.0 × 10−7 J m−3 IP−1 (τ = 0.0125 IP; Figs. 10e–h). Here, total energy consists of more than 95% EW, an indication that upscale energy transfer is again wave induced rather than by the vortices alone (e.g., compared to vortex- and wave-induced upscale transfer cases in Figs. 1d,f). Also, realized wave amplitudes are slightly larger compared to simulations with ΔN2/N2 = 1, consistent with our earlier conclusion that a more energetic vortex field dampens the wave more. Similar to other simulations with upscale energy transfer, spectra reveal a steady increase of low horizontal wavenumber EV throughout the simulation, while EW energy remains relatively steady across the spectrum (not shown). The overall reduced energy levels are consistent with the reduced strength of individual vortices. Furthermore, reducing ΔN2/N2 also reduces the initial Bu of the vortices to 0.025 or RDL. This is a different dynamical regime than our base-case simulations, that is, more strongly dominated by rotation (Lelong and Sundermeyer 2005) and with vortices with smaller Bu tending to be more stable (Griffiths and Linden 1981; Helfrich and Send 1988; Hopfinger and van Heijst 1993; Flor and van Heijst 1996; Lelong and Sundermeyer 2005). Despite the change in Bu, however, the wave still induces upscale energy transfer.

Fig. 10.
Fig. 10.

Time series for four simulations with reduced vortex amplitude (ΔN2/N2 = 0.1), as in Fig. 1.

Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-12-0149.1

4. Discussion

In this study, the effect of a near-inertial internal wave on a field of submesoscale vortices was evaluated. By varying the internal wave amplitude, it was shown that, despite considerations discussed in section 1, a near-inertial internal wave does not limit upscale energy transfer of vortex energy. Rather, if its amplitude is large enough, a wave can induce upscale energy transfer by enhancing vortex–vortex interaction. We now discuss a number of considerations that bear on or are impacted by our major results, including the assumption of stochastic distribution of vortices, the major factors influencing the onset of upscale energy transfer, and the implications of upscale energy transfer to lateral dispersion.

First, regarding our simplification that mixed patches are uniformly distributed in space, in the simulations discussed throughout this paper we have assumed that the occurrence of mixed patches by wave-breaking events is entirely stochastic; that is, the locations of mixed patches follow a uniform distribution. However, we have also conducted simulations (not presented here) that show a single mixed patch can trigger the onset of wave breaking when the wave amplitude is sufficiently large. We speculate that there are two possible mechanisms that may explain this phenomenon: one is that the vortex enhances velocity and shear locally, making overturns of the wave more likely, and the other is that the generation of an initial mixed patch enhances mixing locally, perturbing the ambient stratification and making subsequent overturns more likely. That a vortex can induce wave breaking implies that vortex locations and timings may not be entirely stochastic. This in turn will affect the lateral dispersion generated by such nonrandom fields of mixing events and make comparison to theory more difficult (Jacobs 2012).

Second, as demonstrated by the numerical simulations presented herein, numerous factors determine if and when upscale energy transfer will occur. These include model viscosity and diffusivity, vortex forcing period τ, the strength of mixed patches ΔN2/N2, and the strength of internal wave forcing fw. Using single vortex simulations, we have determined that model viscosity sets the timing of the onset of dipole splitting versus vortex decay. Recall that our simulations suggest dipole splitting is a necessary but not sufficient criterion for wave-induced upscale energy transfer. We suspect that upscale energy transfer would occur at successively lower wave amplitudes if vortices were generated more frequently compared to our base-case simulation (τ = 0.125 IP). Furthermore, beyond a certain threshold, when vortices are generated very frequently (e.g., τ ≤ 0.0125 IP in the present simulations), wave forcing is no longer necessary to spur upscale energy transfer. Conversely, if vortices are generated less frequently compared to our base-case simulation (e.g., τ > 0.125 IP), we expect that larger wave amplitudes would be necessary to spur upscale energy transfer. In the limit of large τ, whether upscale energy transfer occurs may then be dictated by the diffusive time scale. A threshold for the onset of vortex-induced upscale energy transfer was investigated and described by Sundermeyer and Lelong [2005, see their Eq. (20) and their Fig. 6]. They found that the onset of upscale energy transfer depends on a variety of parameters influencing how densely populated vortices are in time and space. Densely populated vortices interact with one another when they last long or occur frequently (or both). Thus, Sundermeyer and Lelong observed upscale energy transfer when the ratio of viscous time scale Tν to vortex generation time τ is larger than a threshold value. The ratio Tν/τ also depends on the characteristics of the vortices such as ΔN2/N2, the net diapycnal diffusivity induced by mixing events κz, and the background viscosity ν. Specifically, they found Tν/τ ∝ 3(N2N2)(κz/ν) ≥ (0.01 − 0.10). The effects of some of these parameters on the onset of upscale energy transfer were evaluated and discussed in sections 3e and 4. Other possible factors that likely influence the energy transfer and interaction between the vortical mode and internal waves are variations in sizes and shapes of mixed patches. For instance, elliptical S vortices go unstable much faster than their circular counterparts (Jacobs 2012). Whether or not a specific vortex field transfers energy upscale depends on any combination of the aforementioned factors.

These considerations notwithstanding, a number of factors can expressly limit upscale energy transfer. For example, strong downscale energy transfer can counteract upscale transfer. Forward energy cascades are known to occur as the result of triad interactions between internal waves. Downscale energy transfer can also result from wave breaking and overturns, that is, three-dimensional turbulence. Additionally, large-scale horizontal shears, including those generated by larger (mesoscale) vortices, could limit upscale transfer by breaking apart growing vortices. In our simulations, when wave forcing was turned off but vortex forcing was maintained, upscale energy transfer continued even in the presence of a large, barotropic dipole. This suggests that mesoscale vortices would not impose a strong limit to upscale energy transfer. However, stronger mesoscale vortices could still destroy smaller vortices. Other sources of shears and strains include fronts and currents. Brunner-Suzuki et al. (2012) found that a jet (for a small range of jet velocities) can, in fact, strain and shear a single S vortex enough to inhibit dipole splitting. This in turn may also inhibit upscale energy transfer. Another limit may be posed by submesoscale frontogenesis. For instance, D’Asaro et al. (2011) observed increased turbulent diffusion near the Kuroshio front where horizontal density gradients and their associated geostrophic currents move energies to smaller scales. Similarly, at the ocean’s surface, submesoscale frontogenesis is enhanced in the presence of a strain field, transferring energy to smaller scales (Capet et al. 2008a,b). Further investigation of the interaction between these mechanisms and the submesoscale inverse energy cascade is needed to address these issues.

Finally, given the potential importance of the vortical mode to submesoscale lateral dispersion in the ocean, we speculate on the implications of our results to such dispersion. In the weakly nonlinear regime, mixing and stirring by submesoscale vortices enhance lateral dispersion (Sundermeyer 1998). This enhanced stirring may explain observational discrepancies described by Sundermeyer et al. (2005). Specifically, they found that observed lateral diffusivity cannot be explained by lateral intrusions or internal wave shear dispersion. Similarly, observations of lateral dispersion during the North Atlantic Tracer Release Experiment could not be explained by internal wave shear dispersion alone (Polzin et al. 2003; Holmes-Cerfon et al. 2011). In our simulations of the strongly nonlinear regime, energies do not equilibrate; thus we do not have an accurate measure of an equilibrium energy spectrum. Hence, traditional passive tracer dispersion diagnostics are not useful. We do know from Okubo (1971) that particle dispersion is dominated by the energy-containing scales. In the presence of upscale energy transfer, larger scales contain more energy. Larger-scale motions also may persist for longer, as they may have larger viscous decay time scales. Consequently, we expect enhanced lateral diffusivity compared to cases where the same energy remains at smaller scales. Unfortunately, parameterizing this enhanced diffusivity is not so simple without a more thorough understanding of what may eventually limit energy growth at the largest scales.

5. Summary

Upscale energy transfer can occur within a field of vortices if these vortices are sufficiently densely populated so as to enable vortex interactions. Such upscale energy transfer is enhanced in the presence of a large-scale internal wave. Even if the vortex field is noninteracting by itself (i.e., there is no upscale energy transfer), a wave can induce upscale transfer. This can be understood by considering the behavior of a single vortex in the presence of a large-scale internal wave (Brunner-Suzuki et al. 2012). Under the influence of a wave, a single stationary S vortex splits into a pair of propagating dipoles. In the present study, we find that, if the wave is weak, it has little or no influence on the vortex field. However, if the wave is stronger, vortices split into dipoles. If dipole splitting is rapid enough that the resulting dipoles travel fast and far enough to enhance vortex interaction, then upscale energy transfer can occur. In these cases, both wave and vortex forcing are required to maintain the upscale energy transfer.

The question of whether energy is transferred upscale or downscale is important to understanding dispersion, mixing, and stirring and nutrient and pollutant transport. The answer depends on the strength of both the background internal wave field and the vortex field. While internal waves are inherently subject to downscale energy transfer via triad interactions, a vortex field by itself can undergo upscale energy transfer if vortices are generated frequently enough in time and space. Further, a vortex field can sustain itself within a strong wave field and contribute to upscale energy transfer even if the waves themselves transfer energy downscale. Note that here waves can even induce upscale energy transfer in the vortex field. Thus, within a certain range of scales and parameters, a large-scale background wave does not limit, but rather excites, upscale energy transfer. What eventually limits the upscale transfer is still an open question. In the coastal ocean, bottom friction may impose such a limit. In the open ocean, characteristics of the flow, or the planetary Rhines scale, are more likely. Also, mesoscale shears and strains could impose a limit on upscale energy transfer. In essence, our results indicate that only the wave amplitude (through the forcing amplitude fw) appears to matter. For the limited range of simulations conducted here, wave frequency, wavelength, and the oscillatory structure of the wave shear seem to be inconsequential. Thus, these results seem to indicate that any large-scale mean flow would have the same effect on the behavior of upscale energy transfer.

Acknowledgments

This work was supported by the National Science Foundation under Grants OCE 0351892 and OCE 0623193 and by the Office of Naval Research under Grant N00014-09-1-0194. We thank J. Jacobs for his insight into the effects and implications of an anisotropic domain for our simulations. Also, we thank corresponding editor E. Kunze and three anonymous reviewers for helpful suggestions and feedback.

APPENDIX

Separation of Wave Field and vortex Field Energy

To estimate how much energy is contained in the wave field versus the vortex field, following Bartello (1995), we apply a normal mode decomposition to the equations of motion [Eqs. (1)(3)]. The linear f-plane equations have three solutions: λ0 = 0, representing the vortical mode, and , representing internal waves. To separate wave and vortex energies, the equations of motion are first Fourier transformed:
ea1
where j = 0, ±; are the three normal-mode amplitudes at wave vector k; are the interaction coefficients; is a forcing term; and is a diffusion term. The right-hand side is assumed to be zero, and the equations are linearized in PV. Vorticity, divergence, and density perturbations can then be defined as
ea2
for kH ≠ 0 and m ≠ 0, leading to
ea3
where
ea4
Expanding Wk into an orthonormal basis, , vortex and wave energies can be computed from the amplitudes , respectively defined as
ea5
ea6
The amplitudes are all normalized by kH resulting in and then expanded for m = 0:
ea7
and for kH = 0 as
ea8
from which Eqs. (9)(11) follow.
Following Polzin et al. (2003), wave EW and vortex energies EV can further be decomposed into their APE and horizontal KE components. Near-inertial internal waves have much larger horizontal than vertical scales and more horizontal KE (and shear variance) than APE (and strain variance). For similar aspect ratios, the opposite (more APE than horizontal KE) is true for the vortical mode. For internal wave and vortex energies, potential and kinetic energies can be computed as
ea9
where
ea10
where W and V indices indicate wave and vortex parts. Note that KEW + KEV ≠ KE (and similar for APE) because f ≠ 0.

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