Direct numerical simulation of a breaking inertia–gravity wave

Abstract

We have performed fully resolved three-dimensional numerical simulations of a statically unstable monochromatic inertia–gravity wave using the Boussinesq equations on an $f$-plane with constant stratification. The chosen parameters represent a gravity wave with almost vertical direction of propagation and a wavelength of 3 km breaking in the middle atmosphere. We initialized the simulation with a statically unstable gravity wave perturbed by its leading transverse normal mode and the leading instability modes of the time-dependent wave breaking in a two-dimensional space. The wave was simulated for approximately 16 h, which is twice the wave period. After the first breaking triggered by the imposed perturbation, two secondary breaking events are observed. Similarities and differences between the three-dimensional and previous two-dimensional solutions of the problem and effects of domain size and initial perturbations are discussed.

On the construction of a direct numerical simulation of a breaking inertia-gravity wave in the upper mesosphere

“…The amplitude for the secondary singular vector A 2 , defined here as the maximum perturbation energy density divided by the maximum basic state energy density, was 0.02. It was shown by Remmler et al [] that only in the fully resolved simulation was the Kolmogorov length never smaller than Δ / π and that the results of the two simulations were otherwise extremely similar (hence grid converged). Figure shows the initial buoyancy field from the full‐resolution simulation and a snapshot at t = 695 s of the buoyancy field together with the kinetic energy dissipation ϵ k .…”

“…(left and right) Rotated coordinate systems for primary and secondary instability analyses [after Remmler et al , ]. c indicates the direction of the phase velocity of the wave.…”

“…One case where DNS is possible is waves in the upper mesosphere (about 80 km altitude), where due to the extremely low ambient density, ν is about 1 m 2 s −1 in the U.S. Standard Atmosphere [ NOAA et al , ]. Remmler et al [] found from simulation of a breaking statically unstable 3 km inertia‐gravity wave a Kolmogorov length of between 1 m and 3 m so that a 3‐D DNS could be achieved with on the order of 10 9 grid cells.…”

“…Instability and breaking are very different for inertia‐gravity waves of different frequencies [ Achatz , ; Lelong and Dunkerton , ], so it is difficult to extrapolate any conclusions from a DNS study to waves with higher or lower frequency. Remmler et al [] produced a DNS of a statically unstable low‐frequency inertia‐gravity wave (referred to as case I later in this paper). The low‐frequency wave decays much less than a high‐frequency wave, only to about three quarters of its initial amplitude within one wave period, about 8 h in that case.…”

“…The present work describes a systematic, hierarchical approach to analyzing an inertia‐gravity wave breaking event, combining linear modal analysis with two‐ and three‐dimensional nonlinear simulation. Aspects of this procedure have already been published in Achatz [], Fruman and Achatz [], and Remmler et al []. Three test cases were chosen, representing waves with different inherent time scales and breaking behavior.…”