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## Abstract

Potential temperature and wind profiles obtained during a case of nighttime low-level deformation frontogenesis are examined. The winds in the lowest kilometer over a region in the central United States are shown to be controlled by thermal wind shear in a stable layer above 400 m, in accordance with the Hoskins–Bretherton semigeostrophic frontogenesis model, and by surface-drag-generated shear in the nearly neutral layer below. In this lower layer, the wind profiles are shown to be in good agreement with a simple baroclinic Ekman–Taylor model. The opposite shear in the two layers produces a low-level jet that appears in soundings taken hundreds of kilometers apart. The agreement of the observed profiles with these models is revealed only after a height-dependent inertial oscillation in both layers is removed from the rapidly evolving hourly wind data.

## Abstract

Potential temperature and wind profiles obtained during a case of nighttime low-level deformation frontogenesis are examined. The winds in the lowest kilometer over a region in the central United States are shown to be controlled by thermal wind shear in a stable layer above 400 m, in accordance with the Hoskins–Bretherton semigeostrophic frontogenesis model, and by surface-drag-generated shear in the nearly neutral layer below. In this lower layer, the wind profiles are shown to be in good agreement with a simple baroclinic Ekman–Taylor model. The opposite shear in the two layers produces a low-level jet that appears in soundings taken hundreds of kilometers apart. The agreement of the observed profiles with these models is revealed only after a height-dependent inertial oscillation in both layers is removed from the rapidly evolving hourly wind data.

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## Abstract

A low-level deformation frontogenesis event that occurred during the Stormscale Operational and Research Meteorology-Fronts Experiment Systems Test (STORM-FEST) field program is analyzed in the context of semigeostrophic theory. The observed evolution and vertical structure of the potential temperature and alongfront wind fields are compared to that predicted by both numerical and analytical solutions of the semigeostrophic equations initialized at the onset of the deformation frontogenesis. The model solutions provide relatively accurate predictions of the surface potential temperature distribution 5 h later, when the frontogenesis ended. The point along the front with the steepest potential temperature gradient is observed to move closer to the point with the highest relative vorticity by an amount that is in rough agreement with the model prediction. Vertical profiles of potential temperature from soundings show a nearly mixed layer below ∼400 m that cannot be predicted by the inviscid solutions, but there is good agreement with inviscid theory above this level. The observed profiles of alongfront wind are characterized by a low-level jet with maximum speed at the level of the inversion, and the vertical shear below the jet maximum is opposite that predicted by the thermal wind equation. The semigeostrophic model does appear to depict this frontogenesis event in the upper layer, while the lower layer is dominated by surface drag and shear-induced turbulent mixing.

## Abstract

A low-level deformation frontogenesis event that occurred during the Stormscale Operational and Research Meteorology-Fronts Experiment Systems Test (STORM-FEST) field program is analyzed in the context of semigeostrophic theory. The observed evolution and vertical structure of the potential temperature and alongfront wind fields are compared to that predicted by both numerical and analytical solutions of the semigeostrophic equations initialized at the onset of the deformation frontogenesis. The model solutions provide relatively accurate predictions of the surface potential temperature distribution 5 h later, when the frontogenesis ended. The point along the front with the steepest potential temperature gradient is observed to move closer to the point with the highest relative vorticity by an amount that is in rough agreement with the model prediction. Vertical profiles of potential temperature from soundings show a nearly mixed layer below ∼400 m that cannot be predicted by the inviscid solutions, but there is good agreement with inviscid theory above this level. The observed profiles of alongfront wind are characterized by a low-level jet with maximum speed at the level of the inversion, and the vertical shear below the jet maximum is opposite that predicted by the thermal wind equation. The semigeostrophic model does appear to depict this frontogenesis event in the upper layer, while the lower layer is dominated by surface drag and shear-induced turbulent mixing.

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## Abstract

Nonlinear geostrophic adjustment is examined with a Boussinesq model. The motion is restricted to a two-dimensional channel in the horizontal and vertical (*x*, *z*) plane; the fluid is in uniform rotation, is stably stratified, inviscid, and incompressible. The flows considered fall under two classes: zero and uniform potential vorticity flows. Steady geostrophic flow fields are determined from initial man imbalances, represented by both symmetric and antisymmetric density anomalies that vary along the *x* axis. The distinguishing characteristic of these solutions is the development of a front, defined as a zero-order discontinuity in both density and geostrophic velocity at one or both vertical boundaries. Frontal formation occurs, as previously discovered by Ou for zero potential vorticity flow, when the initial horizontal density gradient is sufficiently large. The critical values are displayed for different cases in terms of the initial amplitude and initial scale of the density anomaly.

The conversion of initial potential energy into geostrophic kinetic ΔKE and potential ΔPE energies during adjustment is also derived. Ou's result that γ = ΔKE/ΔPE = 1/2, independent of the initial scale is confirmed. It is shown, however, that γ≤1/2 for uniform potential vorticity flow. Large initial scales *a*
^{−1}, large compared to the deformation radius, have the largest values of γ, approaching γ = 1/2 as *a*→0. This limit approaches the solution and energy ratio for zero potential vorticity flow. The energy ratio associated with an antisymmetric density anomaly is characterized by γ→1/3 and *a* →∞: that is, the initial mass imbalance becomes a step function. In the other case, when the initial disturbance is symmetric and vanishes with *a*→∞, γ also vanishes. These results unify previous studies that have not provided the distinction between zero and uniform potential vorticity flows in examinations of the energy conversion process. Yet the reason for this distinction has not been delineated.

## Abstract

Nonlinear geostrophic adjustment is examined with a Boussinesq model. The motion is restricted to a two-dimensional channel in the horizontal and vertical (*x*, *z*) plane; the fluid is in uniform rotation, is stably stratified, inviscid, and incompressible. The flows considered fall under two classes: zero and uniform potential vorticity flows. Steady geostrophic flow fields are determined from initial man imbalances, represented by both symmetric and antisymmetric density anomalies that vary along the *x* axis. The distinguishing characteristic of these solutions is the development of a front, defined as a zero-order discontinuity in both density and geostrophic velocity at one or both vertical boundaries. Frontal formation occurs, as previously discovered by Ou for zero potential vorticity flow, when the initial horizontal density gradient is sufficiently large. The critical values are displayed for different cases in terms of the initial amplitude and initial scale of the density anomaly.

The conversion of initial potential energy into geostrophic kinetic ΔKE and potential ΔPE energies during adjustment is also derived. Ou's result that γ = ΔKE/ΔPE = 1/2, independent of the initial scale is confirmed. It is shown, however, that γ≤1/2 for uniform potential vorticity flow. Large initial scales *a*
^{−1}, large compared to the deformation radius, have the largest values of γ, approaching γ = 1/2 as *a*→0. This limit approaches the solution and energy ratio for zero potential vorticity flow. The energy ratio associated with an antisymmetric density anomaly is characterized by γ→1/3 and *a* →∞: that is, the initial mass imbalance becomes a step function. In the other case, when the initial disturbance is symmetric and vanishes with *a*→∞, γ also vanishes. These results unify previous studies that have not provided the distinction between zero and uniform potential vorticity flows in examinations of the energy conversion process. Yet the reason for this distinction has not been delineated.

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## Abstract

Geostrophic adjustment of fluid in constant rotation is considered. The initial state is characterized by a mass imbalance, the initial velocity field is geostrophic, and zero potential vorticity flow is imposed. The final balanced state is determined from conservation of potential vorticity (zero), linear momentum, and mass. The potential energy released during the adjustment process is partitioned into the kinetic and potential energies of the balanced state in the ratio 1:2, independent of the scale of the initial state. The development of a front (infinite relative vorticity and a zero-order discontinuity in density) does, however, depend on the spatial scales of the initial fields. Moreover, the initial position of the maximum vorticity relative to the initial position of the maximum density gradient significantly affects the balanced flow. Physical interpretations of the principal features are provided.

## Abstract

Geostrophic adjustment of fluid in constant rotation is considered. The initial state is characterized by a mass imbalance, the initial velocity field is geostrophic, and zero potential vorticity flow is imposed. The final balanced state is determined from conservation of potential vorticity (zero), linear momentum, and mass. The potential energy released during the adjustment process is partitioned into the kinetic and potential energies of the balanced state in the ratio 1:2, independent of the scale of the initial state. The development of a front (infinite relative vorticity and a zero-order discontinuity in density) does, however, depend on the spatial scales of the initial fields. Moreover, the initial position of the maximum vorticity relative to the initial position of the maximum density gradient significantly affects the balanced flow. Physical interpretations of the principal features are provided.

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## Abstract

Unbalanced frontogenesis is studied in a two-dimensional, Boussinesq, rotating fluid that is constrained between two rigid, level surfaces. The potential vorticity is zero. The initial state is unbalanced because there is no motion and the potential temperature is given by the error function of *x*. An analytic solution is derived based on the neglect of the barotropic pressure gradient. The solution procedure uses momentum coordinates to obtain nonlinear solutions. When the initial Rossby number (Ro) is less than 1.435 the horizontal wind components display an inertial oscillation. During the first part of the inertial period (0 < *ft* < *π*) the isentropes develop a tilt and frontogenesis occurs, while in the second part (*π* < *ft* < 2*π*) the isentropes return to a vertical orientation and frontolysis brings the temperature gradient back to its original value at *ft* = 2*π*. For larger values of Ro a frontal discontinuity forms before *ft* = *π*.

The importance of the barotropic pressure gradient is determined in a scale collapse problem with a constant potential temperature and no rotation. In this case the inclusion of the barotropic pressure gradient increases the time before the discontinuity forms.

Numerical solutions of the original problem with rotation show that the presence of the barotropic pressure gradient term increases the critical Rossby number from 1.435 to about 1.55. Otherwise the complete solutions are very similar to the analytic solutions, except that the isentropes are no longer straight and the vorticity shows evidence of strong vertical advection by a small-scale vertical jet. Further, shorter timescales are expected with unbalanced fronts as compared with balanced fronts.

## Abstract

Unbalanced frontogenesis is studied in a two-dimensional, Boussinesq, rotating fluid that is constrained between two rigid, level surfaces. The potential vorticity is zero. The initial state is unbalanced because there is no motion and the potential temperature is given by the error function of *x*. An analytic solution is derived based on the neglect of the barotropic pressure gradient. The solution procedure uses momentum coordinates to obtain nonlinear solutions. When the initial Rossby number (Ro) is less than 1.435 the horizontal wind components display an inertial oscillation. During the first part of the inertial period (0 < *ft* < *π*) the isentropes develop a tilt and frontogenesis occurs, while in the second part (*π* < *ft* < 2*π*) the isentropes return to a vertical orientation and frontolysis brings the temperature gradient back to its original value at *ft* = 2*π*. For larger values of Ro a frontal discontinuity forms before *ft* = *π*.

The importance of the barotropic pressure gradient is determined in a scale collapse problem with a constant potential temperature and no rotation. In this case the inclusion of the barotropic pressure gradient increases the time before the discontinuity forms.

Numerical solutions of the original problem with rotation show that the presence of the barotropic pressure gradient term increases the critical Rossby number from 1.435 to about 1.55. Otherwise the complete solutions are very similar to the analytic solutions, except that the isentropes are no longer straight and the vorticity shows evidence of strong vertical advection by a small-scale vertical jet. Further, shorter timescales are expected with unbalanced fronts as compared with balanced fronts.

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## Abstract

Observations of ionospheric disturbances by various investigators have led to the suggestion that auroral energy may be coupled to atmospheric wave motions through joule heating. A linear model of internal gravity-wave generation by joule heating in the region of the auroral electrojet (100–150 km above the earth's surface) is investigated. Heat conduction, viscosity and reflection of wave energy by atmospheric inhomogeneities are not considered. The computed value of the upward wave-energy flux from the source region is of order 0.1–1 erg cm^{−2} sec^{−1} and is of sufficient magnitude to be of importance in the energetics of the F region. Shortcomings of the present model are discussed, with emphasis on how the physical features which have been neglected might affect the present results.

## Abstract

Observations of ionospheric disturbances by various investigators have led to the suggestion that auroral energy may be coupled to atmospheric wave motions through joule heating. A linear model of internal gravity-wave generation by joule heating in the region of the auroral electrojet (100–150 km above the earth's surface) is investigated. Heat conduction, viscosity and reflection of wave energy by atmospheric inhomogeneities are not considered. The computed value of the upward wave-energy flux from the source region is of order 0.1–1 erg cm^{−2} sec^{−1} and is of sufficient magnitude to be of importance in the energetics of the F region. Shortcomings of the present model are discussed, with emphasis on how the physical features which have been neglected might affect the present results.

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## Abstract

Steady, three-dimensional, inviscid flow over orography is examined by means of a semi-geostrophic model. A *nonuniform* basic current, represented by a deformation flow, is employed. A constant Coriolis parameters ƒ and uniform potential vorticity (constant Brunt-Väisälä frequency *N*) characteristic this model. A nondimensional mountain height ε/*D* ≲ 0.5, based on the deformation depth *D* ∼ 3 × 10^{3} m, and a Rossby number Ro ≲ 0.3, based on the mountain breadth *L* ≳ 3.5 × 10^{5} m, provide constraints on the flow field. Analytic solutions are represented in geostrophic coordinate space as the sum of the deformation flow and an anticyclonic mountain vortex. Although the two solutions are independent in geostrophic coordinate space, these flows are coupled nonlinearly in the transformation to physical coordinate space.

A solution is presented for flow over an isolated mountain. The decomposition of the physical space solution into fields of translation, rotation, divergence, and deformation forms the basis of the present analysis. The principal features associated with the solution are a region of relatively strong cyclonic vorticity in the lee of the mountain, accompanied by a region of convergence, and a region of weaker cyclonic vorticity on the windward slope, accompanied by a region of divergence. It is the *ageostrophic* component of the vorticity that provides these cyclonic centers, which are associated with enhanced *deformation* upstream and downstream of the peak. Further, the lee-side cyclonic vorticity enhancement is associated with *the advection of geostrophic deformation*, a feature of semi-geostrophic models that is absent in quasi-geostrophic models. By displacing the basic current's axis of dilatation into the lee of the obstacle, a deformation advection pattern is established that enhances the lee-side cyclonic vorticity center. The *uniform* flow solution is characterized by a single band of cyclonic vorticity north of the peak. This pattern is also established by the advection of geostrophic deformation. The possible relevance of the present model results to physical mechanisms that promote the initiation of lee cyclogenesis is discussed.

## Abstract

Steady, three-dimensional, inviscid flow over orography is examined by means of a semi-geostrophic model. A *nonuniform* basic current, represented by a deformation flow, is employed. A constant Coriolis parameters ƒ and uniform potential vorticity (constant Brunt-Väisälä frequency *N*) characteristic this model. A nondimensional mountain height ε/*D* ≲ 0.5, based on the deformation depth *D* ∼ 3 × 10^{3} m, and a Rossby number Ro ≲ 0.3, based on the mountain breadth *L* ≳ 3.5 × 10^{5} m, provide constraints on the flow field. Analytic solutions are represented in geostrophic coordinate space as the sum of the deformation flow and an anticyclonic mountain vortex. Although the two solutions are independent in geostrophic coordinate space, these flows are coupled nonlinearly in the transformation to physical coordinate space.

A solution is presented for flow over an isolated mountain. The decomposition of the physical space solution into fields of translation, rotation, divergence, and deformation forms the basis of the present analysis. The principal features associated with the solution are a region of relatively strong cyclonic vorticity in the lee of the mountain, accompanied by a region of convergence, and a region of weaker cyclonic vorticity on the windward slope, accompanied by a region of divergence. It is the *ageostrophic* component of the vorticity that provides these cyclonic centers, which are associated with enhanced *deformation* upstream and downstream of the peak. Further, the lee-side cyclonic vorticity enhancement is associated with *the advection of geostrophic deformation*, a feature of semi-geostrophic models that is absent in quasi-geostrophic models. By displacing the basic current's axis of dilatation into the lee of the obstacle, a deformation advection pattern is established that enhances the lee-side cyclonic vorticity center. The *uniform* flow solution is characterized by a single band of cyclonic vorticity north of the peak. This pattern is also established by the advection of geostrophic deformation. The possible relevance of the present model results to physical mechanisms that promote the initiation of lee cyclogenesis is discussed.

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## Abstract

A basic uniform current flows over a two-dimensional finite-amplitude ridge of characteristic male *L* and amplitude ε. The disturbance field is constrained by the geostrophic momentum approximation, by uniform potential vorticity (uniform Brunt-Väisälä frequency *N*) and by the constant Coriolis parameter *f*. Solutions are represented as the sum of a steady disturbance, recently found by Blumen and Gross, and a relatively weak translating disturbance. The translating disturbance is a passive scalar that is advected by the steady mountain circulation. The propagation speed over a ridge in an unbounded atmosphere is shown to increase with the parameter ε/*D*, where *D*=*fL*/*N* is the deformation depth. The steady mountain circulation produces frontolysis in the disturbance field on the windward slope and frontogenesis on the leeward slope. These frontogenetical features are primarily controlled by the steady horizontal velocity, which is divergent on the windward side and convergent in the Ice. The steady mountain circulation also disrupts the initial state of thermal wind balance imposed on the disturbance potential temperature and cross-stream velocity fields. An approximate evaluation of the ageostrophic circulation required to restore thermal wind balance is provided. This circulation, which may be direct or indirect, is related to the spatial structure of the initial disturbance and to its relative position on the ridge. Comparison with a related study by Bannon, and an evaluation of the principal limitations of both models complete the study.

## Abstract

A basic uniform current flows over a two-dimensional finite-amplitude ridge of characteristic male *L* and amplitude ε. The disturbance field is constrained by the geostrophic momentum approximation, by uniform potential vorticity (uniform Brunt-Väisälä frequency *N*) and by the constant Coriolis parameter *f*. Solutions are represented as the sum of a steady disturbance, recently found by Blumen and Gross, and a relatively weak translating disturbance. The translating disturbance is a passive scalar that is advected by the steady mountain circulation. The propagation speed over a ridge in an unbounded atmosphere is shown to increase with the parameter ε/*D*, where *D*=*fL*/*N* is the deformation depth. The steady mountain circulation produces frontolysis in the disturbance field on the windward slope and frontogenesis on the leeward slope. These frontogenetical features are primarily controlled by the steady horizontal velocity, which is divergent on the windward side and convergent in the Ice. The steady mountain circulation also disrupts the initial state of thermal wind balance imposed on the disturbance potential temperature and cross-stream velocity fields. An approximate evaluation of the ageostrophic circulation required to restore thermal wind balance is provided. This circulation, which may be direct or indirect, is related to the spatial structure of the initial disturbance and to its relative position on the ridge. Comparison with a related study by Bannon, and an evaluation of the principal limitations of both models complete the study.

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## Abstract

Three-dimensional, steady and inviscid flow over orography is examined by means of a semigeostrophic model. A constant Coriolis parameter *f*, uniform potential vorticity and a uniform basic flow characterize this model, first used by Merkine and Kálnay-Rivas. It is demonstrated that the neglect of ageostrophic accelerations, which characterize a semigeostrophic model, essentially requires low Rosssby number flow Ro≲0.3, and relatively small values of the nondimensional mountain height ε/*D* ≲ 0.5, where *D* ∼3 × 10^{3} m is the deformation depth. In this parameter range the disturbance potential is a solution of Laplace's equation; the atmosphere is semi-infinite and the finite-amplitude lower boundary is an isentropic surface.

The basic solution, expressed in prolate spheroidal coordinates, provides disturbance flows over an isolated mountain, finite ridges, and the limiting flow over a two-dimensional ridge. A comparison between the quasi-geostrophic and semigeostrophic responses to flow over the isolated mountain shows that (i) a rotational gradient wind correction to quasi-geostrophic flow that is independent of the basic current is introduced; (ii) the ageostrophic response associated with the basic current is an irrotational flow; and (iii) transformation of the semigeostrophic solution from geostrophic coordinate space to physical space displaces the circulation features on level, surfaces radially outward from the vertical axis, and the displacement is proportional to the mountain height.

An anticyclonic bound vortex, representing the sum of the quasi-geostrophic solution and the rotational wind correction, occurs over an isolated circular mountain (mountain parameter *a* = 0) and over long ridges (*a* > 0). However, these circulations are not typically Taylor columns, since they are not two-dimensional: vertical motions occur for 0 < *a* < . The existence of a closed circulation when a basic current is considered is dependent upon Ro, ε/*D* and *a*. In general, relatively larger values of Ro (or smaller ε/*D*) inhibit the formation of a closed circulation, but there is slight dependence on *a*. In agreement with Merkine and Kálnay-Rivas, it is shown that a closed circulation will occur around a mountain at a larger value of Ro when the ridge line is aligned with the basic current than when the ridge fine is normal to the current. The difference is relatively small, and probably unobservable in real flows.

A passive scalar disturbance is advected over ridges, characterized by different values of the mountain parameter *a* and orientation relative to the basic flow. This scalar field is advected anticyclonically around the lip of the ridge north of the current axis, but is retarded at the southern extremity of the ridge where the anticyclonic mountain circulation opposes the basic current. These features of frontal deformation are in qualitative agreement with synoptic analyses in geographic regions of orographic influence. Computed frontogenetical characteristics vary along selected trajectories. The distinctive features of frontogenesis are interpreted and compared with recent results obtained for flew over an infinite ridge.

## Abstract

Three-dimensional, steady and inviscid flow over orography is examined by means of a semigeostrophic model. A constant Coriolis parameter *f*, uniform potential vorticity and a uniform basic flow characterize this model, first used by Merkine and Kálnay-Rivas. It is demonstrated that the neglect of ageostrophic accelerations, which characterize a semigeostrophic model, essentially requires low Rosssby number flow Ro≲0.3, and relatively small values of the nondimensional mountain height ε/*D* ≲ 0.5, where *D* ∼3 × 10^{3} m is the deformation depth. In this parameter range the disturbance potential is a solution of Laplace's equation; the atmosphere is semi-infinite and the finite-amplitude lower boundary is an isentropic surface.

The basic solution, expressed in prolate spheroidal coordinates, provides disturbance flows over an isolated mountain, finite ridges, and the limiting flow over a two-dimensional ridge. A comparison between the quasi-geostrophic and semigeostrophic responses to flow over the isolated mountain shows that (i) a rotational gradient wind correction to quasi-geostrophic flow that is independent of the basic current is introduced; (ii) the ageostrophic response associated with the basic current is an irrotational flow; and (iii) transformation of the semigeostrophic solution from geostrophic coordinate space to physical space displaces the circulation features on level, surfaces radially outward from the vertical axis, and the displacement is proportional to the mountain height.

An anticyclonic bound vortex, representing the sum of the quasi-geostrophic solution and the rotational wind correction, occurs over an isolated circular mountain (mountain parameter *a* = 0) and over long ridges (*a* > 0). However, these circulations are not typically Taylor columns, since they are not two-dimensional: vertical motions occur for 0 < *a* < . The existence of a closed circulation when a basic current is considered is dependent upon Ro, ε/*D* and *a*. In general, relatively larger values of Ro (or smaller ε/*D*) inhibit the formation of a closed circulation, but there is slight dependence on *a*. In agreement with Merkine and Kálnay-Rivas, it is shown that a closed circulation will occur around a mountain at a larger value of Ro when the ridge line is aligned with the basic current than when the ridge fine is normal to the current. The difference is relatively small, and probably unobservable in real flows.

A passive scalar disturbance is advected over ridges, characterized by different values of the mountain parameter *a* and orientation relative to the basic flow. This scalar field is advected anticyclonically around the lip of the ridge north of the current axis, but is retarded at the southern extremity of the ridge where the anticyclonic mountain circulation opposes the basic current. These features of frontal deformation are in qualitative agreement with synoptic analyses in geographic regions of orographic influence. Computed frontogenetical characteristics vary along selected trajectories. The distinctive features of frontogenesis are interpreted and compared with recent results obtained for flew over an infinite ridge.

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## Abstract

Steady-state, two-dimensional disturbances forced by flow over a finite-amplitude ridge are considered. The model represents an extension of the one presented by Robinson (1960). This study is based on the semigeostrophic system of equations for uniform potential vorticity flow. The model equations satisfy the Cauchy-Riemann conditions, and solutions for uniform flow over various shaped ridges may be obtained in terms of a complex potential. The novel result is the determination of solutions for disturbances in a zonal current with linear shear. The boundaries are tilled in the cross-stream direction to coincide with basic state potential temperature surfaces. This simplification, which provides isentropic boundaries, permits the solutions for disturbances in a shear flow to be obtained directly from solutions forced by uniform flow over the same ridge.

Physical properties of the solutions are presented in terms of three parameters: &epsi/*D*, *r* and δ. The amplitude of the ridge is &epsi& and *D* is the deformation depth, based on the characteristic width of the ridge *L*
*r* represents the ratio of &epsi& to the channel depth and δ is the constant shear of the basic current. Solutions corresponding to uniform flow, δ = 0, in an unbounded fluid, *r* = 0, represent a limit that is compared with a previous study (Pierehumbert 1985). The present results confirm Pierrehumbert's conclusion that upstream deceleration is not significant, and that the characteristic vertical depth, over which the disturbances decay, is D. Confinement of the flow by a rigid lid (*r* ≠ 0) and consideration of a shear (δ ≠ 0) do not affect that flow deceleration, nor do these features affect the characteristic decay of the ageostrophic velocity components. However, the presence of a lid causes the geostrophic velocity component to become relatively independent of depth. It is also shown that an increase in the static stability (&epsi&/*D* increasing) enhances the ageostrophic circulation in a manner that is similar to the effect of increasing the shear δ from negative to positive values. Moreover, a linear lower boundary condition may be used in some circumstances because the velocity components on the ridge are relatively insensitive to changes in *r* when 0 > *r* ≳ 0.3 and δ0 However, linearization of the boundary condition cannot be supported when the basic flow changes with height, δ≠ 0. The geostrophic momentum approximation is shown to be valid over most of the domain, but may be violated along the windward slope unless &epsi&/*D* ≤ 0.6, with δ ≤ 0.5. Other considerations that need to be addressed to apply semige-ostrophic theory to mountain flows include a stability analysis of the present solutions and the use of nonisentropic boundary surfaces.

## Abstract

Steady-state, two-dimensional disturbances forced by flow over a finite-amplitude ridge are considered. The model represents an extension of the one presented by Robinson (1960). This study is based on the semigeostrophic system of equations for uniform potential vorticity flow. The model equations satisfy the Cauchy-Riemann conditions, and solutions for uniform flow over various shaped ridges may be obtained in terms of a complex potential. The novel result is the determination of solutions for disturbances in a zonal current with linear shear. The boundaries are tilled in the cross-stream direction to coincide with basic state potential temperature surfaces. This simplification, which provides isentropic boundaries, permits the solutions for disturbances in a shear flow to be obtained directly from solutions forced by uniform flow over the same ridge.

Physical properties of the solutions are presented in terms of three parameters: &epsi/*D*, *r* and δ. The amplitude of the ridge is &epsi& and *D* is the deformation depth, based on the characteristic width of the ridge *L*
*r* represents the ratio of &epsi& to the channel depth and δ is the constant shear of the basic current. Solutions corresponding to uniform flow, δ = 0, in an unbounded fluid, *r* = 0, represent a limit that is compared with a previous study (Pierehumbert 1985). The present results confirm Pierrehumbert's conclusion that upstream deceleration is not significant, and that the characteristic vertical depth, over which the disturbances decay, is D. Confinement of the flow by a rigid lid (*r* ≠ 0) and consideration of a shear (δ ≠ 0) do not affect that flow deceleration, nor do these features affect the characteristic decay of the ageostrophic velocity components. However, the presence of a lid causes the geostrophic velocity component to become relatively independent of depth. It is also shown that an increase in the static stability (&epsi&/*D* increasing) enhances the ageostrophic circulation in a manner that is similar to the effect of increasing the shear δ from negative to positive values. Moreover, a linear lower boundary condition may be used in some circumstances because the velocity components on the ridge are relatively insensitive to changes in *r* when 0 > *r* ≳ 0.3 and δ0 However, linearization of the boundary condition cannot be supported when the basic flow changes with height, δ≠ 0. The geostrophic momentum approximation is shown to be valid over most of the domain, but may be violated along the windward slope unless &epsi&/*D* ≤ 0.6, with δ ≤ 0.5. Other considerations that need to be addressed to apply semige-ostrophic theory to mountain flows include a stability analysis of the present solutions and the use of nonisentropic boundary surfaces.