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## Abstract

The role of the ambient stratification in semigeostrophic surface frontogenesis is examined. Model fronts forming in regions of large static stability 1) are weaker, 2) are tilted more toward the horizontal, and 3) propagate more slowly toward the warm air than fronts forming in regions of small static stability.

These results are discussed in light of the differences between warm and cold fronts.

## Abstract

The role of the ambient stratification in semigeostrophic surface frontogenesis is examined. Model fronts forming in regions of large static stability 1) are weaker, 2) are tilted more toward the horizontal, and 3) propagate more slowly toward the warm air than fronts forming in regions of small static stability.

These results are discussed in light of the differences between warm and cold fronts.

## 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

Dynamic explanations of mountain drag usually invoke viscous effects and/or wave momentum flux by either Rossby or internal gravity waves. This paper explores an alternative mechanism in terms of the unsteadiness of the incident flow. The reaction to acceleration (local time rate of change) of the flow put a stationary obstacle can manifest itself as a contribution to the drag on the flow.

A simple model provides an estimate of this acceleration reaction in a geophysically relevant context. The shallow-water flow of a periodic current around a right-circular cylinder is determined for subinertial periods and arbitrary rotational Froude number. The results of this prototype calculation support the hypothesis that acceleration reaction may provide a substantial contribution to the mountain drag exerted by mesoscale and synoptic-scale obstacles.

## Abstract

Dynamic explanations of mountain drag usually invoke viscous effects and/or wave momentum flux by either Rossby or internal gravity waves. This paper explores an alternative mechanism in terms of the unsteadiness of the incident flow. The reaction to acceleration (local time rate of change) of the flow put a stationary obstacle can manifest itself as a contribution to the drag on the flow.

A simple model provides an estimate of this acceleration reaction in a geophysically relevant context. The shallow-water flow of a periodic current around a right-circular cylinder is determined for subinertial periods and arbitrary rotational Froude number. The results of this prototype calculation support the hypothesis that acceleration reaction may provide a substantial contribution to the mountain drag exerted by mesoscale and synoptic-scale obstacles.

## 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 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

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 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 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

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 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.