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## Abstract

The six-coefficient spectral model (model I) of two-dimensional shallow moist convection discussed by Shirer and Dutton (1979) is extended to an eleven component system (model II) in order that a height-dependent basic wind V(z) could be added to the problem. In this way, some behavior of atmospheric cloud streets that are usually observed in a shearing environment could be studied qualitatively.

In model II, several different types of solutions exist that correspond to a rich variety of physically relevant possibilities. When a basic wind field is present, stationary, advecting or growing and decaying but propagating cloud bands may occur for different magnitudes of the vertical temperature gradient. These two-dimensional rolls are represented by time-dependent or periodic solutions or by an attractor on a two-dimensional torus. At large values of the lapse rate, the invariant set is probably contained on a three-dimensional torus, but whether or not the limit set is composed of a strange attractor (that may be a model of turbulent behavior) is an open question.

With use of a minimizing principle, the expected cloud band orientations are obtained. The resulting formulas provide predictions that generally agree with the linear studies of previous investigators, but our equations apply to an arbitrary height-dependent wind field. When the wind direction does not vary with altitude, the branching two-dimensional rolls are longitudinal or aligned parallel to the wind shear vector. But when the basic wind direction changes with height, some wind profiles support longitudinal rolls; other wind fields load to transverse rolls that are oriented perpendicular to the shear. Two or three coexisting cloud band alignments can also occur.

## Abstract

The six-coefficient spectral model (model I) of two-dimensional shallow moist convection discussed by Shirer and Dutton (1979) is extended to an eleven component system (model II) in order that a height-dependent basic wind V(z) could be added to the problem. In this way, some behavior of atmospheric cloud streets that are usually observed in a shearing environment could be studied qualitatively.

In model II, several different types of solutions exist that correspond to a rich variety of physically relevant possibilities. When a basic wind field is present, stationary, advecting or growing and decaying but propagating cloud bands may occur for different magnitudes of the vertical temperature gradient. These two-dimensional rolls are represented by time-dependent or periodic solutions or by an attractor on a two-dimensional torus. At large values of the lapse rate, the invariant set is probably contained on a three-dimensional torus, but whether or not the limit set is composed of a strange attractor (that may be a model of turbulent behavior) is an open question.

With use of a minimizing principle, the expected cloud band orientations are obtained. The resulting formulas provide predictions that generally agree with the linear studies of previous investigators, but our equations apply to an arbitrary height-dependent wind field. When the wind direction does not vary with altitude, the branching two-dimensional rolls are longitudinal or aligned parallel to the wind shear vector. But when the basic wind direction changes with height, some wind profiles support longitudinal rolls; other wind fields load to transverse rolls that are oriented perpendicular to the shear. Two or three coexisting cloud band alignments can also occur.

## Abstract

The feasibility of developing an objective parameterization technique is examined for general nonlinear hydrodynamical systems. The typical structure of these hydrodynamical systems, regardless of their complexity, is one in which the rates of change of the dependent variables depend on homogeneous quadratic and linear forms, as well as on inhomogeneous forcing terms. As a prototype of the generic problem containing this typical structure, we apply the parameterization technique to various three-component subsets of a five-component nonlinear spectral model of forced, dissipative quasi-geostrophic flow in a channel. The results obtained here lead to specification of the necessary data coverage requirements for applying the technique in general.

The emphasis of this work is on preserving some behavior of the steady states by incorporating in the parameterized models information concerning the topological structure of the original solutions. By formulating the parameterization in terms of the steady states, we intend primarily to illustrate the general technique, but not to suggest that the preservation of temporal behavior can be achieved by addressing the steady solutions alone. The parameterized spectral components are expressed as power series involving the retained components, and it is found that the optimum parameterization is obtained when these series are terminated at quadratic terms. The values of the coefficients in these series are determined from the moments of the original set of spectral components over some range of forcing.

For testing convenience, the moments are computed using the steady solutions to the original five-component model as data. This is accomplished by assuming that the values of the zonal forcing rate obey some standard statistical distributions. In regions of phase space in which multiple steady solutions occur, the likelihood of the occurrence of any one solution may be weighted according to its stability. Thus, the datasets can be viewed as simulating either idealized data, in which both stable and unstable solutions are permitted, or observational data, in which only stable solutions are permitted. Special attention is paid to the sensitivity of the parameterization to data coverage requirements, and to the relation of these requirements to the general structure of the solution surfaces. Significantly, it is shown that with sufficient data coverage, a successful parameterization may be obtained even in the more restrictive case when only stable (observable) solutions are used as data.

## Abstract

The feasibility of developing an objective parameterization technique is examined for general nonlinear hydrodynamical systems. The typical structure of these hydrodynamical systems, regardless of their complexity, is one in which the rates of change of the dependent variables depend on homogeneous quadratic and linear forms, as well as on inhomogeneous forcing terms. As a prototype of the generic problem containing this typical structure, we apply the parameterization technique to various three-component subsets of a five-component nonlinear spectral model of forced, dissipative quasi-geostrophic flow in a channel. The results obtained here lead to specification of the necessary data coverage requirements for applying the technique in general.

The emphasis of this work is on preserving some behavior of the steady states by incorporating in the parameterized models information concerning the topological structure of the original solutions. By formulating the parameterization in terms of the steady states, we intend primarily to illustrate the general technique, but not to suggest that the preservation of temporal behavior can be achieved by addressing the steady solutions alone. The parameterized spectral components are expressed as power series involving the retained components, and it is found that the optimum parameterization is obtained when these series are terminated at quadratic terms. The values of the coefficients in these series are determined from the moments of the original set of spectral components over some range of forcing.

For testing convenience, the moments are computed using the steady solutions to the original five-component model as data. This is accomplished by assuming that the values of the zonal forcing rate obey some standard statistical distributions. In regions of phase space in which multiple steady solutions occur, the likelihood of the occurrence of any one solution may be weighted according to its stability. Thus, the datasets can be viewed as simulating either idealized data, in which both stable and unstable solutions are permitted, or observational data, in which only stable solutions are permitted. Special attention is paid to the sensitivity of the parameterization to data coverage requirements, and to the relation of these requirements to the general structure of the solution surfaces. Significantly, it is shown that with sufficient data coverage, a successful parameterization may be obtained even in the more restrictive case when only stable (observable) solutions are used as data.

## Abstract

Maximally truncated spectral models have been used recently by fluid and atmospheric dynamicists to study nonlinear behavior of the governing partial differential system. However, too few external control parameters may be available in the truncated model to describe adequately the steady states near singular parameter values at which two or more stationary solutions meet. These missing parameters correspond in many cases to small but significant physical effects whose inclusion may be critically important for the model results to be realistic. We apply to truncated spectral models a recently developed contact catastrophe method that allows determination of the crucial physical effects that govern the steady states of a fluid system.

Spectral systems of three different fluid flow models of interest in atmospheric science are considered. Two parameters are necessary for modeling Rayleigh-BÃ©nard convection. One represents the magnitude of the horizontal component, the other the magnitude of the vertical component of the externally imposed heating. Four parameters are required for modeling axisymmetric flow in either a rotating annulus or the atmosphere if the Prandtl number Ïƒ and the aspect ratio *a* are related by Ïƒ*a* < 1. These are the horizontal and vertical components of the external heating, the Coriolis parameter, and either the inclination angle of the vessel (annulus) or the Newtonian heating rate (atmosphere). Four parameters are essential for modeling quasi-geostrophic flow in a channel. They are the three Fourier coefficients of the Newtonian heating rate and the amplitude of a superimposed time-independent zonal current.

## Abstract

Maximally truncated spectral models have been used recently by fluid and atmospheric dynamicists to study nonlinear behavior of the governing partial differential system. However, too few external control parameters may be available in the truncated model to describe adequately the steady states near singular parameter values at which two or more stationary solutions meet. These missing parameters correspond in many cases to small but significant physical effects whose inclusion may be critically important for the model results to be realistic. We apply to truncated spectral models a recently developed contact catastrophe method that allows determination of the crucial physical effects that govern the steady states of a fluid system.

Spectral systems of three different fluid flow models of interest in atmospheric science are considered. Two parameters are necessary for modeling Rayleigh-BÃ©nard convection. One represents the magnitude of the horizontal component, the other the magnitude of the vertical component of the externally imposed heating. Four parameters are required for modeling axisymmetric flow in either a rotating annulus or the atmosphere if the Prandtl number Ïƒ and the aspect ratio *a* are related by Ïƒ*a* < 1. These are the horizontal and vertical components of the external heating, the Coriolis parameter, and either the inclination angle of the vessel (annulus) or the Newtonian heating rate (atmosphere). Four parameters are essential for modeling quasi-geostrophic flow in a channel. They are the three Fourier coefficients of the Newtonian heating rate and the amplitude of a superimposed time-independent zonal current.

## Abstract

The onset and development of both dynamically and convectively forced boundary-layer rolls are studied with linear and nonlinear analyses of a truncated spectral model of shallow Boussinesq flow. Emphasis is given here on the energetics of the dominant roll modes, on the magnitudes of the roll-induced modifications of the initial basic-state wind and temperature profiles, and on the sensitivity of the linear stability results to the use of modified profiles as basic states. It is demonstrated that the roll circulations can produce substantial changes to the cross-roll component of the initial wind profile and that significant changes in orientation angle estimates can result from use of a roll-modified profile in the stability analysis. These results demonstrate that roll contributions must be removed from observed background wind profiles before using them to investigate the mechanisms underlying actual secondary flows in the boundary layer.

The model is developed quite generally to accept arbitrary basic-state wind profiles as dynamic forcing. An Ekman profile is chosen here merely to provide a means for easy comparison with other theoretical boundary-layer studies; the ultimate application of the model is to study observed boundary-layer profiles. Results of the analytic stability analysis are validated by comparing them with results from a larger linear model. For an appropriate Ekman depth, a complete set of transition curves is given in forcing parameter space for roll modes driven both thermally and dynamically. Preferred orientation angles, horizontal wavelengths, and propagation frequencies, as well as energetics and wind profile modifications, are all shown to agree rather well with results from studies on Ekman layers as well as with studies on near-neutral and convective atmospheric boundary layers.

## Abstract

The onset and development of both dynamically and convectively forced boundary-layer rolls are studied with linear and nonlinear analyses of a truncated spectral model of shallow Boussinesq flow. Emphasis is given here on the energetics of the dominant roll modes, on the magnitudes of the roll-induced modifications of the initial basic-state wind and temperature profiles, and on the sensitivity of the linear stability results to the use of modified profiles as basic states. It is demonstrated that the roll circulations can produce substantial changes to the cross-roll component of the initial wind profile and that significant changes in orientation angle estimates can result from use of a roll-modified profile in the stability analysis. These results demonstrate that roll contributions must be removed from observed background wind profiles before using them to investigate the mechanisms underlying actual secondary flows in the boundary layer.

The model is developed quite generally to accept arbitrary basic-state wind profiles as dynamic forcing. An Ekman profile is chosen here merely to provide a means for easy comparison with other theoretical boundary-layer studies; the ultimate application of the model is to study observed boundary-layer profiles. Results of the analytic stability analysis are validated by comparing them with results from a larger linear model. For an appropriate Ekman depth, a complete set of transition curves is given in forcing parameter space for roll modes driven both thermally and dynamically. Preferred orientation angles, horizontal wavelengths, and propagation frequencies, as well as energetics and wind profile modifications, are all shown to agree rather well with results from studies on Ekman layers as well as with studies on near-neutral and convective atmospheric boundary layers.

## Abstract

The possibility of global-scale transitions between atmospheric Hadley and Rossby regimes is investigated with a highly idealized, nonlinear, vertically continuous, rotating, spherical system. Low-order spectral versions of the model are used both to calculate ideal Hadley states and to determine their stabilities to certain three-dimensional baroclinic disturbances of any zonal wavenumber. The flow is forced by an idealized axisymmetric heating pattern based on zonally averaged atmospheric data, and is dissipated using an eddy viscosity formulation.

The dominant modes in the heating pattern force a single meridional cell between the equator and the poles that is compatible with the simple boundary conditions. As the heating rate is increased, these states exchange stability with temporally periodic solutions that have the characteristics of Rossby waves. Although Ekman boundary layer and cumulus friction effects are not included, the transports of heat and momentum by the zonally averaged Rossby flow are reasonable. When all combinations of heating and rotation rates are used, a transition curve separating the ideal Hadley and Rossby regimes is found. The critical values of the heating rates are made more realistic through the use of an effective eddy viscosity that represents energy transports arising from the products of the sub-Hadley and sub-Rossby scale perturbations.

It is shown that a transition from Hadley flow to wavenumber 5 Rossby flow is preferred. This result, which agrees with standard baroclinic instability results, gives a reasonable Rossby wave bifurcation from the Hadley solution. For the cases examined, it is found that the upper symmetric Hadley regime does not exist and that the Hadley to Rossby transition depends on the values of the eddy viscosities. Indeed, the dependence of the preferred zonal wavenumber on the values of the eddy viscosities suggests that small changes in the values of these parameters might result in large changes in the Rossby regime.

## Abstract

The possibility of global-scale transitions between atmospheric Hadley and Rossby regimes is investigated with a highly idealized, nonlinear, vertically continuous, rotating, spherical system. Low-order spectral versions of the model are used both to calculate ideal Hadley states and to determine their stabilities to certain three-dimensional baroclinic disturbances of any zonal wavenumber. The flow is forced by an idealized axisymmetric heating pattern based on zonally averaged atmospheric data, and is dissipated using an eddy viscosity formulation.

The dominant modes in the heating pattern force a single meridional cell between the equator and the poles that is compatible with the simple boundary conditions. As the heating rate is increased, these states exchange stability with temporally periodic solutions that have the characteristics of Rossby waves. Although Ekman boundary layer and cumulus friction effects are not included, the transports of heat and momentum by the zonally averaged Rossby flow are reasonable. When all combinations of heating and rotation rates are used, a transition curve separating the ideal Hadley and Rossby regimes is found. The critical values of the heating rates are made more realistic through the use of an effective eddy viscosity that represents energy transports arising from the products of the sub-Hadley and sub-Rossby scale perturbations.

It is shown that a transition from Hadley flow to wavenumber 5 Rossby flow is preferred. This result, which agrees with standard baroclinic instability results, gives a reasonable Rossby wave bifurcation from the Hadley solution. For the cases examined, it is found that the upper symmetric Hadley regime does not exist and that the Hadley to Rossby transition depends on the values of the eddy viscosities. Indeed, the dependence of the preferred zonal wavenumber on the values of the eddy viscosities suggests that small changes in the values of these parameters might result in large changes in the Rossby regime.

## Abstract

The steady boundary-layer responses that occur over the Great Lakes region during wintertime cold air outbreaks are examined using a two-dimensional, linear, analytic model. The planetary boundary layer (PBL) is modeled as an idealized, constantly stratified, viscous, rotating Boussinesq fluid that moves uniformly between two horizontally infinite, rigid, stress-free plates. The heat from the lakes is parameterized in terms of a specified diabatic forcing function.

Solution of the governing differential equation yields an integral expression for the vertical motion of the general response. Further assessment of the response is gained by examining closed-form analytic solutions to several limiting cases. Four response types are identified that depend upon the values of the Froude number Fr, the mechanical Ekman number E_{x}, the thermal Ekman number E_{x}, and the eddy Prandtl number Pr.

Four different flow regimes are found. When 0 â‰¤ Fr < 1 and Pr â‰¥ 1, there is a purely exponentially damped response that exists over and on both sides of the heating. A flow characterized approximately by 1 â‰¤ Fr^{2} < 1 + E_{r}
^{2} + E_{x} and Pr â‰¥ 1 yields a purely exponentially damped response that exists only over and downstream of the heating, while a flow characterized approximately by ^{2} > 1 + E_{r}^{2} + E_{x}^{2}

The model is used to demonstrate the effects that rotation, stability, mean flow speed, and mechanical and thermal dissipation have on the PBL responses that occur over the Great Lakes during wintertime cold air outbreaks. The simulation of heating by the lakes of strong flow within a moderately cold, shallow PBL produces a model response with ascent and implied clouds and precipitation extending well downstream of the lakes, as are typically observed soon after such a response develops. The simulation of heating by the lakes of weak flow within a very cold, deep PBL produces a model response with ascent and implied clouds and precipitation that are collocated with the lakes, as are typically observed just before such a response decays.

## Abstract

The steady boundary-layer responses that occur over the Great Lakes region during wintertime cold air outbreaks are examined using a two-dimensional, linear, analytic model. The planetary boundary layer (PBL) is modeled as an idealized, constantly stratified, viscous, rotating Boussinesq fluid that moves uniformly between two horizontally infinite, rigid, stress-free plates. The heat from the lakes is parameterized in terms of a specified diabatic forcing function.

Solution of the governing differential equation yields an integral expression for the vertical motion of the general response. Further assessment of the response is gained by examining closed-form analytic solutions to several limiting cases. Four response types are identified that depend upon the values of the Froude number Fr, the mechanical Ekman number E_{x}, the thermal Ekman number E_{x}, and the eddy Prandtl number Pr.

Four different flow regimes are found. When 0 â‰¤ Fr < 1 and Pr â‰¥ 1, there is a purely exponentially damped response that exists over and on both sides of the heating. A flow characterized approximately by 1 â‰¤ Fr^{2} < 1 + E_{r}
^{2} + E_{x} and Pr â‰¥ 1 yields a purely exponentially damped response that exists only over and downstream of the heating, while a flow characterized approximately by ^{2} > 1 + E_{r}^{2} + E_{x}^{2}

The model is used to demonstrate the effects that rotation, stability, mean flow speed, and mechanical and thermal dissipation have on the PBL responses that occur over the Great Lakes during wintertime cold air outbreaks. The simulation of heating by the lakes of strong flow within a moderately cold, shallow PBL produces a model response with ascent and implied clouds and precipitation extending well downstream of the lakes, as are typically observed soon after such a response develops. The simulation of heating by the lakes of weak flow within a very cold, deep PBL produces a model response with ascent and implied clouds and precipitation that are collocated with the lakes, as are typically observed just before such a response decays.

## Abstract

A generalized seven-coefficient model of two-dimensional Rayleigh-BÃ©nard convection is presented. The model simulates successfully one means by which lateral cell expansion can occur as the value of the imposed vertical temperature difference is changed. Such changes in the horizontal wavelengths of the convective rolls are accomplished by the nonlinear transfer of energy from cells to other cells with smaller wavenumbers. The crucial effect is one represented by the advective term vË™âˆ‡v in the equation of motion, and as a consequence an interacting triad of harmonics must be included in the spectral model. Thus, the generalized model has basically the same form as that used by Saltzman or Shirer and Dutton, but in the generalized model the triad of interacting wavenumbers is varied as the vertical heating rate is varied. Actual values of the horizontal wavenumbers are determined by assuming that the first unstable wave will have the largest growth rate, or equivalently that the bifurcation point will have the smallest value. Thus, only the energetically active components are retained; in this way, transitional behavior within two-dimensional convective flow can be simulated properly, and can be interpreted physically as representing the cell expansion process, via successive secondary branching.

When the solutions are compared with those obtained by Clever and Busse from a large three-dimensional spectral model, it is found that for small values of the Prandtl number *P* (âˆ¼0.1), a two-dimensional cell broadening mechanism is likely to operate, but for larger values of *P* (âˆ¼0.7), a three-dimensional mechanism is expected. Consequently, these results suggest that this generalized seven-component model can be used to simulate successfully some transitions in a system having more degrees of freedom, because the seven components apparently form the basic unit by which steady two-dimensional flow develops. Moreover, the modeling philosophy presented here can provide the basis for development of simple atmospheric convection models.

## Abstract

A generalized seven-coefficient model of two-dimensional Rayleigh-BÃ©nard convection is presented. The model simulates successfully one means by which lateral cell expansion can occur as the value of the imposed vertical temperature difference is changed. Such changes in the horizontal wavelengths of the convective rolls are accomplished by the nonlinear transfer of energy from cells to other cells with smaller wavenumbers. The crucial effect is one represented by the advective term vË™âˆ‡v in the equation of motion, and as a consequence an interacting triad of harmonics must be included in the spectral model. Thus, the generalized model has basically the same form as that used by Saltzman or Shirer and Dutton, but in the generalized model the triad of interacting wavenumbers is varied as the vertical heating rate is varied. Actual values of the horizontal wavenumbers are determined by assuming that the first unstable wave will have the largest growth rate, or equivalently that the bifurcation point will have the smallest value. Thus, only the energetically active components are retained; in this way, transitional behavior within two-dimensional convective flow can be simulated properly, and can be interpreted physically as representing the cell expansion process, via successive secondary branching.

When the solutions are compared with those obtained by Clever and Busse from a large three-dimensional spectral model, it is found that for small values of the Prandtl number *P* (âˆ¼0.1), a two-dimensional cell broadening mechanism is likely to operate, but for larger values of *P* (âˆ¼0.7), a three-dimensional mechanism is expected. Consequently, these results suggest that this generalized seven-component model can be used to simulate successfully some transitions in a system having more degrees of freedom, because the seven components apparently form the basic unit by which steady two-dimensional flow develops. Moreover, the modeling philosophy presented here can provide the basis for development of simple atmospheric convection models.

## Abstract

The development of atmospheric boundary layer rolls from the inflection point and parallel instabilities is examined analytically using several three-dimensional linear models of flow in a neutral, rotational fluid. These models are formulated so that either arbitrary or observed background wind profiles can be examined easily to see which roll modes would likely occur. Necessary and sufficient conditions for the development of particular atmospheric modes are determined. These conditions are expressed as polynomials in the critical (eddy) Reynolds number Re_{c} and depend on Fourier coefficients of a given height-dependent background wind profile. The preferred values of orientation angle Î¸ and aspect ratio *A*, which describe the expected roll geometry, are assumed to be those that produce the smallest values of Re_{c}.

The ability of the models to successfully reproduce the modes arising from the inflection point and parallel instability mechanisms is tested by using idealized wind profiles to approximate the mean wind in the boundary layer. For the Ekman profile, the preferred values of Î¸ and *A* are found to agree with those given by larger models, indicating that the simpler analytical models are incorporating the crucial information contained in the wind profile.

Finally, a direct comparison with observations of atmospheric boundary layer rolls is given using a mean wind profile obtained during the 1981 West German KonTur experiment. For this case, the preferred values of Î¸, *A* and Re_{c}, associated with the pure inflection point instability mechanism agree well with their observed values, It is this easy, direct comparison between the model results and observations that is the significant contribution of this analytical modeling approach.

## Abstract

The development of atmospheric boundary layer rolls from the inflection point and parallel instabilities is examined analytically using several three-dimensional linear models of flow in a neutral, rotational fluid. These models are formulated so that either arbitrary or observed background wind profiles can be examined easily to see which roll modes would likely occur. Necessary and sufficient conditions for the development of particular atmospheric modes are determined. These conditions are expressed as polynomials in the critical (eddy) Reynolds number Re_{c} and depend on Fourier coefficients of a given height-dependent background wind profile. The preferred values of orientation angle Î¸ and aspect ratio *A*, which describe the expected roll geometry, are assumed to be those that produce the smallest values of Re_{c}.

The ability of the models to successfully reproduce the modes arising from the inflection point and parallel instability mechanisms is tested by using idealized wind profiles to approximate the mean wind in the boundary layer. For the Ekman profile, the preferred values of Î¸ and *A* are found to agree with those given by larger models, indicating that the simpler analytical models are incorporating the crucial information contained in the wind profile.

Finally, a direct comparison with observations of atmospheric boundary layer rolls is given using a mean wind profile obtained during the 1981 West German KonTur experiment. For this case, the preferred values of Î¸, *A* and Re_{c}, associated with the pure inflection point instability mechanism agree well with their observed values, It is this easy, direct comparison between the model results and observations that is the significant contribution of this analytical modeling approach.

## Abstract

The dynamics of two-dimensional, shallow moist convection is examined with the use of a six-component spectral model. Latent heating effects are incorporated by assuming that upward motion is moist adiabatic and that downward motion is dry adiabatic. The resulting nondimensional system of equations has the same form as that for BÃ©nard convection, with the moist effects included by replacing the Rayleigh number with a modified form.

The six-coefficient model contains a wide variety of multiple solutions with as many as 12 time-independent convective states and one conductive state occurring simultaneously. Temporally periodic solutions are also indicated, and some are found numerically that branch from stationary solutions at critical values of the external parameters. Only some of the solutions are linearly stable and hence observable, and we give a summary of the possible branching orders of these solutions. We find that the development of moist convection proceeds in the model via one of several available sequences of distinct transitions of the flow regime to increasingly complex structures.

## Abstract

The dynamics of two-dimensional, shallow moist convection is examined with the use of a six-component spectral model. Latent heating effects are incorporated by assuming that upward motion is moist adiabatic and that downward motion is dry adiabatic. The resulting nondimensional system of equations has the same form as that for BÃ©nard convection, with the moist effects included by replacing the Rayleigh number with a modified form.

The six-coefficient model contains a wide variety of multiple solutions with as many as 12 time-independent convective states and one conductive state occurring simultaneously. Temporally periodic solutions are also indicated, and some are found numerically that branch from stationary solutions at critical values of the external parameters. Only some of the solutions are linearly stable and hence observable, and we give a summary of the possible branching orders of these solutions. We find that the development of moist convection proceeds in the model via one of several available sequences of distinct transitions of the flow regime to increasingly complex structures.

## Abstract

The accuracies of the usual centered differencing, compact differencing and finite element methods are compared linearly with a geostrophic adjustment problem and nonlinearly with a vorticity advection problem. The finite element method provides the best approximation in the geostrophic adjustment problem on either a staggered or an unstaggered grid. The compact scheme provides the most accurate representation of the wavenumber distribution for the vorticity advection when the Arakawa Jacobian *J*_{7} is used.

## Abstract

The accuracies of the usual centered differencing, compact differencing and finite element methods are compared linearly with a geostrophic adjustment problem and nonlinearly with a vorticity advection problem. The finite element method provides the best approximation in the geostrophic adjustment problem on either a staggered or an unstaggered grid. The compact scheme provides the most accurate representation of the wavenumber distribution for the vorticity advection when the Arakawa Jacobian *J*_{7} is used.