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Peter R. Bannon

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 provided diabatic processes help keep a parcel's entropy close to the base-state entropy at the height of the parcel.

The resulting anelastic set of equations is new but represents a hybrid form of the equations of Dutton and Fichtl and of Lipps and Helmer for deep convection. The advantageous properties of the set include the conservation of energy, available energy, potential vorticity, and angular momentum as well as the accurate incorporation of the acoustic hydrostatic adjustment problem.

A moist version of the equations is developed that conserves energy.

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Peter R. Bannon
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Peter R. Bannon

Abstract

A new derivation of local available energy for a compressible, multicomponent fluid whose base state need not be one of rest that allows for frictional and diabatic processes is presented. The available energy is the sum of the kinetic energy and the available potential and available elastic energies. These energy contributions are defined relative to an arbitrary reference state that can be in motion. Invoking a Lagrangian perspective, it is natural to choose the reference state as the initial state of the parcel. Then the resulting energies are consistent with published formulas for single and binary compressible fluids under inviscid, adiabatic conditions.

When the parcel-theory assumption (that the pressure of the parcel is always that of the environment) is invoked, the available elastic energy is identically zero and a fluid parcel will conserve the sum of its kinetic and available potential energies for inviscid, adiabatic flow. In this case, the parcel's available potential energy is the departure of the parcel's static energy (i.e., the sum of its potential energy and enthalpy) from its initial value. Applications of the theory are made to inertial and symmetric instabilities. Typically the instability is characterized by an increase in kinetic energy at the expense of the available potential energy that becomes negative. In the inertial case, the available potential energy is the negative of the work done by the horizontal pressure gradient force. In the symmetric case, it is the negative of the work done by the horizontal pressure gradient force and the buoyancy force, and it is a modified form of the slantwise convective energy (SCAPE) that includes the work done by the transverse (i.e., perpendicular to the mean flow) Coriolis forces. A convenient method to determine the longitudinal (i.e., parallel to the mean flow) and transverse contributions to the kinetic energy is presented. For upright convection, the decrease in the parcel's available potential energy equals its convective available potential energy. Comparison to traditional energetics is made.

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Peter R. Bannon

Abstract

Earth’s climate system is a heat engine, absorbing solar radiation at a mean input temperature T in and emitting terrestrial radiation at a lower, mean output temperature T out < T in. These mean temperatures, defined as the ratio of the energy to entropy input or output, determine the Carnot efficiency of the system. The climate system, however, does no external work, and hence its work efficiency is zero. The system does produce entropy and exports it to space. The efficiency associated with this entropy production is defined for two distinct representations of the climate system. The first defines the system as the sum of the various material subsystems, with the solar and terrestrial radiation fields constituting the surroundings. The second defines the system as a control volume that includes the material and radiation systems below the top of the atmosphere. These two complementary representations are contrasted using a radiative–convective equilibrium model of the climate system. The efficiency of Earth’s climate system based on its material entropy production is estimated using the two representations.

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Peter R. Bannon

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 Ertel's 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 Lamb's hydrostatic adjustment problem. This prototype problem considers the response of a stably stratified atmosphere to an instantaneous heating that is vertically confined but horizontally uniform. In the anelastic case, the state variables adjust instantaneously to be in hydrostatic balance with the potential temperature perturbation generated by the heating. The anelastic solutions for the pressure, density, and temperature fields are identical to those for the compressible case. In contrast there is a mutual adjustment of the pressure, density, and thermal fields in the compressible case, which is not instantaneous. The total energy in the final state for the two cases is the same.

The other versions of the anelastic approximation are examined for their PV correctness and for their ability to parameterize Lamb's acoustic hydrostatic adjustment process.

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Peter R. Bannon

Abstract

The total potential energy of the atmosphere is the sum of its internal and gravitational energies. The portion of this total energy available to be converted into kinetic energy is determined relative to an isothermal, hydrostatic, equilibrium atmosphere that is convectively and dynamically “dead.” The temperature of this equilibrium state is determined by minimization of a generalized Gibbs function defined between the atmosphere and its equilibrium. Thus, this function represents the maximum amount of total energy that can be converted into kinetic energy and, hence, the available energy of the atmosphere. This general approach includes the effects of terrain, moisture, and hydrometeors. Applications are presented for both individual soundings and idealized baroclinic zones. An algorithm partitions the available energy into available baroclinic and available convective energies. Estimates of the available energetics of the general circulation suggest that atmospheric motions are primarily driven by moist and dry fluxes of exergy from the earth’s surface with an efficiency of about two-thirds.

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Peter R. Bannon

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 tends to zero and the pressure field adjusts to the initial density hold. For larger depths, there is a mutual adjustment of the pressure and density fields.

Use of the one-dimensional analogue of the conservation of Ertel's potential vorticity removes hydrostatic degeneracy and determines the final equilibrium state directly. As a result of the adjustment process, the heated layer has expanded vertically. Since the region below the layer is unaltered, the region aloft is displaced upward uniformly. As a consequence of the expansion, the pressure and temperature anomalies in the layer are reduced from their initial values immediately after the heating. Aloft both the pressure and density fields are increased but there is no change in temperature. Since the base-state atmosphere is isothermal, warm advection is absent; since the vertical displacements of air parcels is uniform aloft, compressional warming is also absent.

The energetics of the adjustment are documented. Initially all the perturbation energy resides in the heated layer with a fraction γ−1 = 71.4% stored as available potential energy, while the remainder is available elastic energy, A fraction κ = R/Cp = (γ − 1)/&gamma = 28.6% of the initial energy is lost to propagating acoustic modes. Here γ = Cp/Cv is the ratio of the specific heats and R is the ideal gas constant. The remainder of the energy is partitioned between the heated layer and the region aloft. The energy aloft appears mostly as elastic energy, and the energy in the layer appears mostly as available potential energy.

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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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Mankin Mak and Peter R. Bannon

Abstract

Abstract not available.

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Mankin Mak and Peter R. Bannon

Abstract

Analytic solutions are obtained for a prototype semigeostrophic frontal model in which cumulus heating is parameterized by applying the conventional wave-CISK scheme and the scheme of Mak in the geostrophic coordinate. Such heating schemes give rise to a vertically tilted heating distribution as suggested by observational evidence of clouds in frontal zones. The wave-CISK scheme produces a larger influence than the Mak scheme and preferentially excites smaller-scale components of the solution. Both schemes generate a significant frontogenetic development at higher levels over the front, although their impact on the temperature near the surface is smaller. The various facets of the model solutions for moist frontogenesis are compared with those for dry frontogenesis. The main differences are the significant enhancement by the heating of frontogenesis aloft and the nongeostrophic component of the circulation. Specifically, the magnitude of the ascending motion over the surface cold front can be readily increased several fold, and its length scale reduced to be comparable to that of the front itself. The propagation of the cold front into the warm sector is also slightly enhanced as a result of the heating.

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