Dale R. Durran

Numerical methods for wave equations in geophysical fluid dynamics

Introduction Basic Finite-Difference Methods Beyond Scalar Wave Equations Series-Expansion Methods Finite Volume Methods 6 Semi-Lagrangian Methods Physically Insignificant Fast Waves Non-reflecting Boundary conditions Appendix: Derivations of two fundamental theorems.

An Upper Boundary Condition Permitting Internal Gravity Wave Radiation in Numerical Mesoscale Models

Abstract A radiative upper boundary condition is proposed for numerical mesoscale models which allows vertically propagating internal gravity waves to pass out of the computational domain with minimal reflection. In this formulation, the pressure along the upper boundary is determined from the Fourier transform of the vertical velocity at that boundary. This boundary condition can easily be incorporated in a wide variety of models and requires little additional computation. The radiation boundary condition is derived from the linear, hydrostatic, Boussinesq equations of motion, neglecting Coriolis effects. However, tests of this radiation boundary condition in the presence of nonhydrostatic, Coriolis, nonlinear and non-Boussinesq effects suggest that it would be effective in many mesoscale modeling applications.

A Compressible Model for the Simulation of Moist Mountain Waves

Abstract A two-dimensional, nonlinear, nonhydrostatic model is described which allows the calculation of moist airflow in mountainous terrain. The model is compressible, uses a terrain-following coordinate system, and employs lateral and upper boundary conditions which minimize wave reflections. The models accuracy and sensitivity are examined. These tests suggest that in numerical simulations of vertically propagating, highly nonlinear mountain waves, a wave absorbing layer does not accurately mimic the effects of wave breakdown and dissipation at high levels in the atmosphere. In order to obtain a correct simulation, the region in which the waves are physically absorbed must generally be included in the computational domain (a nonreflective upper boundary condition should be used as well). The utility of the model is demonstrated in two examples (linear waves in a uniform atmosphere and the 11 January 1972 Boulder windstorm) which illustrate how the presence of moisture can influence propagating waves....

On the Effects of Moisture on the Brunt-Väisälä Frequency

Abstract Expressions are derived for the Brunt- Vaisala frequency Nm, in a saturated atmosphere, which are analogous to commonly-used formulas for the dry Brunt- Vaisala frequency. These formulas are compared with others which have appeared in the literature, and the derivation by Lalas and Einaudi (1974) is found to be correct. The simplifying assumptions, implicit in derivations by Dudis (1972) and Fraser et al. (1973) are clarified. Numerical examples are presented which suggest that these incomplete formulations for Nm are reasonably accurate approximations, except when the saturated static stability is small. A new formula expressing Nm in terms of moist conservative variables is presented, and an accurate approximation is also given which may be useful when evaluating Nm.

Another Look at Downslope Windstorms. Part I: The Development of Analogs to Supercritical Flow in an Infinitely Deep, Continuously Stratified Fluid

Abstract Numerical simulations are conducted to examine the role played by different amplification mechanisms in the development of large-amplitude mountain waves. It is shown that when the static stability has a two-layer structure, the nonlinear response can differ significantly from the solution to the equivalent linear problem when the parameter Nh/U is as small as 0.3. In the cases where the nonlinear waves are much larger than their linear counterparts, the highest stability is found in the lower layer and the flow resembles a hydraulic jump. Simulations of the 11 January 1972 Boulder windstorm are presented which suggest that the transition to supercritical flow, forced by the presence of a low-level inversion, plays an essential role in triggering the windstorm. The similarities between breaking waves and nonbreaking waves which undergo a transition to supercritical flow are discussed.

Improving the Anelastic Approximation

Abstract A new diagnostic equation is presented which exhibits many advantages over the conventional forms of the anelastic continuity equation. Scale analysis suggests that use of this “pseudo-incompressible equation” is justified if the Lagrangian time scale of the disturbance is large compared with the time scale for sound wave propagation and the perturbation pressure is small compared to the vertically varying mean-state pressure. No assumption about the magnitude of the perturbation potential temperature or the strength of the mean-state stratification is required. In the various anelastic approximations, the influence of the perturbation density field on the mass balance is entirely neglected. In contrast, the mass-balance in the “pseudo-incompressible approximation” accounts for those density perturbations associated (through the equation of state) with perturbations in the temperature field. Density fluctuations associated with perturbations in the pressure field are neglected. The pseudo-incompr...

A numerical study of three-dimensional gravity waves triggered by deep tropical convection and their role in the dynamics of the QBO

Abstract A 3D mesoscale model is used to study the structure of convectively triggered gravity waves in the Tropics and their role in the dynamics of the middle atmosphere. Simulations with three stratospheric background zonal wind cases are examined. In the first case the background wind profile is constant; the other two are representative of the easterly and westerly phases of the quasi-biennial oscillation (QBO). Spectral analysis is used to link the structure of the triggered gravity waves to the dominant vertical wavelength of the latent heating within the convection. In the QBO–wind shear cases, upward propagating gravity waves are damped as they approach their critical layer. The signature of critical-layer absorption is clearly visible in the profiles of vertical momentum-flux divergence. In the simulations with open boundary conditions, the response to vertical momentum-flux divergence takes the form of large dynamic pressure differences between the east and west boundaries together with acceler...

Vortex Formation and Vortex Shedding in Continuously Stratified Flows past Isolated Topography

Abstract The flow of a nonrotating atmosphere with uniform stratification and wind speed past an isolated three-dimensional topographic obstacle is investigated with a nonhydrostatic numerical model having a free-slip lower boundary. When the mountain is sufficiently high, the transient development of a quasi-steady flow occurs in two phases. During the first phase, which occurs over a dimensionless time of O(1), the flow is essentially inviscid and adiabatic, and potential vorticity (PV) is conserved. The transient evolution of the flow during the second phase, which occurs over a dimensionless time of O(10) to O(100), is controlled by dissipation and is accompanied by the generation of PV anomalies. Two cases are examined in which the flow is forced to remain left–right symmetric with respect to the axis of the incident flow. In the first, the dimensionless mountain height NH/U is 1.5, and gravity waves break over the mountain. In the second, NH/U = 3, and a quasi-steady recirculating wake containing a ...

Mountain Waves and Downslope Winds

In an effort to enhance the reader’s physical understanding, this chapter begins with a discussion of fundamental concepts including parcel oscillations in a stable atmosphere, wave propagation, and the fluid motions associated with vertically propagating internal gravity waves. This is followed by a brief review of the theory of small-amplitude mountain waves, beginning with the case of air flowing over a series of sinusoidal ridges in a basic state with uniform wind speed and stability. The linear theory is then extended to cover isolated mountains and situations with vertical variations in the basic-state wind speed and stability.

The Dynamics of Mountain-Wave-Induced Rotors

Abstract The development of rotor flow associated with mountain lee waves is investigated through a series of high-resolution simulations with the nonhydrostatic Coupled Ocean–Atmospheric Mesoscale Prediction System (COAMPS) model using free-slip and no-slip lower boundary conditions. Kinematic considerations suggest that boundary layer separation is a prerequisite for rotor formation. The numerical simulations demonstrate that boundary layer separation is greatly facilitated by the adverse pressure gradients associated with trapped mountain lee waves and that boundary layer processes and lee-wave-induced perturbations interact synergistically to produce low-level rotors. Pairs of otherwise identical free-slip and no-slip simulations show a strong correlation between the strength of the lee-wave-induced pressure gradients in the free-slip simulation and the strength of the reversed flow in the corresponding no-slip simulation. Mechanical shear in the planetary boundary layer is the primary source of a she...