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## Abstract

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## Abstract

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## Abstract

Deep quasi-geostrophic theory applies to large-scale flow whose vertical depth scale is comparable to the potential temperature scale height. The appropriate expression for the potential vorticity equation is derived from the general formulation due to Ertel. It is further shown that the *potential* temperature field on a lower boundary acts as a surface charge of potential vorticity.

Deep equivalent barotropic Rossby waves in the presence of a mean zonal wind exhibit an enhanced beta effect but a reduced phase speed. This behavior, analogous to that displayed in shallow water theory, arises due to the inclusion of compressibility effects in the deep theory. These results help clarify the applicability of shallow water theory to barotropic atmospheric flows.

A conceptual model of the role of a surface charge of potential vorticity gradient in generating a change in the relative vorticity of a fluid parcel is described.

## Abstract

Deep quasi-geostrophic theory applies to large-scale flow whose vertical depth scale is comparable to the potential temperature scale height. The appropriate expression for the potential vorticity equation is derived from the general formulation due to Ertel. It is further shown that the *potential* temperature field on a lower boundary acts as a surface charge of potential vorticity.

Deep equivalent barotropic Rossby waves in the presence of a mean zonal wind exhibit an enhanced beta effect but a reduced phase speed. This behavior, analogous to that displayed in shallow water theory, arises due to the inclusion of compressibility effects in the deep theory. These results help clarify the applicability of shallow water theory to barotropic atmospheric flows.

A conceptual model of the role of a surface charge of potential vorticity gradient in generating a change in the relative vorticity of a fluid parcel is described.

## 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 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 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

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.

## 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.

## 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.

## 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.

## 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 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.