Numerical simulation of gravity wave breaking in the lower thermosphere

Authors


Abstract

[1] Numerical simulations are used to study gravity wave (GW) propagation, instability, and breaking in the lower thermosphere. Compressible effects are accounted for via an anelastic formulation of the equations of motion and we employ a realistic description of the background thermodynamic state. An initially low-amplitude, monochromatic GW with horizontal wavelength 60 km and intrinsic frequencyN/3.7 is introduced at the lower boundary and allowed to propagate to higher altitudes. The GW steepens as it propagates upward and displays instability and breaking over the altitude range ∼120–160 km. The effects of momentum deposition due to viscous attenuation and wave breaking are studied by comparing simulations which either include or exclude induced changes to the mean wind. These two cases also bound the range of expected behavior for horizontally localized GW packets. When induced changes to the mean wind are excluded, instability and turbulence occur over a broad altitude range spanning several vertical wavelengths. In contrast, the region of instability and turbulence is confined to a much more limited altitude range when induced mean wind effects are included. Wave breaking and turbulence in this case are largely confined within a shear layer formed by GW momentum transport. In time, the shear layer evolves into a critical level which consumes nearly all of the incident GW energy.

1. Introduction

[2] Observations, theory, and modeling spanning many years have suggested gravity wave (GW) penetration and potentially significant effects extending well into the thermosphere. Early observations of ionospheric irregularities motivated the initial theoretical studies of GWs at high altitudes [see Hines, 1960, 1974], while more recent observations, primarily employing incoherent scatter radars, provided more quantitative descriptions of the vertical scales and frequencies of these motions extending to high altitudes [e.g., Oliver et al., 1997; Djuth et al., 1997, 2004; Nicolls et al., 2004; Vadas and Nicolls, 2008, 2009; Abdu et al., 2009]. Initial linear steady state and ray-tracing models and theory have examined responses to various sources, including convection, body forcing accompanying GW dissipation, and tsunamis, the influences of various wind and temperature fields on GW penetration, and their heating and cooling of the thermosphere [e.g.,Hickey and Cole, 1988; Vadas and Fritts, 2004, 2005, 2006, 2009; Vadas, 2007; Fritts and Vadas, 2008; Fritts et al., 2008; Hickey et al., 2009, 2011; Vadas and Liu, 2009, 2011; Vadas and Crowley, 2010]. The source studies, in particular, suggest that large-scale GWs may achieve large amplitudes extending well into the thermosphere (∼200 to 500 km or higher), despite having nearly undetectable amplitudes in the MLT. Large anticipated GW scales can invalidate the WKB assumption since this approach requires negligible variation in the background state over one vertical wavelength. As discussed byVadas and Fritts [2005] and Vadas [2007], this fact has important implications for the validity of ray tracing methods applied at higher altitudes, while large implied GW amplitudes suggest a potential for similar instability dynamics to those observed and simulated in the MLT and at lower altitudes [see [Fritts and Alexander, 2003; Hecht, 2004, and references therein], and for significant induced mean flows that are not accounted for by linear theory. Finally, the nonlinear GW parameterization initially developed for the atmosphere from the ground to the mesosphere and lower thermosphere region [Medvedev and Klaassen, 1995; Medvedev et al., 1998; Medvedev and Klaassen, 2000] has been extended to the upper thermosphere in attempts to account for broader spectral GW responses at altitudes above the turbopause [Yiğit et al., 2008, 2009; Yiğit and Medvedev, 2009, 2010; Yiğit et al., 2012].

[3] Despite the theoretical potential for large GW amplitudes to contribute instability and neutral turbulence well into the thermosphere, there is no direct observational evidence for such at present. Indeed, the dynamics occurring throughout the lower atmosphere and the MLT have been recognized to lead to turbulence occurring on many scales and extending into the lower thermosphere for many years [Roper, 1966, 1977; Justus, 1967; Bishop et al., 2004]. But these measurements have also yielded indications of a “turbopause” above which “.. molecular diffusion rates are much greater than the wind-induced mixing rates”, with most data suggesting only laminar diffusion occurring above ∼105 and 110 km [Roper, 1977]. We note, however, that this apparent discrepancy between observations and theory may be a matter of the spatial scales on which turbulence is assumed to occur: turbulence at low Reynolds numbers and high altitudes should likely be expected at dramatically larger spatial scales (by ∼1 to 2 orders of magnitude) than observed at lower altitudes, given that the smallest turbulence scales vary as math formula, where ν is the kinematic viscosity and ε is the kinetic energy dissipation rate.

[4] Linear models of GW saturation do not capture nonlinear dynamics of GW processes in the whole atmosphere system. The extended nonlinear parameterization by Yiğit et al., [2008] based on the work by Medvedev and Klaassen [1995]successfully accounts for scale-dependent nonlinear diffusion. In the present work, a finite-volume numerical model solving the anelastic equations described byLipps and Hemler [1982], which yield the dispersion relation appropriate for deep GWs, has been developed in order to shed light into detailed theoretical understanding of nonlinear GW dynamics. The model is applied in direct numerical simulations (DNS) to describe the three-dimensional (3D) evolution of a GW having a high intrinsic frequency,ωi = N/3.72, a large vertical wavelength, λz ≃ 20 km, and exhibiting strong instability dynamics approximately 50 km above the generally accepted “turbopause” altitude. Two cases are considered, one in which the mean flow does not change (yielding instability dynamics accompanying a GW having the specified scales) and a second in which the mean flow is allowed to evolve due to GW momentum transport and deposition.

[5] A brief description of the numerical model is provided in Section 2. Sections 3 and 4 present results from the two cases exhibiting GW breaking and turbulence generation extending to 170 and 140 km, respectively, and Section 5 provides a discussion of the implications of these results and our conclusions.

2. Numerical Model

2.1. Formulation

[6] We simulate the anelastic Navier-Stokes equations in a form originally suggested byLipps and Hemler [1982] and clarified later by Lipps [1990] and Bannon [1996]. This formulation is derived by neglecting density fluctuations everywhere except in the buoyancy term. The thermodynamic definition of the potential temperature fluctuation is also modified slightly in order to arrive at a system of equations that conserves mass, momentum, total energy, and potential vorticity [Bannon, 1996]. In a non-rotating frame, the equations can be written as

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Here the background thermodynamic state is denoted by an overbar and deviations from it are denoted by a prime (e.g. math formula). The solution variables are the velocity, ui (or u, v, w), the pressure fluctuation, p′, and the potential temperature θ. The three coordinate directions are i = 1, 2, 3 or (x, y, z), where gravity (g) is aligned with the z direction. The molecular viscosity and thermal diffusivity are denoted by μ and κ, respectively, and these depend on the temperature through Sutherland's Law [White, 1974]. cp is the specific heat at constant pressure and δij is the Kronecker delta. Note that the hydrostatic balance has been subtracted from the momentum equation, and that the anelastic definition of the potential temperature fluctuation [Bannon, 1996],

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has been used to replace the density fluctuation with the potential temperature and pressure fluctuations. Here H is the density scale height, defined as

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Finally, the thermodynamic temperature is determined through a linearized form of the ideal gas law

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[7] Of particular interest to this work is the fact that the Lipps and Hemler system produces a dispersion relation for low-amplitude GWs in a non-rotating, isothermal atmosphere that agrees with the GW branch of the compressible acoustic-GW dispersion relation [Bannon, 1996], i.e.

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where N is the buoyancy frequency, and where k and m are the horizontal and vertical wave numbers, respectively. The associated linearized GW solution for uniform mean wind is

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where

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Here U is the constant mean wind in the direction of GW propagation, γ is the ratio of specific heats and math formula. The solution is normalized such that, at its most unstable phase, the combination of wave plus background is neutrally stable at the altitude z0 when A = 1. It can be seen that compressible effects enter the solution both directly through scaling of the amplitude with math formula and indirectly through dependence on the parameter ϵ. The latter, which can also be written as ϵ = λz/(4πH), is a measure of the vertical wavelength relative to the density scale height. This parameter will be less than 1 for all but GWs with very large vertical wavelengths.(H is generally in the range of 8–20 km, which means that λz must be in the range of 100–250 km in order to achieve ε = 1.) If ϵ is very small, and a single vertical wavelength is considered, then it is possible to approximate ε ≪ 1 and math formula, in which case the Boussinesq GW solution is recovered.

[8] An important distinction between the Boussinesq and anelastic (or fully compressible) GW solutions is the fact that the former is exact whereas the latter are only approximate. This distinction is apparent if the solution is transformed into a coordinate system aligned with and normal to the phase lines and translating with the mean wind. This transformation results in zero normal velocity for the Boussinesq solution. Thus the velocity vector is perpendicular to the gradient in any quantity and consequently all of the nonlinear advective terms are identically zero. This situation reduces the Boussinesq equations to a linear form which can be solved exactly.

[9] The situation is more complex in the case of anelastic or fully compressible systems since the same transformation results in a non-zero normal velocity component. The nonlinear terms do not vanish in this case and thus a linearization must be undertaken before a wave solution can be obtained. While the linearized approximations can be quite acceptable for small wave amplitude, they becomes increasingly inaccurate as the amplitude approaches 1. Since we are concerned with waves that achieve large amplitudes in this work, we have derived an improved solution for fully compressible or anelastic systems. As we shall show in a future publication, the method of successive approximation can be used to develop the solution in the sequenceu(x, z, t) = ū + Ãu1(x, z, t) + Ã2u2(x, z, t) + …. In this view, the linearized solution u′ = Ãu1 is simply the first term in an infinite series that describes the wave motion. The second term, u2, must be of the form u2 = cif(m/kϵ) sin(2ϕ + ϕ3(m/kϵ)) since it arises from nonlinear interactions between components of the first order solution. More generally, the nth term must be proportional to sin(). This analysis demonstrates that the spectral content of medium to high amplitude waves is composed not only of the parent wave number pair (k, m) but also the harmonics (2 k, 2 m), (3 k, 3 m), …. The amplitude of the nth harmonic decreases like Ãn, so that just a few harmonics may be sufficient to describe waves approaching an amplitude of 1.

[10] It should also be noted that each component of the solution has a unique growth rate with altitude. With à being proportional to math formula, the nth solution component grows like exp[n(z − z0)/(2H)]. Thus while the harmonics may be completely negligible at lower altitudes, their enhanced growth rates render them much more significant at altitudes where the wave amplitude becomes close to 1.

2.2. Numerical Method

[11] The anelastic equation set is discretized using a second-order finite-volume scheme identical in form to the method discussed byFelten and Lund [2006]. This formulation results in exact numerical conservation of mass, momentum, kinetic and thermal energy (in the absence of viscous dissipation) and thus faithfully represents the underlying conservation laws. An important consequence of strict energy conservation is that the scheme has no artificial dissipation.

2.3. Computational Mesh and Background State

[12] Simulations are performed in a computational box having dimensions 60 × 60 × 100 km in the x, y, and z directions, respectively. We imagine the domain to be fixed in the atmosphere with the lower boundary located at an altitude of 100 km and the upper boundary at 200 km. 300 × 300 × 500 mesh points are used, resulting in an isotropic mesh spacing of 200 m. A domain length of 60 km is used in the x direction in order to accommodate a single wavelength of the GW. The same dimension in the lateral direction allows for 3D instability structures of scale up to one horizontal wavelength. The vertical dimensions are set so that instability and turbulence is always displaced at least 20 km from a vertical boundary.

[13] Periodic boundary conditions (BCs) are applied in the horizontal directions and a radiation condition is used at the upper boundary. The radiation condition is an anelastic extension of the Klemp and Durran [1983] method, modified such that the required polarization relations are measured from the computed solution near the upper boundary rather than being specified a priori. This modification allows the radiation condition to indirectly account for viscous and nonlinear effects near the upper boundary, which can lead to polarization relations that differ strongly from the predictions of linear theory.

[14] A GW is introduced into the domain by specifying the vertical velocity according to equation (9) at the lower boundary. The lateral velocity at the lower boundary is taken to be zero so that the GW is two dimensional. The streamwise velocity and potential temperature distributions at the lower boundary are determined via polarization relations. Although we could use equations (8) and (10) for this purpose, we obtained improved results by measuring these relationships directly from the solution at the third grid level, e.g. math formula, where the subscripts 0 and 3 refer to the boundary and third grid level respectively.

[15] As depicted in Figure 1, the flow is initialized with a realistic thermodynamic state for the lower thermosphere [Vadas and Fritts, 2006; Vadas, 2007] and a uniform wind in the x-direction of magnitudeU = −ci0 = 55.32 m/s, where ci0 is the intrinsic phase speed at the lower boundary. This choice for U renders both c and ω zero at the lower boundary.

Figure 1.

Variations in thermodynamic state with altitude. The reference conditions at z0 = 100 km are ρ0 = 1.254 × 10−6 kg/m3, p0 = 9.925 × 10−2 kg/(m-s2), T0 = 275.8 K, θ0 = 275.8 K, N0 = 2.156 × 10−2 s−1.

[16] The atmosphere is assumed to be of fixed composition with constant values of the gas constant, R, and the specific heat ratio, γ. This step must be viewed as an approximation as the atmospheric composition transitions from largely a mixture of N2 and O2 at lower altitudes to one composed mainly of O in the middle and upper thermosphere. As described below, we have chosen a temperature profile which minimizes the variation in R and γ over the altitude range of interest, thereby rendering the fixed composition assumption a reasonable approximation.

[17] The variations in R and γ with altitude can be estimated using the procedure developed by Vadas [2007], who analyzed one month of TIME-GCM data in order to develop curve fit expressions for the composition parameters within the thermosphere. Vadas chose to parameterize the molecular weight,XMW and γ as a function of the local atmospheric density. The specific gas constant is related to the molecular weight via math formula, where math formula is the universal gas constant. Applying the curve fit expressions to the density profile depicted in Figure 1 shows that R and γ deviate from their values in the lower atmosphere by 4% and 1% at 120 km, 10% and 4% at 160 km and 18% and 7% at 200 km, respectively. As shown later in the paper, instability, transition and turbulence are largely contained between 120 and 160 km, an altitude range where the composition parameters vary by less than 10%.

[18] All simulations to be discussed below were run with a fixed time step of Δt = 0.2 s, which arises from the explicit stability limit for the diffusive terms at the upper boundary (where the kinematic viscosity and diffusivity are maximum). This rather small time step affords us exceptional temporal resolution for advective processes as the maximum CFL number (uΔt/Δ) never exceeds 0.2 in any of our simulations.

2.4. Periodic Boundary Conditions

[19] Although periodic boundary conditions have been used in nearly all prior numerical simulations of 3D GW breaking [Fritts et al., 2009a, 2009b] and references therein, their use represents a compromise between numerical efficiency and physical realism - a tradeoff which warrants some careful discussion. From a numerical standpoint, periodic BCs are ideal since they allow for very compact domains, introduce no numerical error, and allow for the use of a fast Fourier transform (FFT)-based Poisson solver in the anelastic pressure-projection method. These efficiencies are critical for high-resolution 3D numerical studies of the wave breaking process as the current simulations using periodic boundary conditions on a domain extending over just one wavelength require 45 million mesh points.

[20] From a physical standpoint, the imposed periodicity imposes some important limitations on the solution and must therefore be interpreted carefully. In reality, GWs are almost always organized into temporarily and spatially localized “packets” comprised of a collection of wave number components. Under a 2D idealization it is possible to simulate the processes leading up to wave breaking from packets and there are several interesting studies where this is done [e.g., Sutherland, 2001; Dosser and Sutherland, 2011a, 2011b]. Such studies shed light on the role of wave-induced changes to the mean flow (self acceleration) and typically allow for a much richer spectrum of wave number components (and hence the opportunity for wave-wave interactions) as compared to simulations of initially monochromatic waves on a compact periodic domain.

[21] Unfortunately horizontally localized wave packets require domains extending over many multiples of the primary GW wavelength. In a three-dimensional setting such large domains translate into an extremely large number of mesh points if the wave breaking process is to be well resolved. Thus for the time being we are effectively constrained to consider an initially monochromatic wave.

[22] The monochromatic wave approach corresponds to an idealized situation where the amplitude of a spatially infinite GW increases in time due to a ramped forcing condition at lower altitudes. As viewed by a stationary observer, the wave-breaking process will progress in time with ever-increasing changes to the mean wind as wave momentum is continually transferred to the mean as a result of viscous dissipation. The monochromatic approach is also loosely representative of a case where the computational box is translated with the mean wind from the quiescent atmosphere into a broad packet spanning many GW wavelengths. As seen in this frame, the wave amplitude will increase in time as the computational domain translates through the leading edge of the packet. Likewise, breaking will progress in time with steadily increasing changes to the mean wind as wave momentum is deposited there.

[23] One limitation of this latter interpretation is that it provides no information on the instability process within the packet as a function of time at a fixed position in space. We attempt to gain insight into this problem by considering a second case where the domain is considered to be fixed in space near the leading edge of the packet. Although the solution is not strictly periodic in the streamwise direction in this case, we assume that systematic changes occurring over a single wavelength are small and may be neglected. Within this context, any GW momentum transferred to the mean flow should advect downstream, away from the computational domain. This will not happen under strict application of periodic boundary conditions, however, since any changes to the mean flow at the downstream boundary are effectively re-introduced at the upstream boundary. We attempt to make up for this inconsistency by numerically constraining the mean wind to be fixed in space and time. Thus, as a further approximation, we neglect the (small) changes to the mean wind that might occur as the flow traverses a single wavelength.Franke and Robinson [1999]used exactly the same approach by considering 2D simulations with and without wave-induced changes to the mean flow.

[24] Operationally, the only difference between the two simulations is whether or not momentum exchange between the GW and the mean is allowed. It turns out that the dynamics are simpler for the case where momentum transfer is excluded (mean wind held fixed) and, for this reason, this case is discussed first in Section 3.

[25] A final important limitation of the periodic boundary conditions is that the wave number spectrum of instability or turbulence-generated GWs is restricted to integer multiples of the primary wave. Additionally, is not possible to generate sub-harmonic motions.

2.5. Gravity Wave Parameters

[26] We consider an initially two-dimensional (2D) GW having horizontal wavelengthλx = 60 km and horizontal intrinsic phase speed ci = −U = −55.32 ms−1, which results in an intrinsic wave period of τw = 2π/ωi = λx/U = 1085 sec. These parameters were chosen to be representative of a GW that is likely to penetrate into the thermosphere. The buoyancy period (τb = 2π/N) varies between 291 and 558 sec between the lower and upper boundaries, and thus the intrinsic frequency ratio (ωi/N = τb/τw) varies between 1/3.72 and 1/1.94. Owing to the decrease in N with altitude, the vertical wavelength (λz) increases from roughly 17 to 36 km from the lower to the upper boundary. This variation can be seen in Figure 2a, which shows the λzprofile as measured from a non wave-breaking simulation (to be discussed below). Also shown is the vertical wavelength distribution predicted by the anelastic dispersion relation,equation (7), for constant horizontal wavelength and intrinsic frequency. The estimate is quite accurate below about 140 km, where the GW amplitude is small and viscous effects are negligible for the GW under consideration. Figure 2b shows the variation in Reynolds number, math formula with altitude. Due to the large changes in atmospheric density, the Reynolds number varies from about 76,000 at the lower boundary to about 60 at the upper boundary. We can anticipate that viscous effects will be rather important above 135 km, where the Reynolds number drops below 1000.

Figure 2.

Variations in (a) vertical wavelength and (b) Reynolds number with altitude.

[27] The GW itself is introduced by imposing the solution at the lower boundary. Starting transients are minimized by increasing the GW amplitude in time from 0 to math formula = 1.74 m/s via tanh(t/τf), where t is time and τf = 3,000 sec (2.77 wave periods). This procedure results in a transient leading edge that progresses upward ahead of the intended steady GW field. The transient is largely consumed by viscous dissipation in the upper portion of the domain and any residual GW energy exits through the radiative upper boundary. 3D instabilities are seeded with small-amplitude random potential temperature perturbations, taken from a uniform distribution over the interval ±0.01 K/sec.

[28] Near the lower boundary (where the kinematic viscosity is small), the GW amplitude grows with altitude as math formula. At higher altitudes, increasing viscous dissipation retards GW growth and ultimately leads to a decrease in amplitude with height. The non-dimensional GW amplitude at the lower boundary is taken to be A = 0.1, which is sufficient to lead toA > 1 at altitudes in the upper portion of our computational domain (see equation (20) for the GW amplitude definition).

3. Results

3.1. Case 1: Constant Mean Wind

[29] In the first case to be studied we exclude mean wind changes due to GW momentum flux divergence in order to focus on the dynamics without the complicating effects of wave refraction due to induced mean shear and GW self-acceleration effects. A companion simulation without this constraint will be discussed in the following section.

3.1.1. Overall Evolution

[30] As described above, we begin the simulation with a uniform wind but no GW. The GW is then introduced gradually at the lower boundary and allowed to propagate naturally to higher altitudes. The wave energy contained in the domain thus grows in time and tends toward a steady state where the energy flux at the lower boundary is offset by both viscous dissipation in the upper portion of the domain and a small residual energy flux through the upper boundary. This behavior is shown by the dashed curve in Figure 3where the domain-averaged GW vertical velocity variance time history is plotted for a non-breaking simulation (run in 2D with no added noise). Results from the 3D wave breaking simulation are also shown, and this case is distinguished by a significant loss in variance once the wave breaks. By comparing the two curves, we see that the breaking simulation obtains about 95% of its expected velocity variance at a time of about 9.5 wave periods (τGW). A large drop in variance then occurs between 9.5 and 13.5 τGW. Beyond this time the variance settles into a new equilibrium level where there is once again a balance between viscous dissipation and the net flux through the lower and upper boundaries. Based on the variance time history, we may divide the flow evolution into three stages. The first, between t = 0 and 9.5 τGWis the initiation phase. It contains the starting transients and the gradual approach to steady state. Although instability structures are likely to be growing near the end of this phase, they are very difficult to detect and are not yet energetically important. The second, or wave-breaking stage between ∼9.5 and 13τGW is much more interesting as it comprises instability, initial wave breaking, and transformation of the flow to a turbulent state. The final stage from ∼13 to 17 τGW is the fully developed turbulent phase where the flow obtains a new equilibrium state with approximately constant GW energy.

Figure 3.

Evolution of the domain-averaged GW vertical velocity variance for Case 1, where momentum exchange with the mean flow is excluded. The variances are normalized by the expected steady state values in the absence of wave breaking.

[31] To probe the dynamics causing the decrease in GW variance seen in Figure 3in greater detail, we show the evolution of the 3D vorticity magnitude in streamwise-vertical (x, z) and spanwise-vertical (y, z) cross sections in Figures 4 and 5, respectively. Figure 4a reveals an approximately linear GW with increasing phase slopes and vertical wavelength due to the decreasing N (and increasing ωi/N) with increasing altitude. Also seen are vertical modulations in the maximum vorticity magnitude, where enhancement occurs below the rising (and least statically stable) GW phases and reduction occurs above the rising GW phases. The modulations are due to the presence of the first harmonic described in Section 2.1, which serves to alternately steepen and flatten the gradients at each half wavelength of the parent wave. We shall refer to the vorticity magnitude modulations as minor and major vorticity bands hereafter.

Figure 4.

Evolution of the vorticity magnitude shown with streamwise-vertical (x, z) cross sections for Case 1. The images are taken at (a) 9.5, (b) 10.5, (c) 11.5, (d) 12.4, (e) 13.4, (f) 14.3, (g) 15.3, and (h) 16.2 τGW. Red and yellow colors depict higher vorticity magnitudes, and the cross sections are located at the center of the spanwise domain shown in Figure 5.

Figure 5.

As in Figure 4, but for spanwise-vertical (y, z) cross sections. The cross sections are located at the center of the streamwise domain shown in Figure 4.

[32] Figure 5a shows that the minor vorticity band centered just above 160 km contains visible modulations in the spanwise direction. The adjacent major vorticity bands are almost undisturbed at this time, indicating that the minor band is the site of the initial instability. The spanwise wavelength of the initial instability is half the domain width, or 30 km. This is close to one vertical wavelength, which is about 27 km at an altitude of 160 km (see Figure 2). 3D visualizations (not shown) reveal that the dominant initial instability structures are spanwise vortices embedded within the vorticity bands. Adjacent pairs undergo something like a Crow instability where a sinuous mode with spanwise wave number two grows rapidly until the vortices connect, forming vortex “hairpins” and vortex rings. Both of these structures are very similar to those observed in previous GW breaking simulations Fritts et al., [1994, 2003, 2009a, 2009b]. While the first set of rings and hairpins are forming, the vortex sheet making up the major vorticity band just below the site of the initial instability rolls up into a system of much stronger vortices, as seen in Figure 4b. These vortices are initially quasi-2D but subsequently undergo a similar sinuous instability, this time with wavelength closer to the full domain width (mode 1). The spanwise character of one of these counter-rotating vortex pairs is seen inFigure 5b.

[33] The vortical structures associated with the secondary instability of the major vorticity band are very similar to the “hairpin” vortices described by Fritts et al. [2009b] for GW breaking in a Boussinesq fluid at similar GW amplitudes and frequencies, but at somewhat higher Reynolds numbers. In this case, however, the instability dynamics evolve significantly more slowly, resulting in a more gradual erosion of the GW amplitude, velocity variance, and momentum flux. These dynamics also evolve with time to lower altitudes and smaller instability scales as the Reynolds number increases and the instability timescale decreases with decreasing kinematic viscosity. The instability dynamics and turbulence generation occurring at decreasing altitudes impose a reduction of the GW amplitude that largely precludes further instability at higher altitudes, despite coherence in the GW field that continues to propagate to higher altitudes at smaller amplitude. These amplitude reductions account for the cessation of instability dynamics at higher altitudes at later stages of the evolution.

3.1.2. Wave Amplitude Evolution

[34] In order to understand the progression of turbulence to lower altitudes more quantitatively, we consider two separate measures of apparent convective instability. The first is the usual GW amplitude,

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where the subscript GW indicates that math formula is computed from just the underlying GW mode and the subscript max indicates the maximum value over the streamwise direction at fixed altitude. The second stability measure is based on the same concept, but considers the turbulent fluctuations in addition to the GW mode. We denote this quantity as the stability index and define it as

display math

Here θ′ is the full potential temperature fluctuation (GW mode plus turbulence) and the subscript cond indicate a conditional average where only the top 10% of the most negative values of ′/dz on any given horizontal plane are included. As demonstrated below, the choice of the top 10% of the distribution renders AGW and ATot nearly identical when no turbulence is present. This result is due to the fact that the trigonometric functions that describe the GW mode have fairly broad maxima, e.g. cos(x) = 0.95 for x = ±π/10. When turbulence is present, sampling the top 10% of the distribution provides a reasonably smooth measure of the stability in the most unstable regions. This is much preferable to choosing the maximum value on any given plane, which results in extreme values of the stability index and can give a false impression of rampant instability.

[35] Note that for either stability measure, A > 1 implies increasing potential temperature with height and hence the potential for local convective instability (negative Richardson number, Ri < 0). AGW > 1 indicates the GW mode itself is locally convectively unstable by this measure, whereas ATot > 1 indicates that the GW mode plus the turbulence is unstable. When turbulence is present we expect ATot > AGW since there are bound to be regions where the turbulent fluctuations reinforce the perturbation due to the GW mode.

[36] We note here that the use of convective overturning is a very simplified approximation to the tendency for GW instability, given the potential for local instability modes with AGW < 1 and Ri ≫ 1 [e.g., Lombard and Riley, 1996; Sonmor and Klaassen, 1997; Fritts and Alexander, 2003; Achatz, 2005; Fritts et al., 2009a, 2009b] and references therein). As such, it understates the expectation of GW instability and turbulence generation at all altitudes. However, given the rapid increase in AGW with time accompanying rapid vertical propagation observed in Case 1, only the most rapid modes of instability have a potential to contribute to GW instability and dissipation. Indeed, we will see below that AGW achieves values well above 1, such that convective instability proves to be a reasonable guide to the onset of instability in this case.

[37] Profiles of AGW and ATot throughout the evolution displayed in Figures 4 and 5 are shown in Figure 6. The first profiles at t = 9.5 τGW reveal that the GW amplitude, AGW, and the more general instability index, ATot, are nearly identical and both exceed AGW = 1 over altitudes from below 140 km to above 180 km. As the wave breaking process proceeds through instability and transition to turbulence, AGW and ATot evolve in different ways. Specific features include (1) a rapid reduction in AGW, to a maximum of ∼1 at 11.5 τGW in the initial region of instability between ∼130 and 160 km, (2) descent of the altitude of maximum AGW accompanying the progression of instabilities and turbulence from higher to lower altitudes, (3) further reductions in maximum AGW to values of ∼0.7 to 0.8 beginning at ∼13.4 τGW and thereafter, (4) overall reductions in AGW relative to AGW prior to instability by factors of ∼3 to 5 above 160 km, (5) increases of ATot, relative to AGW, due to the superposition of the GW and turbulent fluctuations, with maximum values of ∼1.3 to 1.5, and (6) persistence of an excess of ATot, relative to AGW, above the region of most intense turbulence due to residual turbulence at these altitudes and the presence of additional GWs excited by the more intense turbulence below.

Figure 6.

Evolution of the amplitude and stability profiles for Case 1. Times are as shown and each successive pair of profiles is offset by two units on the horizontal scale.

[38] Turning now to more specific details, we see that instabilities arise up to altitudes of only ∼160 km, despite the occurrence of AGW > 1 to altitudes greater than 180 km. We attribute this disparity to the strong variation of Re with altitude. Referring to Figure 2, we see that Re ∼ 300 at 160 km, but Re ∼ 100 at 185 km. Instability dynamics are strongly suppressed at such small values of Re. On the other hand, Re increases rapidly below 160 km, which favors the spread of instability and turbulence in this direction.

[39] As the wave breaks viscous dissipation quickly reduces AGW below 1 at all altitudes. At the same time, turbulent fluctuations superimpose with the reduced GW perturbation to maintain a significant region of convective instability (ATot > 1). Vigorous turbulence is sustained in this manner in spite of the fact that AGW continues to drop well below 1. An equilibrium region of instability and turbulence ultimately develops between about 125 and 150 km. It is interesting to note that nearly all of the turbulence in this zone lies in a region where AGW never exceeded 1. The bottom edge of the turbulent zone correlates very well with the point at which ATotcrosses 1.0. A clear dividing line between turbulent and non-turbulent regions at an altitude close to 125 km can be seen inFigures 4e and 4f.

[40] Once the flow enters a fully developed turbulent state, the AGW profiles become nearly fixed in shape with a maximum less than 0.8 at an altitude of ∼130 km. This behavior is much more apparent in Figure 7, where several AGW profiles are plotted together. The profiles for times 13.4, 15.3, and 17.2 τGW agree reasonably well over the entire altitude range. Much of the GW energy propagating upward is quickly converted to turbulence in the altitude range between 125 and 140 km. A reasonably constant fraction of the incident GW energy also passes through the breaking region, only to be dissipated at higher altitudes.

Figure 7.

Direct comparison of the wave amplitude profiles at various times for Case 1.

[41] The quantitative behavior of the AGW profiles differs significantly from linear saturation theory [Lindzen, 1981], which predicts wave breaking, turbulence, and AGW = 1 at altitudes between ∼140 and ∼180 km (see Figure 6). In reality, we find that AGW saturates closer to 0.75, and produces sustained turbulence at much lower altitudes.

[42] As Figure 6 demonstrates, instability in the fully turbulent regime is maintained through a combination of the disturbances due to the GW and those due to the turbulence. The turbulent fluctuations provide a sufficient assist that it is not necessary for the GW to achieve AGW = 1 in order to maintain instability.

[43] Based on our simulations, we can suggest a simple yet effective improvement to classical saturation theory; simply saturate at AGW = 0.75 rather than 1. We note that similar modifications to Lindzen's [1981] saturation theory have been suggested previously. Medvedev and Klaassen [1995], Medvedev et al. [1998], and Medvedev and Klaassen [2000] used Weinstock's [1976, 1982]nonlinear diffusive theory to show that wave-wave interactions among a spectrum of waves in the middle atmosphere leads to GW saturation at lower altitudes than predicted by theLindzen [1981]theory. The basic assumption behind the nonlinear theory is that shorter wavelength components interact with a given GW in order induce local instabilities on a smaller scale, even though the underlying GW is stable according to its amplitude in isolation. The wave-wave interactions in the nonlinear theory are analogous to the perturbations due turbulence in our study. In either case an external influence acts on the GW to locally trigger instability.Yiğit et al. [2008] extended the work of Medvedev and Klaassen [1995] to develop a GW parameterization for use in general circulation models that describes the nonlinear diffusion as well as thermospheric dissipation of a broad spectrum of waves above the turbopause. This parameterization allows for smoother wave drag profiles with maxima at more appropriate lower altitudes as compared with linear saturation models. As such, the Yiğit et al. [2008] method has enjoyed success in 3D atmospheric general circulation models without need for further artificial wave drag scaling [e.g., Yiğit et al., 2009; Yiğit and Medvedev, 2009, 2010; Yiğit et al., 2012]. Finally, Sonmor and Klaassen [1997] point out several mechanisms that result in GW instability and saturation for AGW < 1.

[44] Here we investigate the benefits of a lower saturation threshold by comparing the results of a very simple saturation model with mean flow acceleration (wave drag per unit mass) profiles measured directly from our simulation. Saturation theory typically starts with an AGWprofile derived from linear theory such as ray-tracing, with or without a viscous correction [Vadas and Fritts, 2005]. For the present illustration, we shall use the AGW profile for t = 9.5 τGW, shown in the first profile of Figure 6, as the input to the saturation model. According to linear theory, we may scale the wave stress as

display math

where the angle brackets denote an average in the horizontal directions, and the subscript “0” denotes a reference position, taken to be the lower boundary of our computational domain. Neglecting viscous effects, the horizontally averaged streamwise momentum equation for a horizontally uniform GW is

display math

Combining the above two equations leads to the following estimate for the mean flow acceleration;

display math

Saturation theory makes use of this estimate in conjunction with a modified amplitude profile. The classical theory would modify the amplitude profile so that it is maximized at 1, i.e., Asat(z) = max(AGW(z), 1.0). Our proposed improvement is to simply change the limit from 1.0 to 0.75.

[45] Predictions of our simple saturation model are compared with the directly measured mean flow acceleration in Figure 8. Noise is reduced in the directly measured data by averaging together five profiles taken from the fully developed regime between 15.3 and 17.2 τGW. These results show that the saturation model with AGW = 1.0 overpredicts the wave-breaking altitude by about 8 km and then overpredicts the mean flow acceleration by a factor of ∼2 over the remainder of the altitude range. In contrast, the saturation model withAGW = 0.75 predicts the wave-breaking altitude accurately and generally estimates the mean flow acceleration within about 20% error. Our proposed improvement is trivial to implement yet appears to greatly improve the accuracy of the mean flow acceleration prediction.

Figure 8.

Predictions of saturation theory compared with directly measured mean flow accelerations for Case1, where momentum exchange with the mean flow is excluded.

3.1.3. Energetics

[46] Freely propagating GWs have limited spectral content. If the wave steepens and breaks, however, a much richer spectrum of length and timescales is produced. Several instability pathways exist and these often have identifiable spectral signatures. For example, wave-wave interactions can lead to turbulence and these are often characterized by resonant interactions between a few frequencies [Klostermeyer, 1991; Müller et al., 1986]. In a clean setting like ours, with a 2D, monochromatic parent wave, we may expect that the strongest interacting waves will be approximately 2D. Thus one way of tracking wave-wave interactions it to monitor the energy contained in the 2D field. This is of course not precise, as interactions with oblique waves may also be present. At the same time, general 3D turbulence may deposit energy in the 2D modes. Realizing these limitations, we will nonetheless followFritts et al. [2009a]and use the 2D energy time history as a reasonable surrogate for wave-wave interactions.

[47] As noted above, there are also more general 3D instability modes that do not necessarily involve wave-wave interactions. Lombard and Riley studied GW instability from a linear stability standpoint and found a wide range of instability scales, orientations, and growth rates. Resonant interactions are predicted by linear theory and these can be either 2D or 3D. 2D resonances have dominant growth rates at low wave amplitudes (AGW < 1), especially at high Re.More general (and perhaps non-resonant) 3D interactions are predicted to dominate at higher wave amplitudes.

[48] In order to better understand the nature of instability and the paths for spectral energy transfer in our simulation, we monitor the domain-averaged kinetic energy contained in various portions of the spectrum. First, we consider energy in the primary GW (k = 1, l = 0), where k and l are streamwise and spanwise wave numbers. The vertical direction is not periodic in our simulation and thus we Fourier transform the velocity fields in just x and y at each vertical grid level. We then sum the appropriate portion of the spectral energy (k = 1, l = 0 in this case) over the vertical direction in order to arrive at the domain average. Two-dimensional energy is defined as (k ≠ 1, l = 0), and 3D energy is defined as (k, l ≠ 0).

[49] Fritts et al. [2009a]used a similar decomposition and found a clear transfer of energy between the primary GW and 3D instability modes. Subsequent to wave breaking, and the generation of a much broader spectral range, wave-wave interactions became visible and these persisted to late times, even as the turbulent energy subsided. They were able to see this behavior quite clearly from the GW, 2D, and 3D energy histories as defined above.

[50] Fritts et al. [2009a]considered a purely monochromatic parent GW within the Boussinesq approximation and made use of periodic boundary conditions in all three directions. Aside from the small-amplitude, broad-band noise seed, there is no mechanism to generate visible 2D motions prior to the nonlinear stages of instability. Our simulation is somewhat more complex in that, prior to breaking, we propagate the wave naturally over several scale heights whereAGW increases beyond 1. The harmonic components of the anelastic solution discussed in Section 2.1become important in this case and are detected in our 2D energy metric. It is important to account for this effect as the GW harmonic solution components arise naturally from self interactions prior to the instability and breaking process. We view the harmonics as part of the “base state” on which the instability acts, not part of the instability itself. Within this view it is useful to subtract the energy associated with the spontaneous generation harmonics from the 2D energy metric, so that the latter is largely a measure of wave-wave interactions resulting from the instability and transition process. As discussed below, a simple, yet effective, correction procedure is constructed by rescaling measurements taken from an auxiliary 2D simulation run at lower amplitude.

[51] The lowest curve in Figure 9shows the 2D energy time history from a noise-free 2D simulation, performed for a low enough amplitude that wave breaking does not occur during the time period shown. There is a clear rise in 2D energy associated with the harmonics between 3 and 9τGW, but beyond this there is little additional growth and the energy appears to approach a steady state. A 2D run performed at higher amplitude (dash-dot curve) shows a similar trend out to ∼9τGW. Beyond this time, however, the energy continues to rise, ultimately experiencing explosive growth as the wave breaks around 11 τGW. Results from the 3D simulation (solid curve) are similar except that rapid growth in the 2D energy begins earlier and is limited to considerably lower values following wave breaking.

Figure 9.

Evolution of the domain-averaged 2D disturbance kinetic energy for Case 1, with 2D and 3D solutions as described in the text.

[52] Spectral analysis reveals that the 2D energy prior to 9 τGW in all three cases is dominated by the first harmonic (k = 2). Since the k = 2 mode must have been created by nonlinear interactions between two fundamental modes (k = 1), each with amplitude AGW, the harmonic should have amplitude proportional to math formula, and energy proportional to math formula. This fact makes it possible to extrapolate the portion of 2D energy associated with the first harmonic from a low amplitude (non-breaking) to a high amplitude (breaking) case. Such an extrapolation is shown inFigure 9 where the data for the AGW = 1.12 case was multiplied by the factor (1.4/1.12)4and replotted as the curve labeled “2D Run A = 1.12, rescaled”. This curve collapses extremely well with the data from the two other simulations run at higher amplitude, prior to a time of about 9 wave periods. We can consider the rescaled curve from the lower amplitude case as a “baseline” for 2D energy and subtract this from the results of wave-breaking simulations in order to better isolate the growth in 2D energy associated with instability, turbulence and associated wave-wave interactions.

[53] Exactly such a correction was applied to the data of Figure 10, which shows the GW, 3D, and corrected 2D energy time histories for the 3D wave-breaking simulation. The 2D baseline energy was added to the GW curve as the harmonic component is really an integral part of the parent wave. All energies in this figure are normalized by the hypothetical domain-averaged KE for an identical wave that did not break (estimated from theA = 1.2 2D case).

Figure 10.

Evolution of the domain-averaged disturbance kinetic energy for Case 1. The 2D energy excludes the GW.

[54] When the harmonic energy is accounted for properly, there is no visible rise in either 2D or 3D energy during the initiation phase from t = 0 to 9.5 τGW. In contrast, the GW energy rises substantially over this time period and achieves about 95% of its expected non-breaking steady state value by 9.5τGW. The wave begins to break at this time and we note a near exclusive transfer of GW energy to 3D modes over the time interval from 9.5 to 10 τGW. This fact is consistent with the observation made earlier that the first energetically important instability structures are 3D in nature.

[55] Referring again to Figure 10, there is a visible increase in 2D energy beginning at ∼10 τGW. The initial increase is mainly due to production of the secondary vortices discussed in connection with Figure 5b. Although these vortices are not strictly 2D, they have a strong projection on the l = 0 mode. The 2D energy reaches a maximum at about 11 τGW, which is during the period when the second high-vorticity band is rolling up. Following the maximum at ∼11τGW, the 2D energy steadily decreases over the remainder of the simulation.

[56] The low level of 2D energy near the end of the simulation implies very little wave-wave interaction activity. In addition, spectral analysis reveals a smooth distribution of energy with wave number, with no tendency to favor a few isolated modes. This is in contrast to the simulations ofFritts et al. [2009a]where the late time behavior was dominated by wave-wave interactions. There are most likely two important reasons for this difference in behavior. FirstFritts et al. [2009a]solve a spin-down problem starting from fixed GW energy that can only decay in time. Any secondary GWs generated during the breaking process will remain until late times since they will generally be dissipated less than the small-scale 3D turbulent motions. Secondly, these secondary GWs remain in the computational domain via the periodic boundary conditions applied in all three directions. In contrast, our simulation provides a steady input of GW energy at the lower boundary, and secondary waves can either escape through the radiation conditions applied at the vertical boundaries or be dissipated in the increasingly viscous region above the breaking zone. The net result is that our simulations produce a sustained strongly turbulent breaking zone at late times with wave-wave interactions that yield secondary GWs that propagate out of the region of strong interaction.

3.2. Case 2: Including GW-Mean Flow Interactions

[57] Case 2 is identical to Case 1 in all aspects except that we now allow momentum transport within the mean flow by the GW. This case will prove to be much more complex, mainly due wave refraction caused by the induced changes to the mean wind.

3.2.1. Mean Wind Evolution

[58] The distinguishing feature of Case 2 is that the mean wind is allowed to vary in response to GW, turbulent, and viscous momentum transports. Since we begin with a uniform mean wind and ramp up the GW amplitude with time, we expect progressively larger changes in the mean wind as the GW penetrates to higher altitudes and lower mean densities. Exactly this type of behavior is seen in Figure 11, which shows mean wind profiles at several times throughout the flow evolution. As discussed below, the GW begins to break shortly after ∼8 τGW. Thus the first three profiles in Figure 11 show only the effects of transient and viscous influences on momentum transport and deposition. Initial transients are diminishing over this interval and should play only a minor role by the time corresponding to the third profile (t = 8.1 τGW). GW transience and viscous dissipation induce large changes in the mean wind at higher altitudes at early times as the (negative) momentum flux increases and the GW experiences viscous dissipation. These cause the mean wind to decelerate rapidly in the upper ∼3/4 of the domain (hence approaching the GW phase speed c = 0 in this reference frame) and to develop an increasingly intense wind shear at the lower edge of this region. By t = 8.1 τGW, the mean wind has been reduced by 70 to 80% at most altitudes above 125 km, and a thin shear layer exists between 120 and 122 km. While the flow exhibits GW breaking and turbulence beyond ∼10 τGW (see below), changes to the mean wind become more subtle during this period. The shear layer continues to descend at lower altitudes and viscous GW dissipation continues to erode the mean wind at the highest altitudes.

Figure 11.

Evolution of the mean wind profiles for Case 2. The initial mean wind is to the east (positive x-direction) with magnitudeU/ci = 1.

[59] The descending shear layer at lower altitudes includes a point where the mean wind is equal to the horizontal phase speed of the excited GW (c = 0). This is a critical level for the incident GW and causes strong refraction, instability, turbulence, and GW dissipation accompanying the reduction of |c − U(z)| as the critical level is approached. These dynamics allow very little of the incident GW to penetrate this region, though other GWs excited by these dynamics that have different horizontal phase speeds may still propagate to much higher altitudes. The implications of this shear layer are discussed further below.

3.2.2. Wave Refraction

[60] The GW is refracted to different vertical scales by both changes in N2 and changes in the mean wind with altitude. This effect can be understood from the dispersion relation together with the definition of the intrinsic frequency, ωi = ω − 2πU/λx. Substitution of this definition into equation (7) and solving for the vertical wavelength yields

display math

Our simulations are designed so that ω = 0 if the GW solution reaches a steady state. Although starting transients from the ramped initial condition as well as adjustments following refraction render ωnon-zero, this is not a dominant effect. For present purposes we can understand the driving mechanisms that lead to wave refraction by neglecting any variations inω in the above relation.

[61] Since the background thermodynamic state chosen for our simulations has an N2 that decreases with altitude above about 120 km (see Figure 1), we expect this to result in an increase in the vertical wavelength above this altitude. This effect was noted earlier and is shown in Figure 2 for Case 1. Equation (25) also indicates that a reduction in the mean wind will lead to a reduction in the vertical wavelength. These competing effects are illustrated in Figure 12which shows vertical wavelength profiles for three times during the non-turbulent portion of the evolution. At t = 4.29 the mean wind has not yet been altered significantly and thus the vertical wavelength is mainly refracted to larger scales above 120 km due to the decrease inN2 above this level (compare this profile with Figure 2). By a time of 6.20 wave periods (middle profile in Figure 12) the mean flow has been reduced significantly above 125 km, and we note a corresponding systematic reduction in the vertical wavelength with time above this altitude. This effect is even more pronounced at a time of 8.11 wave periods, by which time vertical wavelength is generally reduced by at least a factor of 1.67 above 125 km. The maximum reduction occurs at the top of the shear layer (at 126 km) where the vertical wavelength is decreased by a factor of four.

Figure 12.

Evolution of (a) the vertical wavelength and (b) Reynolds number profiles for Case 2.

[62] The mean wind shear-induced refraction to smaller vertical scales has several significant ramifications for the flow evolution. Perhaps most importantly, reduction in the vertical scale results in enhanced vertical gradients. Increased math formula implies an increase in AGW whereas increased math formula implies an increase in the rate of viscous dissipation. Although these two effects work in opposite directions in connection with the tendency for wave breaking, we can identify altitude regions where one or the other dominates. For example, Figure 12 shows a rapid decrease in the vertical wavelength in the vicinity of the shear layer. This results in a corresponding rapid increase in both AGWand in the rate of viscous dissipation. According to linear theory, the four-fold decrease in vertical wavelength noted above can result in up to a four fold increase inAGW. Although viscous dissipation within the thickness of the shear layer will lower this estimate, we can certainly expect the wave to steepen significantly and perhaps break as its amplitude increases sharply.

[63] Above the shear layer, enhanced viscous dissipation is the dominant consequence of reduced vertical wavelength. Since Re depends quadratically on the vertical wavelength, a reduction in λz by the factor ∼1.67 noted previously at nearly all altitudes above ∼125 km results in a factor 1.672 = 2.78 drop in Re over this entire altitude range. This effect is shown in Figure 12b. We see that, by a time of >∼8.1 τGW, Re is well below 300 for nearly all altitudes above 125 km. The low Re implies that the wave will be dissipated much more readily at higher altitudes in this case as compared with Case 1 where the mean flow was held fixed.

3.3. GW Amplitude Evolution and Breaking

[64] Variations of the AGW and stability index accompanying propagation through the large mean wind shear at t ∼ 8.1 τGW and thereafter are shown in Figure 13. AGW profiles and the strong shear layers in which they occur are compared at later times in Figure 14. Instabilities and turbulence due to GW breaking are shown for similar times with streamwise-vertical and spanwise-vertical cross sections of vorticity magnitude inFigures 15 and 16 for comparison with Figures 4 and 5 for Case 1.

Figure 13.

As in Figure 6 Case 2, where momentum by the GW is allowed.

Figure 14.

Direct comparison of the (a) wave amplitude and (b) mean wind profiles at various times for Case 2.

Figure 15.

Evolution of the vorticity magnitude imaged in the x-z plane for Case 2, where GW momentum transport is allowed. The images are taken at (a) 8.11, (b) 9.06, (c) 10.02, (d) 10.97, (e) 11.92, (f) 12.88, (g) 13.83, and (h) 14.79 τGW.

Figure 16.

As in Figure 15, but for vorticity magnitude in the y-z plane.

[65] We consider first the implications of refraction by the strong mean wind shear seen in Figure 12a at t ∼ 8.1 τGW. For a conservative GW not encountering a critical level (a critical level is not yet possible; see the mean wind at t ∼ 8.1 τGW in Figure 11), the momentum flux math formula is constant with altitude. From equations (8) and (9), small math formula implies that w′ ∼ (k/m)u′. Thus math formula, or math formula, and consequently math formula. Thus, the increase in m due to refraction augments the increase in AGW due to density changes alone. The propagation timescale for these combined increases is τp ∼ ∫(1/cgz)dz ∼ ∫(m/ωi)dz, where cgz is the GW vertical group velocity, and integration is over the altitude interval over which these variations occur. Assuming the upper portion of the shear occupies λz ∼ 5 km (see Figure 12a at t ∼ 8.1 τGW) and that ωi ∼ kN/m, this evolution timescale is tpτGW. The corresponding time scale for viscous dissipation is math formula ∼ 4 × 103 s at ∼125 km, which coincidentally is also ∼τGW at this altitude. Referring to Figure 13, we see that AGW ∼ 0.5 at 120 km and increases to ∼1.6 at 125 km at t ∼ 8.1 τGW. This growth by ∼3.2 agrees well with the above estimate (∼2.9) and suggests that amplitude growth allows sufficiently large AGW to enable GW instability and breaking at these altitudes even with strong viscous dissipation. At higher altitudes, however, math formula decreases roughly as ez/H from ∼τGW at ∼125 km, thus ensuring strong viscous reductions of AGW at higher altitudes that preclude instability and breaking, as seen in Figure 13.

[66] AGW profiles shown at t ∼ 10 τGW and later in Figure 14 exhibit maxima that become increasingly localized near the top of the shear layer with time (see Figure 14b). These profiles suggest an approach of the GW at these times to a critical level occurring near where U(z) = 0, with sharp increases in math formula to AGW ∼ 1 because m ∼ |zcz|−1 as a critical level at zc is approached in uniform shear. GW dissipation in such an environment is strong, resulting in nearly complete GW (and momentum flux) attenuation, largely below the critical level, with only GWs arising from these nonlinear dynamics able to propagate to higher altitudes [see Fritts, 1984; Wang et al., 2006, and references therein].

[67] The AGW and stability index evolutions discussed above have provided key clues to the expected evolution of GW breaking and turbulence in Case 2. These are also shown with vorticity magnitude cross sections in Figures 15 and 16 that parallel those shown for Case 1 in Figures 4 and 5. As expected from the discussion of AGW in the evolving mean wind shear descending from ∼125 km after t ∼ 8.1 τGW, instabilities and turbulence arise accompanying the large, but localized, AGW (see Figure 13). Initial instability (Figures 15a, 15b, 16a, and 16b) follows attainment of AGW ∼ ATot ∼ 1.6 at t ∼ 8.1 τGW at ∼125 km. Thereafter, we see rapid reductions of AGW to ∼1 at t ∼ 10 τGW and to ∼0.7 to 0.8 at later times. ATot quickly increases to ∼1.6 AGW accompanying initial instability (see Figure 13); instabilities and turbulence descend in altitude as the shear layer descends; and turbulence scales decrease as Re increases at lower altitudes. This localized GW breaking and the reductions in λz and Re (see Figure 12) accompanying the mean wind accelerations at higher altitudes also ensure small AGW and ATot and preclude further instability at higher altitudes. Importantly, large AGW are restricted to altitudes within, or immediately above, the induced mean shear. This GW breaking closely approximates critical level dynamics at this stage, suggesting very little of the initial GW survives to higher altitudes in this case.

[68] Examination of the vorticity magnitudes in Figures 15 and 16reveals weak structures having higher streamwise wave numbers, some spanwise structure, and/or opposite phase slopes in the streamwise-vertical plane compared to the initial GW at higher altitudes following initial instability. Beginning as early ast ∼ 9 τGW, higher-wave number streamwise modes are seen to modulate the initial GW vorticity field above and below the developing instabilities. These likely arise from the streamwise-localized instability character, which readily projects onto larger scales than characterize the instability vorticity structures themselves. As a transient GW source that is quasi-2D (i.e., larger coherence lengths in the spanwise than in the streamwise direction), we expect these dynamics to excite GWs propagating in both directions (to the left and right) above and below the instability layer. Excited GWs propagating downward and rightward are strongly attenuated by the mean wind shear, whereas those propagating downward and leftward experience increasing intrinsic phase speeds and readily modulate the incident GW vorticity field. Excited GWs propagating upward may have both upstream (to the left) and downstream (to the right) propagation, because mean wind shear above the instability layer is weaker and does not play as strong a filtering role. The vorticity fields seen inFigure 15 at t ∼ 10 τGW and thereafter include a superposition of such motions. At t ∼ 10 and 11 τGW, the additional motions largely modulate the remnants of the initial GW. At later times, however, we see that GWs having upstream and downstream propagation are more prevalent at different altitudes. At t > 12 τGW, for example, we see small-scale GWs propagating downstream (slanted upward to the right) at ∼120 to 130 km, a streamwise wave number 1 GW propagating upstream with a smaller phase slope at ∼130 to 145 km, and intermediate-scale GWs propagating downstream with steeper phase slopes above ∼145 km. Indeed, the upstream-propagating GW at ∼130 to 145 km is likely a remnant of the initial GW, but other GWs can only have been excited by instability dynamics at lower altitudes.

4. Comparisons of GW and Turbulence Character and Energetics in Cases 1 and 2

[69] The domain-averaged kinetic energy time history for the primary GW and other 2D and 3D motions for Case 2 are shown inFigure 17. Also shown for comparison is the GW component for Case 1 from Figure 10. As discussed in connection with Figure 10, the domain-averaged GW energy for Case 1 increases as the GW extends throughout the domain, approaching a steady state until GW instability and breaking occur. GW energy then drops abruptly beginning ∼9.5τGW.

Figure 17.

Evolution of the domain-averaged kinetic energy for Case 2. Data for Case 1 are also shown for comparison. The energy is normalized as inFigure 10.

[70] The corresponding evolution for Case 2 including mean flow changes accompanying GW momentum transport exhibits a nearly identical increase in GW energy until induced mean winds begin to impact GW structure, propagation, and instability. The departure of GW energy for Case 2 from that for Case 1 is seen to occur at t ∼ 4 τGW, which is the time at which non-negligible induced mean winds arise above ∼110 km (seeFigure 11). The maximum GW energy in Case 2 occurs at t ∼ 6.5 τGW. Unlike Case 1, the maximum GW energy, and its initial decline, in Case 2 are not due to GW breaking as these dynamics do not become significant until after t ∼ 8 τGW (see the 3D energy in Figure 17). Rather, the decline is due purely to enhanced viscous dissipation brought on by strong refraction of the GW to smaller λz accompanying reduced U(z) and ωi at higher altitudes. The slope of the GW energy decay is altered very little as the wave breaks at t ∼ 8 τGW, indicating that turbulence plays only a minor role in consuming wave energy. The dominant effect continues to be direct attenuation of the GW due to the enhanced vertical gradients associated with the refraction to smaller λz.

[71] The effect of enhanced viscous dissipation via refraction to smaller vertical scales is impressive. The GW achieves a maximum domain-averaged energy that is only ∼63% of the maximum for Case 1 in the absence of an evolving mean flow. By the time the GW breaks, its energy has decayed by an additional ∼10%, and the resulting dissipation due to small-scale turbulent motions is small compared with the direct viscous attenuation. Indeed, these dynamics comprise a positive feedback in which initial mean flow decelerations induce increased GW dissipation, momentum deposition, and further mean flow decelerations. This cycle continues with increasing intensity, annihilating the GW at higher altitudes, focusing further GW-mean flow interactions at the lowest altitudes experiencing strong accelerations, and driving an intensification and continuing descent of the mean wind shear accompanying the strong gradients inAGW, λz, ωi, and tendency for instability. The upper altitudes of this induced mean wind remain near U(z) ∼ 0, suggesting that critical level instability dynamics ultimately limit the induced mean flow, with a shear layer descent rate dictated by the incident GW momentum flux from below and implying a reduction in the descent rate varying as math formula.

[72] While the temporal evolution of GW kinetic energy for Case 2 differs in important respects from that for Case 1, there are nevertheless some similarities in the evolutions of the 2D and 3D energies. As seen in Figure 17 Case 2 exhibits initial generation of 2D and 3D structures distinct from the primary GW accompanying instability of the flow beginning at t ∼ 8 τGW. The 2D and 3D energies increase from ∼5 to 10% of the GW energy, respectively, shortly after initiation of instability to maxima as large as ∼10 to 30% of the GW energy at later times. These estimates are in reasonable agreement with the fractions seen in Figure 10 for Case 1, though with variations throughout their respective temporal evolutions. One might anticipate a common ratio of 3D to GW energy in regions of instability if the dynamics accounting for instability and turbulence generation are similar. While these regions have very different GW scales and vertical extents in Cases 1 and 2, turbulence generation appears to be confined in each case to regions where the AGW is sufficiently large to sustain instability (and a stability index ATot > 1).

[73] In both cases, we also observe an approximate quadrature relationship between the GW and 3D energies, with the 3D energy having maxima as the GW energy is declining most rapidly. This suggests that more aggressive instabilities and turbulence have a direct influence on AGW, with 3D “turbulence” energy more dependent on the source than on slow turbulence decay at high Re and later times.

[74] Major differences in the energetics of the two cases are likely due to the very different roles played by viscosity with and without GW refraction to smaller λz and ωi. In Case 1, viscosity plays a much smaller role in the dynamics, except at the highest altitudes, where it suppresses the tendency for GW instability despite GW amplitudes of AGW > 0.7 between ∼160 and 190 km prior to instability and persisting to 160 km or above at later times. Indeed, Re remains less than 300 below ∼160 km throughout the Case 1 simulation. Viscosity thus plays essentially no role in the suppression of instability at lower altitudes and later times in Case 1. In contrast, Case 2 exhibits strong refraction to smaller GW λz and ωi at all altitudes at which we see GW instability in Case 1. The consequences of this refraction are strong viscous constraints on AGW at essentially all altitudes at which instability was observed in Case 1. This is even apparent in the initial instability in Case 2, where we saw above that the timescales for GW propagation through the shear layer and for viscous dissipation are comparable, and instability only arises accompanying GW refraction to AGW > 1 in strong mean shear.

[75] Viscosity also affects the turbulent phase of the flow evolution differently in the two cases. In Case 1, turbulence exists over the fairly broad altitude range 125–145 km. Viscosity is relatively large at these altitudes, resulting in low Reynolds number turbulence with a moderate dynamic range. In contrast, Case 2 produces higher Reynolds number turbulence with increased dynamic range in the narrow altitude range between 117–125 km. These differences are readily apparent in comparing Figures 4 and 15. The differences in turbulent Reynolds number have important implications for the relative resolution of the two simulations. Measurements of Kolmogorov scale, η shows that the relative mesh spacing (Δ/η)max (with η averaged over horizontal planes and max taken over all grid levels) remains below 1.5 at all times for Case 1, indicating ideal DNS resolution. In case 2 (Δ/η)max steadily increases following wave breaking and exceeds 3.0 at a time close to 12 τGW. Beyond this time the turbulent dynamics may be affected by the inability to resolve an adequate portion of the dissipation range. This defect is not of great concern, however, since most of the important effects of refraction, instability, and wave breaking, have occurred well before a time of 12 τGW.

[76] Another difference between the two cases lies in the initial increase in 2D versus 3D energy. Comparing Figures 10 and 17, we see that the rise in 2D energy lags the rise in 3D energy for Case 1, but that the two rise simultaneously in Case 2. The occurrence of instability in the presence of strong mean shear in Case 2 appears to promote a Kelvin-Helmholtz-type instability having more nearly 2D character (with spanwise-aligned vorticity) than seen in Case 1, in which initial instabilities immediately exhibit strong 3D character.

5. Summary and Conclusions

[77] We employ a finite-volume numerical model describing theLipps and Hemler [1982]anelastic equations to simulate GW propagation and instability dynamics over the altitude range 100–200 km in the Earth's lower thermosphere. The model uses periodic boundary conditions in the horizontal directions, with radiation boundary conditions at the upper and lower boundaries to allow GWs to exit the domain to avoid artificial reflections. Realistic variations of density and viscosity allow the GW amplitude to increase naturally as they propagate to higher altitudes and exhibit nonlinear interactions, GW instability, breaking, and turbulence over broad regions spanning multiple scale heights. These capabilities significantly advance the state-of-the-art of nearly all prior GW instability studies that employed triply periodic boundary conditions with the Boussinesq approximation (e.g.,Fritts et al. [2009a, 2009b] and previous studies cited therein).

[78] Two very different cases are considered. One assumes no GW momentum transport or mean flow modifications in order to approximate GW wavelengths and frequencies observed in the MLT for which the corresponding mean flow interactions have already occurred. A second simulation addresses the consequences for the same initial GW, but explicitly includes momentum transport and mean flow modifications accompanying GW transience and dissipation. We consider these two simulations to represent the extremes of GW-mean flow interactions expected in observations of such dynamics throughout the atmosphere. Both simulations assume a 2D GW having a horizontal wavelength of 60 km, an intrinsic period of 1085 s, and a horizontal phase speed of 55.32 m/s, propagating westward against an initially constant eastward mean wind of 55.32 m/s, such that the GW is stationary in the ground-based frame and the initial frequency is zero. These parameters and the mean atmospheric structure simulated, yield a vertical wavelength that varies from ∼17 to 36 km from the lower to the upper boundary, with a Reynolds number that varies from ∼76,000 to 60 from the lower to the upper boundary. The GW in each case is ramped from zero to a constant amplitude at the lower boundary that enables GW instability and breaking occurring initially near ∼125 or 160 km altitude, depending on whether momentum transport is included or excluded.

[79] Case 1, in which GW momentum transport is suppressed (i.e., having already occurred in MLT observations), exhibits initial GW instability and breaking at ∼160 km, with instability progressing rapidly to lower altitudes as instability and turbulence erode the GW amplitude at higher altitudes. Initial instability accompanies GW amplitudes, math formula, but persists and extends to lower altitudes for AGW ∼ 0.7 to 0.8. Initial instability scales are very large at the highest altitudes because these dynamics are strongly influenced by the large kinematic viscosity and low Re at these altitudes. As GW instability descends to lower altitudes, however, Re increases strongly and the instabilities and resulting turbulence form at, and extend to, much smaller spatial scales. Importantly, GW instabilities reduce, but do not destroy, the GW, thus allowing it to continue propagating to higher altitudes with reduced amplitude.

[80] Case 2, in which GW momentum transport is allowed to drive changes in the mean flow, differs markedly by exhibiting dramatic, and deep, mean flow accelerations in the direction of GW propagation at higher altitudes. These impose rapid and increasing refraction to smaller GW λz and ωi for trailing GWs. These dynamics both (1) enhance the potential for GW instability and breaking at lower altitudes and (2) enable viscous dissipation to erode the potential for GW instability at higher altitudes. The resulting instabilities in Case 2 are much more confined vertically and limited to much lower altitudes. Mean flow modifications accompanying the GW momentum transport in Case 2 play a key role in defining the evolution of strong mean wind shears at the transition from weak to strong mean flow accelerations. A positive feedback occurs between mean flow acceleration and GW refraction to smaller λz and ωithat confine mean flow accelerations and instabilities to lower and more limited altitudes. Ultimately, this leads to a sharp wind shear that blocks upward propagation of the primary GW, achieves effective critical-level trapping of GW energy and momentum fluxes, and penetrates to lower altitudes at a rate dependent on the GW momentum flux. Instability dynamics and mean flow accelerations in this case both suppress propagation of the initial GW to higher altitudes and enable excitation of additional GWs that propagate upward and downward from the instability layer thereafter. Implications for GWs entering the thermosphere include the following: (1) instabilities and turbulence arising from deep GWs at high altitudes are likely intermittent and will have large spatial scales; (2) GWs having large momentum fluxes and exhibiting strong mean flow accelerations will induce mean flow variations that directly influence the GW structure, including refraction to smaller GWλz and ωiwith time; (3) strong GW mean flow accelerations for quasi-steady forcing will involve strong vertical gradients that will yield strong mean wind shears that intensify with time, concentrate instability and turbulence dynamics within the wind shear, and exhibit a descent with time; and (4) coincident large wind shears and layers of strong turbulence may be direct evidence of strong GW forcing of the MLT at larger spatial scales.

[81] Our simulation results are consistent with several earlier works. Franke and Robinson [1999] considered 2D simulations with and without mean flow interaction and found a similar tendency for more vigorous instability at lower altitudes when mean flow interactions were included. Similarly, Dosser and Sutherland [2011b] note a similar tendency for instability at lower altitudes when mean flow interactions are included in the simulation of wave packets. Our finding that saturation occurs below AGW = 1 is consistent with nonlinear diffusive theory [Medvedev and Klaassen, 1995, 2000; Medvedev et al., 1998; Yiğit et al., 2008, 2009].

[82] It should be noted that the current simulations were performed for a non-reacting, neutral atmosphere of fixed molecular composition. While we believe this is an acceptable approximation for the lower portion of the domain, the neglected ionospheric and chemical processes may bias the results for the upper altitudes. With this in mind, we ascribe more confidence to the results of Case 2, which has GW instability and turbulence confined to the region below 130 km. Case 1, on the other hand, shows initial instability at altitudes as high as 160 km, and thus is likely to be more strongly affected by the simplifying assumptions. In either case, confidence in the GW processes studied here can be increased by rescaling the results to lower thermospheric or mesospheric altitudes.

[83] Our ongoing research efforts with the finite-volume model are aimed at including a more realistic description of the thermospheric atmosphere. We are also in the process of considering various GW sources, GW packets localized in space and time, GW superpositions, and GW interactions with complex environments.

Acknowledgments

[84] The research described here was supported by AFOSR under contract FA9550-09-C-0197, NASA under contract NNH09CF40C, and NSF under grant ATM-0836407. Computational resources were provided by the DoD High Performance Computing Modernization Program.