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## 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.

## 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.

## 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 Î³^{âˆ’1} = 71.4% stored as available potential energy, while the remainder is available elastic energy, A fraction Îº = *R*/*C _{p}* = (Î³ âˆ’ 1)/&gamma = 28.6% of the initial energy is lost to propagating acoustic modes. Here Î³ =

*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 Î³^{âˆ’1} = 71.4% stored as available potential energy, while the remainder is available elastic energy, A fraction Îº = *R*/*C _{p}* = (Î³ âˆ’ 1)/&gamma = 28.6% of the initial energy is lost to propagating acoustic modes. Here Î³ =

*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 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

Earthâ€™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 radiativeâ€“convective equilibrium model of the climate system. The efficiency of Earthâ€™s climate system based on its material entropy production is estimated using the two representations.

## Abstract

Earthâ€™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 radiativeâ€“convective equilibrium model of the climate system. The efficiency of Earthâ€™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 pressureâ€“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 pressureâ€“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 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.

## 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.

## Abstract

The problem of quasi-geostrophic frontogenesis due to a horizontal deformation field is re-examined. Exact analytic solutions of all flow fields for all times are found for the case of a vertically semi-infinite, uniformly stratified, Boussinesq atmosphere. The imposed horizontal deformation field is assumed independent of height but may translate horizontally relative to the initial potential temperature distribution and to the variable bottom topography. Only straight, infinitely long fronts and ridge-like topographies are considered. The solutions in the absence of orography confirm and extend earlier investigations for surface and occluded fronts.

It is shown that the presence of monotonically sloping topography below a region of deformation leads to the formation of a surface discontinuity in potential temperature in the absence of an initial horizontal thermal gradient. The associated secondary circulation is the sum of a closed thermally direct and indirect component.

The analysis for a translating deformation field interacting with an isolated orographic feature yields many interesting features. A cyclone-anticyclone couplet initially forms over the high ground. The cyclonic low pressure disturbance of reduced static stability can descend the leeside of the mountain before the arrival of the deformation field. The cold anticyclone remains fixed over the orography. A surface front translating with the imposed deformation field experiences a reduction in static stability before and after its passage over the mountain. An increase in static stability occurs while the front is over the mountain. The horizontal temperature gradient of a cold front is temporarily weakened as it approaches the mountain and strengthened after climbing the mountain peak. The ageostrophic vertical deformation field associated with the mountain acts to retard and weaken a surface cold front and to tilt its frontal zone (i.e., axis of maximum horizontal potential temperature gradient) toward the vertical on the upslope side of the mountain. The converse holds on the downslope. The subsequent interaction of a surface cold front with the leeside orographic cyclone leads to an increase in the low-level baroclinicity.

## Abstract

The problem of quasi-geostrophic frontogenesis due to a horizontal deformation field is re-examined. Exact analytic solutions of all flow fields for all times are found for the case of a vertically semi-infinite, uniformly stratified, Boussinesq atmosphere. The imposed horizontal deformation field is assumed independent of height but may translate horizontally relative to the initial potential temperature distribution and to the variable bottom topography. Only straight, infinitely long fronts and ridge-like topographies are considered. The solutions in the absence of orography confirm and extend earlier investigations for surface and occluded fronts.

It is shown that the presence of monotonically sloping topography below a region of deformation leads to the formation of a surface discontinuity in potential temperature in the absence of an initial horizontal thermal gradient. The associated secondary circulation is the sum of a closed thermally direct and indirect component.

The analysis for a translating deformation field interacting with an isolated orographic feature yields many interesting features. A cyclone-anticyclone couplet initially forms over the high ground. The cyclonic low pressure disturbance of reduced static stability can descend the leeside of the mountain before the arrival of the deformation field. The cold anticyclone remains fixed over the orography. A surface front translating with the imposed deformation field experiences a reduction in static stability before and after its passage over the mountain. An increase in static stability occurs while the front is over the mountain. The horizontal temperature gradient of a cold front is temporarily weakened as it approaches the mountain and strengthened after climbing the mountain peak. The ageostrophic vertical deformation field associated with the mountain acts to retard and weaken a surface cold front and to tilt its frontal zone (i.e., axis of maximum horizontal potential temperature gradient) toward the vertical on the upslope side of the mountain. The converse holds on the downslope. The subsequent interaction of a surface cold front with the leeside orographic cyclone leads to an increase in the low-level baroclinicity.

## 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.

## 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.

## Abstract

Three finite-difference global grids [the original Kurihara (OK), a modified Kurihara (MK) and latitude- longitude (LL)] are tested by comparing numerical solutions with a barotrpic free-surface model to a high-resolution control run and by comparing forecasts with a general circulation model to observations.

With the free surface model, 30-day integrations are made for three different resolutions of each grid and with three initial conditions two mathematical patterns and one 500 mb observed field. The LL grid performed well on the mathematical patterns, especially the case with zonal wavenumber 4. The numerical solutions with the high resolution MK grid also performed satisfactorily for both mathematical patterns. The OK grid did not perform as well, particularly on the case with zonal wavenumber 1. For the observed case, the LL grid in general had lower rms errors although the solutions did not depend as strongly on the three different grid types as the solutions for the mathematical patterns.

For the three-dimensional cases, the GFDL nine-level model was used for 14-day forecasts for observed conditions in March and 3-day forecasts in November. Forecast sensitivity to the different grids is low for short range. The MK grid had the lowest 500 mb rms errors for the duration of both forecasts. Both the MK and LL grids were free of the problem of anomalously high geopotential heights over the North Pole that occurred with the OK grid.

## Abstract

Three finite-difference global grids [the original Kurihara (OK), a modified Kurihara (MK) and latitude- longitude (LL)] are tested by comparing numerical solutions with a barotrpic free-surface model to a high-resolution control run and by comparing forecasts with a general circulation model to observations.

With the free surface model, 30-day integrations are made for three different resolutions of each grid and with three initial conditions two mathematical patterns and one 500 mb observed field. The LL grid performed well on the mathematical patterns, especially the case with zonal wavenumber 4. The numerical solutions with the high resolution MK grid also performed satisfactorily for both mathematical patterns. The OK grid did not perform as well, particularly on the case with zonal wavenumber 1. For the observed case, the LL grid in general had lower rms errors although the solutions did not depend as strongly on the three different grid types as the solutions for the mathematical patterns.

For the three-dimensional cases, the GFDL nine-level model was used for 14-day forecasts for observed conditions in March and 3-day forecasts in November. Forecast sensitivity to the different grids is low for short range. The MK grid had the lowest 500 mb rms errors for the duration of both forecasts. Both the MK and LL grids were free of the problem of anomalously high geopotential heights over the North Pole that occurred with the OK grid.