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Scott R. Fulton

Abstract

In semigeostrophic theory one can recover the balanced wind and mass fields from the potential vorticity by solving a nonlinear elliptic boundary-value problem. This paper describes the efficient solution of this invertibility relation in two dimensions by a multigrid algorithm. Numerical results show this method is competitive with a generalized Buneman algorithm for the linearized cases to which the latter applies. The fully nonlinear problem is solved to well below the level of truncation error 50–80 times faster than by simple relaxation on a single grid.

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Scott R. Fulton

Abstract

This paper describes the application of adaptive multigrid techniques to the problem of tropical cyclone track prediction. Based on the nondivergent barotropic vorticity equation, the model uses an adaptive multigrid method to refine the mesh around the moving vortex. Like conventional nested-grid models, this model achieves nonuniform resolution by superimposing uniform grids of different mesh sizes. Unlike nested-grid models, multigrid processing uses the interplay between solutions on fine and coarse grids—in regions where they overlap—to 1) solve the implicit problem for the streamfunction with optimum efficiency, 2) automatically achieve two-way interaction at the grid interfaces, and 3) provide accurate truncation error estimates for use in determining where to refine or coarsen the grids. An exchange rate algorithm accomplishes the latter task, approximately optimizing the grid selection based on a user-specified trade-off between accuracy and computational work.

Numerical results demonstrate that the model chooses reasonable grids with minimal user intervention. Using adaptive mesh refinement is at least an order of magnitude more efficient than using a single uniform grid, and the overhead cost of adaptive regridding is less than 2% of the total execution time. The adaptive multigrid approach allows track prediction errors due to discretization to be essentially eliminated from the problem at a reasonable computational cost.

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Scott R. Fulton
and
Wayne H. Schubert

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.

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Scott R. Fulton
and
Wayne H. Schubert

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.

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Scott R. Fulton
and
Wayne H. Schubert

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.

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Scott R. Fulton
and
Wayne H. Schubert

Abstract

The semigeostrophic equations take a particularly simple form when isentropic and geostrophic coordinates are used simultaneously: the horizontal ageostrophic velocities are entirely implicit, and the entire dynamics reduces to a predictive equation for the potential pseudodensity (inverse Ertel potential vorticity) and an invertibility relation. However, a perceived disadvantage of isentropic coordinates is the difficulty of treating a lower boundary that is not an isentropic surface.

Here we present the massless layer method, which allows isentropic surfaces to intersect the lowerboundary, and show that this extends the applicability of potential vorticity modeling in isentropic/geostrophic coordinates. When applied to the classic problem of surface frontogenesis by a vertically independent deformation field, the model produces realistic fronts with a surface discontinuity in finite time and tropopause folding, without the need for special treatment of the lower boundary.

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Scott R. Fulton
,
Wayne H. Schubert
, and
Scott A. Hausman

Abstract

Observational evidence indicates that upper-tropospheric and lower-stratospheric anticyclones occur in mesoscale convective systems, possibly resulting from the vertical redistribution of mass. The authors examine the gradient adjustment process that occurs when mass from the lower troposphere is impulsively injected between isentropic levels in the vicinity of the tropopause. Formulating the quasi-static primitive equations for inviscid, adiabatic, axisymmetric flow on an f plane using entropy and potential radius coordinates allows us to compute the final state in gradient balance by solving a single nonlinear elliptic problem. Solutions of this elliptic problem illustrate the development of an anticyclonic lens at the level of mass injection, with accompanying cold and warm temperature anomalies above and below, respectively. For a given amount of injected mass, a lower-stratospheric injection results in a stronger anticyclone than does an upper-tropospheric injection. Mass injections at low latitudes result in anticyclonic lens structures that are of larger horizontal extent and smaller vertical extent. The entrainment of stratospheric air into the mesoscale convective anvil is also shown to have an effect on the structure of the anticyclone. The theoretical results presented here are in substantial agreement with recent observations of the structure of upper-level anticyclones produced by mesoscale convective systems.

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Paul E. Ciesielski
,
Scott R. Fulton
, and
Wayne H. Schubert

Abstract

We consider the multigrid solution of the transverse circulation equation for a tropical cyclone. First we develop a standard multigrid scheme (SMG) which cycles between different levels of discretization (grids) to efficiently reduce the error in the solution on all scales. Whereas relaxation is inefficient as a solution method, it is used within the multigrid approach as a smoother to reduce the high-wavenumber errors on each grid. The added cost of the coarse grids is small because they contain relatively few points. The efficiency of the SMG scheme is compared to more conventional methods. Gauss-Seidel and successive-over-relaxation (SOR). Results show that the SMG scheme solves to the level of truncation error 26 times faster than an optimal SOR method.

In the SMG scheme an unbounded domain is approximated with a wall at a finite radius which leads to significant errors in the numerical solution. To better simulate an unbounded domain, we develop a second scheme (LRMG) which naturally combines local mesh refinement with multigrid processing. In this scheme, the lateral boundary is moved far enough out that the wall boundary condition is realistic, and the grid is coarsened in the outer region so that little additional work is required. Since finer grids are introduced only where needed, the LPMG scheme maintains the usual multigrid efficiency.

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Scott R. Fulton
,
Paul E. Ciesielski
, and
Wayne H. Schubert

Abstract

Multigrid methods solve a large class of problems very efficiently. They work by approximating a problem on multiple overlapping grids with widely varying mesh sizes and cycling between thew approximations, using relaxation to reduce the error on the scale of each grid. Problems solved by multigrid methods include general elliptic partial differential equations, nonlinear and eigenvalue problems, and systems of equations from fluid dynamics. The efficiency is optimal: the computational work is proportional to the number of unknowns.

This paper reviews the basic concepts and techniques of multigrid methods, concentrating on their role as fast solvers for elliptic boundary-value problems. Analysis of simple relaxation schemes for the Poisson problem shows that their slow convergence is due to smooth error components; approximating these components on a coarser grid leads to a simple multigrid Poisson solver. We review the principal elements of multigrid methods for more general problems, including relaxation schemes, grids, grid transfers, and control algorithms, plus techniques for nonlinear problems and boundary conditions. Multigrid applications, current research, and available software are also discussed.

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Wayne H. Schubert
,
Scott R. Fulton
, and
Rolf F. A. Herttenstein

Abstract

When a Squall line propagates through the atmosphere, it not only excite transient gravity–inertia wave motion but also produces more permanent modifications to the large-scale balanced flow. Here we calculate this balanced response using the is isentropic/geostrophic coordinate version of semigeostrophic theory. This approach results in a simple mathematical form in which the horizontal ageostrophic velocities am completely implicit and the entire dynamics reduces to a predictive equation for the potential pseudodensity and an invertibility relation. For a two-dimemional squall line, the potential pseudoderisity equation is simple enough to be solved analytically. The solutions illustrate how the squall line leaves in its wake a region of low potential pseudodensity in the lower troposphere and a region of potential pseudodensity in the upper troposphere. The solutions also show that the character of the potential pseudodensity modification by the squall line depends on the ratio of the convective overturning time to the squall line leaves passage time. This allows us to dynamically distinguish intensely mining, wide. slowly moving squau lines from weakly raining, narrow, fast-moving squall lines. After the potential pseudensity is determined, it can be used in the invertibility relation to yield balanced wind and mass fields which capture some of the observed large-scale features associated with squall lines.

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