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Peter R. Bannon

Abstract

A Hamiltonian formulation for the dynamics and thermodynamics of a compressible, rotating, binary fluid subject to gravity is developed. Here, binary refers to the presence of two components of the fluid, such as solids dissolved in a liquid or gaseous and liquid water existing along with dry air. These fluids are idealized in that the influences of diffusion processes are ignored and the binary flow is restricted to a single velocity.

The equations are presented in generic form applicable to an arbitrary binary geophysical flow. The relevant Poisson bracket satisfies Jacobi's identity. Three distinct Casimir invariants are described. The first reflects the conservation of entropy and concentration of the minor component. The second is a consequence of the conservation of the absolute circulation on curves formed by the intersection of surfaces of constant entropy with surfaces of constant concentration. The third is a generic potential vorticity of the form (ω  ·  ∇λ)/ρ. Here, ω is the absolute vorticity, ρ is the total density of the fluid, and λ is any thermodynamic variable. For example, λ can be the pressure, density, temperature, or mixing ratio as well as the more common choice of potential temperature.

Available energy of the system is defined locally in the finite-amplitude as well as in the small-amplitude limit. Both definitions are partitioned into available potential and available elastic energies.

A linear stability analysis indicates that the fluid is statically stable provided the square of the sound speed is positive, the total density decreases with height, and the square of a suitably defined buoyancy frequency is positive.

The formulation is applicable to a salty ocean and to a moist atmosphere. For the atmosphere, the full theory holds in the presence of either liquid water or ice in equilibrium with its vapor.

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Peter R. Bannon

Abstract

The effect of a vertical incident wind shear on rotating airflow over a mountain ridge is discussed physically from a variety of perspectives. The apparent paradox that the shear reduces both the vertical displacement of fluid parcels aloft and the mountain anticyclone is resolved. The importance of meridional displacements in representing the static stability field is also demonstrated.

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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 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 shown to be the correct extension of Ertel's potential vorticity to moist flows. This virtual potential vorticity, along with the density field of the hydrometeors, can be inverted to obtain the other flow variables for a balanced flow.

In their most general form the equations include prognostic equations for the hydrometeors' temperature and velocity. Diagnostic equations for these fields are shown to be valid provided the diffusive timescales of heat and momentum are small compared to the dynamic timescales of interest. As a consequence of this approximation, the forces and heating acting on the hydrometeors are added to those acting on the moist air. Then the momentum equation for the moist air contains a drag force proportional to the weight of the hydrometeors, a hydrometeor loading. Similarly, the thermal energy equation for the moist air contains the heating of the hydrometeors. This additional heating of the moist air implies a diabatic loading for which the heating of the hydrometeors is realized by the moist air.

The validity of the diagnostic equations fails for large raindrops, hail, and graupel. In these cases the thermal diffusive timescales of the hydrometeors can be several minutes, and prognostic rather than diagnostic equations for their temperatures must be solved. However, their diagnostic momentum equations remain valid.

Anelastic and Boussinesq versions of the equations are also described.

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Peter R. Bannon

Abstract

Several theories of the planetary boundary layer that retain the flow accelerations in approximate form are compared. Two special test cases focus on the role of either local or convective accelerations. The semigeotriptic theory of Cullen predicts the boundary layer pumping most accurately for the cases and parameter range considered here.

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Peter R. Bannon

Abstract

No abstract available.

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Peter R. Bannon

Abstract

The linear Eady model of baroclinic instability with the geostrophic momentum (GM) approximation is solved analytically in physical space and shown to be identical to linear three-dimensional semigeostrophic theory. Both the growth rates and the wavenumber of the short-wave cutoff are larger than those predicted by quasi-geostrophic (QG) theory. This behavior arises because the effective static stability is reduced in the GM case. These results are opposite to those using standard nongeostrophic (NG) theory, and the discrepancy increases with decreasing Richardson number. Energetically, the unstable GM normal modes enhance the conversion of available potential energy compared to the QG modes and also convert available kinetic energy to eddy kinetic energy. With regards to the structure of the unstable modes, the northward tilt with height in the GM case is more consistent with NG theory than is the QG solution which displays no meridional tilt.

Additional analysis addresses the effect of assuming that either the meridional or zonal component of the perturbation wind field is geostrophic.

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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

An alternative derivation of the available energy for a geophysical fluid system is presented. It is shown that determination of the equilibrium temperature of the system by the minimization of an energy availability function is equivalent to that found by the vanishing of the entropy difference between the fluid and its equilibrium state. Applications to the atmosphere and the ocean are presented.

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Peter R. Bannon
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