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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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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 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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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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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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Peter R. Bannon and Thomas L. Salem Jr.

Abstract

The Ekman-Taylor boundary layer model is solved for the case of a linear variation of the geosptophic wind with height. The two-layer model couples a Monin–Obukhov similarity layer to an Ekman layer with a vertically constant eddy diffusivity. The presence of the thermal wind contributes both an along-isotherm and a cross-isotherm component to the boundary layer flow. The along-isotherm flow is supergeostrophic and results from the net downward transport of geostrophic momentum by the eddies. The cross-isotherm flow is toward the warm air and results from the Coriolis deflection of the geostrophic momentum-rich air aloft that has been mixed downward. The effect of the baroclinity (i.e., the thermal wind shear) on the wind field is conveniently summarized geometrically.

The model predicts that the surface vorticity increases in regions of cyclonic thermal vorticity (i.e., the vorticity of the thermal wind). However, anticyclonic thermal vorticity produces convergence of the low-level warmward flow and rising motion. Thus, a warm core cyclone experiences increased boundary layer convergence.

The effects of horizontal gradients in the turbulent momentum mixing on the surface vorticity, convergence, and rising motion are ascertained. For example, there is convergence of the Ekman mass transport and an upward contribution to the boundary layer pumping for mixing gradients directed downstream or to the right of the surface geostrophic wind and directed upstream or to the left of the surface thermal wind. The mixing gradients appear most sensitive to variations in the surface stability (i.e., the air - surface temperature difference).

A case study estimates the influence of these processes on the surface vorticity in a frontal zone. The surface vorticity is shown to be displaced behind (i.e., coldward of) its geostrophic location, in agreement with observations.

An appendix provides justification for the generalized Prandtl boundary layer approximation that, to lowest order, the pressure and thermal fields (and their vertical variations) in the boundary layer are those associated with the large-scale interior flow.

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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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Paul F. Fanelli and Peter R. Bannon

Abstract

A nonlinear, numerical model of a compressible atmosphere is used to simulate the hydrostatic and geostrophic adjustment to a localized prescribed heating applied over five minutes with a size characteristic of an isolated, deep, cumulus cloud. This thermal forcing generates both buoyancy waves and a horizontally propagating Lamb wave packet as well as a steady state rich in potential vorticity. The adjustments in three model atmospheres (an isothermal, a constant lapse rate, and one with a stratosphere) are studied.

The Lamb wave packet and the two lowest-order buoyancy waves are relatively unaffected by nonlinearities but the higher-order modes and the steady state are. The heating generates a vertically stacked dipole of potential vorticity with a cyclonic perturbation below an anticyclonic perturbation. In contrast to the linear results, the nonlinear dipole is severely distorted by vertical and horizontal advections. In addition, the Lamb wave packet contains some weak positive perturbation potential vorticity.

The energetics is examined using traditional and Eulerian available energetics. Traditional energetics consists of kinetic, internal, and potential energies. It is shown that the Lamb wave packet contains more total traditional energy than that input to the atmosphere by the heating. The traditional energy in the packet resides primarily in the form of internal energy and only secondarily in the form of potential energy. The passage of the Lamb wave packet produces an atmosphere that, overall, is cooler, less dense, and with less total traditional energy than the initial atmosphere. Eulerian available energetics consists of kinetic, available potential, and available elastic energies. The heating generates both available elastic and potential energy that is then converted into kinetic energy. Most of the available elastic energy projects onto the Lamb packet, while almost all of the available potential energy is associated with the buoyancy waves and the steady state.

The effects of varying the spatial and temporal scale of the heating, while keeping the net heating the same, are examined. As the duration of the heating decreases, the amount of energy projected onto the waves increases. Increasing the size of the heating decreases the amount of energy projected onto the waves.

The adjustment is kinetically more vigorous in the nonisothermal atmospheres because of the reduction in the base-state static stability. The presence of a stratosphere produces large anomalies at and above the tropopause that are linked to the vertical motions of the buoyancy wave field.

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Peter R. Bannon and Joseph A. Zehnder

Abstract

Flow incident on a mountain ridge with a linear vertical windshear is studied for a Boussinesq, adiabatic, inviscid fluid on the f-plane. A scale analysis indicates that the semigeostrophic approximation of a geostrophic mountain-parallel wind holds for sufficiently shallow mountain slopes if the Rossby number squared is small. In such a limit, the equation for the vertical displacement of a fluid parcel is elliptic if there is forward shear (wind increasing with height) or weak backward shear (wind decreasing with height) but hyperbolic if there is strong backward shear such that the incident wind vanishes at some level in the flow.

Steady-state results indicate that forward shear weakens the cold-core geostrophic mountain anticyclone predicted by barotropic theory while weak backshear strengthens it. This behavior arises from the warm- (cold-) air advection in the forward (backward) shear case. While the total ageostrophic flux of mass across the mountain peak is greater for the forward shear case, the maximum ageostrophic cross-mountain wind is less.

Results for the semigeostrophic initial-value problem with a critical level depict the development of a stronger and narrower baroclinic lee trough than for quasi-geostrophic theory.

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