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## Abstract

The adjustment of a compressible, stably stratified atmosphere to sources of hydrostatic and geostrophic imbalance is investigated using a linear model. Imbalance is produced by prescribed, time-dependent injections of mass, heat, or momentum that model those processes considered “external” to the scales of motion on which the linearization and other model assumptions are justifiable. Solutions are demonstrated in response to a localized warming characteristic of small isolated clouds, larger thunderstorms, and convective systems.

For a semi-infinite atmosphere, solutions consist of a set of vertical modes of continuously varying wavenumber, each of which contains time dependencies classified as steady, acoustic wave, and buoyancy wave contributions. Additionally, a rigid lower-boundary condition implies the existence of a discrete mode—the Lamb mode— containing only a steady and acoustic wave contribution. The forced solutions are generalized in terms of a temporal Green's function, which represents the response to an instantaneous injection.

The response to an instantaneous warming with geometry representative of a small, isolated cloud takes place in two stages. Within the first few minutes, acoustic and Lamb waves accomplish an expansion of the heated region. Within the first quarter-hour, nonhydrostatic buoyancy waves accomplish an upward displacement inside of the heated region with inflow below, outflow above, and weak subsidence on the periphery—all mainly accomplished by the lowest vertical wavenumber modes, which have the largest horizontal group speed. More complicated transient patterns of inflow aloft and outflow along the lower boundary are accomplished by higher vertical wavenumber modes. Among these is an outwardly propagating rotor along the lower boundary that effectively displaces the low-level inflow upward and outward.

A warming of 20 min duration with geometry representative of a large thunderstorm generates only a weak acoustic response in the horizontal by the Lamb waves. The amplitude of this signal increases during the onset of the heating and decreases as the heating is turned off. The lowest vertical wavenumber buoyancy waves still dominate the horizontal adjustment, and the horizontal scale of displacements is increased by an order of magnitude. Within a few hours the transient motions remove the perturbations and an approximately trivial balanced state is established.

A warming of 2 h duration with geometry representative of a large convective system generates a weak but discernible Lamb wave signal. The response to the conglomerate system is mainly hydrostatic. After several hours, the only signal in the vicinity of the heated region is that of inertia–gravity waves oscillating about a nontrivial hydrostatic and geostrophic state.

This paper is the first of two parts treating the transient dynamics of hydrostatic and geostrophic adjustment. Part II examines the potential vorticity conservation and the partitioning of total energy.

## Abstract

The adjustment of a compressible, stably stratified atmosphere to sources of hydrostatic and geostrophic imbalance is investigated using a linear model. Imbalance is produced by prescribed, time-dependent injections of mass, heat, or momentum that model those processes considered “external” to the scales of motion on which the linearization and other model assumptions are justifiable. Solutions are demonstrated in response to a localized warming characteristic of small isolated clouds, larger thunderstorms, and convective systems.

For a semi-infinite atmosphere, solutions consist of a set of vertical modes of continuously varying wavenumber, each of which contains time dependencies classified as steady, acoustic wave, and buoyancy wave contributions. Additionally, a rigid lower-boundary condition implies the existence of a discrete mode—the Lamb mode— containing only a steady and acoustic wave contribution. The forced solutions are generalized in terms of a temporal Green's function, which represents the response to an instantaneous injection.

The response to an instantaneous warming with geometry representative of a small, isolated cloud takes place in two stages. Within the first few minutes, acoustic and Lamb waves accomplish an expansion of the heated region. Within the first quarter-hour, nonhydrostatic buoyancy waves accomplish an upward displacement inside of the heated region with inflow below, outflow above, and weak subsidence on the periphery—all mainly accomplished by the lowest vertical wavenumber modes, which have the largest horizontal group speed. More complicated transient patterns of inflow aloft and outflow along the lower boundary are accomplished by higher vertical wavenumber modes. Among these is an outwardly propagating rotor along the lower boundary that effectively displaces the low-level inflow upward and outward.

A warming of 20 min duration with geometry representative of a large thunderstorm generates only a weak acoustic response in the horizontal by the Lamb waves. The amplitude of this signal increases during the onset of the heating and decreases as the heating is turned off. The lowest vertical wavenumber buoyancy waves still dominate the horizontal adjustment, and the horizontal scale of displacements is increased by an order of magnitude. Within a few hours the transient motions remove the perturbations and an approximately trivial balanced state is established.

A warming of 2 h duration with geometry representative of a large convective system generates a weak but discernible Lamb wave signal. The response to the conglomerate system is mainly hydrostatic. After several hours, the only signal in the vicinity of the heated region is that of inertia–gravity waves oscillating about a nontrivial hydrostatic and geostrophic state.

This paper is the first of two parts treating the transient dynamics of hydrostatic and geostrophic adjustment. Part II examines the potential vorticity conservation and the partitioning of total energy.

The common statement that the surface pressure in a hydrostatic atmosphere is equal to the weight per unit area of the air aloft is shown to be true only for a Cartesian world. Here the unit area is the surface area of the base of the atmospheric column. For either a cylindrical or a spherical planet the surface pressure is always less than the weight per unit area of the overlying atmosphere. In these curved geometries, lateral pressure forces help support an individual column, thereby reducing the load carried by the surface pressure at the column's base. It is estimated that the surface pressure is a factor of 0.25% less than the weight per unit area of a resting atmosphere similar to that on Earth.

The common statement that the surface pressure in a hydrostatic atmosphere is equal to the weight per unit area of the air aloft is shown to be true only for a Cartesian world. Here the unit area is the surface area of the base of the atmospheric column. For either a cylindrical or a spherical planet the surface pressure is always less than the weight per unit area of the overlying atmosphere. In these curved geometries, lateral pressure forces help support an individual column, thereby reducing the load carried by the surface pressure at the column's base. It is estimated that the surface pressure is a factor of 0.25% less than the weight per unit area of a resting atmosphere similar to that on Earth.

## Abstract

Numerical anelastic models solve a diagnostic elliptic equation for the pressure field using derivative boundary conditions. The pressure is therefore determined to within a function proportional to the base-state density field with arbitrary amplitude. This ambiguity is removed by requiring that the total mass be conserved in the model. This approach enables one to determine the correct temperature field that is required for the microphysical calculations. This correct, mass-conserving anelastic model predicts a temperature field that is an accurate approximation to that of a compressible atmosphere that has undergone a hydrostatic adjustment in response to a horizontally homogeneous heating or moistening. The procedure is demonstrated analytically and numerically for a one-dimensional, idealized heat source and moisture sink associated with moist convection. Two-dimensional anelastic simulations compare the effect of the new formulation on the evolution of the flow fields in a simulation of the ascent of a warm bubble in a conditionally unstable model atmosphere.

In the Boussinesq case, the temperature field is determined uniquely from the heat equation despite the fact that the pressure field can only be determined to within an arbitrary constant. Boussinesq air parcels conserve their volume, not their mass.

## Abstract

Numerical anelastic models solve a diagnostic elliptic equation for the pressure field using derivative boundary conditions. The pressure is therefore determined to within a function proportional to the base-state density field with arbitrary amplitude. This ambiguity is removed by requiring that the total mass be conserved in the model. This approach enables one to determine the correct temperature field that is required for the microphysical calculations. This correct, mass-conserving anelastic model predicts a temperature field that is an accurate approximation to that of a compressible atmosphere that has undergone a hydrostatic adjustment in response to a horizontally homogeneous heating or moistening. The procedure is demonstrated analytically and numerically for a one-dimensional, idealized heat source and moisture sink associated with moist convection. Two-dimensional anelastic simulations compare the effect of the new formulation on the evolution of the flow fields in a simulation of the ascent of a warm bubble in a conditionally unstable model atmosphere.

In the Boussinesq case, the temperature field is determined uniquely from the heat equation despite the fact that the pressure field can only be determined to within an arbitrary constant. Boussinesq air parcels conserve their volume, not their mass.

## Abstract

This study explores the diurnal variation in the movement and structure of the dryline using a one-dimensional shallow-water model. The model is adapted to test some common theories of dryline motion including the diurnal variation in surface friction, static stability, inversion erosion, and momentum mixing aloft.

These mechanisms of diurnal variation are first studied individually and then in unison. A diurnal variation in the surface friction produces a model dryline that moves westward during the day (in disagreement with observations) and has a southerly wind maximum near midnight. A diurnal variation in the static stability produces a model dryline that steepens in slope and moves eastward during the day and then surges westward at night with a southerly wind maximum 6 to 9 h after the minimum stability. Inversion erosion during the day produces a nearly vertical model dry front that moves eastward during the day with surface southwesterlies. At sunset the model dryline surges westward with a southerly wind maximum before midnight. A diurnal variation of the momentum mixing aloft has no significant effect on the model dryline. Results show that the combined case with a diurnal variation of surface friction, inversion erosion, and static stability with terrain most accurately describes the observed dryline system. The westward surge depicted in the model is compared to the flow evolution of the corresponding dam-break problem for a rotating fluid.

## Abstract

This study explores the diurnal variation in the movement and structure of the dryline using a one-dimensional shallow-water model. The model is adapted to test some common theories of dryline motion including the diurnal variation in surface friction, static stability, inversion erosion, and momentum mixing aloft.

These mechanisms of diurnal variation are first studied individually and then in unison. A diurnal variation in the surface friction produces a model dryline that moves westward during the day (in disagreement with observations) and has a southerly wind maximum near midnight. A diurnal variation in the static stability produces a model dryline that steepens in slope and moves eastward during the day and then surges westward at night with a southerly wind maximum 6 to 9 h after the minimum stability. Inversion erosion during the day produces a nearly vertical model dry front that moves eastward during the day with surface southwesterlies. At sunset the model dryline surges westward with a southerly wind maximum before midnight. A diurnal variation of the momentum mixing aloft has no significant effect on the model dryline. Results show that the combined case with a diurnal variation of surface friction, inversion erosion, and static stability with terrain most accurately describes the observed dryline system. The westward surge depicted in the model is compared to the flow evolution of the corresponding dam-break problem for a rotating fluid.