A low-order model investigation of the analysis of gravity waves in the ensemble Kalman filter.

The behavior of the ensemble Kalman filter (EnKF) is examined in the context of a model that exhibits a nonlinear chaotic (slow) vortical mode coupled to a linear (fast) gravity wave of a given amplitude and frequency. It is shown that accurate recovery of both modes is enhanced when covariances between fast and slow normal-mode variables (which reflect the slaving relations inherent in balanced dynamics) are modeled correctly. More ensemble members are needed to recover the fast, linear gravity wave than the slow, vortical motion. Although the EnKF tends to diverge in the analysis of the gravity wave, the filter divergence is stable and does not lead to a great loss of accuracy. Consequently, provided the ensemble is large enough and observations are made that reflect both time scales, the EnKF is able to recover both time scales more accurately than optimal interpolation (OI), which uses a static error covariance matrix. For OI it is also found to be problematic to observe the state at a frequency that is a subharmonic of the gravity wave frequency, a problem that is in part overcome by the EnKF.However, error in themodeled gravity wave parameters can be detrimental to the performance of the EnKF and remove its implied advantages, suggesting that a modified algorithm or a method for accounting for model error is needed.

Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

Analysis of convectively-generated gravity waves in mesoscale model simulations and wind profiler observations

The characteristics of convectively-generated gravity waves during an episode of deep convection near the coast of Wales are examined in both high resolution mesoscale simulations [with the (UK) Met Oce Unified Model] and in observations from a Mesosphere-Stratosphere-Troposphere (MST) wind profiling Doppler radar. Deep convection reached the tropopause and generated vertically propagating, high frequency waves in the lower stratosphere that produced vertical velocity perturbations O(1 m/s). Wavelet analysis is applied in order to determine the characteristic periods and wavelengths of the waves. In both the simulations and observations, the wavelet spectra contain several distinct preferred scales indicated by multiple spectral peaks. The peaks are most pronounced in the horizontal spectra at several wavelengths less than 50 km. Although these peaks are most clear and of largest amplitude in the highest resolution simulations (with 1 km horizontal grid length), they are also evident in coarser simulations (with 4 km horizontal grid length). Peaks also exist in the vertical and temporal spectra (between approximately 2.5 and 4.5 km, and 10 to 30 minutes, respectively) with good agreement between simulation and observation. Two-dimensional (wavenumber-frequency) spectra demonstrate that each of the selected horizontal scales contains peaks at each of preferred temporal scales revealed by the one- dimensional spectra alone.

Inertia-Gravity Waves Emitted from Balanced Flow: Observations, Properties, and Consequences

This paper describes laboratory observations of inertia–gravity waves emitted from balanced fluid flow. In a rotating two-layer annulus experiment, the wavelength of the inertia–gravity waves is very close to the deformation radius. Their amplitude varies linearly with Rossby number in the range 0.05–0.14, at constant Burger number (or rotational Froude number). This linear scaling challenges the notion, suggested by several dynamical theories, that inertia–gravity waves generated by balanced motion will be exponentially small. It is estimated that the balanced flow leaks roughly 1% of its energy each rotation period into the inertia–gravity waves at the peak of their generation. The findings of this study imply an inevitable emission of inertia–gravity waves at Rossby numbers similar to those of the large-scale atmospheric and oceanic flow. Extrapolation of the results suggests that inertia–gravity waves might make a significant contribution to the energy budgets of the atmosphere and ocean. In particular, emission of inertia–gravity waves from mesoscale eddies may be an important source of energy for deep interior mixing in the ocean.

A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances

Impact of mass lumping on gravity and Rossby waves in 2D finite-element shallow-water models

The goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finite-element velocity/surface-elevation pairs that are used to approximate the linear shallow-water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P0-P1, RT0 and P-P1 pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results.

Spontaneous generation and impact of inertia-gravity waves in a stratified, two-layer shear flow

Inertia-gravity waves exist ubiquitously throughout the stratified parts of the atmosphere and ocean. They are generated by local velocity shears, interactions with topography, and as geostrophic (or spontaneous) adjustment radiation. Relatively little is known about the details of their interaction with the large-scale flow, however. We report on a joint model/laboratory study of a flow in which inertia-gravity waves are generated as spontaneous adjustment radiation by an evolving large-scale mode. We show that their subsequent impact upon the large-scale dynamics is generally small. However, near a potential transition from one large-scale mode to another, in a flow which is simultaneously baroclinically-unstable to more than one mode, the inertia-gravity waves may strongly influence the selection of the mode which actually occurs.