1) Definitions

We now consider geostrophic–ageostrophic energy transfers. The transfer functions are given by

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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 k_z).

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 (N_υ, N_b) onto the normal-mode variables. Since the projection is linear and the basis orthonormal and complete, the total transfer is unchanged. Defining T(k_h) := T_υ(k_h) + T_b(k_h),

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where [N_A⁽⁰⁾k, N_A⁽⁺⁾k, N_A⁽⁻⁾k] denotes the projected nonlinear terms. The geostrophic and ageostrophic contributions to the transfer are given by

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The nonlinear terms, which represent convolutions, may be further decomposed. Clearly,

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where N_GG is the purely geostrophic convolution, N_AA the purely ageostrophic convolution, and N_AG the mixed nonlinear term. The hat denotes a projection onto geostrophic modes, and N_GG and N_AA can be obtained by filtering v and b. Substituting

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and

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with the cross term

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Here T_GG−G is associated with the GGG triads, T_AA−G with the AAG triads, and T_GA−G with the GAG triads; T_AA−G and T_GA−G cannot be represented with quasigeostrophic processes. There are analogous expressions for T_GG−G, T_GA−A, and T_AA−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 T_GA−G, T_AA−G, and T_GG−G for Fr = 100. Since log-linear axes are used, the k_h = 0 transfer cannot be shown; however, it is at least several orders of magnitude smaller than the k_h = 1 transfer. To maintain area preservation the transfer functions are scaled by k_h.

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

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

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In the unstratified case, by contrast, transfer between the 2D and 3D modes can be effected only by 3D–3D–2D triads, namely by

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where U is the 2D component, u the 3D component, and the hat denotes the 2D projection.⁶ Assuming v_g ∼ U and v_a ∼ u, the structure of T_AA−G should then resemble that of T_3D3D−2D.

Figure 6a confirms this for weak stratification: T_AA−G has a strong negative peak at small k_h. 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 k_h. Note that this effective damping should not be confused with explicit dissipation.

As expected, T_GA−G is nonzero: the transfer between the base flow and the perturbation cannot be ascribed solely to AAG triads. Nevertheless, T_GA−G is smaller in magnitude than T_AA−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 T_3D3D−2D and T_net. There is good agreement between them for Fr = 100. The deviations between the curves are larger, at a given value of k_h, for smaller Fr (not shown). Note that Σ_khT_net < 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 T_net exceeds that of the quasigeostrophic GGG transfer.

Time averaging obscures fluctuations on a given time slice. For example, T_net can be positive at small k_h. 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 T_net 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 T_net resembles T_G, which we omit for brevity.

Figure 7b, which shows T_A, 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 k_h implies that the injection of geostrophic energy is balanced by the loss of energy toward small scales.

The extraction at small k_h 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 (T_net < 0 for k_h ≲ 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, |T_net|/|T_GG−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 N_h = N_υ = 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,

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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).

1) Ageostrophic energy

Time series of E_ageo for L_d/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, E_ageo/E_geo ≲ 0.1 for L_d/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 ε_a (or H₀); this is analogous to the scaling estimate u ∼ ε_aU 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., L_d/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):

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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 E_ageo(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 L_d/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 L_d/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 E_ageo 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 L_d/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, H_b/Δz ∼ 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 k_z > k_b [cf. (A15)]; therefore adequate resolution of the buoyancy scale may be crucial. The influence of H_b 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-k_h peak in T_net persists and is separated from the dissipation range by an inertial range (Fig. 10a); with respect to the latter, T_A increases sharply toward small scales, though the baseline is not as flat as before (Fig. 10b). The purely geostrophic transfer, T_GG−G, is independent of L_d/L₀ for L_d/L₀ > 0.21 (not shown). By contrast with the results for weak stratification (Fig. 6), GAG triads make a greater contribution to T_net. There is qualitatively similar behavior when the average is defined using the integral time, τ (not shown).

The ratio of the T_net peak to the T_GG−G peak remains O(1) or greater. Given that Ro and Fr are small and E_geo ≫ E_ageo, 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 L_d cannot be easily discerned from Fig. 10. The trend with respect to L_d/L₀ is quantified in Fig. 11. The net geostrophic damping,

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decreases with L_d/L₀, but only down to L_d/L₀ ∼ 0.5 (Fig. 11a). This is consistent with the maximization of the transfer around L_d/L₀ = 0.67 (cf. Fig. 10a). The trend is nonmonotonic: D_g increases for L_d/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 L_d/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, L_d/L₀ ∼ 1, at which D_g is maximized; D_g reflects the transition from subsynoptic to synoptic flow.

This is confirmed by Fig. 11b, which plots E_geo(k_z = 0)/E_geo versus L_d/L₀. This diagnostic characterizes the two-dimensionality of the geostrophic basic flow. There is a kink around L_d/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 > H_b (cf. Table 2). The buoyancy scale is comparable to the height of the computational box for

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implying the crossover occurs at ε_c ∼ 0.01. From Table 2, ε_a > 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 L_d/L₀ ∼ 1, the net damping induced by D_g over t ∈ [20, 60] is about 10% of E_geo.

These results vindicate the choice H ∼ H₀, L ∼ L₀ 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 ε_a (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 ν_eddy 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:

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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

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where the base-flow energy E_base(k_h) = ½Σ_|k|=khU_lU*_l(k).

From this equation an eddy viscosity can be derived. The eddy viscosity represents perturbation terms in the base-flow momentum equation as ν_eddy∇²_hU. Thus

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where Nⁱ_U is the ith perturbation term in (19). Defining Nⁱ_U with the perturbation advection terms,

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[Note that there is a misprint in the corresponding equation of P2, Eq. (14).] Obviously ν_eddy 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 ν_eddy 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 v_g and v_a, respectively, yielding

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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, ν_eddy represents perturbation terms in the geostrophic momentum equation as ν_eddy∇²_h v_g. Both AAG and GAG triads contribute to ν_eddy, as indicated by the two terms in the numerator. Again, note that the terminology is somewhat misleading: ν_eddy need not represent the effect of eddies.

Figure 12 plots eddy viscosities for L_d/L₀ = 3.33 and L_d/L₀ = 0.21. We show the individual contributions from the GAG and AAG triads, as well as ν_eddy. The magnitudes of the GAG and AAG contributions are comparable for the subsynoptic flow (L_d/L₀ = 3.33), but ν_eddy,AAG, which is mostly positive, dominates for synoptic flow (L_d/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-k_h peak in T_net (Fig. 10) is now manifested as a positive, low-k_h peak in ν_eddy. This is analogous to unstratified turbulence (P2, section VI) and weak stratification (omitted for brevity from section 3). The contributions from low k_h dominate ν_eddy, even with strong stratification.

The importance of ν_eddy can be assessed by comparing it to the effective viscosity, ν_eff. Letting Nⁱ_U = ν∇⁸U,

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if the dominant contribution to the numerator comes from k_z ∼ 0, ν_eff ∼ k⁶_h.

In Fig. 12, the dissipation due to ν_eddy is several orders of magnitudes larger than hyperviscosity at small k_h; that is, ν_eddy ≫ ν_eff. (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 ν_eddy and ν_eff are reversed, while at intermediate scales |ν_eddy| ∼ ν_eff. 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): ν_eddy increases at small scales.

In P2 the large, low-k_h peak is coupled with smaller values for high-k_h. Here ν_eddy can be modeled as

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where f (x) is a monotonically decreasing function and k_cross is a crossover wavenumber. This is a good approximation for synoptic flow, though there is slight undershoot at large k_h (Fig. 12b); that is, ν_eddy < 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 L_d/L₀, consider the eddy viscosities of Fig. 13. The peak magnitude of ν_eddy is maximized for L_d/L₀ = 0.67 (Fig. 13a), in agreement with Fig. 11. The structure of ν_eddy also changes. This can be seen more clearly in Fig. 13b, which shows ν̂_eddy, the eddy viscosity normalized by its peak value. For smaller L_d/L₀ there is less undershoot, though the change is most pronounced for the smallest value, L_d/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 ν_eddy to be qualitatively different. For L_d/L₀ ≳ 1, a more broadband structure would ensue if geostrophic–ageostrophic interactions did not remain confined to small k_h (or small ε_a). Indeed, growth via random straining (section b of appendix B) is favored in a thin domain because there are fewer small-k_z modes that depart from quasi-2D motion and modify interactions with small-scale modes below H_b.

Figure 14 depicts the resolution dependence for the subsynoptic value, L_d/L₀ = 3.33 (Fig. 14a), and the synoptic value, L_d/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-k_h 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 H_b is adequately resolved; that is, Δz ≲ H_b. As the resolution increases, ν_eddy loses some of its broadband structure: for L_d/L₀ = 3.33, the low-k_h peak begins to emerge at N_h ≳ 150 (Fig. 14a), from which H_b 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 N_h = 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 k_b is not adequately resolved. In section 4b(2) we explicitly examine the influence of contributions from large k_z to geostrophic–ageostrophic interactions.

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

1) Ageostrophic energy

The growth of E_ageo (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 (E_ageo 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 H_b, that is, k_z > k_b, may experience exponential growth. To investigate this we subsample the data:

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where k_s denotes the vertical wavenumber above which ageostrophic modes are retained. Thus there is no filtering for k_s = 0.

Figure 15b shows the effect of subsampling for Fr = 0.1. The maximum value of E_ageo decreases as large vertical scales are filtered out, that is, as k_s increases; here k_b ∼ 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 k_s ≳ 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 v_g 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 σ_Ea (k_z; t = 3, t₀ = 0.39) increases toward large k_z (Fig. 16a), as with weak stratification (cf. Fig. 5). By contrast, σ_Ea (k_h; 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 k_h. 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 k_z and small k_h. 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 k_z ≳ k_b, 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 σ_Ea (k_h, k_z) 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 ε_a ≪ 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 k_h (Fig. 17a), while the undershoot of ν_eddy persists (Fig. 17b). Again, defining the basic state with the ω equation does not alter these conclusions.

More interestingly, the resolution dependence of ν_eddy can be examined more carefully with an isotropic domain. Here ν_eddy appears to converge for marginally synoptic flow, L_d/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, L_d/L₀ = 10 (Fig. 18b). The structure of ν_eddy 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 ν_eddy changes in the latter. Although there is minimal undershoot when H_b is resolved, as in the synoptic case (Fig. 18a), it is unclear whether the undershoot will also disappear for subsynoptic flow when H_b is adequately resolved (cf. Fig. 18b and Table 3).

There are two explanations for the resolution dependence of ν_eddy. The geostrophic energy spectrum or the geostrophic–ageostrophic interactions could change qualitatively as the resolution increases; alternatively, small-scale vertical modes below k_b 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 k_z > k_b, we subsample the data. Redefining the transfer spectra as

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where à denotes subsampled ageostrophic modes, that is, A_k^(±) = 0 for k_z > k_s, we can examine the influence of the cutoff scale, k_s.

Figure 19 shows ν_eddy (k_h; k_s) for Fr = 0.1 and Fr = 0.01. In the marginally synoptic case (Fig. 19a), ν_eddy (k_h; k_s) converges for k_s ≳ 10. This agrees with the estimate k_b ∼ 4 and explains the rapid convergence of ν_eddy 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: ν_eddy (k_h; k_s) increases for k_s > k_b ∼ 40. This explains why the structure of ν_eddy changes with resolution in Fig. 18b: the “rectification” of ν_eddy at small k_h, that is, the decreased undershoot of the baseline at higher resolution, follows from the increase in ν_eddy (k_h; k_s). In this way agreement with (23) is improved.

These results indicate that small-scale modes with k_z > k_b 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 ν_eddy (k_h; k_s) > 0 for k_s > k_b. This is not observed in Fig. 19 because of decay of the ageostrophic energy. At early times, ν_eddy (k_h; k_s) > 0 for k_h ≲ 10 and the structure of ν_eddy (k_h; k_s) does not change significantly with k_s; at late times, the perturbation may return energy to the base flow so that ν_eddy(k_h; k_s) < 0 for k_s > k_b.