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  • View in gallery

    Schematic representation of the procedure.

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    Vertical vorticity isosurfaces for the geostrophic modes at t ∼ 0. These fields correspond to the initial conditions of the runs analyzed in Figs. 3 and 15; Ro = 0.1. The isosurface values are defined by the rms vertical vorticity. For clarity, vertical slices have been extracted from the isosurfaces. The panels correspond to different values of the deformation radius, Ld: (a) Ld/L0 = 0.01, (b) Ld/L0 = 0.1, (c) Ld/L0 = 1, (d) Ld/L0 = 10.

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    Ageostrophic energy Eageo vs t for weak stratification (Ro = 0.1, Fr ≥ 1). The 2 curves are obtained by averaging data over the time interval t ∈ [1, 5]; for clarity only the Fr = 1, 10, and 100 curves are shown: (a) t ≤ 60, (b) t ≤ 8; Ld/L0 < 1 in all cases. Note the exponential scaling prior to saturation.

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    Illustration of the effect of the initial vertical velocity. Comparison between time series of the ageostrophic energy, Eageo and square of the rms vertical velocity, w2rms. Here, Fr = 10.

  • View in gallery

    Spectral growth rates, σEa(kh, kz), corresponding to Fig. 3: t0 = 0.8 (a) Fr = 10, t ∼ 4; (b) Fr = 1, t ∼ 1.7. The axes range from kh = 1 to kh = Kh and kz = 1 to kz = Kz. Growth is maximized for the “effectively pressureless” modes, kz > kh.

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    Transfer spectra for Ro = 0.1 and Fr = 100: (a) components of the total geostrophic transfer, TG; (b) comparison of the net ageostrophic–geostrophic transfer, Tnet, with the net 3D–2D transfer, T3D3D–2D.

  • View in gallery

    Transfer spectra corresponding to the runs of Fig. 3: (a) net geostrophic transfer, Tnet; (b) total ageostrophic transfer, TA.

  • View in gallery

    (a) Ageostrophic energy Eageo vs t for strong stratification in an anisotropic domain. (b) An expanded view is shown. Note the rapid adjustment. Here N/f = 10.

  • View in gallery

    Evolution of Eageo from a basic state defined with the ω equation (strong stratification, anisotropic domain). (a) Time series corresponding to Fig. 8b. The 2 curve for Ld/L0 = 0.33 is plotted for reference. (b) Comparison with the initial evolution defined using the regular procedure, that is, geostrophic modes with w = 0. Here, Ld/L0 = 1.33.

  • View in gallery

    Transfer spectra corresponding to the runs of Fig. 8 (strong stratification, anisotropic domain). (a) Net geostrophic transfer, Tnet; (b) total ageostrophic transfer, TA.

  • View in gallery

    The Ld/L0 dependence of the transfer in Fig. 10 (strong stratification, anisotropic domain). (a) Net geostrophic damping, Dg, vs Ld/L0. (b) Two-dimensionality of the geostrophic flow, Eg(kz = 0)/Eg, vs Ld/L0. The error bars are obtained from the standard error of a 4-member ensemble, each member corresponding to a different basic state.

  • View in gallery

    Eddy viscosity components for strong stratification in an anisotropic domain. The AAG and GAG components are shown, as well as the total eddy viscosity, νeddy, and the effective viscosity, νeff. (a) Subsynoptic flow, Ld/L0 = 3.33; (b) synoptic flow, Ld/L0 = 0.21.

  • View in gallery

    Eddy viscosities for strong stratification in an anisotropic domain. (a) Total eddy viscosity, νeddy; (b) normalized eddy viscosity, ν̂eddy = νeddy/νmax, where νmax is the peak value.

  • View in gallery

    Resolution dependence of νeddy (strong stratification, anisotropic domain). (a) Subsynoptic flow, Ld/L0 = 3.33; (b) synoptic flow, Ld/L0 = 0.33.

  • View in gallery

    Ageostrophic energy Eageo vs t for strong stratification, isotropic domain; (a) Fr ≤ 0.1. For comparison, the Fr = 1 curve is also shown. (b) Subsampled results for Fr = 0.1. Only modes satisfying kzks are retained.

  • View in gallery

    Spectral growth rates, σEa, corresponding to Fig. 15, expressed in terms of kh and kz: (a) σEa(kz; t = 3); (b) σEa(kh; t = 3). Reference time t0 = 0.39.

  • View in gallery

    Transfer spectra and eddy viscosities (strong stratification, isotropic domain). (a) Net geostrophic transfer, Tnet; (b) total eddy viscosity, νeddy.

  • View in gallery

    Resolution dependence of νeddy (strong stratification, isotropic domain); (a) Fr = 0.1, (b) Fr = 0.01.

  • View in gallery

    Subsampled νeddy (strong stratification, isotropic domain). Only modes satisfying kzks are used in the calculations; (a) Fr = 0.1, (b) Fr = 0.01.

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Dissipation of Synoptic-Scale Flow by Small-Scale Turbulence

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  • 1 McGill University, Montréal, Québec, Canada
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Abstract

Although it is now accepted that imbalance in the atmosphere and ocean is generic, the feedback of the unbalanced motion on the balanced flow has not received much attention. In this work the parameterization problem is examined in the context of rotating stratified turbulence, that is, with a nonhydrostatic Boussinesq model. Using the normal modes as a first approximation to the balanced and unbalanced flow, the growth of ageostrophic perturbations to the quasigeostrophic flow and the associated feedback are studied. For weak stratification, there are analogies with the three-dimensionalization of decaying 2D turbulence: the growth rate of the ageostrophic perturbation follows a linear estimate, geostrophic energy is extracted from the base flow, and the associated damping on the geostrophic base flow (the “eddy viscosity”) is peaked at large horizontal scales. For strong stratification, the transfer spectra and eddy viscosities maintain this structure if there is synoptic-scale motion and the buoyancy scale is adequately resolved. This has been confirmed for global Rossby and Froude numbers of O(0.1). Implications for atmospheric and oceanic modeling are discussed.

Corresponding author address: Dr. Keith Ngan, Department of Atmospheric and Oceanic Sciences, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montréal QC H3A 2K6, Canada. Email: kngan@meteo.mcgill.ca

This article included in the Spontaneous Imbalance special collection.

Abstract

Although it is now accepted that imbalance in the atmosphere and ocean is generic, the feedback of the unbalanced motion on the balanced flow has not received much attention. In this work the parameterization problem is examined in the context of rotating stratified turbulence, that is, with a nonhydrostatic Boussinesq model. Using the normal modes as a first approximation to the balanced and unbalanced flow, the growth of ageostrophic perturbations to the quasigeostrophic flow and the associated feedback are studied. For weak stratification, there are analogies with the three-dimensionalization of decaying 2D turbulence: the growth rate of the ageostrophic perturbation follows a linear estimate, geostrophic energy is extracted from the base flow, and the associated damping on the geostrophic base flow (the “eddy viscosity”) is peaked at large horizontal scales. For strong stratification, the transfer spectra and eddy viscosities maintain this structure if there is synoptic-scale motion and the buoyancy scale is adequately resolved. This has been confirmed for global Rossby and Froude numbers of O(0.1). Implications for atmospheric and oceanic modeling are discussed.

Corresponding author address: Dr. Keith Ngan, Department of Atmospheric and Oceanic Sciences, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montréal QC H3A 2K6, Canada. Email: kngan@meteo.mcgill.ca

This article included in the Spontaneous Imbalance special collection.

1. Introduction

In computational fluid dynamics, models with limited spatial resolution are employed. For this reason, the parameterization of unresolved motion has attracted much attention. In the popular large-eddy simulation (LES) approach (e.g., Fröhlich and Rodi 2002; Mason 1994), one obtains an accurate simulation with a coarse grid by modeling interactions between resolved and unresolved modes. As an alternative to subgrid-scale modeling, numerical data from high-resolution simulations can be analyzed; the resulting eddy viscosity, which represents the effects of unresolved interactions, is then appended to the kinematic viscosity (Domaradzki et al. 1987, 1993). LES is a standard technique in the modeling of industrial flows. It also lies behind turbulence schemes used in atmospheric and oceanic models (e.g., the well-known Smagorinsky model).

Nevertheless, the problem is more intricate in geophysical fluid dynamics. Here we desire information beyond the total contribution of unresolved, small-scale processes, namely, the separate contributions from different modes. For practical and theoretical reasons, it may be desirable to partition the “feedback” into distinct contributions from, say, gravity waves and vortical modes, or more generally into interactions between balanced (slow) and unbalanced (fast) modes.

The feedback of the unbalanced modes onto the balanced ones is a quantity of great interest. By definition this quantity is zero in a balance model. Even so, this feedback is crucial in real geophysical flows. It has been suggested that ageostrophic circulations play a role in, for example, frontogenesis and troposphere folding (cf. O’Sullivan and Dunkerton 1995).

There have been several studies of this problem. For example, Dewar and Killworth (1995) showed analytically that the effect of inertia–gravity waves vanishes to leading order in the Rossby number and confirmed this prediction numerically with reduced-gravity layer models. For rotating shallow water, minimal energy transfer between the quasigeostrophic flow and the inertia–gravity waves was observed in the numerical simulations of Farge and Sadourny (1989). In the study whose objectives are closest to our own, Errico (1982) argued, using a low-order version of a two-layer, primitive equation model, that the energy cascade to small scales and the rate of energy dissipation may be affected by weak ageostrophic processes. Less is known about balanced–unbalanced interactions for larger Rossby and Froude numbers. If we accept that imbalance in the atmosphere and ocean is generic and nonnegligible, quantifying interactions between (nominally) balanced and unbalanced modes at realistic Rossby and Froude numbers should be a useful line of research.

In a previous publication of ours this problem was addressed in the context of the three-dimensionalization of decaying 2D turbulence (Ngan et al. 2005, hereafter P2). We showed that the 3D modes (the perturbation) extract energy from the 2D modes (the base flow), which is then cascaded from large scales to small scales. The spectral eddy viscosity, which represents the effects of 3D modes on 2D modes, (i) peaks strongly at low wavenumbers, (ii) approaches zero above a crossover wavenumber, and (iii) decreases in magnitude as the aspect ratio of the domain is decreased. Although these results were obtained for unstratified, nonrotating flow, they may hold more generally. On synoptic scales the 2D modes may be analogous to the geostrophic motion and 3D modes to the ageostrophic motion. Thus the effects of gravity waves could be straightforward to parameterize: the accuracy of the parameterization would not hinge on the modeling of small-scale dynamics, about which uncertainty is greatest.

Using a numerical model of the nonhydrostatic Boussinesq equations, we examine the growth of ageostrophic perturbations to a quasigeostrophic base flow and quantify the interactions between base flow and perturbation for global Rossby numbers of O(0.1). The quasigeostrophic base flow may be viewed as a proxy for balanced motion, and the ageostrophic motion as a proxy for unbalanced motion. Obviously, more sophisticated definitions of balance exist, but this is a reasonable starting point for synoptic-scale flow: if the Rossby and Froude numbers are small, then only a small fraction of the motion will be associated with a higher-order balance. Following P2, transfer spectra and eddy viscosities are calculated, and the spectral structure of balanced–unbalanced interactions is considered in detail.

Geostrophic and ageostrophic modes are defined using normal modes of the nonhydrostatic Boussinesq equations. This facilitates connections with rotating stratified turbulence phenomenology. Scale analysis of the nonhydrostatic Boussinesq equations (Lilly 1983) implies that geostrophic modes decouple from ageostrophic modes as Ro, Fr → 0, in agreement with quasigeostrophic theory. It has been shown that the energy transfers for Ro, Fr ≪ 1 can be understood by analyzing the dynamics of the triads (Bartello 1995), geostrophic–geostrophic–geostrophic (GGG), geostrophic–ageostrophic–geostrophic (GAG), and ageostrophic–ageostrophic–geostrophic (AAG). GGG triads yield quasigeostrophic dynamics; AAG triads are responsible for nonlinear geostrophic adjustment in the limit Ro, Fr → 0, that is, the downscale cascade of ageostrophic energy, the geostrophic mode acting as a catalyst on account of potential vorticity conservation; GAG triads, while nonresonant, effect geostrophic–ageostrophic energy transfer for finite Ro and Fr.

Many studies of balance have employed normal modes (cf. Errico 1981; Leith 1980). This suggests connections with recent work on “spontaneous imbalance.” Following the pioneering work of Warn (1986) and Warn and Menard (1986), it has become clear that unbalanced motion may be generated through the action of the slow, balanced flow: balance is intrinsically “fuzzy.” In principle, spontaneous imbalance may occur even for unbalanced motion of arbitrarily small amplitude (Ford et al. 2000; Vanneste and Yavneh 2004), by contrast with the classic Rossby adjustment (or “dam break”) problem (e.g., Gill 1982), in which the adjustment occurs on the fast time scale and depends crucially on the (unbalanced) initial conditions. Indeed the phenomenon of spontaneous imbalance is essentially independent of the definition of balance that one chooses to adopt [Vanneste and Yavneh (2004) consider a special flow for which balance may be defined exactly].

We examine the extent to which geostrophic–ageostrophic interactions can be attributed to a specific mechanism of spontaneous imbalance. The three-dimensionalization of decaying 2D turbulence occurs via random straining of small-aspect-ratio 3D perturbations by the 2D base flow (specifically through a time-dependent, hyperbolic instability; see (Ngan et al. 2004, hereafter P1, and appendix B for details); if an analogous mechanism applied to rotating stratified turbulence, this would represent a route by which energy could be extracted from the balanced modes and transferred to unbalanced modes, whereupon a forward cascade to dissipation scales may occur (cf. Straub 2003). In other words, the unbalanced motion may induce an effective dissipation on the balanced flow. Since large-scale horizontal random straining is ubiquitous in the upper troposphere and lower stratosphere (cf. Shepherd et al. 2000), this mechanism may be of wide applicability. The generation of an essentially 3D response to a quasi-2D flow is reminiscent of the vertical tracer cascade in stratospheric dynamics (Haynes and Anglade 1997).

We argue that the behavior is analogous (in some important respects) to that of unstratified turbulence if the stratification is (locally) weak and the global Rossby number comparable to or smaller than the global Froude number; in particular, the growth of the ageostrophic energy follows a linear estimate and the damping of the (quasi-) geostrophic base flow by ageostrophic motion occurs preferentially at large horizontal scales. Even for global Froude numbers as small as O(0.1), the geostrophic–ageostrophic transfer is maximized at large horizontal scales.

Although the dynamics of a triply periodic Bousssinesq model may be broadly representative of those of the real atmosphere and ocean, our results cannot be interpreted literally. Phase information and temporal dependence are discarded in the calculation of the eddy viscosity and transfer spectra. Nevertheless, the eddy viscosity may still delineate constraints that a successful parameterization should respect. We emphasize, however, that our eddy viscosity is not the standard subgrid eddy viscosity, for it represents the effects of imbalance: the dissipation described in this paper arises from neither molecular nor eddy processes. “Wave drag” may be a more appropriate term, but as the risk of confusion would be even greater, “eddy viscosity” is retained.

The organization of the paper follows P2. After reviewing the numerical formulation and procedure (section 2), numerical results for weak stratification (section 3) and strong stratification (section 4) are considered in turn, transfer spectra and eddy viscosities being highlighted. We argue that the results are analogous to those for unstratified turbulence if the perturbation grows on vertical scales smaller than the buoyancy (or overturning) scale and the base flow has horizontal length scale greater than the deformation radius; we underscore the role of the buoyancy scale in determining the resolution dependence of the results. Implications for atmospheric and oceanic modeling are discussed in section 5. Background material on the definition of the basic state (appendix A) and the time-dependent, hyperbolic instability (appendix B) is deferred to the appendixes.

2. Numerical formulation and procedure

a. Model

We consider the nonhydrostatic Boussinesq equations for rotating stratified flow:
i1520-0469-65-3-766-e1
where b := ′/θ0 is the (perturbation) buoyancy, θ = θ0(dθ/dz)z + θ′ is the potential temperature, Ω = Ω is the angular velocity of the rotating frame, N = [(g/θ0)(dθ/dz)]1/2 is the constant Brunt–Väisälä frequency, v is the 3D velocity with vertical component w, and D denotes the diffusion operator. Our numerical model is triply periodic, cylindrically truncated, pseudospectral, and dealiased with the two-thirds rule; a weak Robert filter (Asselin 1972) is applied to suppress the computational mode. The grid is isotropic with N2h × Nυ collocation points and dimensions L0 = 2π and H0 = 2πNυ/Nh. The domain may be either isotropic or anisotropic. For an isotropic domain, the aspect ratio, εa := Nυ/Nh, is unity; for an anisotropic, thin domain, εa < 1.

We use ∇8 hyperviscosity for D. The hyperviscosity coefficient is chosen so that the dissipation at the truncation scale stays constant (ν = ν0/N8h, where ν0 = 2.62 × 104). The time step is chosen so that buoyancy oscillations are explicitly resolved for all values of N: the time step is at least 200 times smaller than the buoyancy time scale. Here Nh is chosen so that the buoyancy scale (see section 2d) is resolved.

b. Procedure

The procedure is a variant of that employed in P1 and P2. For each (N, f ) pair, a basic state is generated from random initial conditions by setting the amplitude of the wave modes to zero (cf. Errico 1984). See appendix A for details. At t = 0, the geostrophic basic state is perturbed with a random, small-amplitude perturbation. The total energy of the basic state is identical for all runs, irrespective of N and f. The kinetic energy of the perturbation is a specified fraction of the basic state’s. Figure 1 summarizes the procedure.

The base-flow energy is chosen so that the (initial) eddy turnover time, defined by the rms vertical vorticity, is O(1). Likewise, the (initial) velocity scale, defined by the rms horizontal velocity, satisfies U ∼ 0.6.1 For the runs described herein, f = 1.

The Rossby and (vertical) Froude numbers are defined as Ro := U/fL and Fr := U/NH, respectively. Some freedom exists in the choice of H and L: a small-scale, “micro” (or local) definition may be adopted based on the vorticity; alternatively, the large-scale, “macro” (or global) definitions, LL0, HH0, may be used. Neither is appropriate in all cases: whereas the former is suited to the growth of small-scale perturbations, the latter best describes the effect of small-scale perturbations on the large-scale base flow (the geostrophic energy spectra are fairly steep; initially k−4 or steeper). Given the concerns of this paper, and its widespread use in atmospheric science, the macrodefinitions
i1520-0469-65-3-766-e2
seem natural. This yields a Rossby number of O(0.1), which is representative of synoptic-scale, tropospheric flow, though it may be an overestimate.

Small errors are incurred in approximating the Rossby and Froude numbers with nominal, time-independent values based on H0 and L0. In our simulations, the Rossby and Froude numbers typically vary over an order of magnitude. It is also possible to estimate H and L with the first moments, H1 and L1, of the geostrophic energy spectrum. In an isotropic domain, L1 agrees with L0 to within a factor of 2.

c. Normal modes

The time-dependent base flow and perturbation, that is, the nominal balanced and unbalanced modes, are defined using the linear normal modes of the nonhydrostatic Boussinesq equations (Bartello 1995):
i1520-0469-65-3-766-e3
where the scaled variables ζk = i(kxυkkyuk), Dk = k/kz(ikxuk + ikyυk), and Tk = khbk/N. The modal frequency σk = f2k2z + N2k2h/k, and it is assumed that kh, kz ≠ 0.2 Here A(0)k denotes vortical modes, while Ak(±) denotes gravity waves. We shall refer to the former as “geostrophic” and the latter as “ageostrophic”; in the initial spinup Ak(±) = 0 (see appendix A). The ageostrophic and geostrophic energies are given by
i1520-0469-65-3-766-e4
where the asterisk denotes complex conjugation.
Some of the calculations require geostrophic or ageostrophic velocities. They are obtained by projection. For example, in the former case,
i1520-0469-65-3-766-e5
from which the geostrophic velocities (ug, υg) follow; wg = 0.

Note that while the normal modes are exact solutions only for linear dynamics, that is, for Ro, Fr → 0, the [A(0)k, A(+)k, A(−)k] basis can be used generally, for it is complete. The normal-mode basis has been useful in the study of unbalanced turbulence, where Ro and Fr might not be small (e.g., Bartello et al. 1996; Bartello 1995).

At finite Ro and Fr, the geostrophic solution3 does not remain an exact solution of the Boussinesq equations. Departures from exact quasigeostrophy enable ageostrophic perturbations to grow. For the Rossby and Froude numbers described herein, the evolution is insensitive to the structure of the (external) perturbation: runs with small temperature perturbations are nearly indistinguishable from runs with small vorticity perturbations, or even with no perturbation at all.

Although the normal-mode basis is formally valid for all Ro and Fr, the use of linear normal modes is debatable for larger values. In this case geostrophic–ageostrophic interactions might not be representative of balanced–unbalanced interactions. This is addressed in section 3.

Using the normal-mode basis, the velocity field is decomposed into a geostrophic base flow, vg, and an ageostrophic perturbation, va. The full velocity v := vg + va. Diagnostics will be defined as needed.

d. Key length scales

In analyzing the results, reference will be made to two important length scales: the buoyancy scale and the deformation radius.

At the buoyancy scale, Hb, the vertical advective time scale matches the buoyancy time scale, N−1. With the definition
i1520-0469-65-3-766-e6
Fr > 1 for H < Hb; in other words, the stratification is relatively weak. Recently it has been shown that Hb does indeed define an overturning scale for stably stratified turbulence (Waite and Bartello 2006). If
i1520-0469-65-3-766-e7
we say there is weak stratification.

Other definitions exist. For example, it is possible to use the characteristic vertical velocity, W (Hopfinger 1987), though the previous definition makes sense when characterizing the transition from quasi-horizontal motion to motion with strong buoyancy fluctuations. In phenomenological theories of stratified turbulence (e.g., Lumley 1964), much attention has been devoted to the so-called Osmidov scale, Hos := (ϵ/N3)1/2, where ϵ is the turbulent dissipation rate.

The deformation radius, Ld, characterizes the importance of rotation. It may be defined as
i1520-0469-65-3-766-e8
where H0 is used for the vertical length scale (Gill 1982). For horizontal length scales that are large compared to Ld, that is, Ld/L < 1, rotation dominates, and, according to the Taylor–Proudman theorem, there is quasi-2D motion for Ro ≪ 1. The flow is nearly barotropic, but baroclinic effects are retained. If
i1520-0469-65-3-766-e9
we say there is synoptic flow. This condition is equivalent to εaN/f < 1 or Ro/Fr < 1. In the “synoptic limit” Ld/L0 → 0.

The structure of the base flow changes with Ld/L0. As Ld/L0 is increased (at fixed, small Ro) there is a transition from “tubes” to “pancakes,” in accord with rotating stratified turbulence phenomenology (e.g., Riley and Lelong 2000). This can be seen in Fig. 2, which plots vertical vorticity isosurfaces at t ∼ 0.4 For synoptic flow and Ld/L0 < 1 (Figs. 2a,b), there are coherent vortex tubes; for subsynoptic flow and Ld/L0 > 1 (Fig. 2d), there are thin “pancakes.” For Ld/L0 = 1 (Fig. 2c), vortex tubes are present but less coherent than before.

Intuitively, the numerical results are expected to be analogous to those in the unstratified case (cf. P2) if there is weak stratification and quasi-2D motion. Connections with three-dimensionalization and the time-dependent, hyperbolic instability are discussed in appendix B. In sections 3 and 4 we investigate the effects of varying the buoyancy scale and the deformation radius: we progress from a regime in which both (7) and (9) are satisfied, through to one in which neither holds.

3. Weak stratification, synoptic flow (N/f < 1)

In this section we examine weak stratification, H0Hb, and synoptic flow, Ld/L0 < 1. The base flow is dominated by “tubes” (cf. Figs. 2a,b). Table 1 lists relevant parameters and length scales for the different runs; resolution checks have been excluded for brevity.

Although N/f ∼ 100 in the atmosphere and ocean, N/f < 1 in this section. This allows the qualitative arguments of section 2d to be examined in a regime where the unstratified results should extend. Weak stratification might, however, be relevant to the atmospheric boundary layer or the oceanic mixed layer.

a. Ageostrophic energy

We begin by analyzing the growth of Eageo, (4). Figure 3 plots Eageo against time for Fr ≥ 1. In all cases roughly exponential growth is established before Eageo saturates and decays for t ≳ 10. This is analogous to P1, in which there is exponential growth of the 3D perturbation.

Although we are more interested in the long-time interactions between geostrophic and ageostrophic modes than in the short-time growth of the ageostrophic energy, it is instructive to pursue the analogy. In P1, the growth rate of the 3D energy, E3D, was compared to the linear growth estimates, Γ and Γ0 (see appendix B for definitions). More specifically, the actual growth rate, γ, agrees well with the domain average of Γ0, , where the overbar denotes a time average and the angle brackets denote a domain average. (The agreement is not exact, but given the averaging and the turbulent nature of the flow, agreement within 50% is surprisingly good; see P1, section IV.B for discussion.)

In the present case, Eageo would grow at a rate 2 if the linear growth rate estimates applied. In Fig. 3b, exp(2t) is plotted for the three largest stratifications; for clarity Fr = 1000 is omitted and each curve is shifted so as to be adjacent to the corresponding Eageo curve. There is very good agreement up to Fr = 1: 2 ∼ 1.2 and γ ∼ 1.4 for Fr = 100. Here 2 underpredicts the actual growth for Fr = 1, the strongest stratification (though γ remains bracketed by 2 and 2, as in the unstratified case). Evidently the perturbation growth may differ for stronger stratifications; this is discussed in section 4a(1). Recall that Fr = 1 is a borderline case, lying between weak and strong stratification, and that dHb/dt ≤ 0 for decaying flow, meaning that not all modes remain within the “weakly stratified regime.”

For t ∈ [1, 5], in which there is exponential growth of the perturbation, γ is approximately independent of Fr. Instability mechanisms that are not strain dominated may show a stronger dependence on the stratification. For example, the so-called ageostrophic, anticyclonic instability shuts off for weak stratification (McWilliams et al. 2004; Molemaker et al. 2005).

On very short time scales, Eageo grows much faster than for Fr = 10 and Fr = 1. Note the change in slope at t ∼ 1. This rapid adjustment is likely related to the fact that w = 0 for geostrophic flow [cf. (5)]. Figure 4 compares time series of Eageo and w2rms, with wrms being the rms vertical velocity. Like Eageo, w2rms undergoes a rapid adjustment for t ≲ 1. A similar adjustment may also be seen in Bartello (1995), where the adjustment arises on account of initial conditions with b = 0 rather than w = 0. Since the adjustment occurs on the fast, inertial time scale, Ti ∼ 1, rather than the slow, advective time scale, Tadv ∼ 10, it cannot be attributed to spontaneous imbalance and the time-dependent, hyperbolic instability.

The initial adjustment can be mitigated with a more judicious choice of basic state, for example, a higher-order balance with w ≠ 0. This is demonstrated explicitly in section 4a(1).

The ageostrophic energy Eageo is a coarse-grained diagnostic. To characterize the spectral signature of the growth another diagnostic is needed. We compute the spectral growth rate, σEa. Following P1,
i1520-0469-65-3-766-e10
where t0 is a reference time. The results are sensitive to t and t0. We choose t0 = 0.8, which roughly corresponds to the onset of exponential scaling for Fr = 1 and Fr = 10. Here t is chosen to lie inside the exponential regime.

Figure 5 shows σEa snapshots taken after the onset of exponential growth but before the saturation of the ageostrophic perturbation. The key point is that σEa is maximized near kh = 1 and decreases below the line kh = kz, just as in the three-dimensionalization of decaying 2D turbulence (cf. P1, Fig. 11): the growth is preferentially determined by small-aspect-ratio modes with small kh and large |kz|. This may be clearly seen for Fr = 10 (Fig. 5a). Maximization of the growth rate near kh = 0 has also been observed in stratified mixing layers (Caulfield and Peltier 2000). At later times the structure becomes diffuse, but growth for kz > kh continues to be favored. For Fr = 1 (Fig. 5b) the largest growth rates remain localized around kh = 1.

In both panels there are large growth rates in the vicinity of the vertical truncation scale. This arises from the initial decay of the kinetic energy at large kz. Growth is initiated near kh = 1, where there is weak horizontal dissipation, before spreading to larger kh (not shown). See section 4b(1).

b. Transfer spectra

1) Definitions

We now consider geostrophic–ageostrophic energy transfers. The transfer functions are given by
i1520-0469-65-3-766-eq1
where the former is associated with the advection of velocity and the latter with advection of buoyancy. Here ℜ denotes the real part. Note that the sum is taken over a cylinder in spectral space (i.e., there is an implicit sum over kz).
It is convenient to reexpress these transfer functions in terms of the normal-mode variables. This is done by projecting the state variables (ω, b) and the nonlinear terms (Nυ, Nb) onto the normal-mode variables. Since the projection is linear and the basis orthonormal and complete, the total transfer is unchanged. Defining T(kh) := Tυ(kh) + Tb(kh),
i1520-0469-65-3-766-e11
where [NA(0)k, NA(+)k, NA(−)k] denotes the projected nonlinear terms. The geostrophic and ageostrophic contributions to the transfer are given by
i1520-0469-65-3-766-eq2
The nonlinear terms, which represent convolutions, may be further decomposed. Clearly,
i1520-0469-65-3-766-eq3
where NGG is the purely geostrophic convolution, NAA the purely ageostrophic convolution, and NAG the mixed nonlinear term. The hat denotes a projection onto geostrophic modes, and NGG and NAA can be obtained by filtering v and b. Substituting
i1520-0469-65-3-766-eq4
and
i1520-0469-65-3-766-eq5
with the cross term
i1520-0469-65-3-766-e12
Here TGG−G is associated with the GGG triads, TAA−G with the AAG triads, and TGA−G with the GAG triads; TAA−G and TGA−G cannot be represented with quasigeostrophic processes. There are analogous expressions for TGG−G, TGA−A, and TAA−A.

By contrast with section 3a, we focus on the long-time dynamics, which should be independent of the initial conditions. The results are temporally averaged over the interval 20 ≤ t ≤ 60. Similar results have been obtained with other time intervals (e.g., [10, 20] or [50, 60]).

2) Spectral structure

Figure 6a shows TGA−G, TAA−G, and TGG−G for Fr = 100. Since log-linear axes are used, the kh = 0 transfer cannot be shown; however, it is at least several orders of magnitude smaller than the kh = 1 transfer. To maintain area preservation the transfer functions are scaled by kh.

The expression TGG−G, which describes quasigeostrophic dynamics, is positive at kh = 1 and approximately zero for large kh; this is consistent with an inverse energy cascade. Detailed balance is satisfied: Σk TGG−G ≈ 0.5 At early times, before the ageostrophic perturbation has experienced much growth, TGTGG−G (not shown). At the late times corresponding to Fig. 6, the nonlinear transfer by the GAG and AAG triads is larger at small kh.

In resonant interaction theory, the GAG triads are solely responsible for geostrophic–ageostrophic transfers (Bartello 1995). For Fr → 0 geostrophic modes catalyze transfer between ageostrophic modes in AAG triads. But, in the present case, Fr is large and the AAG triads can make an important contribution to the geostrophic–ageostrophic transfer. Thus, the net transfer is given by
i1520-0469-65-3-766-e13
In the unstratified case, by contrast, transfer between the 2D and 3D modes can be effected only by 3D–3D–2D triads, namely by
i1520-0469-65-3-766-e14
where U is the 2D component, u the 3D component, and the hat denotes the 2D projection.6 Assuming vgU and vau, the structure of TAA−G should then resemble that of T3D3D−2D.

Figure 6a confirms this for weak stratification: TAA−G has a strong negative peak at small kh. Just as 3D perturbations represent a strong damping on 2D decaying turbulence, ageostrophic perturbations represent a damping on geostrophic turbulence: there is extraction of base-flow energy at small kh. Note that this effective damping should not be confused with explicit dissipation.

As expected, TGA−G is nonzero: the transfer between the base flow and the perturbation cannot be ascribed solely to AAG triads. Nevertheless, TGA−G is smaller in magnitude than TAA−G. This is reassuring inasmuch as, strictly speaking, Fr ≪ 1 in order for the normal-mode basis to be applicable.

The analogy between geostrophic motion and 2D flow, on the one hand, and ageostrophic motion and 3D flow, on the other, is reinforced by Fig. 6b, which plots T3D3D−2D and Tnet. There is good agreement between them for Fr = 100. The deviations between the curves are larger, at a given value of kh, for smaller Fr (not shown). Note that ΣkhTnet < 0.

Extraction of energy continues after the saturation of the ageostrophic perturbation. Even though the perturbation energy decays for t ≳ 10 (cf. Fig. 3), energy transfers between geostrophic and ageostrophic modes persist. This is consistent with the 2D–3D transfers described in P2. The maximum amplitude of Tnet exceeds that of the quasigeostrophic GGG transfer.

Time averaging obscures fluctuations on a given time slice. For example, Tnet can be positive at small kh. While such contributions are important in subgrid-scale modeling (Mason 1994; Fröhlich and Rodi 2002), they lie beyond the scope of this study.

3) 3) Fr dependence

Figure 7a shows Tnet for the runs analyzed in section 3a. The structure remains more or less unchanged, though the magnitude of the transfer (i.e., extraction) is noticeably smaller for Fr = 1. Recall that Fr = 1 lies at the boundary between weak and strong stratification. Here Tnet resembles TG, which we omit for brevity.

Figure 7b, which shows TA, indicates that energy extracted from the geostrophic flow by the ageostrophic perturbation cascades toward small scales. This is analogous to the forward cascade of 3D energy in P2. The magnitude of the ageostrophic transfer increases for weaker stratification. The absence of a peak at low kh implies that the injection of geostrophic energy is balanced by the loss of energy toward small scales.

The extraction at small kh is not an artifact of the initial conditions. Although the basic state is dominated by tubes, only a single (cyclonic) tube survives in the geostrophic base flow at later times; however, it becomes small and amorphous for stronger stratification. Moreover, the extraction occurs over a range of horizontal scales (Tnet < 0 for kh ≲ 10).

4) Robustness

In these runs the global Rossby number is O(0.1); a smaller Rossby number has been obtained from a less-energetic basic state. With Ro = 0.01, the transfer spectra closely resemble those depicted in Fig. 7: the extraction of energy from the large-scale base flow is robust. Because the horizontal momentum equation is dominated by inertial oscillations for Ro ≪ 1, |Tnet|/|TGG−G| is much smaller. For the weakest stratifications, the maximum amplitude of the ageostrophic perturbation growth is several orders of magnitude smaller for Ro = 0.01.

A single realization is used for each Fr in Fig. 7. We may therefore question the robustness of the results; however, ensemble calculations (not shown) indicate that statistical variations are small.

The results are not strongly dependent on resolution. The inertial range expands for Nh = Nυ = 180, but the structure of the transfer remains the same.

The comparisons between different Fr can also be made using a dimensionless, integral time. For example,
i1520-0469-65-3-766-e15
In fact, integral times do not differ significantly among the different Fr. The results shown in Fig. 7 are qualitatively unchanged when transfer spectra are averaged over a long time interval defined by τ (not shown).

4. Strong stratification, synoptic and subsynoptic flow (N/f ≥ 1)

In section 3 there is weak stratification and N/f < 1. This is a sufficient condition for synoptic flow in an isotropic domain; that is, εaN/f < 1. With strong stratification, however, synoptic motion is not guaranteed. Here Ld/L0 < 1 only if εa is sufficiently small.

a. Anisotropic domain, synoptic flow

Anisotropic, strongly stratified flow represents the natural extension of weak stratification (section 3): for εa ≪ 1, the geostrophic base flow has a tubelike structure because Ld/L0 < 1 (cf. Figs. 2a,b). We fix N and f and vary εa; equivalently, Ro is fixed and Fr varied. See Table 2. We choose N/f = 10 because it is less computationally demanding than the more realistic N/f = 100.

1) Ageostrophic energy

Time series of Eageo for Ld/L0 ∈ [0.21, 3.33] are plotted in Fig. 8. There is rapid adjustment followed by saturation and slow decay. The flow stays nearly geostrophic: for large t, Eageo/Egeo ≲ 0.1 for Ld/L0 = 3.33 (the ratio is even smaller in the synoptic limit). The growth and saturation of the ageostrophic perturbation agrees with low-resolution, hydrostatic Boussinesq simulations (Errico 1984), and the notion that generation of inertia–gravity waves by balanced flow is ubiquitous and inescapable (cf. Warn 1986; Warn and Menard 1986). The “saturation level” decreases for smaller εa (or H0); this is analogous to the scaling estimate u ∼ εaU of P2.

Although the initial adjustment occurs on the O(1) inertial (or buoyancy) time scale, there is an important difference with respect to Fig. 3: it is not followed by a transition toward a regime with clean exponential scaling. The expanded view of the initial growth confirms this (Fig. 8b). However, there is a hint of exponential scaling for synoptic flow (e.g., Ld/L0 = 0.21).

In section 3a, we discussed the influence of the initialization on the rapid adjustment for t ≲ 1. The rapid adjustment may arise from the use of a basic state in which w = 0. As is well known, however, the vertical velocity is implicit in quasigeostrophic theory even though it does not appear explicitly in the potential vorticity equation. Thus for approximately quasigeostrophic flows, the (ageostrophic) vertical velocity may be estimated with the ω equation (Hoskins et al. 1978):
i1520-0469-65-3-766-e16
Initial conditions defined in this way correspond to a more accurate balance (cf. appendix A).

Figure 9a shows the effect of defining w(t = 0) with the ω equation. To facilitate comparison the vertical axis is identical to that of Fig. 8b. The rapid adjustment is largely suppressed, as may be expected from the discussion of section 3a; however, it is not suppressed completely. Indeed, there is quasi-exponential scaling.

Because Eageo(t = 0) is nonnegligible when using the ω equation, there are systematic differences when the time evolution is compared to Fig. 8a. These systematic differences do not vanish at large t: Fig. 9b shows this for Ld/L0 = 1.33. Crucially, however, the decay rates show much better agreement: memory of the initial conditions is lost and we may expect energy transfer rates to be comparable. There is similar behavior for other values of Ld/L0.

The ω equation could be used for all calculations. However, the same could be said about any higher-order balance, and moreover the differences are not very important on long time scales. The key point is that, in accord with present understanding of spontaneous imbalance (Vanneste and Yavneh 2004), growth of the unbalanced motion persists.

Even with the modified initial conditions, the growth of Eageo occurs more rapidly than predicted by . There are several possible explanations for this discrepancy. First, even with the ω equation the rapid adjustment is not entirely eliminated; consequently, the subsequent growth may be affected (i.e., masked). Second, strong stratification yields markedly nonstationary flow: for Ld/L0 = 0.33, increases by almost two orders of magnitude from t = 0.2 to t = 0.6. Averaging Γ0 in space and time cannot be justified as before. A final reason relates to the vertical resolution. From Table 2, Hbz ∼ 1 − 3, meaning that the buoyancy scale is only marginally resolved. This may be significant if, by analogy with the three-dimensionalization of decaying 2D turbulence, growth of the ageostrophic perturbation were influenced by the time-dependent hyperbolic instability. As explained in section b of appendix B, pressureless random straining occurs for kz > kb [cf. (A15)]; therefore adequate resolution of the buoyancy scale may be crucial. The influence of Hb is considered in section 4b.

A number of authors have described the growth of gravity waves in flows for which conventional mechanisms, like stratified shear instability, cannot be invoked. Charron and Brunet (1999) speculated that resonant triad interactions could be responsible for the growth of a gravity wave mode diagnosed from the SKYHI general circulation model; in their numerical simulations of a baroclinic life cycle, O’Sullivan and Dunkerton (1995) observed the generation of inertia–gravity waves via processes engendered by geostrophic adjustment. Geostrophic adjustment, that is, removal of wave energy, seems distinct from spontaneous imbalance, that is, a mechanism for the growth of the wave energy; however, generation of the ageostrophic perturbation may continue while there is a forward cascade of ageostrophic energy [cf. section 4a(2)]. Below it will be shown that the AAG triad, which is responsible for nonlinear geostrophic adjustment in the Ro, Fr → 0 limit, plays an important role in the extraction of large-scale geostrophic energy.

2) Transfer spectra

The transfer spectra, Fig. 10, generally resemble those for weak stratification (section 3b). There is extraction of geostrophic energy and a forward cascade of ageostrophic energy. With respect to the former, the (negative) low-kh peak in Tnet persists and is separated from the dissipation range by an inertial range (Fig. 10a); with respect to the latter, TA increases sharply toward small scales, though the baseline is not as flat as before (Fig. 10b). The purely geostrophic transfer, TGG−G, is independent of Ld/L0 for Ld/L0 > 0.21 (not shown). By contrast with the results for weak stratification (Fig. 6), GAG triads make a greater contribution to Tnet. There is qualitatively similar behavior when the average is defined using the integral time, τ (not shown).

The ratio of the Tnet peak to the TGG−G peak remains O(1) or greater. Given that Ro and Fr are small and EgeoEageo, this may seem counterintuitive. As emphasized, however, by Errico (1982), correlations between the geostrophic and ageostrophic motion may render products involving geostrophic and ageostrophic variables comparable to products involving the geostrophic motion alone, at least in a time-averaged sense.

The influence of Ld cannot be easily discerned from Fig. 10. The trend with respect to Ld/L0 is quantified in Fig. 11. The net geostrophic damping,
i1520-0469-65-3-766-e17
decreases with Ld/L0, but only down to Ld/L0 ∼ 0.5 (Fig. 11a). This is consistent with the maximization of the transfer around Ld/L0 = 0.67 (cf. Fig. 10a). The trend is nonmonotonic: Dg increases for Ld/L0 ≲ 0.5.

This behavior is the product of two competing effects. One has been described already, the well-known decoupling between geostrophic and ageostrophic modes for strongly stratified flows (Lilly 1983). The other pertains to synoptic flow. As Ld/L0 decreases, so does the aspect ratio of the domain; but from P2, the damping should weaken in a thinner domain, for it is more two-dimensional. Putting these trends together, there is a critical value, Ld/L0 ∼ 1, at which Dg is maximized; Dg reflects the transition from subsynoptic to synoptic flow.

This is confirmed by Fig. 11b, which plots Egeo(kz = 0)/Egeo versus Ld/L0. This diagnostic characterizes the two-dimensionality of the geostrophic basic flow. There is a kink around Ld/L0 = 3. The two diagnostics in Fig. 11 yield slightly different estimates of the transition from subsynoptic to synoptic flow.

This transition cannot be attributed to the thinness of the domain. Even for our thinnest domain, H0 > Hb (cf. Table 2). The buoyancy scale is comparable to the height of the computational box for
i1520-0469-65-3-766-eq6
implying the crossover occurs at εc ∼ 0.01. From Table 2, εa > 0.03.

Previous results on the noninteraction between gravity waves and balanced flow (e.g., Dewar and Killworth 1995) may appear to be at odds with Figs. 10 and 11. However, there is no contradiction, for those results were obtained with discrete, reduced-gravity layer models. Moreover, the net energy transfer from geostrophic to ageostrophic modes is quite weak: at Ld/L0 ∼ 1, the net damping induced by Dg over t ∈ [20, 60] is about 10% of Egeo.

These results vindicate the choice HH0, LL0 used to define Ro and Fr. A “micro-Froude” number defined by the first moment of the vertical energy spectrum would not capture this behavior, for it is approximately independent of εa (not shown). Since the large-scale structure of the flow obviously plays an important role, Ro and Fr should be defined accordingly.

We note also that, while the results described above were obtained with a geostrophic basic state, they appear insensitive to the initial vertical velocity. Calculations with the ω equation yield similar long-time statistics; that is, Fig. 10 is qualitatively unchanged, though the magnitude of the transfer is considerably smaller in the synoptic limit (not shown). The νeddy magnitudes (see Fig. 13 below) show much less sensitivity to the basic state.

3) Eddy viscosity

In subgrid-scale modeling the eddy viscosity models the effect of unresolved scales on resolved scales (e.g., Domaradzki et al. 1987, 1993). Calculation of the eddy viscosity is a standard problem in turbulence modeling. For an application to the Boussinesq equations see Bartello et al. (1996); for a discussion of the GCM problem see Koshyk and Boer (1995). Here we consider a slightly modified procedure in which ageostrophic modes rather than small-scale ones are filtered (see P2 for background). In this way the effects of ageostrophic motion (i.e., gravity waves) can be modeled. This procedure can also be applied to other definitions of balance, provided they can be formulated unambiguously.

We recall the procedure for unstratified, nonrotating flow. Letting U and u denote, respectively, the 2D base flow and 3D perturbation, we obtain a closed set of nonlinear equations:
i1520-0469-65-3-766-e18
where the hat denotes a projection onto the 2D base flow and the prime a projection onto the perturbation; D/Dt = ∂/∂t + U · . These equations amount to a Reynolds stress decomposition. In spectral space the energy equation for the base flow is
i1520-0469-65-3-766-e19
where the base-flow energy Ebase(kh) = ½Σ|k|=khUlU*l(k).
From this equation an eddy viscosity can be derived. The eddy viscosity represents perturbation terms in the base-flow momentum equation as νeddy2hU. Thus
i1520-0469-65-3-766-eq7
where NiU is the ith perturbation term in (19). Defining NiU with the perturbation advection terms,
i1520-0469-65-3-766-e20
[Note that there is a misprint in the corresponding equation of P2, Eq. (14).] Obviously νeddy recasts information that is already present in the transfer spectra; however, it has an attractive physical interpretation and can be more conveniently employed in a parameterization. Furthermore νeddy is normalized with respect to U. This ensures that the eddy viscosity reflects interactions between the base flow and the perturbation.
For rotating stratified flow there is a slight change of notation. Here we replace U and u with vg and va, respectively, yielding
i1520-0469-65-3-766-e21
where the hat now denotes a projection onto the geostrophic modes. This is the definition that will be used for all the calculations described below. Here, νeddy represents perturbation terms in the geostrophic momentum equation as νeddy2h vg. Both AAG and GAG triads contribute to νeddy, as indicated by the two terms in the numerator. Again, note that the terminology is somewhat misleading: νeddy need not represent the effect of eddies.

Figure 12 plots eddy viscosities for Ld/L0 = 3.33 and Ld/L0 = 0.21. We show the individual contributions from the GAG and AAG triads, as well as νeddy. The magnitudes of the GAG and AAG contributions are comparable for the subsynoptic flow (Ld/L0 = 3.33), but νeddy,AAG, which is mostly positive, dominates for synoptic flow (Ld/L0 = 0.21).

According to resonant interaction theory, only GAG triads effect geostrophic–ageostrophic transfer for Fr ≪ 1. The large AAG contributions in Fig. 12 may arise from a preference for interactions at small vertical scales, where Fr might not be small.

In section 4a(2), the extraction of geostrophic energy was described; here there is damping of large-scale geostrophic modes. The negative, low-kh peak in Tnet (Fig. 10) is now manifested as a positive, low-kh peak in νeddy. This is analogous to unstratified turbulence (P2, section VI) and weak stratification (omitted for brevity from section 3). The contributions from low kh dominate νeddy, even with strong stratification.

The importance of νeddy can be assessed by comparing it to the effective viscosity, νeff. Letting NiU = ν8U,
i1520-0469-65-3-766-e22
if the dominant contribution to the numerator comes from kz ∼ 0, νeffk6h.

In Fig. 12, the dissipation due to νeddy is several orders of magnitudes larger than hyperviscosity at small kh; that is, νeddyνeff. (It is, however, small in absolute terms: it occurs on a time scale that is about an order of magnitude greater than that associated with the decay of the total geostrophic energy, which is dominated by small-scale contributions). At small scales the roles of νeddy and νeff are reversed, while at intermediate scales |νeddy| ∼ νeff. For homogeneous turbulence the subgrid-scale eddy viscosity has a completely different structure: there is a “cusp” at the truncation scale (Kraichnan 1976; Chollet and Lesieur 1981): νeddy increases at small scales.

In P2 the large, low-kh peak is coupled with smaller values for high-kh. Here νeddy can be modeled as
i1520-0469-65-3-766-e23
where f (x) is a monotonically decreasing function and kcross is a crossover wavenumber. This is a good approximation for synoptic flow, though there is slight undershoot at large kh (Fig. 12b); that is, νeddy < 0. For subsynoptic flow, (23) does not apply as well: there is broadband structure and the baseline deviates more strongly from zero (Fig. 12a). Again, the effects of the ageostrophic perturbation on the geostrophic base flow do not average out.

To investigate the influence of Ld/L0, consider the eddy viscosities of Fig. 13. The peak magnitude of νeddy is maximized for Ld/L0 = 0.67 (Fig. 13a), in agreement with Fig. 11. The structure of νeddy also changes. This can be seen more clearly in Fig. 13b, which shows ν̂eddy, the eddy viscosity normalized by its peak value. For smaller Ld/L0 there is less undershoot, though the change is most pronounced for the smallest value, Ld/L0 = 0.21. In the synoptic limit geostrophic–ageostrophic interactions resemble 2D–3D interactions in unstratified turbulence (cf. P2, Fig. 15).

If the base flow is not 2D, we might expect νeddy to be qualitatively different. For Ld/L0 ≳ 1, a more broadband structure would ensue if geostrophic–ageostrophic interactions did not remain confined to small kh (or small εa). Indeed, growth via random straining (section b of appendix B) is favored in a thin domain because there are fewer small-kz modes that depart from quasi-2D motion and modify interactions with small-scale modes below Hb.

Figure 14 depicts the resolution dependence for the subsynoptic value, Ld/L0 = 3.33 (Fig. 14a), and the synoptic value, Ld/L0 = 0.33 (Fig. 14b). Because the grid is isotropic, the horizontal and vertical resolutions increase in step. There is significant undershoot for the coarsest resolutions—the low-kh peak disappears in Fig. 14a. This phenomenon is robust and has been observed for other parameters. The structure is qualitatively unchanged when the time averages are defined using the integral time, τ (not shown).

We speculate that the growth of the perturbation cannot be properly modeled unless Hb is adequately resolved; that is, ΔzHb. As the resolution increases, νeddy loses some of its broadband structure: for Ld/L0 = 3.33, the low-kh peak begins to emerge at Nh ≳ 150 (Fig. 14a), from which Hb is marginally resolved (cf. Table 2). Even for synoptic flow (Fig. 14b), convergence is difficult to ascertain in an anisotropic domain: the ensemble variation for Nh = 240 (not shown) is comparable to the variation between the curves. The inconclusive convergence may be a by-product of the limited number of vertical modes employed in these anisotropic simulations.

The sensitivity to the buoyancy scale may be attributed to geostrophic–ageostrophic interactions [i.e., the numerator in (21)] or the geostrophic energy spectrum [i.e., the denominator]. Waite and Bartello (2006) have shown that energy spectra exhibit a bumpy structure at large scales if kb is not adequately resolved. In section 4b(2) we explicitly examine the influence of contributions from large kz to geostrophic–ageostrophic interactions.

For strong stratification and synoptic flow, the dominant contribution to νeddy comes from the smallest kh, in agreement with (23). This supports the idea that νeddy may be amenable to parameterization.

b. Isotropic domain, subsynoptic flow

Strong stratification in an isotropic domain is a case without obvious geophysical relevance. Yet this case is worth considering anyway to elucidate the influence of synoptic flow. Table 3 lists parameters. For Ld/L0 > 1 the geostrophic base flow consists of pancakes rather than tubes (see Fig. 2d) and quasi two-dimensionality cannot be assumed.

1) Ageostrophic energy

The growth of Eageo (Fig. 15a) follows the same pattern established by anisotropic, subsynoptic flow. Rapid initial adjustment is followed by a brief transitional regime before the perturbation saturates. The Fr = 1 curve, taken from the results for weak stratification, is shown for comparison, to underscore the rapidity of the growth.

As before, the rapid adjustment may be partly attributed to the initial conditions. Using the ω equation (16), to define w(t = 0) greatly reduces the magnitude of the adjustment (by about five orders of magnitude; not shown), but the long-time behavior is qualitatively unchanged (Eageo differs by ∼10% for Fr = 0.1 and 0.01).

The accuracy of the linear growth estimates is poor. For Fr = 0.1, the growth rate is over 8 times larger than ; for Fr = 0.02 and Fr = 0.01, the agreement breaks down completely as ≈ 0 (more precisely, < 0).

Nevertheless, the random straining mechanism may still be relevant. We hypothesize that vertical modes below Hb, that is, kz > kb, may experience exponential growth. To investigate this we subsample the data:
i1520-0469-65-3-766-eq8
where ks denotes the vertical wavenumber above which ageostrophic modes are retained. Thus there is no filtering for ks = 0.

Figure 15b shows the effect of subsampling for Fr = 0.1. The maximum value of Eageo decreases as large vertical scales are filtered out, that is, as ks increases; here kb ∼ 4 if the long-time value U ∼ 0.3 is used. Furthermore, the rapid adjustment appears to be followed by quasi-exponential growth for t ≳ 1. The quasi-exponential scaling is cleanest for ks ≳ 20. Qualitatively, the change is greater than that yielded by the ω equation. Although the growth rate does not match 2, which is considerably smaller, a modified linear growth rate estimate could be obtained by taking the vertical structure of vg into account. Subsampling data generated from initial conditions defined by the ω equation yields qualitatively similar results (not shown).

In spectral space the growth rate tends to be maximized near the vertical truncation scale, irrespective of Fr. Here σEa (kz; t = 3, t0 = 0.39) increases toward large kz (Fig. 16a), as with weak stratification (cf. Fig. 5). By contrast, σEa (kh; t = 3, t0 = 0.39) does not always peak at the smallest horizontal scales: for Fr = 0.1 the growth rate is maximized at large horizontal scales, that is, small kh. The choice of t is arbitrary; similar trends are obtained with smaller t.

This behavior is consistent with the anisotropic results (cf. Fig. 5). In the synoptic limit, the structure of geostrophic–ageostrophic interactions resembles that for unstratified turbulence and the growth of the ageostrophic perturbation is favored at large kz and small kh. This agrees with the behavior of decaying 2D turbulence (cf. P2 and appendix Ba). This is also consistent with properties of inertia–gravity waves in the upper troposphere and lower stratosphere, which have large horizontal scales (∼1000–2000 km) and small vertical scales (∼1–4 km; cf. O’Sullivan and Dunkerton 1995 and references therein). The random straining mechanism could be relevant to the initial growth of the ageostrophic perturbation on scales kzkb, even for strong stratification. Nevertheless, the evidence is circumstantial: the (rough) exponential scaling in Fig. 15b does not extend over a wide range and the structure of σEa (kh, kz) is sensitive to t.

2) Transfer spectra and eddy viscosities

The anisotropic simulations of sections 4a(2) and 4a(3) suffer from a reduced number of vertical modes for εa ≪ 1. Although, in principle, anisotropic and isotropic simulations are equivalent—the same values of Ro and Fr can be examined with either configuration—it is useful to verify the robustness of the previous results. Figure 17 confirms that, for subsynoptic flow, there continues to be extraction of geostrophic energy at small kh (Fig. 17a), while the undershoot of νeddy persists (Fig. 17b). Again, defining the basic state with the ω equation does not alter these conclusions.

More interestingly, the resolution dependence of νeddy can be examined more carefully with an isotropic domain. Here νeddy appears to converge for marginally synoptic flow, Ld/L0 = 1 (Fig. 18a): the persistent undershoot in Fig. 14 could be an artifact of the limited number of vertical modes. By contrast, the convergence is slower for subsynoptic flow, Ld/L0 = 10 (Fig. 18b). The structure of νeddy is different in the two cases: whereas increasing the resolution primarily serves to extend and displace the baseline in the former case, the structure of νeddy changes in the latter. Although there is minimal undershoot when Hb is resolved, as in the synoptic case (Fig. 18a), it is unclear whether the undershoot will also disappear for subsynoptic flow when Hb is adequately resolved (cf. Fig. 18b and Table 3).

There are two explanations for the resolution dependence of νeddy. The geostrophic energy spectrum or the geostrophic–ageostrophic interactions could change qualitatively as the resolution increases; alternatively, small-scale vertical modes below kb should make a significant contribution to geostrophic–ageostrophic interactions if the random straining mechanism is relevant. To examine the contribution of small-scale modes with kz > kb, we subsample the data. Redefining the transfer spectra as
i1520-0469-65-3-766-eq9
where à denotes subsampled ageostrophic modes, that is, Ak(±) = 0 for kz > ks, we can examine the influence of the cutoff scale, ks.

Figure 19 shows νeddy (kh; ks) for Fr = 0.1 and Fr = 0.01. In the marginally synoptic case (Fig. 19a), νeddy (kh; ks) converges for ks ≳ 10. This agrees with the estimate kb ∼ 4 and explains the rapid convergence of νeddy with resolution (Fig. 18a). Because of the way we have chosen to define our numerical viscosity, higher resolution implies a smaller viscous scale and a wider weakly stratified range. In the subsynoptic case (Fig. 19b), convergence is not achieved: νeddy (kh; ks) increases for ks > kb ∼ 40. This explains why the structure of νeddy changes with resolution in Fig. 18b: the “rectification” of νeddy at small kh, that is, the decreased undershoot of the baseline at higher resolution, follows from the increase in νeddy (kh; ks). In this way agreement with (23) is improved.

These results indicate that small-scale modes with kz > kb may exert an important influence on geostrophic–ageostrophic interactions. They are also consistent with growth of the ageostrophic perturbation via random straining in the weakly stratified range (cf. Fig. 15b), which implies a damping on the large-scale base flow. If the growth of small-scale ageostrophic modes were responsible for dissipation of the synoptic base flow, we might expect νeddy (kh; ks) > 0 for ks > kb. This is not observed in Fig. 19 because of decay of the ageostrophic energy. At early times, νeddy (kh; ks) > 0 for kh ≲ 10 and the structure of νeddy (kh; ks) does not change significantly with ks; at late times, the perturbation may return energy to the base flow so that νeddy(kh; ks) < 0 for ks > kb.

5. Summary and discussion

This paper has examined the extent to which our understanding of the three-dimensionalization of decaying 2D turbulence can be used to analyze and interpret the growth and feedback of ageostrophic perturbations in rotating stratified turbulence. For weak stratification, the growth rate of the ageostrophic perturbation is well predicted by a linear estimate (Fig. 3), the ageostrophic perturbation extracts energy from the geostrophic base flow (Fig. 6), and the eddy viscosity, νeddy, assumes a “characteristic structure” in which there is a preference for interactions at large horizontal scales. For strong stratification, applicability is more subtle. The growth rate departs from the linear estimate (Fig. 15) and νeddy develops broadband structure for subsynoptic flow (Fig. 17b). But with a small aspect ratio, εa, there is synoptic-scale motion, NH0/fL0 < 1, and νeddy reverts to its previous form in the limit of high resolution (Fig. 14), that is, for wavenumbers kzkb, where kb is the buoyancy wavenumber.

These results suggest that the time-dependent, hyperbolic instability responsible for the three-dimensionalization of decaying 2D turbulence (P1, P2) may apply to rotating, stratified turbulence, at least in some important respects. The mechanism generalizes (appendix Bb) if (i) the buoyancy scale, Hb, is large compared to the vertical length scale (Hb > H0) and (ii) the deformation radius is small compared to the horizontal length scale (L0 > Ld). In addition to the evidence adduced above, this claim is also supported by the anisotropy of geostrophic–ageostrophic interactions. According to appendix Ba, the perturbation and base flow should interact most strongly at large horizontal scales (small kh) and small vertical scales (large kz). For weak stratification, spectral growth rates, σEa, confirm that the ageostrophic perturbation is maximized for small kh/kz (Fig. 5). For strong stratification, the σEa increase toward large kz (Fig. 16a); if there is synoptic flow, they also increase toward small kh (Fig. 16b).

We have considered parameters that are representative of large-scale tropospheric flow. The global Rossby number Ro = O(0.1) with global Froude numbers ranging from synoptic values (Fr ≤ 0.1, Ld/L0 < 1) to subsynoptic values (Fr > 0.1, Ld/L0 > 1). Although we have tried to interpret the results for weak and strong stratification in terms of the same mechanism, there are some important differences. Interactions between the large-scale geostrophic flow and the small-scale ageostrophic perturbation depend on Ld and Hb. With respect to Ld, there are large-scale buoyancy oscillations for subsynoptic flow, implying that the geostrophic base flow cannot be modeled with horizontal random strain; with respect to Hb, small-scale perturbation modes cannot be captured if it is not adequately resolved. The eddy viscosity curves show persistent “undershoot” at small kh, that is, νeddy < 0 (Figs. 14 and 18) for subsynoptic flow with marginal resolution of Hb. Calculations with subsampled data indicate that the “rectification” of the νeddy at higher resolution can indeed be partly ascribed to small-scale vertical modes with kz > kb (Fig. 19). Similarly, quasi-exponential growth of the ageostrophic energy, Eageo, is obtained if the data are subsampled to restrict attention to large kz (Fig. 15b). The idea that the small-scale flow can exert a strong feedback on the largest scales also appears in other contexts; this is the defining property of the alpha effect in magnetohydrodynamics (e.g., Gilbert 2003).

A subtle point concerns the balanced basic state. Using the geostrophic modes yields quasigeostrophic dynamics but a vanishing vertical velocity (the latter does not appear explicitly in quasigeostrophic theory). Consequently a rapid adjustment on inertial time scales ensues (Fig. 4). This adjustment can be mitigated by defining the basic state with the quasigeostrophic ω equation (Fig. 9a). Robustness tests indicate that long-time statistics are qualitatively insensitive to the ω equation (e.g., Fig. 9b). Of course, any definition of balanced flow may be questioned; nevertheless, the one adopted here is a reasonable starting point for tropospheric flow.

From a theoretical perspective, random straining by the geostrophic base flow may be a generic mechanism for the generation of imbalance via a nominally balanced flow (i.e., spontaneous imbalance). This mechanism could be important in the middle atmosphere, where there is horizontal random straining (e.g., Shepherd et al. 2000). In the geophysical literature, the hyperbolic instability, which underlies the pressureless growth mechanism, has received less attention than other instability mechanisms; however, an analysis of solvability conditions for the “balance equations” (McWilliams et al. 1998) supports the claim that it can lead to a loss of balance [cf. their Eq. (7) and our (A13)].

From a practical perspective, there are several implications. First, the effects of imbalance might not be difficult to parameterize. A νeddy that is strongly peaked at small kh is good news for parameterization efforts because it means that high resolution (in the horizontal) might not be required in a parameterized model. If the buoyancy and dissipation scales are widely separated, the feedback of the ageostrophic modes on the geostrophic flow simplifies.

These results are consistent with recent results on the parameterization of gravity wave drag in middle atmosphere models (McLandress and Scinocca 2005). It has been shown that the precise details of the nonlinear dissipation mechanism are secondary as long as the momentum balance is respected. Although the gravity wave–drag problem is much more complicated than our idealized one—we do not need to contend with the propagation of gravity waves, and the resulting deposition of (pseudo) momentum, from a source region to a dissipation region—the simplified interaction seems analogous.

Second, our results provide guidance about appropriate resolution for direct numerical simulation. While there are no special requirements for the horizontal resolution, the vertical resolution should be sufficient to resolve the buoyancy scale. This suggests an alternative view of synoptic-scale flow. Small vertical scales, with kz > kb, can have an important influence on geostrophic–ageostrophic interactions. If the vertical grid spacing is coarser than the buoyancy scale, vertical overturning might not be properly represented, and an inverse energy cascade could develop (Bartello 2000; Waite and Bartello 2006). In state-of-the-art numerical weather prediction models, the buoyancy scale is marginally resolved (ΔzHb ∼ 1 km). In climate models, however, resolution of the buoyancy scale could be an important issue.

Acknowledgments

We are grateful to an anonymous referee for many valuable criticisms and suggestions. Helpful comments were received from M.-P. Lelong, R. B. Scott, K. S. Smith, J. Sukhatme, and M. L. Waite. Financial support was provided by NSERC and the Canadian Foundation for Climate and Atmospheric Science. Computational resources were provided by CLUMEQ.

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APPENDIX A

GGG Triads and Quasigeostrophy

The basic states are generated by setting the amplitude of the wave modes to zero. At each time step, the dynamical fields (ω, bk) are reconstructed from [A(0)k, Ak(±) = 0]. In particular, (ζk, Dk, Tk) = (ζk, Dk, Tk)G.

A number of authors have shown that zeroing the wave modes yields quasigeostrophic dynamics. This has been done for the shallow-water equations by Salmon (1998) and for the hydrostatic Boussinesq equations by Leith (1980). Leith also shows that a single iteration of nonlinear normal-mode iteration yields quasigeostrophic [i.e., GGG flow plus the ω equation, (16)]. Recall that the vertical velocity has no influence on QG dynamics.

We now demonstrate this equivalence for nonhydrostatic, Boussinesq flow. We only consider modes satisfying (kz ≠ 0, kh ≠ 0). The GGG flow is obtained by truncating the Boussinesq equations to
i1520-0469-65-3-766-ea1
It was shown in Bartello (1995) that the linear quasigeostrophic potential vorticity, q, can be expressed as
i1520-0469-65-3-766-ea2
where F is Fourier transform and the inviscid, real space quasigeostrophic equation
i1520-0469-65-3-766-ea3
Recall some properties of the flow truncated to the zero modes. From Eq. (5a) in Bartello (1995)
i1520-0469-65-3-766-ea4
(to within a factor of N). The velocity is perfectly geostrophic, the flow is hydrostatic and therefore there is thermal wind balance. From thermal wind,
i1520-0469-65-3-766-ea5
since ∂p′/∂z = −′/ρo = b. Since we have zero horizontal divergence, we can introduce a vertically varying streamfunction such that uk = −ikyψk = ikyζk/2h and υk = −ikyψk = ikxζk/2h. Both components of the thermal wind relation imply that ifkzζk = −k2hbk, which is satisfied by the above expressions for ζk and bk.
Now, from the Boussinesq equations
i1520-0469-65-3-766-ea6
Recall that the linear term is zero for this mode. The stretching term is also zero, since w is zero for geostrophic modes. Also, the latter implies that u · = uh · h. That means we have
i1520-0469-65-3-766-ea7
The last term can be obtained by taking the scalar product of the thermal wind relation (A5) and ∂uh/∂z; that is,
i1520-0469-65-3-766-ea8
whence the quasigeostrophic potential vorticity Eq. (A3) is recovered.

APPENDIX B

Time-Dependent Hyperbolic Instability

Unstratified flow

In P1 we showed that decaying two-dimensional turbulence three-dimensionalizes through a time-dependent version of the hyperbolic instability (e.g., Klaassen and Peltier 1985; Caulfield and Kerswell 2000). Physically, the instability arises from straining of the 3D perturbation by the 2D base flow—horizontal vorticity, in the form of so-called rib vortices, is generated. Mathematically, the hyperbolic instability may be analyzed with a “pressureless” approximation (e.g., Leblanc and Cambon 1997). From the linearized perturbation equations for unstratified, nonrotating flow,
i1520-0469-65-3-766-eb1
where we have taken the constant density ρ0 = 1. The horizontal pressure gradient may be neglected if the horizontal scale of the perturbation is large compared to the vertical scale, that is, if εa < 1, εa := H/L = kh/kz being the ratio of vertical and horizontal scales, or alternatively, a spectral aspect ratio. More precisely,
i1520-0469-65-3-766-eb2
and εa → 0 corresponds to the hydrostatic limit, which suggests applicability to large-scale geophysical flows. On occasion we refer to “pressureless modes.”
In the hydrostatic limit the perturbation equation reduces to the compact form
i1520-0469-65-3-766-eb3
where D/Dt = ∂/∂t + U · . We emphasize that there is no inconsistency between “pressureless dynamics” and the existence of a vorticity equation, in which pressure terms do not appear explicitly. A similar analysis could be performed on the perturbation vorticity equation by taking εa → 0; however, on account of the vortex tilting term, this equation is not as convenient to analyze.
The perturbation Eq. (B3) is formally analogous to the equation for a material line element
i1520-0469-65-3-766-eb4
For slowly varying (i.e., frozen) strain, it is clear that there is exponential stretching. Following the pioneering work of Cocke (1969) on the stretching of material lines in statistically isotropic flows, Kraichnan proved that material lines stretch exponentially if the correlation time of the turbulent fluctuations is infinitesimally small compared to the eddy turnover time of the turbulence (Kraichnan 1974). As is intuitively clear (the minus sign should not make a difference), and as shown explicitly by Kraichnan, this analysis also applies to equations of the form (B3). Therefore uh—and hence the 3D perturbation, u—grows exponentially if there is a separation in time scales between the 2D and 3D motion. This is “pressureless growth.”
The analogy with material line stretching can be pursued further. Following work on the characterization of tracer gradients (Lapeyre et al. 2001; Klein et al. 2000), we can obtain estimates of the (linear) growth rate along particle trajectories (Straub 2003):
i1520-0469-65-3-766-eb5
where σ = [(UxVy)2 + (Vx + Uy)2]1/2 and ω = Vx − Uy denote the strain and vorticity of the base flow, and −ϕ̇ is the rotation rate of the strain axes in the Lagrangian frame. These estimates are spatially local: variations of the strain and vorticity along particle trajectories are neglected. Here Γ0 is the well-known Weiss criterion for the existence of coherent structures in 2D turbulence, while Γ is an extension that takes the rotation of the strain axes into account. With background rotation of angular velocity 2Ω or Coriolis parameter f,
i1520-0469-65-3-766-eb6
With strong rotation, Ro ≪ 1, growth via random straining is suppressed.

If > 0 the flow is strain dominated and characteristic solutions are hyperbolic, as in the steady hyperbolic instability. This condition also defines one of the limits to balance described by McWilliams et al. (1998).

Applicability to decaying 2D turbulence was verified numerically in P1 and P2. In particular, the growth of the perturbation is strongly determined by the (effectively) pressureless modes and the initial growth rate is bracketed by the linear estimates, Γ and Γ0. The analysis applies to flow in isotropic and anisotropic domains.

Nonlinear dynamics were studied in P2. We showed that the “return to isotropy” is extremely slow in thin domains. We also showed that the eddy viscosity is strongly peaked at low wavenumbers, which indicates that the perturbation–base flow interaction is confined to the largest (horizontal) scales, reflecting the relevance of the pressureless growth mechanism even after the saturation of the perturbation. Growth of the perturbation via random straining by the base flow represents a damping on large horizontal scales, that is, an energy sink or a positive eddy viscosity.

Extension to rotating stratified turbulence

Although the discussion in appendix Ba is specific to unstratified flow, the time-dependent hyperbolic instability could represent a route by which unbalanced motion is generated by the balanced flow. For the analysis to generalize it is necessary that there be (i) dominance of the production term, −va · hvg; and (ii) advection via random straining by a two-dimensional base flow. Here we relate the pressureless growth mechanism to the qualitative arguments of section 2d.

The validity of the pressureless approximation (B2) depends on the weakness of the stratification: the condition εa < 1 does not guarantee (effectively) pressureless dynamics. In other words, how weak must the stratification be in order for there to be pressureless dynamics at a given vertical scale, H? Stratified turbulence phenomenology suggests that buoyancy effects are relatively weak below the buoyancy scale (cf. section 2d). In spectral space, the buoyancy wavenumber kb := N/U (it is conventional to exclude the factor of 2π) and there are effectively pressureless dynamics for vertical wavenumbers satisfying
i1520-0469-65-3-766-eb7
(For brevity we generally omit the absolute value.) This defines the “weakly stratified range.”
With respect to the second condition, the basic state must be quasi two-dimensional if the problems are to be isomorphic. Departures from the previous, unstratified results may arise from baroclinicity in the base flow. For rapid rotation, there is quasi-2D flow on scales where rotation dominates, that is,
i1520-0469-65-3-766-eb8
for given H and L. There is some freedom in the definition of Ld: it is possible to choose a vertical scale corresponding to the perturbation or the base flow, that is, to different baroclinic Rossby radii (Gill 1982, section 7.5). Choosing LL0, HH0 suggests a connection with the first baroclinic deformation radius of a continuously stratified fluid. It yields the condition (9) and ensures that base flow is quasi 2D when Ld/L0 < 1.

It is important to keep in mind that (B8) is a sufficient condition: the ageostrophic perturbation could still grow if it were violated. However, without random straining by a quasi-2D flow, the geostrophic–ageostrophic interactions might differ qualitatively from those described in P1 and P2.

Although synoptic flow is one of the sufficient conditions for the previous results to be recovered, the instability occurs irrespective of Ld and the global Froude number. Thus it may complement asymptotic analyses of spontaneous imbalance that require small Fr (e.g., Ford et al. 2000).

Fig. 1.
Fig. 1.

Schematic representation of the procedure.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 2.
Fig. 2.

Vertical vorticity isosurfaces for the geostrophic modes at t ∼ 0. These fields correspond to the initial conditions of the runs analyzed in Figs. 3 and 15; Ro = 0.1. The isosurface values are defined by the rms vertical vorticity. For clarity, vertical slices have been extracted from the isosurfaces. The panels correspond to different values of the deformation radius, Ld: (a) Ld/L0 = 0.01, (b) Ld/L0 = 0.1, (c) Ld/L0 = 1, (d) Ld/L0 = 10.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 3.
Fig. 3.

Ageostrophic energy Eageo vs t for weak stratification (Ro = 0.1, Fr ≥ 1). The 2 curves are obtained by averaging data over the time interval t ∈ [1, 5]; for clarity only the Fr = 1, 10, and 100 curves are shown: (a) t ≤ 60, (b) t ≤ 8; Ld/L0 < 1 in all cases. Note the exponential scaling prior to saturation.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 4.
Fig. 4.

Illustration of the effect of the initial vertical velocity. Comparison between time series of the ageostrophic energy, Eageo and square of the rms vertical velocity, w2rms. Here, Fr = 10.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 5.
Fig. 5.

Spectral growth rates, σEa(kh, kz), corresponding to Fig. 3: t0 = 0.8 (a) Fr = 10, t ∼ 4; (b) Fr = 1, t ∼ 1.7. The axes range from kh = 1 to kh = Kh and kz = 1 to kz = Kz. Growth is maximized for the “effectively pressureless” modes, kz > kh.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 6.
Fig. 6.

Transfer spectra for Ro = 0.1 and Fr = 100: (a) components of the total geostrophic transfer, TG; (b) comparison of the net ageostrophic–geostrophic transfer, Tnet, with the net 3D–2D transfer, T3D3D–2D.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 7.
Fig. 7.

Transfer spectra corresponding to the runs of Fig. 3: (a) net geostrophic transfer, Tnet; (b) total ageostrophic transfer, TA.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 8.
Fig. 8.

(a) Ageostrophic energy Eageo vs t for strong stratification in an anisotropic domain. (b) An expanded view is shown. Note the rapid adjustment. Here N/f = 10.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 9.
Fig. 9.

Evolution of Eageo from a basic state defined with the ω equation (strong stratification, anisotropic domain). (a) Time series corresponding to Fig. 8b. The 2 curve for Ld/L0 = 0.33 is plotted for reference. (b) Comparison with the initial evolution defined using the regular procedure, that is, geostrophic modes with w = 0. Here, Ld/L0 = 1.33.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 10.
Fig. 10.

Transfer spectra corresponding to the runs of Fig. 8 (strong stratification, anisotropic domain). (a) Net geostrophic transfer, Tnet; (b) total ageostrophic transfer, TA.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 11.
Fig. 11.

The Ld/L0 dependence of the transfer in Fig. 10 (strong stratification, anisotropic domain). (a) Net geostrophic damping, Dg, vs Ld/L0. (b) Two-dimensionality of the geostrophic flow, Eg(kz = 0)/Eg, vs Ld/L0. The error bars are obtained from the standard error of a 4-member ensemble, each member corresponding to a different basic state.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 12.
Fig. 12.

Eddy viscosity components for strong stratification in an anisotropic domain. The AAG and GAG components are shown, as well as the total eddy viscosity, νeddy, and the effective viscosity, νeff. (a) Subsynoptic flow, Ld/L0 = 3.33; (b) synoptic flow, Ld/L0 = 0.21.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 13.
Fig. 13.

Eddy viscosities for strong stratification in an anisotropic domain. (a) Total eddy viscosity, νeddy; (b) normalized eddy viscosity, ν̂eddy = νeddy/νmax, where νmax is the peak value.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 14.
Fig. 14.

Resolution dependence of νeddy (strong stratification, anisotropic domain). (a) Subsynoptic flow, Ld/L0 = 3.33; (b) synoptic flow, Ld/L0 = 0.33.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 15.
Fig. 15.

Ageostrophic energy Eageo vs t for strong stratification, isotropic domain; (a) Fr ≤ 0.1. For comparison, the Fr = 1 curve is also shown. (b) Subsampled results for Fr = 0.1. Only modes satisfying kzks are retained.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 16.
Fig. 16.

Spectral growth rates, σEa, corresponding to Fig. 15, expressed in terms of kh and kz: (a) σEa(kz; t = 3); (b) σEa(kh; t = 3). Reference time t0 = 0.39.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 17.
Fig. 17.

Transfer spectra and eddy viscosities (strong stratification, isotropic domain). (a) Net geostrophic transfer, Tnet; (b) total eddy viscosity, νeddy.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 18.
Fig. 18.

Resolution dependence of νeddy (strong stratification, isotropic domain); (a) Fr = 0.1, (b) Fr = 0.01.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Fig. 19.
Fig. 19.

Subsampled νeddy (strong stratification, isotropic domain). Only modes satisfying kzks are used in the calculations; (a) Fr = 0.1, (b) Fr = 0.01.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2265.1

Table 1.

Parameters for weak stratification (section 3).

Table 1.
Table 2.

Parameters for strong stratification, anisotropic domain (section 4a).

Table 2.
Table 3.

Parameters for strong stratification, isotropic domain (section 4b).

Table 3.

1

For convenience we use the rms total velocity rather than the rms horizontal velocity. The latter depends weakly on Fr.

2

See Bartello (1995) for details on the treatment of the special cases kh, kz = 0.

3

Hereafter, to conform with conventional usage, we drop the qualifier.

4

In fact, we plot the geostrophic projection of the total vorticity. However, the difference is immaterial at t ∼ 0, because the ageostrophic component starts small. At later times the picture is more complicated, but broadly similar.

5

The sum is many orders of magnitude smaller than the peak, but detailed balance is not satisfied exactly on account of roundoff errors.

6

In P2 this transfer was labeled as T3D→2D.

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