Generalized modulation theory for strongly nonlinear gravity waves in a compressible atmosphere

This study investigates strongly nonlinear gravity waves in the compressible atmosphere from the Earth’s surface to the deep atmosphere. These waves are effectively described by Grimshaw’s dissipative modulation equations which provide the basis for finding stationary solutions such as mountain lee waves and testing their stability in an analytic fashion. Assuming energetically consistent boundary and far-field conditions, that is no energy flux through the surface, free-slip boundary, and finite total energy, general wave solutions are derived and illustrated in terms of realistic background fields. These assumptions also imply that the wave-Reynolds number must become less than unity above a certain height. The modulational stability of admissible, both non-hydrostatic and hydrostatic, waves is examined. It turns out that, when accounting for the self-induced mean flow, the wave-Froude number has a resonance condition. If it becomes 1/ 1 / 2 1/\sqrt 2 , then the wave destabilizes due to perturbations from the essential spectrum of the linearized modulation equations. However, if the horizontal wavelength is large enough, waves overturn before they can reach the modulational stability condition.

On the Local Available Energetics in a Moist Compressible Atmosphere

A new derivation of local available energetics for a fully compressible, nonhydrostatic, moist atmosphere is presented. The available energetics is defined relative to an arbitrary dry reference state in hydrostatic balance with stable stratification. By introducing the modified potential temperature, a positive-definite expression of the moist available potential energy (APE) is derived. The change of the moist APE must include the role of convection to function both as a source of latent heat and as an atmosphere dehumidifier. The sum of this moist APE and the available elastic energy (AEE) is the moist available energy. In the local energy cycle, the moist available energy is partly used to generate kinetic energy (KE) and partly used to lift the water vapor to the higher level where it precipitates, resulting in the increase of gravitational energy of moist species. The moist APE is converted into vertical KE through the buoyancy term; the vertical KE is converted into the AEE through the vertical perturbation pressure gradient term; and the AEE is converted into horizontal KE through the horizontal divergence/convergence term. In addition, there exist two adiabatic nonconservative processes, which act on the AEE and APE, respectively. A suitable choice of the reference state should make these two processes much less significant than the conversions between the available energy and KE. An alternative method is presented to construct such a reference state. Application to the idealized baroclinic atmosphere shows that this reference state is much more relevant to the local available energy analysis than the isothermal one.

Theory of the norm-induced metric in atmospheric dynamics

We suggest that some metrics for quantifying distances in phase space are based on linearized flows about unrealistic reference states and hence may not be applicable to atmospheric flows. A new approach of defining a norm-induced metric based on the total energy norm is proposed. The approach is based on the rigorous mathematics of normed vector spaces and the law of energy conservation in physics. It involves the innovative construction of the phase space so that energy (or a certain physical invariant) takes the form of a Euclidean norm. The metric can be applied to both linear and nonlinear flows and for small and large separations in phase space. The new metric is derived for models of various levels of sophistication: the 2-D barotropic model, the shallow-water model and the 3-D dry, compressible atmosphere in different vertical coordinates. Numerical calculations of the new metric are illustrated with analytic dynamical systems as well as with global reanalysis data. The differences from a commonly used metric and the potential for application in ensemble prediction, error growth analysis and predictability studies are discussed.

Nonlinear Atmospheric Adjustment to Momentum Forcing

A nonlinear, numerical model of a dry, compressible atmosphere is used to simulate the hydrostatic and geostrophic adjustment to a localized prescribed injection of momentum applied over 5 min. with a size characteristic of an isolated, deep, cumulus cloud. This theoretical study is relevant to the initialization of updrafts in compressible numerical weather prediction models. The four different forcings studied are vertical, divergent horizontal, and nondivergent horizontal momentum forcings, and a prescribed transverse circulation. These forcings are applied to an isothermal atmosphere, a nonisothermal atmosphere, and an atmosphere with a nonisothermal troposphere capped by an isothermal stratosphere. These scenarios are studied by analyzing the resulting perturbation fields and the energetics of the system. Potential vorticity is used to determine the possibility of steady atmospheric states. The energetics of the system are examined to observe the creation and propagation of atmospheric waves. Both traditional and available energetics are used to determine the presence and strength of these waves. Traditional energetics consist of kinetic, internal, and potential energies while available energetics consist of kinetic, available potential, and available elastic energies. The efficiencies are similar for these different energetics, though they represent different phenomena. The traditional energetics show a strong dependence on the presence of a Lamb wave, whereas in the available energetics the Lamb wave has little or no effect.