Scott R. Fulton

Polygonal Eyewalls, Asymmetric Eye Contraction, and Potential Vorticity Mixing in Hurricanes

Hurricane eyewalls are often observed to be nearly circular structures, but they are occasionally observed to take on distinctly polygonal shapes. The shapes range from triangles to hexagons and, while they are often incomplete, straight line segments can be identified. Other observations implicate the existence of intense mesovortices within or near the eye region. Is there a relation between polygonal eyewalls and hurricane mesovortices? Are these phenomena just curiosities of the hurricane’s inner-core circulation, or are they snapshots of an intrinsic mixing process within or near the eye that serves to determine the circulation and thermal structure of the eye? As a first step toward understanding the asymmetric vorticity dynamics of the hurricane’s eye and eyewall region, these issues are examined within the framework of an unforced barotropic nondivergent model. Polygonal eyewalls are shown to form as a result of barotropic instability near the radius of maximum winds. After reviewing linear theory, simulations with a high-resolution pseudospectral numerical model are presented to follow the instabilities into their nonlinear regime. When the instabilities grow to finite amplitude, the vorticity of the eyewall region pools into discrete areas, creating the appearance of polygonal eyewalls. The circulations associated with these pools of vorticity suggest a connection to hurricane mesovortices. At later times the vorticity is ultimately rearranged into a nearly monopolar circular vortex. While the evolution of the finescale vorticity field is sensitive to the initial condition, the macroscopic end-states are found to be similar. In fact, the gross characteristics of the numerically simulated end-states are predicted analytically using a generalization of the minimum enstrophy hypothesis. In an effort to remove some of the weaknesses of the minimum enstrophy approach, a maximum entropy argument developed previously for rectilinear shear flows is extended to the vortex problem, and end-state solutions in the limiting case of tertiary mixing are obtained. Implications of these ideas for real hurricanes are discussed.

Geostrophic Adjustment in an Axisymmetric Vortex

Abstract A linearized system of equations for the atmospheres first internal mode in the vertical is derived. The system governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height. When the basic flow is at rest, solutions for the transient and final adjusted state are found by the method of Hankel transforms. Two examples are considered, one with an initial top hat potential vorticity and one with an initial Gaussian-type potential vorticity. These two examples, which extend the work of Fischer (1963) and Obukhov (1949), indicate that the energetical efficiency of cloud-cluster-scale heating in producing balanced vortex flow is very low, on the order of a few percent. The vast majority of the energy is simply partitioned to gravity-inertia waves. In contrast the efficiency of cloud-cluster-scale vorticity transport is very high. When the basic state possesses positive relative vorticity in an inner region, the energy partition can be subst...

Multigrid Methods for Elliptic Problems: A Review

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 ...

Vertical normal mode transforms: theory and application

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.

A Classification of Binary Tropical Cyclone–Like Vortex Interactions*

Abstract The interaction between two tropical cyclones with different core vorticities and different sizes is studied with the aid of a nondivergent barotropic model, on both the f plane and the sphere. A classification of a wide range of cases is presented, using the Dritschel–Waugh scheme, which subdivides vortex interactions into five types: elastic interaction, partial straining out, complete straining out, partial merger, and complete merger. The type of interaction for a vortex pair on the f plane, and the same pair on the sphere, was the same for 77 out of 80 cases studied. The primary difference between the results on the f plane and those on the sphere is that the vorticity centroid of the pair is fixed on the f plane but can drift a considerable distance poleward and westward on the sphere. In the spherical case, the interaction between the cyclone pair and the associated β-induced cyclonic and anticyclonic circulations can play an important role. The “partial merger” regime is studied in detail...

Improved spectral multigrid methods for periodic elliptic problems

Abstract The spectral multigrid method for periodic elliptic problems is examined. Several modifications are introduced, including a midpoint pseudospectral discretization which eliminates the need for filtering the highest Fourier mode and new relaxation schemes for isotropic and anisotropic problems. Numerical results are presented demonstrating substantial increases in efficiency and accuracy over previous methods.

Chebyshev Spectral Methods for Limited-Area Models. Part I: Model Problem Analysis

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 ...

Balanced Atmospheric Response to Squall Lines

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 convecti...

Fourier Analysis of Multigrid Methods on Hexagonal Grids

This paper applies local Fourier analysis to multigrid methods on hexagonal grids. Using oblique coordinates to express the grids and a dual basis for the Fourier modes, the analysis proceeds essentially the same as for rectangular grids. The framework for one- and two-grid analyses is given and then applied to analyze the performance of multigrid methods for the Poisson problem on a hexagonal grid. Numerical results confirm the analysis. Uniform hexagonal grids provide an approximation to spherical geodesic grids; numerical results for the latter show similar performance. While the analysis is similar to that for rectangular grids, the results differ somewhat: full weighting is superior to injection for restriction, Jacobi relaxation performs about as well as Gauss-Seidel relaxation, and underrelaxation is not required for good performance. Also, coarse-fine or four-color ordering (both analogues of red-black ordering on the rectangular grid) improves the performance of Jacobi relaxation, with the latter achieving a smoothing factor of approximately 0.25. An especially simple compact fourth-order discretization works well, and the full multigrid algorithm produces the solution to the level of truncation error in work proportional to the number of unknowns.

Surface Frontogenesis in Isentropic Coordinates

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.