Absorption of waves by large-scale winds in stratified turbulence

The atmosphere is a nonlinear stratified fluid in which internal gravity waves are present. These waves interact with the flow, resulting in wave turbulence that displays important differences with the turbulence observed in isotropic and homogeneous flows. We study numerically the role of these waves and their interaction with the large-scale flow, consisting of vertically sheared horizontal winds. We calculate their space- and time-resolved energy spectrum (a four-dimensional spectrum) and show that most of the energy is concentrated along a dispersion relation that is Doppler shifted by the horizontal winds. We also observe that when uniform winds are let to develop in each horizontal layer of the flow, waves whose phase velocity is equal to the horizontal wind speed have negligible energy. This indicates a nonlocal transfer of their energy to the mean flow. Both phenomena, the Doppler shift and the absorption of waves traveling with the wind speed, are not accounted for in current theories of stratified wave turbulence.

Figure 1

Reduced parallel energy spectrum $E\left(k_{\parallel }\right)$ for the three simulations with three-dimensional forcing. A $k_{\parallel }^{-3}$ slope is shown only as a reference.

Figure 2

Space-resolved axisymmetric energy spectrum $e(k_{\perp },k_{\parallel })$ for two runs with (a) $\text{Fr}\approx 0.02$ and (b) $\text{Fr}\approx 0.01$. Note the spectral anisotropy resulting from stratification. The solid line indicates the modes with wave period equal to the eddy turnover time, ${\tau }_{\omega }={\tau }_{\mathrm{NL}}$. In (b) a ridge is formed close to this curve, indicating energy is transferred towards modes with lower $k_{\perp }$ but the transfer is halted when ${\tau }_{\omega }\approx {\tau }_{\mathrm{NL}}$. Modes with ${\tau }_{\omega }<{\tau }_{\mathrm{NL}}$ (i.e., modes below the solid line) are often called wave modes, as these modes have the wave period as the fastest time scale. However, a proper characterization of waves requires space- and time-resolved spectra.

Figure 3

Space- and time-resolved spectrum of the potential energy $E_{\theta }(k_x=0,k_y,k_z,\omega )$ (normalized by $E\left(\mathbf{k}\right)$, with 0 corresponding to white and 1 corresponding to black), for $\text{Fr}=0.01$ and for two different values of $k_z$: (a) $k_z=0$ and (b) $k_z=10$. The fundamental dispersion relation ${\omega }_0\left(\mathbf{k}\right)$ from Eq. (3) is given by the thin solid curve, along with two Doppler-shifted branches with $U_y=\pm 0.4$ (dashed curves). Energy is mostly concentrated in the fan defined by the two shifted branches, although not uniformly distributed. Waves that travel with the flow (upper half fan) concentrate most of the power. The area shaded with light gray (transparent orange in the online version) corresponds to $\omega <U_y{k}_y$ with $U_y=0.4$; note there is almost no power in this region. The defect of energy in all modes in this area indicates these waves are absorbed by critical layers (CL), with their energy being transferred to the flow. Inset: Fraction of the energy $F$ that is contained within the two Doppler-shifted branches as a function of the wave number. In the inertial range, $\approx 80\%$ of the energy corresponds to Doppler-shifted waves.

Figure 4

Probability density function for horizontal gradients of the vertical velocity being aligned or antialigned with the mean horizontal flow, minus the probability of having uniformly distributed alignment between the fields [$\theta$ is uniformly distributed in $[0,2\pi )$]. The PDF indicates (independently of Fig. 3) that there is an excess of waves traveling with the mean horizontal flow (compared with against the flow) and also a defect of waves propagating perpendicular to the flow. Inset: Probability distribution of the Cartesian components of the mean horizontal velocity $\mathbf{U}$ and of the pointwise horizontal velocity ${\mathbf{u}}_{\perp }$. The dotted vertical lines indicate the values $\pm 0.4$, used to draw the Doppler-shifted branches in Fig. 3.

Figure 5

Space- and time-resolved spectrum of vertical kinetic energy $E_z(k_x=0,k_y,k_z=10,\omega )$, normalized by $E_z(\mathbf{k}$), and for $\text{Fr}=0.01$. There is no discernible difference with the space and time spectrum of the potential temperature (see Fig. 3). The solid line corresponds to the linear dispersion relation of gravity waves, while the dashed lines indicate the two Doppler-shifted dispersion relations with $U_y=\pm 0.4$. The region shaded with light gray (transparent orange in the online version) indicates that modes that can have a CL absorption; note the lack of energy in that region. Inset: Fraction of the energy $F$ that is contained within the two Doppler-shifted branches as a function of the wave number.

Figure 6

Space- and time-resolved spectrum of the potential energy $E_{\theta }(k_x=0,k_y,k_z=10,\omega )$ [normalized by $E_{\theta }(\mathbf{k}$)] for $\text{Fr}=0.02$ and for two different forcing functions: (a) three-dimensional forcing and (b) Taylor-Green forcing. Labels are as in Fig. 5. The insets show in each case the fraction of the energy $F$ that is contained within the two Doppler-shifted branches. In (b), the inset on the right also shows a schematic representation of the Taylor-Green forcing. In (a), note the decrease in the level of stratification of the system still preserves the main features of the spectrum. In (b), the Taylor-Green forcing injects most of the energy in vortical modes ($\omega \approx 0$), the energy in the waves is therefore smaller, and the flow geometry prevents the CL absorption from developing.

Figure 7

Vertical cuts (at constant $x=\pi$) of the potential temperature $\theta$, and of the $y$ component of the velocity field $u_y$, for simulations with the same $\text{Fr}=0.02$ but with different forcing functions. Panels (a) and (b) correspond to Taylor-Green forcing: (a) $u_y(y,z)$ and (b) $\theta (y,z)$. Panels (c) and (d) correspond to three-dimensional forcing: (c) $u_y(y,z)$ and (d) $\theta (y,z)$. Note the more evident horizontal winds in the simulation with three-dimensional forcing, the different nature of the fluctuations in the temperature in both flows, and that, despite the overturning motions, horizontal structures and vertical shear are still present in the Taylor-Green simulation.

Figure 8

Space- and time-resolved spectrum of the potential energy $E_{\theta }(k_x=0,k_y,k_z,\omega )$ [normalized by $E_{\theta }(\mathbf{k}$)] for two values of $k_z$: (a) $k_z=0$ and (b) $k_z=10$. There is no mechanical forcing in this simulation but a randomly generated, isotropic, and constant-in-time external source of temperature fluctuations. The flow is then dominated by gravity waves, with an almost negligible large-scale horizontal flow. Note the absence of CL absorption and the negligible Doppler shift; most of the energy is concentrated along the dispersion relation for the waves.