A non-linear investigation of critical levels for internal atmospheric gravity waves

Abstract

The behaviour of internal gravity waves near a critical level is investigated by means of a transient two dimensional finite difference model. All the important non-linear, viscosity and thermal conduction terms are included, but the rotational terms are omitted and the perturbations are assumed to be incompressible.For Richardson numbers greater than 2-0 the interaction of the incident wave and the mean flow is largely as predicted by the linear theory-very little of the incident wave penetrates through the critical level and almost all of the wave's energy and momentum are absorbed by changes in the original wind. However, these changes in the wind are centred above the critical level, so that the change in the wind has only a small effect on the height of the critical level. For Richardson numbers less than 2.0 and greater than 0-25 a significant fraction of the incident wave is reflected, part of which could have been predicted by the linear theory. For these stable Richardson numbers a steady state is apparently reached where the maximum wind change continues to grow slowly, but the minimum Richardson number and wave magnitudes remain constant. This condition represents a balance between the diffusion outward of the added momentum and the rate at which it is absorbed. For Richardson numbers less than 0.25, overreflexion, predicted from the linear theory, is observed, but because the system is dynamically unstable no over-reflecting steady state is ever reached.

The physics of orographic gravity wave drag

“…Booker and Bretherton's results were for low-amplitude waves. In the case of larger-amplitude, nonlinear mountain waves, Breeding [71] showed that total critical levels behave in a similar way to that predicted from linear theory (i.e., leading to nearly total momentum flux absorption) for Ri c ≥ 5, but reflect a substantial amount of wave energy when 0.25 < Ri < 1. They estimated the amount of reflection as 35% for Ri c = 0.53 and 7% for Ri c = 2.12, but were unable to establish a relation between the reflection coefficient and the wave amplitude (i.e., nonlinearity).…”

Some Influences of Background Flow Conditions on the Generation of Turbulence due to Gravity Wave Breaking above Deep Convection

“…For example, at 15 km and ϭ 3 km the standard deviation (calculated over the entire 100-km length of the domain) of the x component of the horizontal velocity is 2.69 m s Ϫ1 for the 2D model and 1.04 m s Ϫ1 for the 3D model (along y ϭ 25 km); the 2D model has wave amplitudes about 2.5 times larger than the 3D model. Larger amplitude waves will break down more readily because of nonlinear effects and wave-induced critical levels (e.g., Breeding 1971;Fritts 1982). Specifically, if the wind is written as u ϭ U ϩ uЈ, where U is the background wind and uЈ is the wave perturbation, a wave-induced critical level will occur if (U ϩ uЈ) Ϫ c ϭ 0 (i.e., the larger the wave-induced amplitude uЈ, the smaller the change in U required to induce wave breaking).…”

Full‐wave modeling of small‐scale gravity waves using Airborne Lidar and Observations of the Hawaiian Airglow (ALOHA‐93) O(1S) images and coincident Na wind/temperature lidar measurements

“…Our simulations were performed using the winds measured by the Na w/T lidar, as discussed in section 2.2, and as represented in our model using equation (1) We do so at the higher altitudes only to emphasize the 40 differences between the wave propagation into the thermosphere, while noting that the results we obtain at 110 a0 these two levels exceeds unity, and as theory predicts [e.g., Breeding, 1971], the attenuation of the wave through the critical level is large. Between about 91 and 102.5 km the intrinsic wave period never falls below about 20 min, so that the intrinsic phase speed is about 60% of that given in Table 1 showed that at lower altitudes (-60 to 80 km) the wave amplitudes were small but nonetheless perhaps detectable.…”