Peter R. Bannon

On the Anelastic Approximation for a Compressible Atmosphere

Abstract The equations of motion for a compressible atmosphere under the influence of gravity are reexamined to determine the necessary conditions for which the anelastic approximation holds. These conditions are that (i) the buoyancy force has an O (1) effect in the vertical momentum equation, (ii) the characteristic Vertical displacement of an air parcel is comparable to the density scale height, and (iii) the horizontal variations of the thermodynamic state variables at any height are small compared to the static reference value at that height. It is shown that, as a consequence of these assumptions, two additional conditions hold for adiabatic flow. These ancillary conditions are that (iv) the spatial variation of the base-state entropy is small, and (v) the Lagrangian time scale of the motions must be lager than the inverse of the buoyancy frequency of the base state. It is argued that condition (iii) is more fundamental than (iv) and that a flow can be anelastic even if condition (iv) is violated pr...

Theoretical Foundations for Models of Moist Convection

Abstract The equations describing the dynamics and thermodynamics of cloudy air are derived using the theories of multicomponent fluids and multiphase flows. The formulation is completely general and allows the hydrometeors to have temperatures and velocities that differ from those of the dry air and water vapor. The equations conserve mass, momentum, and total thermodynamic energy. They form a complete set once terms describing the radiative processes and the microphysical processes of condensation, sublimation, and freezing are provided. An equation for the total entropy documents the entropy sources for multitemperature flows that include the exchange of mass, momentum, and energy between the hydrometeors and the moist air. It is shown, for example, that the evaporation of raindrops in unsaturated air need not produce an increase in entropy when the drops are cooler than the air. An expression for the potential vorticity in terms of the density of the moist air and the virtual potential temperature is ...

Hydrostatic Adjustment: Lamb's Problem

Abstract The prototype problem of hydrostatic adjustment for large-scale atmospheric motions is Presented. When a horizontally infinite layer of compressible fluid, initially at rest, is instantaneously heated, the fluid is no longer in hydrostatic balance since its temperature and pressure in the layer have increased while its density remains unchanged. The subsequent adjustment of the fluid is described in detail for an isothermal base-state atmosphere. The initial imbalance generates acoustic wave fronts with trailing wakes of dispersive acoustic gravity waves. There are two characteristic timescales of the adjustment. The first is the transit time it takes an acoustic front to travel from the source region to a particular location. The second timescale, the acoustic cutoff frequency, is associated with the trailing wake. The characteristic depth scale of the adjustment is the density scale height. If the depth of the heating is small compared with the scale height, the final pressure perturbation tend...

The carbonate-silicate cycle and CO2/climate feedbacks on tidally locked terrestrial planets.

Atmospheric gaseous constituents play an important role in determining the surface temperatures and habitability of a planet. Using a global climate model and a parameterization of the carbonate-silicate cycle, we explored the effect of the location of the substellar point on the atmospheric CO(2) concentration and temperatures of a tidally locked terrestrial planet, using the present Earth continental distribution as an example. We found that the substellar points location relative to the continents is an important factor in determining weathering and the equilibrium atmospheric CO(2) level. Placing the substellar point over the Atlantic Ocean results in an atmospheric CO(2) concentration of 7 ppmv and a global mean surface air temperature of 247 K, making ∼30% of the planets surface habitable, whereas placing it over the Pacific Ocean results in a CO(2) concentration of 60,311 ppmv and a global temperature of 282 K, making ∼55% of the surface habitable.

On the dynamics of the East African Jet. I. Simulation of mean conditions for July

Abstract The East African jet, also popularly called the Somali jet, is viewed as a western boundary current of the East African highlands. Inertial and Coriolis forces. bottom friction and orography are believed important in the jet dynamics. A barotropic, primitive equation model on an equatorial beta plane is used to test this hypothesis. The flow is driven by a mass source term representing the subsidence in the southern branch of the monsoon Hadley cell. Steady, zonally symmetric solutions indicate that the combination of inertial forces, surface friction and weak subsidence can provide an adequate description of the southeast trades over the South Indian Ocean. It is deduced that, in order for the easterly flow to change into westerlies south of the equator, convergence of the flow must occur at the transition latitude, and the meridional mass flux must vanish. A two-dimensional numerical model successfully simulates most of the major large-scale features of the climatological low-level flow over th...

Frontogenesis over a Mountain Ridge

Abstract The interaction Of developing two-dimensional cold and warm frontal systems with a mesoscale mountain ridge is examined. The flow of the rotating model atmosphere is assumed to be inviscid, adiabatic, and Boussinesq. The geostrophic momentum approximation is made. An imposed horizontal deformation field forces the frontogenesis. The nonlinear model equations are solved numerically in physical space using terrain-following Coordinates which incorporate the fully nonlinear, lower boundary condition. Comparison of the model results with and without topography enables assessment of the impact of the mountain on the frontogenesis. A scale analysis indicates that the most general problem of frontal interaction with an infinite mountain ridge encompasses a seven-dimensional parameter space. The scale analysis provides justification for die two-dimensional geostrophic momentum approximation and defines an inverse Richardson number as a measure of the importance of the ageostrophic advection. Sensitivity ...

Quasi-Geostrophic Frontogenesis over Topography

Abstract The problem of quasi-geostrophic frontogenesis due to a horizontal deformation field is re-examined. Exact analytic solutions of all flow fields for all times are found for the case of a vertically semi-infinite, uniformly stratified, Boussinesq atmosphere. The imposed horizontal deformation field is assumed independent of height but may translate horizontally relative to the initial potential temperature distribution and to the variable bottom topography. Only straight, infinitely long fronts and ridge-like topographies are considered. The solutions in the absence of orography confirm and extend earlier investigations for surface and occluded fronts. It is shown that the presence of monotonically sloping topography below a region of deformation leads to the formation of a surface discontinuity in potential temperature in the absence of an initial horizontal thermal gradient. The associated secondary circulation is the sum of a closed thermally direct and indirect component. The analysis for a tr...

Wave Response during Hydrostatic and Geostrophic Adjustment. Part I: Transient Dynamics

Abstract The adjustment of a compressible, stably stratified atmosphere to sources of hydrostatic and geostrophic imbalance is investigated using a linear model. Imbalance is produced by prescribed, time-dependent injections of mass, heat, or momentum that model those processes considered “external” to the scales of motion on which the linearization and other model assumptions are justifiable. Solutions are demonstrated in response to a localized warming characteristic of small isolated clouds, larger thunderstorms, and convective systems. For a semi-infinite atmosphere, solutions consist of a set of vertical modes of continuously varying wavenumber, each of which contains time dependencies classified as steady, acoustic wave, and buoyancy wave contributions. Additionally, a rigid lower-boundary condition implies the existence of a discrete mode—the Lamb mode— containing only a steady and acoustic wave contribution. The forced solutions are generalized in terms of a temporal Greens function, which repres...

Potential vorticity conservation, hydrostatic adjustment, and the anelastic approximation

Abstract An examination of the anelastic equations of Lipps and Hemler shows that the approximation requires the temperature and potential temperature scale heights of the base state are large compared to the pressure and density scale heights. As a consequence the fractional changes of the temperature and potential temperature fields relative to their base state values are equivalent. Alternatively this equivalency requires that the ratio of the ideal gas constant to the specific heat capacity at constant pressure is small. The anelastic equations are examined for their ability to conserve potential vorticity (PV). The equations are shown to be “PV correct” in the sense that they conserve potential vorticity in a manner consistent with Ertels theorem and with the assumptions of the anelastic approximation. The ability to conserve potential vorticity helps the anelastic system capture the integrated effect of the acoustic modes in Lambs hydrostatic adjustment problem. This prototype problem considers th...

A Model of Rocky Mountain Lee Cyclogenesis

Abstract A quasi-geostrophic model of cyclogenesis in the lee of the Rocky Mountains treats the cyclogenesis as a forecasting problem and uses an initial value approach. The model consists of the interaction of a growing baroclinic wave with an infinitely long mountain ridge. This transient interaction simulates many of the observed features of the phenomena, including the formation of a lee trough concurrent with the poleward displacement of the incident low on the upstream side of the mountain and the development of a lee cyclone equatorward of the unperturbed storm track. Despite this development, the low is weakened by its interaction with the orography.These results are explained physically and compared with those using a normal-mode approach to lee cyclogenesis.