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Louisa B. Nance

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Louisa B. Nance
and
Dale R. Durran

Abstract

The generation of nonstationary trapped mountain lee waves through nonlinear wave dynamics without any concomitant change in the background flow is investigated by conducting two-dimensional mountain wave simulations. These simulations demonstrate that finite-amplitude lee-wave patterns can exhibit temporal variations in local wavelength and amplitude, even when the background flow is perfectly steady. For moderate amplitudes, a nonlinear wave interaction involving the stationary trapped wave and a pair of nonstationary waves appears to be responsible for the development of nonstationary perturbations on the stationary trapped wave. This pair of nonstationary waves consists of a trapped wave and a vertically propagating wave, both having horizontal wavelengths approximately twice that of the stationary trapped wave. As the flow becomes more nonlinear, the nonstationary perturbations involve a wider spectrum of horizontal wavelengths and may dominate the overall wave pattern at wave amplitudes significantly below the threshold required to produce wave breaking. Sensitivity tests in which the wave propagation characteristics of the basic state are modified without changing the horizontal wavelength of the stationary trapped wave indicate these nonstationary perturbations are absent when the background flow does not support nonstationary trapped waves with horizontal wavelengths approximately twice that of the stationary trapped mode. These sensitivity tests also show that a second nonstationary trapped wave can assume the role of the nonstationary vertically propagating wave when the Scorer parameter in the upper layer is reduced below the threshold that will support the vertically propagating wave. In this case, a resonant triad composed of three trapped waves appears to be responsible for the development of nonstationary perturbations.

The simulations suggest that strongly nonlinear wave dynamics can generate a wider range of nonstationary trapped modes than that produced by temporal variations in the background flow. It is suggested that the irregular variations in lee-wave wavelength and amplitude observed in real atmospheric flows and the complex fluctuations above a fixed point that are occasionally found in wind profiler observations of trapped lee waves are more likely to be generated by nonlinear wave dynamics than changes in the background flow.

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Louisa B. Nance
and
Dale R. Durran

Abstract

The impact of mean-flow variability on finite-amplitude trapped mountain lee waves is investigated by conducting two-dimensional mountain wave simulations for a set of idealized, time-dependent background flows. The lee-wave patterns generated by these time-dependent flows depend on two factors: 1) the degree to which the transition in the background flow changes the amplitude of the stationary trapped lee wave and 2) the difference between the group velocities of the trapped waves generated before and after the transition. When the transition in the background flow significantly reduces the amplitude of the stationary lee wave, the lee-wave pattern generated prior to the transition gradually drifts downstream away from the mountain or back over the mountain, depending on the sign of this wave packet’s group velocity after the transition. When the transition in the background flow changes the resonant wavelength while leaving the lee-wave amplitude relatively unchanged, the lee-wave train develops either 1) a smooth transition in horizontal wavelength or 2) a region of irregular variations in wavelength and amplitude, depending on the difference between the group velocities of the waves generated before and after the transition. Although linear theory is able to predict how changes in the background flow will affect the group velocities of the trapped waves, it is not able to predict whether the temporal variations in the large-scale environmental flow will amplify or dampen the resonant waves when the waves are no longer linear.

Regions of irregular variations in wavelength and amplitude may develop when stationary trapped waves generated after a transition in the background flow overtake the trapped waves generated before the transition. The fluctuations in the vertical velocities associated with such numerically simulated lee waves are compared with wind profiler observations. Estimates of the time required for the trapped waves generated after the transition to overtake those generated before the transition suggest that the temporal changes in the background flow required to qualitatively reproduce the observed vertical velocity variations are not likely to occur on a realistic timescale. In addition, the observed temporal variations in lee-wave vertical velocities appear to be the superposition of at least two distinct frequencies, whereas the temporal variations in the simulated waves are dominated by one distinct frequency.

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Louisa B. Nance
and
Dale R. Durran

Abstract

The accuracy of three anelastic systems (Ogura and Phillips; Wilhelmson and Ogura; Lipps and Hemler) and the pseudo-incompressible system is investigated for small-amplitude and finite-amplitude disturbances. Based on analytic solutions to the linearized, hydrostatic mountain wave problem, the accuracy of the Lipps and Hemler and pseudo-incompressible systems is distinctly superior to that of the other two systems. The linear dispersion relations indicate the accuracy of the pseudo-incompressible system should improve and the accuracy of the Lipps and Hemler system should decrease as the waves become more nonhydrostatic.

Since analytic solutions are not available for finite-amplitude disturbances, five nonlinear, nonhydrostatic numerical models based on these four systems and the complete compressible equations are constructed to determine the ability of each “sound proof” system to describe finite-amplitude disturbances. A comparison between the analytic solutions and numerical simulations of the linear mountain wave problem indicate the overall quality of the simulations is good, but the numerical errors are significantly larger than those associated with the pseudo-incompressible and Lipps and Hemler approximations. Numerical simulations of flow past a steady finite-amplitude heat source for an isothermal atmosphere and an atmosphere with an elevated inversion indicate the Lipps and Hemler and pseudo-incompressible systems also produce the most accurate approximations to the compressible solutions for finite-amplitude disturbances.

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