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Timothy Sotack and Peter R. Bannon

Abstract

Lamb’s hydrostatic adjustment problem for the linear response of an infinite, isothermal atmosphere to an instantaneous heating of infinite horizontal extent is generalized to include the effects of heating of finite duration. Three different time sequences of the heating are considered: a top hat, a sine, and a sine-squared heating. The transient solution indicates that heating of finite duration generates broader but weaker acoustic wave fronts. However, it is shown that the final equilibrium is the same regardless of the heating sequence provided the net heating is the same.

A Lagrangian formulation provides a simple interpretation of the adjustment. The heating generates an entropy anomaly that is initially realized completely as a pressure excess with no density perturbation. In the final state the entropy anomaly is realized as a density deficit with no pressure perturbation. Energetically the heating generates both available potential energy and available elastic energy. The former remains in the heated layer while the latter is carried off by the acoustic waves.

The wave energy generation is compared for the various heating sequences. In the instantaneous case, 28.6% of the total energy generation is carried off by waves. This fraction is the ratio of the ideal gas constant R to the specific heat at constant pressure c p. For the heatings of finite duration considered, the amount of wave energy decreases monotonically as the heating duration increases and as the heating thickness decreases. The wave energy generation approaches zero when (i) the duration of the heating is comparable to or larger than the acoustic cutoff period, /N A ∼ 300 s, and (ii) the thickness of the heated layer approaches zero. The maximum wave energy occurs for a thick layer of heating of small duration and is the same as that for the instantaneous case.

The effect of a lower boundary is also considered.

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Peter R. Bannon and Mankin Mak

Abstract

The temporal evolution of the frontogenetic forcing of the horizontal gradients of temperature and longfront velocity is discussed for the moist semigeostrophic frontal model of Mak and Bannon where CISK schemes formulated in geostrophic coordinates parameterize the precipitation due to conditional symmetric instability/slantwise convection.

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Peter R. Bannon and Sukyoung Lee

Abstract

A heat-engine analysis of a climate system requires the determination of the solar absorption temperature and the terrestrial emission temperature. These temperatures are entropically defined as the ratio of the energy exchanged to the entropy produced. The emission temperature, shown here to be greater than or equal to the effective emission temperature, is relatively well known. In contrast, the absorption temperature requires radiative transfer calculations for its determination and is poorly known.

The maximum material (i.e., nonradiative) entropy production of a planet’s steady-state climate system is a function of the absorption and emission temperatures. Because a climate system does no work, the material entropy production measures the system’s activity. The sensitivity of this production to changes in the emission and absorption temperatures is quantified. If Earth’s albedo does not change, material entropy production would increase by about 5% per 1-K increase in absorption temperature. If the absorption temperature does not change, entropy production would decrease by about 4% for a 1% decrease in albedo. It is shown that, as a planet’s emission temperature becomes more uniform, its entropy production tends to increase. Conversely, as a planet’s absorption temperature or albedo becomes more uniform, its entropy production tends to decrease. These findings underscore the need to monitor the absorption temperature and albedo both in nature and in climate models.

The heat-engine analyses for four planets show that the planetary entropy productions are similar for Earth, Mars, and Titan. The production for Venus is close to the maximum production possible for fixed absorption temperature.

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Peter R. Bannon and Mankin Mak

Abstract

Diabatic processes are included in a quasi-geostrophic model of surface frontogenesis due to an imposed horizontal deformation field. Analytic solutions are found for prescribed heating and for conditional instability of the second kind (CISK) parameterizations of cumulus convection.

The solutions for prescribed heating show that condensational heating has no direct effect on the surface potential temperature field. Indirectly this heating aloft may alter the surface frontogenesis by its induced ageostrophic horizontal divergences, but such an effect is estimated to be small Condensational heating does, however, increase the strength of the mid-tropospheric frontogenesis and intensifies the vertical velocity above the surface front.

In contrast prescribed boundary layer heating (e.g., surface heat transfer, subcloud evaporative cooling) modifies the surface temperature field directly and can also create a strong ageostrophic convergence near the ground.

Solutions for CISK heating parameterizations indicate that the heating and hence the ascending motion becomes concentrated in a narrow region on the warm side of the surface front. The dynamically induced heating is greater in magnitude and narrower in halfwidth for wave-CISK than for a CISK scheme proposed by Mak. An intermediate scheme which has features of both parameterizations is also studied.

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Adam R. Edson and Peter R. Bannon

Abstract

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.

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Jeffrey M. Chagnon and Peter R. Bannon

Abstract

This study compares the response to injections of mass, heat, and momentum during hydrostatic and geostrophic adjustment in a compressible atmosphere. The sensitivity of the adjustment to these different injection types is examined at varying spatial and temporal scales through analysis of the transient evolution of the fields as well as the partitioning of total energy between acoustic waves, buoyancy waves, Lamb waves, and the steady state.

The effect of a cumulus cloud on its larger-scale environment may be represented as a vertical mass source/sink and a localized warming. To examine how the response to such injections may differ, injections of mass and heat that generate identical potential vorticity (PV) distributions and, hence, identical steady states, are compared. When the duration of the injection is very short (e.g., a minute or less), the injection of mass generates a very large acoustic wave response relative to the PV-equivalent injection of heat. However, the buoyancy wave response to these two injection types is quite similar.

The responses to injections of divergent momentum in the vertical and horizontal directions are also compared. It is shown that neither divergent momentum injection generates any PV and, thus, there is no steady-state response to these injections. The waves excited by these injections generally propagate their energy in the direction of the injection. Consequently, an injection of vertical momentum is an efficient generator of vertically propagating, horizontally trapped, high-frequency buoyancy waves. Such waves have a short time scale and are therefore very sensitive to the injection duration. Analogously, an injection of divergent horizontal momentum is an efficient generator of horizontally propagating, vertically trapped low-frequency buoyancy waves that are relatively insensitive to the injection duration. Because of this difference in the response to horizontal and vertical injections of momentum, the response to the injection of an isolated updraft differs depending on whether a compensating horizontal inflow/outflow is also specified. This additional specification of inflow/outflow helps filter acoustic waves and encourages a stronger updraft that is not removed as rapidly by the buoyancy waves. This finding is relevant to the initialization of updrafts in compressible numerical weather prediction models.

Injection of nondivergent momentum generates waves in the regions of convergence/divergence produced by the deflection of the current by Coriolis forces. The energy partitioning for such an injection is sensitive to the width and depth of the current relative to the Rossby radius of deformation, but the response is insensitive to the duration of injection for time scales shorter than several hours.

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Joseph A. Zehnder and Peter R. Bannon

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 of the model to variations in the orographic and frontal Richardson numbers and the ratio of the orographic and frontal length scales is examined.

Flow over the mountain ridge results in retardation of the surface cold front on the upstream side, while rapid advection of the front across the mountain top yields an advancement of the frontal position downstream. The combination of acceleration and deceleration produces a net 100 km advancement of the front far downstream compared with the front-only case. The front is significantly weakened on the upslope side, but reappears stronger in the lee. Aloft, the upper-level front advances a similar distance in the presence of the mountain and undergoes a slight weakening. The general character of the interaction is relatively independent of the initial frontal strength.

Superposition of a warm front with the orographic disturbance results in strengthening on the upstream side and weakening in the lee. The retardation and advancement are similar to those of the cold front.

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Peter R. Bannon and Joseph A. Zehnder

Abstract

The linear problem of rotating, stratified, adiabatic, hydrostatic, Boussinesq airflow over a mountain ridge is solved analytically for the case where the spatially uniform, normally incident airflow is the sum of a steady and sinusoidally varying component. The mountain generates a response at the fundamental frequency of the wind and all higher harmonics.

During flow acceleration, the evanescent (vertically decaying) modes deepen and broaden the high-low pressure asymmetry across the ridge and increase the mountain drag. In contrast, the evanescent modes for steady airflow product only a symmetric mountain anticyclone that generates no drag. The influence of the acceleration is more pronounced for mesoscale and synoptic-scale ridges (i.e., ridges whose Rossby number is order unity or smaller) and when the fundamental period is near the inertial period.

The transience also amplifies the magnitude of the maximum wave drag over its value predicted from steady airflow theory using the instantaneous wind speed. The total acceleration reaction due to both evanescent and wave modes can be larger than this steady airflow drag.

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Jeffrey M. Chagnon and Peter R. Bannon

Abstract

The initial and steady-state response of a compressible atmosphere to an instantaneous, localized heat source is investigated analytically. Potential vorticity conservation removes geostrophic and hydrostatic degeneracy and provides a direct method for obtaining the steady-state solution. The heat source produces a vertical potential vorticity dipole that induces a hydrostatically and geostrophically balanced cyclone–anticyclone structure in the final state. For a typical deep mesoscale heating, the net displacements required for the adjustment to the final steady state include a small, O(100 m) ascent of the core of the heated air with weak far-field descent and a large, O(10 km) outward/inward lateral displacement at the top/base of the heating.

The heating initially generates available elastic and potential energy. The energy is then exchanged between kinetic, elastic, potential, and acoustic and gravity wave energy. In the final state, after the acoustic and gravity wave energy has dispersed, the remaining energy is partitioned between kinetic, and available potential and elastic energy. The fraction of wave energy increases with increasing horizontal wavenumber.

The effect of several vertical boundary conditions is assessed. It is shown that a rigid lid suppresses the vertical expansion of the heated layer and reduces the fraction of wave energy. The impact of the rigid lid on the steady-state solution is maximized for the horizontal wavenumber zero solution and when the heating takes place close to the rigid upper boundary.

The compressible solution is used as a prototype for comparing and evaluating several compressibility approximations: the anelastic, pseudo-incompressible, and modified-compressible approximations. The anelastic model omits the available elastic energetics entirely, but the pseudo-incompressible and modified-compressible models omit either its generation or storage. The result is an ambiguous projection of heating energy onto the remaining energy terms. The errors associated with these approximations are only significant on synoptic scales. Furthermore, the modified-compressible set does not conserve potential vorticity globally.

The initial response to the heating differs for each approximation. Although the initial compressible response consists of pressure and potential temperature anomalies confined to the heated layer, the modified-compressible atmosphere generates density and potential temperature anomalies but no pressure anomaly. The anelastic atmosphere undergoes an instantaneous acoustic adjustment in which pressure and density anomalies exist inside and outside of the heated region. The pseudo-incompressible atmosphere generates an instantaneous, net divergence characterized by a residual velocity remaining after the heating and an instantaneous pulse in the pressure and velocity fields.

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Peter R. Bannon and Pe-Cheng Chu

Abstract

Scale analysis indicates that five nondimensional parameters (R 0 2 ε, μ λ and kλ) characterize the disturbance generated by the steady flow of a uniform wind (U 0, V 0) incident on a mountain ridge of width a in an isothermal, uniformly rotating, uniformly stratified, vertically semi-infinite atmosphere. Here μ = h 0/H R is the ratio of the mountain height h 0 to the deformation depth H R = fa/N where f is the Coriolis parameter and N is the static buoyancy frequency. The parameters λ = H R/H and kλ are the ratios of HR to the density scale height H and the potential temperature scale height H/k respectively. There are two Rossby numbers: One based on the incident flow that is parallel to the mountain. ε = V 0/fa, and one normal to the mountain, R 0 = U 0/fa. If R 0 2 ≪1, then the mountain-parallel flow is in approximate geostrophic balance and the flow is semigeostrophic.

The semigeostrophic case reduces to the quasi-geostrophic one in the limit as μ and ε tend to zero. If the flow is Boussinesq (λ = 0), then the semigeostrophic solutions expressed in a streamfunction coordinate can be derived from the quasi-geostrophic solutions in a geometric height coordinate.

If the flow is anelastic (λ ≈ 1), no direct correspondence between the two approximations was found. However the anelastic effects are qualitatively similar for the two and lead to: (i) an increase in the strength of the mountain anticyclone, (ii) a reduction in the extent (and possible elimination) of the zone of blocked, cyclonic flow, (iii) a permanent turning of the flow proportional to the mass of air displaced by the mountain, and (iv) an increase in the ageostrophic cross-mountain flow. The last result implies an earlier breakdown of semigeostrophic theory for anelastic flow over topography.

Apart from a strengthening of the cold potential temperature anomaly over the mountain, the presence of a finite potential temperature scale height (i.e., k nonzero) does not significantly alter the flow solution.

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