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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

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 virtual temperature of a moist air parcel is defined as the temperature of a dry air parcel having the same mass, volume, and pressure. It is shown here that a virtual air parcel can be formed diabatically by warming the parcel to its virtual temperature while replacing its water vapor with the equivalent mass of dry air under isobaric, isochoric conditions. Conversely a saturated virtual air parcel can be formed diabatically by cooling the parcel to its saturated virtual temperature while replacing some of its dry air with the equivalent mass of water vapor under isobaric, isochoric conditions. These processes of virtualization can be represented on a vapor pressure–temperature diagram. This diagram facilitates the comparison of the relative density of two moist air parcels at the same pressure. The effects of liquid and/or solid water can also be included.

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Peter R. Bannon

Abstract

The final equilibrium state of Lamb's hydrostatic adjustment problem is found for finite amplitude heating. Lamb's problem consists of the response of a compressible atmosphere to an instantaneous, horizontally homogeneous heating. Results are presented for both isothermal and nonisothermal atmospheres.

As in the linear problem, the fluid displacements are confined to the heated layer and to the region aloft with no displacement of the fluid below the heating. The region above the heating is displaced uniformly upward for heating and downward for cooling. The amplitudes of the displacements are larger for cooling than for warming.

Examination of the energetics reveals that the fraction of the heat deposited into the acoustic modes increases linearly with the amplitude of the heating. This fraction is typically small (e.g., 0.06% for a uniform warming of 1 K) and is essentially independent of the lapse rate of the base-state atmosphere. In contrast a fixed fraction of the available energy generated by the heating goes into the acoustic modes. This fraction (e.g., 12% for a standard tropospheric lapse rate) agrees with the linear result and increases with increasing stability of the base-state atmosphere.

The compressible results are compared to solutions using various forms of the soundproof equations. None of the soundproof equations predict the finite amplitude solutions accurately. However, in the small amplitude limit, only the equations for deep convection advanced by Dutton and Fichtl predict the thermodynamic state variables accurately for a nonisothermal base-state atmosphere.

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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

Abstract

A new derivation of local available energy for a compressible, multicomponent fluid that allows for frictional, diabatic, and chemical (e.g., phase changes) processes is presented. The available energy is defined relative to an arbitrary isothermal atmosphere in hydrostatic balance with uniform total chemical potentials. It is shown that the available energy can be divided into available potential, available elastic, and available chemical energies. Each is shown to be positive definite.

The general formulation is applied to the specific case of an idealized, moist, atmospheric sounding with liquid water and ice. The available energy is dominated by available potential energy in the troposphere but available elastic energy dominates in the upper stratosphere. The available chemical energy is significant in the lower troposphere where it dominates the available elastic energy. The total available energy increases with increasing water content.

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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 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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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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