Mark D. Fruman

Symmetric Stability of Compressible Zonal Flows on a Generalized Equatorial β Plane

Abstract Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial β plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector. A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Co...

Equatorially Bounded Zonally Propagating Linear Waves on a Generalized β Plane

Abstract Meridionally confined zonally propagating wave solutions to the linear hydrostatic Boussinesq equations on a generalized equatorial β plane that includes the “nontraditional” Coriolis force terms associated with the poleward component of planetary rotation are calculated. Kelvin, Rossby, inertia–gravity, and mixed Rossby–gravity modes generalize from the traditional model with the dispersion relation unchanged. The effects of the nontraditional terms on all waves are the curving upward with latitude of the surfaces of constant phase and the equatorial trapping width of the solutions (the equatorial radius of deformation) increasing by order (Ω/N)2 compared to the traditional case, where Ω is the planetary rotation rate and N the buoyancy frequency. In addition, for the Rossby, inertia–gravity, and mixed Rossby–gravity modes, there is a phase shift of O(Ω/N) in the zonal and vertical velocity components relative to the meridional component, and their spatial structures are further modified by diff...

Secondary Instabilities in Breaking Inertia-Gravity Waves

AbstractThe three-dimensionalization of turbulence in the breaking of nearly vertically propagating inertia–gravity waves is investigated numerically using singular vector analysis applied to the Boussinesq equations linearized about three two-dimensional time-dependent basic states obtained from nonlinear simulations of breaking waves: a statically unstable wave perturbed by its leading transverse normal mode, the same wave perturbed by its leading parallel normal mode, and a statically stable wave perturbed by a leading transverse singular vector. The secondary instabilities grow through interaction with the buoyancy gradient and velocity shear in the basic state. Which growth mechanism predominates depends on the time-dependent structure of the basic state and the wavelength of the secondary perturbation. The singular vectors are compared to integrations of the linear model using random initial conditions, and the leading few singular vectors are found to be representative of the structures that emerge...

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

A systematic approach to the direct numerical simulation (DNS) of breaking upper mesospheric inertia-gravity waves of amplitude close to or above the threshold for static instability is presented. Normal mode or singular vector analysis applied in a frame of reference moving with the phase velocity of the wave (in which the wave is a steady solution) is used to determine the most likely scale and structure of the primary instability and to initialize nonlinear “2.5-D” simulations (with three-dimensional velocity and vorticity fields but depending only on two spatial coordinates). Singular vector analysis is then applied to the time-dependent 2.5-D solution to predict the transition of the breaking event to three-dimensional turbulence and to initialize three-dimensional DNS. The careful choice of the computational domain and the relatively low Reynolds numbers, on the order of 25,000, relevant to breaking waves in the upper mesosphere, makes the three-dimensional DNS tractable with present-day computing clusters. Three test cases are presented: a statically unstable low-frequency inertia-gravity wave, a statically and dynamically stable inertia-gravity wave, and a statically unstable high-frequency gravity wave. The three-dimensional DNS are compared to ensembles of 2.5-D simulations. In general, the decay of the wave and generation of turbulence is faster in three dimensions, but the results are otherwise qualitatively and quantitatively similar, suggesting that results of 2.5-D simulations are meaningful if the domain and initial condition are chosen properly.

Direct Numerical Simulation of Breaking Atmospheric Gravity Waves

We present the results of fully resolved direct numerical simulations of monochromatic gravity waves breaking in the middle atmosphere. The simulations are initialized with optimal perurbations of the gives waves. Given a wavelength of 3 km, the required grid sizes range up to 3.6 billion computational cells, depending on the necessary domain size and the turbulence intensity. Our results provide an insight into the mechanics of gravity wave breaking they will be of great value for the validation of lower order methods for the prediction of wave breaking.

Numerical Simulation of Breaking Gravity Waves

Geophysical flows including stable stratification and system rotation are a special challenge for turbulence subgrid-sclae models for large-eddy simulation (LES) and hence require validation with suitable test cases. We validate different subgrid-scale models for this kind of flows using a breaking monochromatic inertia-gravity wave that has been studied before. We find that the standard Smagorinsky model cannot be recommended while the dynamic Smagorinsky model and the implicit turbulence model ALDM are suitable to simulate this complex flow with high accuracy.

Validation of Large-Eddy Simulation Methods for Gravity Wave Breaking

Abstract To reduce the computational costs of numerical studies of gravity wave breaking in the atmosphere, the grid resolution has to be reduced as much as possible. Insufficient resolution of small-scale turbulence demands a proper turbulence parameterization in the framework of a large-eddy simulation (LES). The authors validate three different LES methods—the adaptive local deconvolution method (ALDM), the dynamic Smagorinsky method (DSM), and a naive central discretization without turbulence parameterization (CDS4)—for three different cases of the breaking of well-defined monochromatic gravity waves. For ALDM, a modification of the numerical flux functions is developed that significantly improves the simulation results in the case of a temporarily very smooth velocity field. The test cases include an unstable and a stable inertia–gravity wave as well as an unstable high-frequency gravity wave. All simulations are carried out both in three-dimensional domains and in two-dimensional domains in which th...