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  • View in gallery

    Horizontal slices of nondimensional vertical vorticity, , at z = 0 and t = 15, for (left) QG and (right) BO simulations, with and without a tropopause.

  • View in gallery

    Instantaneous kinetic energy wavenumber spectra for U = 1 m s−1, z = 0, and t = 15 (as in the snapshots of Fig. 1). The subplot shows the vertical structure of these spectra in the shaded area, . This wavenumber range is used throughout the paper to compute spectral slopes.

  • View in gallery

    Slope of the kinetic energy spectrum as a function of the flow strength U. The slope is fitted between wavenumbers 10 and 70, the shaded region in Fig. 2, and subsequently averaged between t = 5 and 10.

  • View in gallery

    Time-averaged (t = 5–10) kinetic energy horizontal wavenumber spectra for (a) weak- and (b) intermediate-flow regimes near the tropopause, (c) weak and intermediate flows without a tropopause, and (d) strong flows, both with and without a tropopause. Red and blue curves are associated with the variable-N and constant-N cases, respectively. Note also that for each panel, the flow strength increases as the color gets darker. The shaded area corresponds to the wavenumber range over which spectral slopes were measured.

  • View in gallery

    Kinetic energy spectra at three altitudes: the tropopause (z = 0; red), the midtroposphere (z = −Dz/4; green), and the midstratosphere (z = Dz/4; purple).

  • View in gallery

    Horizontal slices of nondimensional vertical vorticity at z = 0 and t = 15 (as in Figs. 1 and 2). (top left)–(top right) Near-tropopause roll-ups rapidly vanish with increasing U. (bottom) By contrast, there is no significant qualitative change in the flow structure of the constant-N case. The scale is normalized by U for the sake of comparison.

  • View in gallery

    As in Fig. 6, but for stronger flows and at t = 7.5 (the middle of the averaging period).

  • View in gallery

    Horizontal slices of vertical velocity at z = 0 for the case U = 10 m s−1 with a tropopause. Qualitatively similar snapshots are obtained at other heights with or without a tropopause.

  • View in gallery

    Total energy as a function of time for various flow strengths (different colors). Energy is defined up to an arbitrary constant. Here, this constant was set to the QG potential energy, the volume-integrated b2/2N2. The shaded region shows the time interval over which spectra are averaged and slopes are computed.

  • View in gallery

    Frequency spectra at z = 0 for (low resolution: 2563) Boussinesq simulations at U = 5 m s−1, with (solid) and without (dashed) a tropopause. Each color represents a different set of horizontal wavenumbers. For clarity, cases kH = 8 and 2 are vertically offset by two and six decades, respectively. Note that frequency spectra are typically computed from real-space fields, which implicitly entails an averaging of frequency spectra over all wavenumbers. The typical frequency spectrum, then, is closely approximated by our k = 2, 3, or 4 spectra—the wavenumbers containing most of the energy—and the distinctive features of higher-wavenumber spectra are averaged out.

  • View in gallery

    Frequency spectra and time-lagged correlations in the case U = 20 m s−1, for a few horizontal wavenumbers (different colors) in low-resolution (2563) Boussinesq simulations. Only the variable-N case is shown, but the constant-N case is qualitatively similar. Correlations are computed for the fast (ω > f) component of velocity fields except for the supplementary dashed line, for which all frequencies above 0.5f were included to capture the bulk of the near-inertial peak apparent for wavenumber kH = 1. The vertical line stands for a quarter of the inertial period, within which maximum correlation is expected for inertia–gravity waves. Correlations are computed over the usual time-averaging period, t = 5–10.

  • View in gallery

    The ratios of advection to the other terms of the inviscid momentum equation for u. The ratios are averaged over the usual time window for low-resolution (2563) Boussinesq simulations. Triangles and dots denote wavenumbers at which fast-time-scale motion exhibits time-lagged correlations above and below 0.5, respectively.

  • View in gallery

    Decomposition of the total kinetic energy spectrum into its rotational and divergent (green and gold solid lines, respectively) and slow and fast components (green and gold dots and triangles, respectively) for low-resolution (2563) Boussinesq simulations (top) with (red) and (bottom) without (blue) a tropopause. Gold triangles and dots indicate time-lagged correlations above and below 0.5, respectively (cf. section 4c).

  • View in gallery

    Time-averaged kinetic energy spectra (red) decomposed into slow and fast (green and gold dots and triangles, respectively) and rotational and divergent (green and gold solid lines, respectively) components. The thin and thick solid lines are from low- and high-resolution runs, respectively. The fast and slow decompositions were only calculated for low-resolution runs. Also shown is a comparison between kinetic energy spectra (red) and buoyancy variance spectra normalized by (purple). These were shifted right by a decade for clarity.

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On Boussinesq Dynamics near the Tropopause

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  • 1 McGill University, Montreal, Quebec, Canada
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Abstract

The near-tropopause energy spectrum closely follows a −5/3 power law at mesoscales. Most theories addressing the mesoscale spectrum assume unbalanced dynamics but ignore the tropopause (near which the bulk of the data were collected). Conversely, it has also been proposed that the mesoscale spectrum results from tropopause-induced alterations of geostrophic turbulence. This paper seeks to reconcile these a priori mutually exclusive theories by presenting simulations that permit both unbalanced motion and tropopause-induced effects. The model integrates the nonhydrostatic Boussinesq equations in the presence of a rapidly varying background stratification profile (an idealized tropopause). Decaying turbulence simulations were performed over a wide range of Rossby numbers. In the limit of weak flow (U ≲ 1 m s−1), the essential features of the Boussinesq simulations are well captured by a quasigeostrophic version of the model: secondary roll-ups of filaments and shallow spectral slopes are observed near the tropopause but not elsewhere. However, these tropopause-induced effects rapidly disappear with increasing flow strength. For flow strengths more typical of the tropopause (U ~ 10 m s−1), the spectrum develops a shallow, near −5/3 tail associated with fast-time-scale, unbalanced motion. In contrast to weak flows, this spectral shallowing is evident at any altitude and regardless of the presence of a tropopause. Diagnostics of the fast component of motion reveal significant inertia–gravity wave activity at large horizontal scales (where the balanced flow dominates). However, no evidence points to such activity in the shallow range. That is, the mesoscale of the model is dominated by unbalanced turbulence, not waves. Implications and limitations of these findings are discussed.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Olivier Asselin, olivier.asselin@mail.mcgill.ca

Abstract

The near-tropopause energy spectrum closely follows a −5/3 power law at mesoscales. Most theories addressing the mesoscale spectrum assume unbalanced dynamics but ignore the tropopause (near which the bulk of the data were collected). Conversely, it has also been proposed that the mesoscale spectrum results from tropopause-induced alterations of geostrophic turbulence. This paper seeks to reconcile these a priori mutually exclusive theories by presenting simulations that permit both unbalanced motion and tropopause-induced effects. The model integrates the nonhydrostatic Boussinesq equations in the presence of a rapidly varying background stratification profile (an idealized tropopause). Decaying turbulence simulations were performed over a wide range of Rossby numbers. In the limit of weak flow (U ≲ 1 m s−1), the essential features of the Boussinesq simulations are well captured by a quasigeostrophic version of the model: secondary roll-ups of filaments and shallow spectral slopes are observed near the tropopause but not elsewhere. However, these tropopause-induced effects rapidly disappear with increasing flow strength. For flow strengths more typical of the tropopause (U ~ 10 m s−1), the spectrum develops a shallow, near −5/3 tail associated with fast-time-scale, unbalanced motion. In contrast to weak flows, this spectral shallowing is evident at any altitude and regardless of the presence of a tropopause. Diagnostics of the fast component of motion reveal significant inertia–gravity wave activity at large horizontal scales (where the balanced flow dominates). However, no evidence points to such activity in the shallow range. That is, the mesoscale of the model is dominated by unbalanced turbulence, not waves. Implications and limitations of these findings are discussed.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Olivier Asselin, olivier.asselin@mail.mcgill.ca

1. Introduction

The near-tropopause horizontal wavenumber energy spectrum has a remarkably simple, double-power-law shape, with a steep −3 slope at synoptic scales breaking to a shallower −5/3 slope at mesoscales (Nastrom and Gage 1985). Synoptic-scale dynamics are typically interpreted in the light of Charney (1971)’s theory of geostrophic turbulence, which predicts a forward enstrophy cascade along a −3 spectrum below the baroclinic injection scale. By contrast, there is yet to be a consensus on the dynamics underlying the shallower mesoscale spectrum.

The most vividly debated theories addressing the mesoscale power spectrum rely on unbalanced dynamics to make sense of its −5/3 slope. One of the earliest hypotheses contends that the shallow range is dominated by weakly nonlinear inertia–gravity waves [see Dewan (1979), VanZandt (1982), and most recently, Callies et al. (2016)]. Others have argued that the mesoscale range is strongly turbulent, while emphasizing the role of rotation to a greater or lesser extent (Lindborg 2006, 2015; Bartello 2010; Vallgren et al. 2011; Deusebio et al. 2013; Kafiabad and Bartello 2016a,b). Interestingly, none of these unbalanced theories explicitly take into account the near-discontinuous jump in stratification characterizing the tropopause (e.g., Birner 2006) even though the bulk of the data supporting a −5/3 slope were gathered by commercial aircraft flying in its vicinity (e.g., Nastrom and Gage 1985).

A simple model of how the tropopause might impact the mesoscale energy spectrum is captured by surface quasigeostrophic (SQG) dynamics [see Johnson (1978), Blumen (1978), and for a recent review, Lapeyre (2017)]. Under the quasigeostrophic (QG) approximation, a discontinuity in the background stratification leads to a forward cascade of buoyancy variance along a −5/3 energy spectrum (Juckes 1994; Held et al. 1995). More recently, Tulloch and Smith (2006, 2009) studied a finite-depth version of the original semi-infinite SQG model. At scales large compared to the Rossby deformation radius, the top and bottom boundaries interact in a manner familiar from Eady (1949)’s classic study of baroclinic instability, thereby producing a −3 spectral slope. At smaller scales, however, the bottom boundary is not felt at tropopause heights, and the dynamics reduces to that of the original SQG model. As a result, the finite-depth SQG model produces a spectrum similar to that observed near the tropopause, with its characteristic −3 to −5/3 slope break.

Simply put, theories exploiting the tropopause to explain the mesoscale slope break ignore unbalanced dynamics, whereas unbalanced theories ignore the tropopause. In this paper, we aim to reconcile these two sets of theories. To do so, we perform idealized simulations that permit both unbalanced motion and tropopause-induced effects. Our model, outlined in section 2, integrates the nonhydrostatic Boussinesq equation set and its QG equivalent. The impact of the tropopause is assessed by comparing two sets of simulations: one with constant stratification (no tropopause) and another with a rapid change in stratification mimicking the tropopause. We neither neglect rotation nor assume that unbalanced motion is quasi linear. As such, none of the aforementioned theories of the mesoscale spectrum are a priori excluded. Our simulations span a wide range of flow strengths. For very weak flows, one expects the Boussinesq and QG dynamics to converge. As pointed out by Asselin et al. (2016b), however, it is unclear that SQG-like dynamics will persist in Boussinesq simulations when the flow becomes strong enough that a vertical Froude number based on the thickness of the stratification jump surpasses unity. For more vigorous flows, significant ageostrophic, fast-time-scale motion is anticipated. It is unclear, however, whether the presence of a tropopause might influence the production of such fast motion. Finally, it will also be of interest to clarify in what circumstances this motion might be characterized as inertia–gravity waves.

Section 3 describes the weak-flow regime and gives an overview of how the mesoscale slope of the kinetic energy spectrum varies as a function of flow strength in our Boussinesq simulations. These simulations are further explored in section 4. We find that SQG-like dynamics dominate when the flow is very weak (U ≲ 1 m s−1) but that both the spectral shallowing and related physical-space features rapidly become less evident with increasing root-mean-square wind magnitude U. For stronger flows (U ≳ 10 m s−1), a shallow mesoscale spectrum is recovered, both near the tropopause and elsewhere. However, in this case, the spectral shallowing is unambiguously related to fast-time-scale motion. We use three diagnostics to characterize the extent to which the fast motion might be considered wavelike or weakly nonlinear. Significant wavelike activity is normally associated with distinct peaks in the frequency spectrum. For inertia–gravity waves, one also expects a lagged correlation between the two components of horizontal velocity (e.g., for near-inertial motion, the northward velocity component leads the eastward component by one-quarter cycle in the Northern Hemisphere). Finally, nonlinear terms should remain relatively small in the governing equations. Using these diagnostics, we find that large-scale ageostrophic motion might reasonably be characterized as waves. However, no evidence supports weak nonlinearity or wavelike motion in the mesoscale part of the spectrum (i.e., the wavenumber band over which the total kinetic energy spectrum exhibits a −5/3 slope). Implications and limitations of our findings are discussed in section 5.

2. Formulation of the problem

To explore near-tropopause dynamics, we conduct numerical integrations of the nonhydrostatic Boussinesq equations for a wide range of flow strengths. The equation set may be written as (e.g., Vallis 2006)
e1
e2
e3
where is the wind vector, is the three-dimensional material derivative, b is the buoyancy perturbation, f and N are the Coriolis and Brunt–Vaïsälä frequencies, ϕ plays the role of pressure, and represents dissipative processes.
We consider two stratification profiles: with and without a tropopause. Following Smith and Bernard (2013), the tropopause is modeled as a continuous yet rapid change in stratification; specifically,
e4
where and . That is, the value of α is set such that N undergoes a rapid transition between its tropospheric and stratospheric values (Nt and Ns) over a small vertical scale h. In the case without a tropopause, and .
To isolate the role of unbalanced motions, we also perform control runs using a QG version of our model. Let U, L, and H be the characteristic velocity and horizontal and vertical length scales of the flow. In the limit of small Rossby and Froude numbers,
e5
the nonhydrostatic equations reduce to their quasigeostrophic equivalent in regions of slowly varying background stratification. That is, (1)(3) simplifies to
e6
where is the streamfunction, q is the quasigeostrophic potential vorticity, and represents advection by the horizontal, geostrophic wind. If the stratification profile varies over a small vertical scale, ; however, (6) is strictly valid only if (5) is replaced with the more stringent condition, Ro, Fr ≪ ε (Asselin et al. 2016b).
In order for the two equation sets, (1)(3) and (6), to precisely correspond one to another in the appropriate limit, the dissipation operator must take the form
e7
In other words, vertical diffusion is neglected, and the velocity, buoyancy, and potential vorticity fields share the same viscosity coefficient ν and hyperviscosity order γ. Here, we set γ = 8. The domain is doubly periodic in the horizontal and vertically bounded by rigid lids placed at (see Table 1). At the lids, vertical velocity is set to zero and u, υ, and b evolve freely. The simulations are all initialized with the same horizontally isotropic, random-phase geostrophic streamfunction:
e8
where the vertical wavenumber is slaved to the horizontal wavenumber by the QG relation, , the random-phase Φ lies between 0 and 2π, and the amplitude
e9
is a Gaussian centered on wavenumber k0 = 5, with and .
Table 1.

Key parameters of the numerical simulations.

Table 1.

Table 1 outlines the key parameters used in the simulations. Importantly, the length scales (including the tropopause thickness, h) and the Coriolis and Brunt–Vaïsälä frequencies are fixed to realistic values. The only free parameter is the flow strength, or equivalently, the Rossby number. Specifically, the tropopause-level horizontal wind field is normalized to have a root-mean-square U ranging from 1 to 100 m s−1, thus bracketing observed values in the atmosphere. This corresponds to Rossby numbers ranging from 0.025 to 2.5 or, equivalently, from ε/2 to 50ε, where ε = h/H ≈ 0.05 is the nondimensional tropopause thickness.

The flow is unforced and thus decays freely under the influence of hyperviscosity. The transform method is used to compute horizontal derivatives with spectral accuracy. Time and vertical derivatives are approximated by second-order centered finite differences. The model was run at two different resolutions: 5123 (unless noted) and 2563 for complementary diagnostics requiring transforms to frequency space.

3. An overview of results

a. Weak-flow limit

Our exploration of near-tropopause dynamics begins with the limiting case of weak flow, U = 1 m s−1. Figure 1 shows snapshots of vertical vorticity at z = 0 after t = 15 turnover times.1 Near the tropopause (Fig. 1, top panels), both QG and Boussinesq (BO) simulations produce secondary roll-ups of filaments, as is typical of SQG dynamics (e.g., Held et al. 1995). By contrast, the roll-ups are absent in the constant-stratification setup (Fig. 1, bottom panels). This is consistent with the QG simulations of Smith and Bernard (2013), in which the SQG features gradually disappear with distance from the tropopause.

Fig. 1.
Fig. 1.

Horizontal slices of nondimensional vertical vorticity, , at z = 0 and t = 15, for (left) QG and (right) BO simulations, with and without a tropopause.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

The kinetic energy spectra of Fig. 2 also reveal an enhanced small-scale activity near the tropopause. That is, spectra are significantly shallower in the presence (red) than in the absence (blue) of a tropopause. Near the tropopause, the slope of the spectra is closer to −2 than to −5/3, as is typical of decaying SQG simulations (e.g., Sukhatme and Pierrehumbert 2002; Capet et al. 2008). On the other hand, simulations without a tropopause produce a steeper −3 slope in agreement with Charney (1971)’s theory of geostrophic turbulence.

Fig. 2.
Fig. 2.

Instantaneous kinetic energy wavenumber spectra for U = 1 m s−1, z = 0, and t = 15 (as in the snapshots of Fig. 1). The subplot shows the vertical structure of these spectra in the shaded area, . This wavenumber range is used throughout the paper to compute spectral slopes.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

The subplot of Fig. 2 displays spectra with varying distance below the tropopause over the horizontal wavenumber range 10–70 (shaded area). Clearly, spectra are shallow only in the vicinity of the tropopause. At a distance of 5h (750 m) below, spectra (darker reds) are essentially as steep as those found in the absence of a tropopause (blue). Again, Charney (1971)’s argument holds away from rapid changes in stratification, and a steep −3 spectrum results.

Overall, the QG simulations agree very well with the more realistic Boussinesq simulations in this weak-flow regime. The agreement, however, is better in the absence of a tropopause. This is not surprising since the strict validity of the quasigeostrophic approximation requires the Rossby number to be a factor ε = h/H smaller when a tropopause is present (Asselin et al. 2016b). In this weak-flow limit, the Rossby number is about 0.025. Since this is much smaller than unity, QG is valid in the interior. However, since 0.025 corresponds to a Rossby number of about ε/2, QG is only marginally valid near the tropopause. Lower-resolution runs were performed to confirm that the QG and Boussinesq near-tropopause spectra indeed converge as the Rossby number is further decreased (not shown).

b. Spectral slope versus flow strength

What if the flow is stronger? Or equivalently, what if the Rossby number is more typical of the atmosphere? Figure 3 shows the spectral slopes2 at z = 0 for a wide range of flow strengths, both in the presence (red) and in the absence (blue) of a tropopause. That is, each point along the solid lines represents a single Boussinesq simulation. Slopes were fitted between wavenumbers 10 and 70 (shaded region in Fig. 2) and averaged over t = 5–10 turnover times to minimize the impact of both initial conditions and subsequent decay. The colored dashed lines are the slopes computed from the QG control runs, which are independent of U.

Fig. 3.
Fig. 3.

Slope of the kinetic energy spectrum as a function of the flow strength U. The slope is fitted between wavenumbers 10 and 70, the shaded region in Fig. 2, and subsequently averaged between t = 5 and 10.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

Thus far, we have only examined the weak-flow limit, U = 1 m s−1, corresponding to the leftmost border of Fig. 3. In the absence of a tropopause, a Charney-like regime is recovered in both the QG and the Boussinesq simulations (bottom left of Fig. 3; the blue curves converge to steep slopes3). Near a tropopause, both sets of simulations produce SQG-like features. However, the convergence of Boussinesq and QG runs requires even weaker U (top left of Fig. 3; red curve approaching shallower slopes). For somewhat larger U, the near-tropopause spectrum steepens until it reaches a minimum at U = 5 m s−1, thereafter becoming increasingly shallow with increasing U. For our constant-N simulations, the spectrum shows a monotonic shallowing with increasing U. Slopes for both sets of simulations converge around U = 20 m s−1 and appear to asymptotically approach −5/3 at high values of U.

The main aim of this paper is to interpret the key features of Fig. 3. The following section reports the major insights provided by diagnostics of our simulations over the whole range of Rossby numbers. In a nutshell, we find that SQG-like dynamics produce the shallowing in the weak-flow limit but that these tropopause-induced effects rapidly vanish with increasing U. From U = 5 m s−1 onward, increasing amounts of unbalanced motion cause the spectrum to shallow again irrespective of the tropopause.

4. Diagnostics

a. Wavenumber spectra

We begin with consideration of wavenumber spectra. Figure 4 displays time-averaged kinetic energy spectra at z = 0 for a wide range of flow strengths. In the weak-flow regime (U < 5 m s−1; Fig. 4a), significant shallowing is observed over a wide wavenumber band, roughly between kH = 10 and 100. This is consistent with SQG-like dynamics operating between the Rossby and jump deformation radii.4 This shallowing, however, becomes increasingly less pronounced with increasing U.

Fig. 4.
Fig. 4.

Time-averaged (t = 5–10) kinetic energy horizontal wavenumber spectra for (a) weak- and (b) intermediate-flow regimes near the tropopause, (c) weak and intermediate flows without a tropopause, and (d) strong flows, both with and without a tropopause. Red and blue curves are associated with the variable-N and constant-N cases, respectively. Note also that for each panel, the flow strength increases as the color gets darker. The shaded area corresponds to the wavenumber range over which spectral slopes were measured.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

As U is further increased, a spectral shallowing reappears. In this intermediate regime (U = 5–10 m s−1; Fig. 4b), however, this is associated with a gradual buildup of a shallow tail. That is, shallow, approximately −5/3 slopes are observed over a range of kH that extends farther to the left with increasing U. Moreover, this shallow tail is evident at any height (Fig. 5), as well as in our constant-N simulations (Fig. 4c). This contrasts the SQG-like behavior seen in weak flows, for which shallowing occurs in the vicinity of the tropopause only.

Fig. 5.
Fig. 5.

Kinetic energy spectra at three altitudes: the tropopause (z = 0; red), the midtroposphere (z = −Dz/4; green), and the midstratosphere (z = Dz/4; purple).

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

For strong flow, the tropopause appears to have no significant impact on the spectra. Indeed, those produced by the constant-N and variable-N setups are virtually indistinguishable for U ≥ 20 m s−1 (Fig. 4d). As U is further increased, the shallow tail builds up to such an extent that essentially the entire spectrum approaches a −5/3 slope. Note that the high-wavenumber end of the spectrum is influenced by a bottleneck effect (Falkovich 1994; Spyksma et al. 2012). As a result, there tends to be a plateau near dissipative scales at early times; however, this plateau later vanishes (not shown).

b. Physical-space fields

Physical-space fields also provide valuable insight. As we have seen above, very weak near-tropopause flows produce secondary roll-ups of filaments, as is typical of SQG dynamics (cf. Fig. 1). These roll-ups, however, become fewer and weaker as the flow gets stronger (top panels of Fig. 6, from left to right). This is consistent with the work of Bembenek et al. (2015), who found ageostrophic motion to inhibit the instability of filaments in a surface-intensified elliptic vortex. At U = 5 m s−1, roll-ups have virtually vanished, and the cases with and without a tropopause begin to look more similar. This loss of small-scale activity is consistent with the steepening of the spectrum in the weak-flow regime (Fig. 4a).5

Fig. 6.
Fig. 6.

Horizontal slices of nondimensional vertical vorticity at z = 0 and t = 15 (as in Figs. 1 and 2). (top left)–(top right) Near-tropopause roll-ups rapidly vanish with increasing U. (bottom) By contrast, there is no significant qualitative change in the flow structure of the constant-N case. The scale is normalized by U for the sake of comparison.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

Stronger flows, on the other hand, are characterized by less coherent and much-finer-scale structures (Fig. 7). These are associated with fast-time-scale, unbalanced motion. It is not obvious the extent to which this unbalanced motion is attributable to the initial adjustment versus subsequent balanced-to-unbalanced transfers. That is, substantial amounts of unbalanced motion are produced during the initial adjustment of the flow. Indeed, for strong-enough U, the initial (or early time) Rossby and Froude numbers locally reach values above unity, implying inertial and static instabilities in anticyclonic regions. Figure 8 shows early time snapshots of vertical velocity for U = 10 m s−1. Strong isolated centers of instability are conspicuous after only one turnover time, and waves are seen to radiate away from these centers. This initial adjustment is accompanied by substantial dissipation (Fig. 9). This dissipation, however, falls off rapidly after about t = 5. Our choice of t = 5–10 for the averaging period thus attempts to minimize the impact of this initial adjustment.

Fig. 7.
Fig. 7.

As in Fig. 6, but for stronger flows and at t = 7.5 (the middle of the averaging period).

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

Fig. 8.
Fig. 8.

Horizontal slices of vertical velocity at z = 0 for the case U = 10 m s−1 with a tropopause. Qualitatively similar snapshots are obtained at other heights with or without a tropopause.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

Fig. 9.
Fig. 9.

Total energy as a function of time for various flow strengths (different colors). Energy is defined up to an arbitrary constant. Here, this constant was set to the QG potential energy, the volume-integrated b2/2N2. The shaded region shows the time interval over which spectra are averaged and slopes are computed.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

For U ≥ 20 m s−1, we do not observe a significant difference in the finescale structure between cases with and without a tropopause. Indeed, the real-space fields appear qualitatively similar (Fig. 7). Furthermore, not only do slopes converge (Fig. 3), but also, the entire spectra are nearly indistinguishable for U ≥ 20 m s−1 (Fig. 4d). In other words, although we cannot exclude the possibility that tropopause effects lead to increased production of unbalanced motion, in our simulations, this appears to be weak.

c. Inertia–gravity wave diagnostics

Unbalanced theories of the mesoscale spectrum disagree on the nature of mesoscale motion. Is it dominated by weakly nonlinear inertia–gravity waves or by more strongly turbulent unbalanced motions? In this section, we present three diagnostics attempting to quantify, as a function of horizontal wavenumber, the extent to which this motion should be considered weakly nonlinear or wavelike.

1) Frequency spectra

We first consider frequency spectra—but with a twist. Typically, real-space fields are transformed to frequency space; here instead, transforms are performed from wavenumber space. More concretely, we begin with time series of horizontal winds, , at some horizontal wavevector (kx, ky). A Fourier transform6 in time yields the frequency–space fields, . With this, one can compute the frequency spectrum of horizontal kinetic energy:
e10
where . The process may be repeated for any wavevector. Here, we did so for , with n = 0, 1, 2, … . For each , we randomly selected 10 wave-vectors such that rounds to . This yielded M = 10 frequency spectra to be averaged for each ; specifically,
e11

Figure 10 displays a few of these frequency spectra at z = 0 for the case U = 5 m s−1. Each color is associated with a different wavenumber. The solid vertical line marks the Coriolis frequency, which we take as a convenient threshold separating “slow” and “fast” motion. Spectra for simulations with a tropopause (solid) and without (dashed) have qualitatively similar shapes. They appear flat at the lowest resolved frequencies but then fall off for higher frequencies. How far these elevated energy levels extend in frequency space and whether or not peaks emerge in the fN band is wavenumber dependent. At low kH, one or several distinct peaks are seen near or above the Coriolis frequency, consistent with significant inertia–gravity wave activity. However, no such peaks are evident at higher horizontal wavenumbers. For instance, wavenumbers 2 (brown) and 8 (purple) have clear wave signatures, both in the presence and the absence of a tropopause. By contrast, for wavenumber 32 (green), the relatively flat spectrum at low frequencies extends almost to the Coriolis frequency, so that there is no clear time-scale separation between the slow and fast modes.

Fig. 10.
Fig. 10.

Frequency spectra at z = 0 for (low resolution: 2563) Boussinesq simulations at U = 5 m s−1, with (solid) and without (dashed) a tropopause. Each color represents a different set of horizontal wavenumbers. For clarity, cases kH = 8 and 2 are vertically offset by two and six decades, respectively. Note that frequency spectra are typically computed from real-space fields, which implicitly entails an averaging of frequency spectra over all wavenumbers. The typical frequency spectrum, then, is closely approximated by our k = 2, 3, or 4 spectra—the wavenumbers containing most of the energy—and the distinctive features of higher-wavenumber spectra are averaged out.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

2) Time-lagged correlations

Distinct peaks in the fN band are not the only signature of inertia–gravity waves. If the flow contains significant wave activity, then one also expects the fast-time-scale portion of the horizontal velocity fields to be lag correlated. For example, assuming positive f and near-inertial motion, the component of wind υ leads the component u by about one-fourth of an inertial period. More generally, for a given vertical wavenumber (or mode), wave solutions obey a relation of the form
e12
where A and τ are real constants and the label “fast” denotes the fast component of the fields (with all frequencies below f cut off). Another way, then, to quantify wave activity is to compute the time-lagged correlation:
e13
where is a time-averaging operator and is the inner product (here, we define ). If significant inertia–gravity wave activity is present, one expects a strong negative correlation for some τ near or below a quarter of the inertial period (i.e., because such waves have frequencies greater than or comparable to f).

Figure 11 shows frequency spectra (Fig. 11, left panels) and time-lagged correlations (Fig. 11, right panels) at U = 20 m s−1 for a few horizontal wavenumbers. In the frequency spectra, only the lowest wavenumber (black) exhibits a distinct near-inertial peak, whereas evidence of wavelike activity at higher horizontal wavenumbers is difficult to discern. As expected, the near-inertial peak seen for kH = 1 is associated with a strong negative time-lagged correlation peaked at a value of τ around one-quarter of an inertial period. Using a lower cutoff to delineate fast and slow time scales (e.g., so the bulk of the spectral peak is captured) shifts the peak closer to the expected value for near-inertial motion. For higher values of kH, this negative correlation weakens and shifts to lower τ. That is, inertia–gravity waves appear to be less dominant and to have higher frequencies at larger kH. Interestingly, however, these correlations remain significant up to about kH = 16 despite there being no obvious signature of wavelike activity in the frequency spectrum of these wavenumbers. We take correlations of 0.5 as a convenient threshold for “significant wavelike activity.” In the case of U = 20 m s−1, then, only wavenumbers below 16 are ascribed to inertia–gravity waves.

Fig. 11.
Fig. 11.

Frequency spectra and time-lagged correlations in the case U = 20 m s−1, for a few horizontal wavenumbers (different colors) in low-resolution (2563) Boussinesq simulations. Only the variable-N case is shown, but the constant-N case is qualitatively similar. Correlations are computed for the fast (ω > f) component of velocity fields except for the supplementary dashed line, for which all frequencies above 0.5f were included to capture the bulk of the near-inertial peak apparent for wavenumber kH = 1. The vertical line stands for a quarter of the inertial period, within which maximum correlation is expected for inertia–gravity waves. Correlations are computed over the usual time-averaging period, t = 5–10.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

3) Linearity

As a third and final diagnostic of inertia–gravity wave activity, we measure the degree of linearity of the flow. More specifically, we compute the ratios of the nonlinear advective term to the other linear terms in the inviscid momentum equation for u:
e14
where is advection, is the Coriolis term, is the pressure gradient term (which includes both linear and nonlinear contributions), and is the tendency term.

Figure 12 shows how these ratios vary as of function of scale for different flow strength. In all cases shown, the Coriolis and pressure gradient terms dominate over the advection and tendency terms at large scales. This is consistent with quasigeostrophic theory, in which the advection and tendency terms are on the order of Ro in the expansion. As one goes downscale, however, these terms gain importance. For U = 10–20 m s−1, smaller scales are instead characterized by an approximate equilibrium between the advective, tendency, and pressure gradient terms. Clearly, nonlinearity cannot be assumed weak at these scales.

Fig. 12.
Fig. 12.

The ratios of advection to the other terms of the inviscid momentum equation for u. The ratios are averaged over the usual time window for low-resolution (2563) Boussinesq simulations. Triangles and dots denote wavenumbers at which fast-time-scale motion exhibits time-lagged correlations above and below 0.5, respectively.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

The linearity and lag-correlation diagnostics agree well. To show this, we added triangles and dots on wavenumbers for which the fast-time-scale motion exhibits a time-lag correlation above and below 0.5, respectively. Remarkably, time-lag correlations plunge below 0.5 just as advection becomes one of the dominant terms.

Another interesting feature of Fig. 12 is that the advection and tendency terms always have comparable magnitude. This indicates that the bulk of the flow evolves on the advective time scale at all wavenumbers. The latter increases with flow strength and wavenumber and approaches the Coriolis time scale, implying there is no meaningful separation of time scales in the mesoscale range.

d. Balanced and unbalanced spectra

QG and SQG dynamics are characterized by slowly evolving, balanced motion. Unbalanced motion, on the other hand, evolves on fast, typically near- or superinertial time scales. Here, we exploit this time-scale separation to define proxies for the balanced and unbalanced components of the flow. Specifically, we begin with the frequency spectra computed above in (11). Then, for every kH, we integrate the frequency spectra over two distinct frequency bands, from 0 to f and from f to ∞, giving the slow and fast components of the wavenumber spectrum, respectively:
e15
e16
Balanced motion is also predominantly rotational. By contrast, unbalanced motion typically has rotational and divergent components of comparable magnitude. Thus, as a second proxy to balance, we decompose kinetic energy into its rotational and divergent parts, where
e17
e18
and .

Figure 13 shows these two decompositions of the kinetic energy spectrum, both with (Fig. 13, top panels) and without (Fig. 13, bottom panels) a tropopause. Slow and fast components appear as large dots (or triangles), whereas rotational and divergent components are displayed as solid lines.

Fig. 13.
Fig. 13.

Decomposition of the total kinetic energy spectrum into its rotational and divergent (green and gold solid lines, respectively) and slow and fast components (green and gold dots and triangles, respectively) for low-resolution (2563) Boussinesq simulations (top) with (red) and (bottom) without (blue) a tropopause. Gold triangles and dots indicate time-lagged correlations above and below 0.5, respectively (cf. section 4c).

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

As seen above, the spectrum is shallower in the presence of a tropopause when the flow is weak (U = 1 m s−1; Fig. 13, left column). Here, we also see that both fast and divergent motions are very weak, such that the slow, rotational, and total spectra are virtually indistinguishable. Additionally, we noted earlier that this weak-flow limit is weakly dissipative (cf. Fig. 9). All of this evidence suggests that tropopause-induced, SQG-like dynamics are causing the spectral shallowing in the weak-flow regime.

The picture changes around U = 5 m s−1 as the fast and divergent components of the flow begin to impact slightly on the total spectrum. The effect becomes clearer around U = 10–20 m s−1: fast/divergent motion causes the total energy spectrum to develop a shallow tail. That is, for strong-enough flows, the shallow fast/divergent spectra cross the steeper slow/rotational spectra, leading to a kink in the total spectrum. The kink occurs at a wavenumber that depends on U: spectra produced by stronger flows become shallow at lower wavenumbers.

An analogous result was obtained without a tropopause by Bartello (2010). In that study, a forced-dissipative triply periodic Boussinesq flow was projected onto the linear normal (i.e., geostrophic and wave) modes. At high-enough Rossby number, the steep geostrophic spectrum was found to cross the shallow ageostrophic spectrum, resulting in a kink in the total spectrum. Our results are consistent with this picture except that, here, we use slow or fast and rotational or divergent decompositions rather than linear normal modes to distinguish ostensibly balanced and unbalanced flow.

For all cases shown, the total kinetic energy spectrum is very well approximated by both the slow and rotational components at low wavenumbers. This suggests that balance is robust at those scales. However, for sufficiently large kH and U, the slow spectrum falls off more rapidly than does the rotational spectrum. This is an indication of imbalance. Indeed, referring back to frequency spectra at those scales (e.g., left panel of Fig. 11), there is simply no time-scale separation allowing for a meaningful distinction between slow and fast.

Using the diagnostics outlined in the previous section, we decorated wavenumbers at which the fast motion might reasonably be considered as wavelike or not with triangles and dots, respectively. More specifically, correlations of 0.5 or higher were taken as indicative of significant wavelike activity (cf. section 4c). As seen from Fig. 13, waves are only present at horizontal wavenumbers for which the slow motion dominates. That is, the mesoscale kink roughly delineates the wavelike larger scales from the more turbulent small scales. In particular, the range of scales where the total spectrum shows an approximate −5/3 appears to correspond to strongly nonlinear unbalanced motions, not to waves. The same conclusion was reached, among others, by Deusebio et al. (2013) and Kafiabad and Bartello (2016a).

Crucially, the imbalance-induced shallowing mechanism described above takes place irrespective of the presence of a tropopause. Indeed, for intermediate and strong flows (U ≥ 5 m s−1), the balanced–unbalanced decomposition is similar between the variable-N and constant-N cases (cf. top and bottom panels of Fig. 13). It remains to be clarified whether the small differences in spectral slope seen in the intermediate-flow regime (U = 5–10 m s−1; Fig. 3) stem from residual SQG-like effects or tropopause-induced imbalance.

To facilitate comparison between our simulations and observations, Fig. 14 shows near-tropopause kinetic (red) and buoyancy variance spectra (purple) in dimensional units for cases U = 10 and 20 m s−1. Compared with observations [e.g., Fig. 3 of Nastrom and Gage (1985)], there is an overall agreement in the shapes of wind and buoyancy variance spectra. Not only is the −3 to −5/3 slope break captured, but also, the magnitude of the spectral densities and location of the transition wavenumber agree reasonably well with observations. While these results are encouraging, they should nonetheless be interpreted with caution. The most important caveats are covered in the following section.

Fig. 14.
Fig. 14.

Time-averaged kinetic energy spectra (red) decomposed into slow and fast (green and gold dots and triangles, respectively) and rotational and divergent (green and gold solid lines, respectively) components. The thin and thick solid lines are from low- and high-resolution runs, respectively. The fast and slow decompositions were only calculated for low-resolution runs. Also shown is a comparison between kinetic energy spectra (red) and buoyancy variance spectra normalized by (purple). These were shifted right by a decade for clarity.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0097.1

5. Discussion

In this contribution, we presented numerical integrations of the nonhydrostatic Boussinesq equations in the presence of rapidly varying background stratification. As expected, our QG control runs captured the essential dynamics of the flow in the limit of vanishing Rossby number: a Charney-like dynamics in the flow interior (or in the absence of a tropopause) and a Blumen-like dynamics in the vicinity of the tropopause. In the latter case, however, a much smaller Rossby number—Ro ≲ ε, where εh/H is the nondimensional height scale of the stratification change—was necessary for the QG and Boussinesq simulations to converge, in agreement with the theoretical prediction of Asselin et al. (2016b). In this very weak-flow regime (U ≲ 1 m s−1), the tropopause modifies the characteristics of geostrophic turbulence, leading to the formation of secondary roll-ups of filaments and associated shallow spectral slopes.

These tropopause-induced effects, however, rapidly lose their importance with increasing Rossby number: roll-ups disappear, and the spectrum steepens accordingly. At U = 5 m s−1 (ε ≲ Ro ≪ 1), the near-tropopause spectrum hits a minimum, and the variable-N and constant-N simulations yield qualitatively similar vorticity fields. This weak-flow regime is interesting for its own sake, and we wish to report on it in a forthcoming paper;7 however, it fails to explain the observed mesoscale spectrum in three major regards. First, climatological near-tropopause winds (e.g., Hoinka 1999) are much stronger than the low 1 m s−1 required to produce a significant SQG-like shallowing. Second, observations indicate substantial amounts of divergent motion in the mesoscale (e.g., Callies et al. 2016), whereas weak-flow simulations are completely dominated by rotational modes (cf. left panels of Fig. 13). Finally, the steep interior spectra simulated (cf. left panel of Fig. 5) are at odds with unambiguous evidence of near −5/3 mesoscale slopes throughout the troposphere (Frehlich and Sharman 2010; Cho et al. 1999a).

The imbalance-induced shallowing mechanism appears to be a better candidate for explaining the observed mesoscale spectrum. That is, for flow strengths more typical of the atmosphere, U ≳ 10 m s−1, the energy spectrum develops a shallow tail associated with unbalanced motion. This forward energy cascade of fast-time-scale motion appears to be fed from lower-wavenumber, unbalanced motion where theories rooted in weak nonlinearity may well apply. In other words, a working hypothesis would have that (i) balanced-to-unbalanced energy transfers occur at relatively large horizontal scales, (ii) this energy is transferred forward by a mechanism similar to that described in Bartello (1995), and (iii) this weakly nonlinear energy cascade at synoptic scales feeds into a strongly nonlinear cascade in the mesoscale. Interestingly, to the extent we have observed in our simulations, this overall picture is unaffected by the presence of a tropopause (cf. top and bottom panels in Fig. 13) and is consistent with the results produced by triply periodic models (e.g., Ngan et al. 2008; Kafiabad and Bartello 2016a,b).

However, a few caveats prevent us from making definitive claims about the atmosphere. First, our model is missing potentially important sources of waves, such as rough topography (Nastrom et al. 1987; Cho et al. 1999b) and convective activity (Karoly et al. 1996; Choi et al. 2006). Including these processes could plausibly extend to smaller scales the range over which fast motion displays wavelike signatures. Moreover, in the real atmosphere, waves gain amplitude with altitude to preserve momentum as background density decays (a feature neglected in the Boussinesq approximation). It is likely, therefore, that our model not only underestimates the amount of wavelike activity but also its variability with altitude.

Furthermore, our initial conditions come from a vertically smooth wind profile, whereas the midlatitude atmosphere is typically characterized by a strong localized jet and significant troposphere–stratosphere asymmetries. In our simulations, the spectral kink is essentially independent of altitude (cf. realistic wind strengths in Fig. 5), consistent with balanced and unbalanced motion being uniformly distributed in the vertical. This is at odds with observations (e.g., Frehlich and Sharman 2010) and more realistic models (e.g., Hamilton et al. 2008; Waite and Snyder 2009, 2013; Burgess et al. 2013), which reveal significant vertical structure in horizontal wavenumber spectra. An outstanding question is whether this discrepancy could be removed by initializing or forcing the simulations with a more realistic wind profile.

Our current setup also makes it difficult to estimate how much of the unbalanced mesoscale variability is attributable to the initial adjustment of the flow versus subsequent balanced-to-unbalanced transfers. Fully addressing this issue would involve explicitly calculating the balanced-to-unbalanced transfers and is beyond the scope of this contribution. These calculations were recently performed in a simpler context (constant stratification in a triply periodic domain) using a combination of frequency analysis and a higher-order definition of balance (Kafiabad and Bartello 2016a,b).

To conclude, our idealized simulations of the near-tropopause flow reconcile two families of theories for interpreting the mesoscale spectrum. For very weak flows, the tropopause indeed modifies the characteristics of geostrophic turbulence, leading to an SQG-like dynamics. For the stronger flows more typical of the tropopause, the synoptic-scale flow can be thought of as a superposition of balanced rotational modes and weaker inertia–gravity waves, whereas the mesoscale is dominated by unbalanced turbulence. The basic character and level of this unbalanced motion does not appear to be qualitatively influenced by the tropopause.

Acknowledgments

We thank Jörn Callies, Erik Lindborg, and an anonymous reviewer for helpful comments and suggestions. We also acknowledge the support from the Natural Sciences and Engineering Research Council of Canada (NSERC). The numerical calculations were performed using Compute Canada’s computer servers, Colosse and Guillimin.

REFERENCES

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1

Throughout this paper, time is normalized by the turnover time of the initial field, τ = L/U.

2

Kinetic energy and buoyancy variance spectra are found to have similar shapes (and thus slopes) in our simulations (e.g., Fig. 14). For the sake of conciseness, we focus mainly on the former.

3

Note that weak constant-N flows need more time to converge to a −3 slope (about t = 15 turnover times; cf. Fig. 2). At later times, then, the low-U portion of the blue curve of Fig. 3 rises up to −3, somewhat reducing the discrepancy between the cases with and without a tropopause. The other simulations have well-converged slopes, and Fig. 3 is otherwise qualitatively unaltered by changes of the time-averaging period.

4

The analysis in appendix I of Asselin et al. (2016b) suggests that SQG-like dynamics operate at scales much smaller than the maximum of the inverse of the tropospheric and stratospheric Rossby radii. In our case, then, we expect shallow spectra at scales kH ≫ 2f/NtDz ~ 2–3. Furthermore, Smith and Bernard (2013) showed that this SQG-like range extends down to the jump deformation radius, N0h/f, which corresponds to wavenumber 106 in our simulations.

5

Note that the spectra and slopes of Figs. 3 and 4 were averaged over t = 5–10, which slightly precedes the formation of roll-ups (t ≈ 15; as in Figs. 1 and 6). The near-tropopause energy spectra, however, undergo little variation at low U from t = 5 onward. In particular, the red curve of Fig. 3 is qualitatively unaffected by the averaging period.

6

To minimize high-frequency noise, the beginning and the end of time series are smoothed with a Hann function prior to the transform.

7

The interested reader is referred to Asselin et al. (2016a) for a preliminary study of this weak-flow regime, ε ~ Ro ≪ 1.

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