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## 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*/*C _{p}* = (γ − 1)/&gamma = 28.6% of the initial energy is lost to propagating acoustic modes. Here γ =

*C*/

_{p}*C*is the ratio of the specific heats and

_{v}*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.

## 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*/*C _{p}* = (γ − 1)/&gamma = 28.6% of the initial energy is lost to propagating acoustic modes. Here γ =

*C*/

_{p}*C*is the ratio of the specific heats and

_{v}*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.

## 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.

## 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.

## 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.

## 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.

## Abstract

Barotropic simulations of the East African jet are extended to include the Arabian Sea branch of the flow and to allow for flow over the mountains of Africa. Large-scale mass source-sink forcing, present to the east of the model orography, drives the low-level circulation.

Many features of the southeast trades, cross-equatorial flow and southwest monsoon are simulated. Among them are the separation of the jet from the African highlands, a wind speed maximum over the Arabian Sea and a reinforcement of the southwest monsoon by the Arabian northerlies. Splitting of the jet over the Arabian Sea is not simulated.

Starting from a state of rest, a well-developed southwest monsoon is achieved in a week of simulated time. Inclusion of a prescribed Southern Hemisphere midlatitude disturbance excites a significant response in the cross-equatorial flow, even though flow is permitted over the African mountains. Downstream, the surges excite a response over both the Arabian Sea and the Bay of Bengal. The bay response lags that over the sea by one to two days and is a factor of 2 weaker. Despite the satisfaction of the necessary condition for barotropic instability, no signs of instability appear during the onset, surge or steady-state phases of the simulations.

## Abstract

Barotropic simulations of the East African jet are extended to include the Arabian Sea branch of the flow and to allow for flow over the mountains of Africa. Large-scale mass source-sink forcing, present to the east of the model orography, drives the low-level circulation.

Many features of the southeast trades, cross-equatorial flow and southwest monsoon are simulated. Among them are the separation of the jet from the African highlands, a wind speed maximum over the Arabian Sea and a reinforcement of the southwest monsoon by the Arabian northerlies. Splitting of the jet over the Arabian Sea is not simulated.

Starting from a state of rest, a well-developed southwest monsoon is achieved in a week of simulated time. Inclusion of a prescribed Southern Hemisphere midlatitude disturbance excites a significant response in the cross-equatorial flow, even though flow is permitted over the African mountains. Downstream, the surges excite a response over both the Arabian Sea and the Bay of Bengal. The bay response lags that over the sea by one to two days and is a factor of 2 weaker. Despite the satisfaction of the necessary condition for barotropic instability, no signs of instability appear during the onset, surge or steady-state phases of the simulations.

## 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.

## 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.

## 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.

## 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.

## Abstract

The East African jet, also popularly called the Somali jet, is viewed as a western boundary current of the East African highlands. Inertial and Coriolis forces. bottom friction and orography are believed important in the jet dynamics. A barotropic, primitive equation model on an equatorial beta plane is used to test this hypothesis. The flow is driven by a mass source term representing the subsidence in the southern branch of the monsoon Hadley cell.

Steady, zonally symmetric solutions indicate that the combination of inertial forces, surface friction and weak subsidence can provide an adequate description of the southeast trades over the South Indian Ocean. It is deduced that, in order for the easterly flow to change into westerlies south of the equator, convergence of the flow must occur at the transition latitude, and the meridional mass flux must vanish.

A two-dimensional numerical model successfully simulates most of the major large-scale features of the climatological low-level flow over the South Indian Ocean and cast coast of Africa during the northern summer. It is shown that while the broad outer flank of the jet is inertially controlled, with bottom friction playing a secondary role, the narrow inner flank is the result of orographically enhanced bottom friction. The mountain backbone of Madagascar is demonstrated to be essential to the development of a relative wind speed maximum at the northern tip of the island and of an upstream ridge-downstream trough pressure distribution over the island.

The sensitivity of the model jet to variations in the upstream forcing and in the friction parameterization is also examined.

## Abstract

The East African jet, also popularly called the Somali jet, is viewed as a western boundary current of the East African highlands. Inertial and Coriolis forces. bottom friction and orography are believed important in the jet dynamics. A barotropic, primitive equation model on an equatorial beta plane is used to test this hypothesis. The flow is driven by a mass source term representing the subsidence in the southern branch of the monsoon Hadley cell.

Steady, zonally symmetric solutions indicate that the combination of inertial forces, surface friction and weak subsidence can provide an adequate description of the southeast trades over the South Indian Ocean. It is deduced that, in order for the easterly flow to change into westerlies south of the equator, convergence of the flow must occur at the transition latitude, and the meridional mass flux must vanish.

A two-dimensional numerical model successfully simulates most of the major large-scale features of the climatological low-level flow over the South Indian Ocean and cast coast of Africa during the northern summer. It is shown that while the broad outer flank of the jet is inertially controlled, with bottom friction playing a secondary role, the narrow inner flank is the result of orographically enhanced bottom friction. The mountain backbone of Madagascar is demonstrated to be essential to the development of a relative wind speed maximum at the northern tip of the island and of an upstream ridge-downstream trough pressure distribution over the island.

The sensitivity of the model jet to variations in the upstream forcing and in the friction parameterization is also examined.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.