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- Author or Editor: Wayne H. Schubert x
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Abstract
The geopotential tendency form of semigeostrophic theory is derived and compared with the potential vorticity form. The tendency form is compact and particularly convenient for non-Boussinesq, nonuniform potential vorticity flows.
Abstract
The geopotential tendency form of semigeostrophic theory is derived and compared with the potential vorticity form. The tendency form is compact and particularly convenient for non-Boussinesq, nonuniform potential vorticity flows.
Abstract
Lilly's model of a horizontally homogeneous cloud-topped mixed layer is studied. The model is closed by taking a weighted (weighting factor or entrainment parameter k) average of Lilly's maximum and minimum entrainment cases. The dependence of steady-state solutions on large-scale divergence, sea surface temperature and entrainment parameter k is investigated. By numerical integration the response of the mixed layer to a diurnally varying radiative flux is investigated. Significant variations in the state of the mixed layer and in the convective fluxes are found.
Abstract
Lilly's model of a horizontally homogeneous cloud-topped mixed layer is studied. The model is closed by taking a weighted (weighting factor or entrainment parameter k) average of Lilly's maximum and minimum entrainment cases. The dependence of steady-state solutions on large-scale divergence, sea surface temperature and entrainment parameter k is investigated. By numerical integration the response of the mixed layer to a diurnally varying radiative flux is investigated. Significant variations in the state of the mixed layer and in the convective fluxes are found.
Some statistical measures of growth of American Meteorological Society technical journals have been compiled. A general upward trend in total number of articles, pages, and an increase (nearly doubling during the past 20 years) in the average length of articles is found. Approximately half of this growth appears to be attributable to the increasing figure content of papers and half to the expansion of text apart from figures. Growth causes and impacts are discussed.
Some statistical measures of growth of American Meteorological Society technical journals have been compiled. A general upward trend in total number of articles, pages, and an increase (nearly doubling during the past 20 years) in the average length of articles is found. Approximately half of this growth appears to be attributable to the increasing figure content of papers and half to the expansion of text apart from figures. Growth causes and impacts are discussed.
Abstract
This study considers how spectral methods can be applied to limited-area models using Chebyshev polynomials as basis functions. We review the convergence of Sturm–Liouville series to motivate the use of the Chebyshev polynomials, and describe the tau and collocation projections which allow the use of general (nonperiodic) boundary conditions. These methods are illustrated for a simple model problem, the linear advection equation in one dimension, and numerical results confirm their high accuracy.
Time differencing and efficiency are considered in detail using both asymptotic analysis and numerical result from the model problem. The stability condition for Chebyshev methods with explicit time differencing, often thought to be severe, is shown to be less severe than that for finite difference methods when high accuracy is desired. Fourth-order Runge-Kutta time differencing is the most efficient of the many schemes considered. When the accuracy desired is high enough, Chebyshev spectral methods are more efficient than finite difference methods; numerical results suggest that this may be true in practice even for very modest accuracies.
Abstract
This study considers how spectral methods can be applied to limited-area models using Chebyshev polynomials as basis functions. We review the convergence of Sturm–Liouville series to motivate the use of the Chebyshev polynomials, and describe the tau and collocation projections which allow the use of general (nonperiodic) boundary conditions. These methods are illustrated for a simple model problem, the linear advection equation in one dimension, and numerical results confirm their high accuracy.
Time differencing and efficiency are considered in detail using both asymptotic analysis and numerical result from the model problem. The stability condition for Chebyshev methods with explicit time differencing, often thought to be severe, is shown to be less severe than that for finite difference methods when high accuracy is desired. Fourth-order Runge-Kutta time differencing is the most efficient of the many schemes considered. When the accuracy desired is high enough, Chebyshev spectral methods are more efficient than finite difference methods; numerical results suggest that this may be true in practice even for very modest accuracies.
Abstract
Chebyshev spectral methods were studied in Part I for the linear advection equation in one dimension. Here we extend these methods to the nonlinear shallow water equations in two dimensions. Numerical models are constructed for a limited domain on a β-plane, using open (characteristic) boundary conditions based on Rieman invariants to simulate an unbounded domain. Reflecting boundary conditions (wall and balance) are also considered for comparison. We discuss the formulation of the Chebyshev–tau and Chebyshev–collocation discretizations for this problem. The tau discretization avoids aliasing error in evaluating quadratic nonlinear terms, while the collocation method is simpler to program.
Numerical results from a linearized one-dimensional test problem demonstrate that with the characteristic boundary conditions the stability properties for various explicit time differencing schemes an essentially the same as obtained in Part I for the linear advection equation. These open boundary conditions also give much more accurate results than the reflecting boundary conditions. In two dimensions, numerical results from the nonlinear models indicate that the Chebyabev–tau discretization should be based on the rotational form of the equations for efficiency, while the Chebyshev–collocation discretization should be based on the advective form for accuracy. Little difference is seen between the tau and collocation solutions for the test cases considered, other than efficiency: with explicit time differencing, the collocation model requires an order of magnitude less computer time.
Abstract
Chebyshev spectral methods were studied in Part I for the linear advection equation in one dimension. Here we extend these methods to the nonlinear shallow water equations in two dimensions. Numerical models are constructed for a limited domain on a β-plane, using open (characteristic) boundary conditions based on Rieman invariants to simulate an unbounded domain. Reflecting boundary conditions (wall and balance) are also considered for comparison. We discuss the formulation of the Chebyshev–tau and Chebyshev–collocation discretizations for this problem. The tau discretization avoids aliasing error in evaluating quadratic nonlinear terms, while the collocation method is simpler to program.
Numerical results from a linearized one-dimensional test problem demonstrate that with the characteristic boundary conditions the stability properties for various explicit time differencing schemes an essentially the same as obtained in Part I for the linear advection equation. These open boundary conditions also give much more accurate results than the reflecting boundary conditions. In two dimensions, numerical results from the nonlinear models indicate that the Chebyabev–tau discretization should be based on the rotational form of the equations for efficiency, while the Chebyshev–collocation discretization should be based on the advective form for accuracy. Little difference is seen between the tau and collocation solutions for the test cases considered, other than efficiency: with explicit time differencing, the collocation model requires an order of magnitude less computer time.
Abstract
The separation of the vertical structure of the, solutions of the primitive (hydrostatic) meteorological equations is formalized as a vertical normal-mode transform. The transform is implemented for arbitrary static stability profiles by the Rayleigh-Ritz method, which is based on a variational formulation closely connected with energetics. With polynomial basis functions the order of accuracy is exponential. When vertical transforms of observed fields are computed, energy may be aliased onto the wrong vertical modes; this aliasing may be reduced substantially by a careful choice of sampling levels. The spectral distributions of observed tropical forcings of the wind and mass fields are presented.
Abstract
The separation of the vertical structure of the, solutions of the primitive (hydrostatic) meteorological equations is formalized as a vertical normal-mode transform. The transform is implemented for arbitrary static stability profiles by the Rayleigh-Ritz method, which is based on a variational formulation closely connected with energetics. With polynomial basis functions the order of accuracy is exponential. When vertical transforms of observed fields are computed, energy may be aliased onto the wrong vertical modes; this aliasing may be reduced substantially by a careful choice of sampling levels. The spectral distributions of observed tropical forcings of the wind and mass fields are presented.
Abstract
Under certain circumstances a large fraction of the energy generated by the release of latent heat in a tropical cyclone cab be partitioned to gravity-inertia wave motion rather than to balanced flow. In this way most of the generated energy is radiated away to the far field. If a primitive equation tropical cyclone model is to successfully simulate this process, its lateral boundary condition must be able to transmit the gravity-inertia wave energy generated in the interior of the model. Most present models seem deficient in this regard. As an improvement we explore the possibility of using a cylindrical, pure gravity wave radiation condition. Since there is a wide range of gravity wave phase velocities in a stratified atmosphere, it is necessary to apply the radiation condition vertical mode by vertical mode rather than level by level. The usefulness of this radiation condition and several other conditions in present use is tested both by a reflectivity analysis and by simple numerical time integrations.
Abstract
Under certain circumstances a large fraction of the energy generated by the release of latent heat in a tropical cyclone cab be partitioned to gravity-inertia wave motion rather than to balanced flow. In this way most of the generated energy is radiated away to the far field. If a primitive equation tropical cyclone model is to successfully simulate this process, its lateral boundary condition must be able to transmit the gravity-inertia wave energy generated in the interior of the model. Most present models seem deficient in this regard. As an improvement we explore the possibility of using a cylindrical, pure gravity wave radiation condition. Since there is a wide range of gravity wave phase velocities in a stratified atmosphere, it is necessary to apply the radiation condition vertical mode by vertical mode rather than level by level. The usefulness of this radiation condition and several other conditions in present use is tested both by a reflectivity analysis and by simple numerical time integrations.
