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Abstract
The prototype problem of hydrostatic adjustment for large-scale atmospheric motions is Presented. When a horizontally infinite layer of compressible fluid, initially at rest, is instantaneously heated, the fluid is no longer in hydrostatic balance since its temperature and pressure in the layer have increased while its density remains unchanged. The subsequent adjustment of the fluid is described in detail for an isothermal base-state atmosphere.
The initial imbalance generates acoustic wave fronts with trailing wakes of dispersive acoustic gravity waves. There are two characteristic timescales of the adjustment. The first is the transit time it takes an acoustic front to travel from the source region to a particular location. The second timescale, the acoustic cutoff frequency, is associated with the trailing wake. The characteristic depth scale of the adjustment is the density scale height. If the depth of the heating is small compared with the scale height, the final pressure perturbation tends to zero and the pressure field adjusts to the initial density hold. For larger depths, there is a mutual adjustment of the pressure and density fields.
Use of the one-dimensional analogue of the conservation of Ertel's potential vorticity removes hydrostatic degeneracy and determines the final equilibrium state directly. As a result of the adjustment process, the heated layer has expanded vertically. Since the region below the layer is unaltered, the region aloft is displaced upward uniformly. As a consequence of the expansion, the pressure and temperature anomalies in the layer are reduced from their initial values immediately after the heating. Aloft both the pressure and density fields are increased but there is no change in temperature. Since the base-state atmosphere is isothermal, warm advection is absent; since the vertical displacements of air parcels is uniform aloft, compressional warming is also absent.
The energetics of the adjustment are documented. Initially all the perturbation energy resides in the heated layer with a fraction γ−1 = 71.4% stored as available potential energy, while the remainder is available elastic energy, A fraction κ = R/Cp = (γ − 1)/&gamma = 28.6% of the initial energy is lost to propagating acoustic modes. Here γ = Cp /Cv is the ratio of the specific heats and 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/Cp = (γ − 1)/&gamma = 28.6% of the initial energy is lost to propagating acoustic modes. Here γ = Cp /Cv is the ratio of the specific heats and 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
A quasi-geostrophic model of cyclogenesis in the lee of the Rocky Mountains treats the cyclogenesis as a forecasting problem and uses an initial value approach. The model consists of the interaction of a growing baroclinic wave with an infinitely long mountain ridge. This transient interaction simulates many of the observed features of the phenomena, including the formation of a lee trough concurrent with the poleward displacement of the incident low on the upstream side of the mountain and the development of a lee cyclone equatorward of the unperturbed storm track. Despite this development, the low is weakened by its interaction with the orography.These results are explained physically and compared with those using a normal-mode approach to lee cyclogenesis.
Abstract
A quasi-geostrophic model of cyclogenesis in the lee of the Rocky Mountains treats the cyclogenesis as a forecasting problem and uses an initial value approach. The model consists of the interaction of a growing baroclinic wave with an infinitely long mountain ridge. This transient interaction simulates many of the observed features of the phenomena, including the formation of a lee trough concurrent with the poleward displacement of the incident low on the upstream side of the mountain and the development of a lee cyclone equatorward of the unperturbed storm track. Despite this development, the low is weakened by its interaction with the orography.These results are explained physically and compared with those using a normal-mode approach to lee cyclogenesis.
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
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
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.
Abstract
Solutions for steady inviscid quasi- and semi-geostrophic flow over a mountain ridge on the f-plane are degenerate in the sense that an arbitrary constant mountain-parallel flow can be added to the solution. It is shown that consideration of the problem as an initial value one removes this degeneracy. The quasi-geostrophic results presented here for a semi-infinite atmosphere vary for different initial conditions according to whether the flow is Boussinesq, anelastic, or deep. We enumerate conditions for which a mountain drag and an upstream influence exists.
Abstract
Solutions for steady inviscid quasi- and semi-geostrophic flow over a mountain ridge on the f-plane are degenerate in the sense that an arbitrary constant mountain-parallel flow can be added to the solution. It is shown that consideration of the problem as an initial value one removes this degeneracy. The quasi-geostrophic results presented here for a semi-infinite atmosphere vary for different initial conditions according to whether the flow is Boussinesq, anelastic, or deep. We enumerate conditions for which a mountain drag and an upstream influence exists.
Abstract
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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.
