Robert Gall

A Comparison of Linear Baroclinic Instability Theory with the Eddy Statistics of a General Circulation Model

Abstract The linear instability of two zonal mean flows, one computed by a general circulation model and the other corresponding to the observed winter zonal mean flow, is presented. For these calculations, we utilize a numerical primitive equation model, where the spherical geometry of the earth has been retained. By comparing the waves predicted by linear theory with the eddies that appear in the general circulation model, it is determined that significant discrepancies exist. For the wavenumber range 1 through 15, the linear theory predicts the maximum growth rate to be for wavenumbers 12-15. The wavenumbers that dominate the intermediate-scale transient eddies in the general circulation model are much longer (5–7). In addition, linear theory predicts the maximum amplitude of the geopotential perturbation for wavenumbers 5–7 to be near the earths surface, while in the general circulation model, the maximum amplitude of this quantity for wavenumbers 5–7 is at the tropopause level. Also, the phase speed...

Structural Changes of Growing Baroclinic Waves

Abstract The nonlinear interaction between a single zonal wave and the zonal mean flow is simulated with a primitive equation model. It is determined that as the wave evolves and modifies the zonal flow, the wave growth rate diminishes more rapidly near the earths surface than it does aloft, allowing the upper portions of the wave to grow to a larger amplitude than the surface disturbance. The more rapid reduction in growth rate near the surface is accomplished primarily by an increase of static stability. It is proposed that this mechanism accounts for some of the differences in wave structures between linear baroclinic instability theory (where the, maximum amplitudes of the geopotential perturbation for wavenumbers 5-7 is at the earths surface) and the eddies in a general circulation model (where the maximum amplitude for wavenumbers 5–7 is at the tropopause) that were noted by Gall (1976). In addition, the amount of kinetic energy within an individual wave at the time that the increase of kinetic en...

A Linear Analysis of the Multiple Vortex Phenomenon in Simulated Tornadoes

Abstract A simple vertically-integrated axisymmetric model is used to Calculate axisymmetric flows for different swirl ratios(s) in tornado simulators. Thew axisymmetric states are then tested for stability using a primitive-equation linear model where the waves have both an azimuthal and a vertical wavenumber. For S high enough for there to be a central downdraft in the axisymmetric vortex, the vortex is unstable; otherwise it is stable. For relatively lows only azimuthal waves 1 and 2 are unstable, with wave 1 most unstable at lows followed by 2 at somewhat highers As S is further increased, the most unstable wave shifts to 4, then 5, and so forth. With some tuning, the model predicts the transitions from 0–1 and 1–2 secondary vortices to occur at about the observed value ofs Vertical wavelength are about 3 m, but they increase with increases There are two modes of instability: one in which only waves 1 or 2 are unstable and which appears at lows and a second mode where waves 4, 5 or 6 are most unstable...

The Effects of Released Latent Heat in Growing Baroclinic Waves

Abstract The effects of released latent heat on the development of baroclinic waves are explored using two numerical experiments in which these waves are allowed to grow from small perturbations on a flow that initially was zonally constant. In one experiment the effects of released latent heat were excluded; in the other, these effects were included and the initial zonally constant flow was considered saturated everywhere. In the moist experiment the growth rates of all wavelengths were found to be significantly increased over the corresponding growth rates in the dry experiment. However, the wavelength of maximum growth rate (wavenumber 15) was the same in both the dry and moist experiments. At the time of maximum development of wavenumber 15, the kinetic energy structure in the moist experiment was quite different from that in the dry experiment. In the moist experiment there was a distinct double maximum in the vertical; while in the dry experiment most of the kinetic energy of wavenumber 15 was near ...

Baroclinic Instability and the Selection of the Zonal Scale of the Transient Eddies of Middle Latitudes

Abstract Because the linear growth rates of baroclinic waves on realistic zonal flows are largest at relatively high zonal wavenumbers (e.g., 15), the observed peaks in the transient kinetic energy spectrum cannot be explained simply by peaks in the linear growth-rate spectrum. When the growth-rate spectrum is fairly flat, as suggested by recent studies, then as the waves evolve, the decrease of the instability of the zonal flow and the increase of dissipation in the developing waves become important in determining which wavelength will dominate after the waves are fully developed. In particular, the stabilization of the zonal flow because of northward and upward eddy transport (which is primarily confined to the lower troposphere in all baroclinic waves) causes the instability of the short baroclinic waves (wavenumber > 10) to decrease more rapidly than that of the intermediate-scale waves (wavenumber <10). In addition, as it is usually modeled, dissipation increases with time more rapidly in the short w...

Cyclone-Scale Forcing of Ultralong Waves

Abstract A numerical experiment is carried out with a simplified general circulation model. In this experiment, instabilities of all wavelengths are allowed to develop simultaneously from small perturbations on a zonally symmetric flow. The initial development of the ultralong waves in this experiment is apparently forced by the interaction between the cyclone-scale waves and the basic flow in which they are embedded. Because the spectrum of the developing baroclinic waves is not monochromatic, the interaction between the cyclones and the basic flow varies with longitude, and waves longer than the cyclone scale are forced. The structure of the ultralong waves in the numerical experiment is consistent with this forcing mechanism. One implication for numerical weather prediction is that errors in forecasts of ultralong waves may be due in part to errors in the cyclone scale.

Some Effects of Momentum Diffusion on Axisymmetric Vortices

Abstract In the absence of momentum diffusion (viscous or turbulent) in a steady state axially bounded vortex, such as that produced by a Ward-type tornado simulator, a two-celled vortex configuration in which the inner cell is entirely stagnant is to be expected. Therefore, diffusion is directly responsible or at least highly influential in producing other vortex phenomena, including a single-cell structure, a vortex breakdown, a central downdraft, and a subhydrostatic central surface pressure. The means by which these are brought about by diffusion are discussed in the context of the tornado simulator. Some of the given arguments are extended in a more speculative manner to atmospheric vortices.

A Two-Dimensional Linear Stability Analysis of the Multiple Vortex Phenomenon

Abstract Two numerical models have been constructed and used to investigate the formation of secondary vortices in axisymmetrically forced rotating flows. The vortex flow examined is that developed in a laboratory vortex simulator where secondary vortices have been produced and extensively studied. The first numerical model generated a collection of steady-state, axisymmetric, two-dimensional vortex flows for a range of swirl ratios. The second model tested those flows for instability by simulating the behavior of small-amplitude, linear perturbations superimposed on the flows: amplification of the perturbations indicated instability, whereas damping indicated stability. The results of the instability study show that the vortex is stable for the lowest swirl ratios but that, above a certain value, instability persists indefinitely. The most rapidly growing wavenumber shifts steadily with increasing swirl from 1 to approximately 5 in the swirl range investigated. Growth rates were found to be high enough f...

A linear analysis of the transition curve for the baroclinic annulus

Abstract A rotating cylindrical annulus of an incompressible fluid with horizontal density gradients is studied by the use of numerical models. Steady axisymmetric states are calculated using the full Navier-Stokes equations for a broad range of thermal Rossby number (Ror) and Taylor number (Ta). These states are tested for stability to nonaxisymmetric perturbations by the use of a model based upon the linearized hydrostatic primitive equations. The results include a prediction of the transition curve, the curve separating axisymmetric flow and nonaxisymmetric flow. This predicted curve is very close to that observed in the laboratory. The structure and energetics of the fastest growing eigenmodes are examined. It is found that the structure of the linear wave, for one point in the nonaxisymmetric regime, has only small differences from the nonlinear wave calculated by Williams. The structures of the waves at this and other points are similar to the classic Eady wave, except near the extreme lower part of...

Thermally Driven Flow in a Rotating Spherical Shell: Axisymmetric States

Abstract Numerical models are utilized to study a spherical analogue of the rotating annulus experiments modeling atmospheric motion. Motivation for this work is partially provided by NASAs proposal to conduct such an experiment on Spacelab (the Atmospheric General Circulation Experiment). A liquid is contained between two rigid, co-rotating, concentric hemispheres, with thermal gradients imposed upon both spheres. Temperature are lower on the inner sphere than on the outer sphere, and decrease towards the pole. A constant radial body force (inward) is assumed. Utilizing the Navier-Stokes equations assuming symmetry about the polar axis, finite-difference numerical models obtain steady-state solutions to the equations. The differences in solutions for case of varying rotation rates and latitudinal thermal gradients are discussed and explained. Hydrostatic and nonhydrostatic solutions are compared for cylindrical and spherical cases. For the spherical shell, it is found that the differences between hydros...