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

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

## Abstract

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

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

## 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

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

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

Analytic solutions are obtained for a prototype semigeostrophic frontal model in which cumulus heating is parameterized by applying the conventional wave-CISK scheme and the scheme of Mak in the geostrophic coordinate. Such heating schemes give rise to a vertically tilted heating distribution as suggested by observational evidence of clouds in frontal zones. The wave-CISK scheme produces a larger influence than the Mak scheme and preferentially excites smaller-scale components of the solution. Both schemes generate a significant frontogenetic development at higher levels over the front, although their impact on the temperature near the surface is smaller. The various facets of the model solutions for moist frontogenesis are compared with those for dry frontogenesis. The main differences are the significant enhancement by the heating of frontogenesis aloft and the nongeostrophic component of the circulation. Specifically, the magnitude of the ascending motion over the surface cold front can be readily increased several fold, and its length scale reduced to be comparable to that of the front itself. The propagation of the cold front into the warm sector is also slightly enhanced as a result of the heating.

## Abstract

Analytic solutions are obtained for a prototype semigeostrophic frontal model in which cumulus heating is parameterized by applying the conventional wave-CISK scheme and the scheme of Mak in the geostrophic coordinate. Such heating schemes give rise to a vertically tilted heating distribution as suggested by observational evidence of clouds in frontal zones. The wave-CISK scheme produces a larger influence than the Mak scheme and preferentially excites smaller-scale components of the solution. Both schemes generate a significant frontogenetic development at higher levels over the front, although their impact on the temperature near the surface is smaller. The various facets of the model solutions for moist frontogenesis are compared with those for dry frontogenesis. The main differences are the significant enhancement by the heating of frontogenesis aloft and the nongeostrophic component of the circulation. Specifically, the magnitude of the ascending motion over the surface cold front can be readily increased several fold, and its length scale reduced to be comparable to that of the front itself. The propagation of the cold front into the warm sector is also slightly enhanced as a result of the heating.

## 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.