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Harshvardhan and David A. Randall

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David A. Randall

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Numerical simulation of geostrophic adjustment in shallow water is discussed for the case of an unstaggered grid for vorticity, divergence, and mass. The dispersion equation is shown to be very well behaved and superior to that obtained with the Arakawa grids A–E.

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Sandro Rambaldi and David A. Randall

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Thermals are modeled by considering their boundaries as self-advecting vortex sheets. Both slab-symmetric and axisymmetiic geometries are considered. Discussion is restricted to the case of a neutral environment, and turbulent processes are not considered.

For thermals of circular cross section, the initial accelerations are obtained analytically.

A numerical method is developed to simulate the evolution of thermals by time-marching. The vortex sheet is divided into finite segments, whose positions are tracked in a quasi-Lagrangian fashion. Self-advection is considered. A redistribution procedure is adopted to prevent the segments from bunching unmanageably. The induced field of motion is fully determined both inside and outside the thermal.

Results show that the axisymmetric thermal rises more quickly than the slab-symmetric thermal, that both thermals develop concave bases, that vorticity maxima occur both within the concavities and on the trailing edges, and that the leading edges are remarkably smooth.

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Ross Heikes and David A. Randall

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The finite-difference scheme for the Laplace and flux-divergence operators described in the companion paper (Part I) is consistent when applied on a grid consisting of perfect hexagons. The authors describe a necessary and sufficient condition for this finite-difference scheme to be consistent when applied on a grid consisting of imperfect hexagons and pentagons, and present an algorithm for generating a spherical geodesic grid on a sphere that guarantees that this condition is satisfied. Also, the authors qualitatively describe the error associated with the operators and estimate their order of accuracy when applied on the new grid.

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Ross Heikes and David A. Randall

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The streamfunction-velocity potential form of shallow-water equations, implemented on a spherical geodesic grid, offers an attractive solution to many of the problems associated with fluid-flow simulations in a spherical geometry. Here construction of a new type of spherical geodesic grid is outlined, and discretization of the equations is explained. The model is subjected to the NCAR suite of seven test cases for shallow-water models.

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Todd D. Ringler and David A. Randall

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Using the shallow water equations, a numerical framework on a spherical geodesic grid that conserves domain-integrated mass, potential vorticity, potential enstrophy, and total energy is developed. The numerical scheme is equally applicable to hexagonal grids on a plane and to spherical geodesic grids. This new numerical scheme is compared to its predecessor and it is shown that the new scheme does considerably better in conserving potential enstrophy and energy. Furthermore, in a simulation of geostrophic turbulence, the new numerical scheme produces energy and enstrophy spectra with slopes of approximately K −3 and K −1, respectively, where K is the total wavenumber. These slopes are in agreement with theoretical predictions. This work also exhibits a discrete momentum equation that is compatible with the Z-grid vorticity-divergence equation.

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Maike Ahlgrimm, David A. Randall, and Martin Köhler

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A strategy for model evaluation using spaceborne lidar observations is presented. Observations from the Geoscience Laser Altimeter System are recast onto the model grid to assess the ability of two versions of the Integrated Forecasting System to model marine stratocumulus clouds. The two model versions differ primarily in their treatment of clear and cloudy boundary layers. For each grid column, a representative cloud fraction and cloud-top height are derived from the observations, as well as from the model. By applying the same threshold criteria for cloud fraction and cloud-top height independently to model and observations, samples containing marine stratocumulus clouds can be identified. The frequency of occurrence, cloud fraction, and cloud-top height distributions for all samples thus identified are compared. The evaluation shows improvements in the frequency of occurrence and cloud-top height of marine stratocumulus, though modeled cloud tops remain lower than observed. Additional runs reveal a sensitivity to the strength of the environmental mixing that occurs during the test parcel ascent of the boundary layer parameterization. With a more aggressive parcel, the modeled clouds agree even better with observations.

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Patrick T. Haertel and David A. Randall

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A method for simulating fluid motions that shows promise for application to the oceans is explored. Incompressible inviscid fluids with free surfaces are represented as piles of slippery sacks. A system of ordinary differential equations governs the motions of the sacks, and this system is solved numerically in order to simulate a nonlinear deformation, internal and external gravity waves, and Rossby waves. The simulations are compared to analytic and finite-difference solutions, and the former converge to the latter as the sizes of the sacks are decreased.

The slippery-sack method appears to be well suited to ocean modeling for the following reasons: 1) it perfectly conserves a fluid's distributions of density and tracers; 2) unlike existing isopycnic models the slippery-sack method is capable of representing a continuum of fluid densities and vertically resolving neutral regions; 3) the inclusion of continuous topography adds no numerical complexity to the slippery-sack method; 4) the slippery-sack method conserves energy in the limit as the time step approaches zero; and 5) the slippery-sack method is computationally efficient.

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Todd D. Ringler and David A. Randall

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Shallow-water equations discretized on a perfect hexagonal grid are analyzed using both a momentum formulation and a vorticity-divergence formulation. The vorticity-divergence formulation uses the unstaggered Z grid that places mass, vorticity, and divergence at the centers of the hexagons. The momentum formulation uses the staggered ZM grid that places mass at the centers of the hexagons and velocity at the corners of the hexagons. It is found that the Z grid and the ZM grid are identical in their simulation of the physical modes relevant to geostrophic adjustment. Consistent with the continuous system, the simulated inertia–gravity wave phase speeds increase monotonically with increasing total wavenumber and, thus, all waves have nonzero group velocities.

Since a grid of hexagons has twice as many corners as it has centers, the ZM grid has twice as many velocity points as it has mass points. As a result, the ZM-grid velocity field is discretized at a higher resolution than the mass field and, therefore, resolves a larger region of wavenumber space than the mass field. We solve the ∇2 f = λf eigenvalue problem with periodic boundary conditions on both the Z grid and ZM grid to determine the modes that can exist on each grid. The mismatch between mass and momentum leads to computational modes in the velocity field. Two techniques that can be used to control these computational modes are discussed. One technique is to use a dissipation operator that captures or “sees” the smallest-scale variations in the velocity field. The other technique is to invert elliptic equations in order to filter the high wavenumber part of the momentum field.

Results presented here lead to the conclusion that the ZM grid is an attractive alternative to the Z grid, and might be particularly useful for ocean modeling.

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Michael D. Toy and David A. Randall

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The isentropic system of equations has particular advantages in the numerical modeling of weather and climate. These include the elimination of the vertical velocity in adiabatic flow, which simplifies the motion to a two-dimensional problem and greatly reduces the numerical errors associated with vertical advection. The mechanism for the vertical transfer of horizontal momentum is simply the pressure drag acting on isentropic coordinate surfaces under frictionless, adiabatic conditions. In addition, vertical resolution is enhanced in regions of high static stability, which leads to better resolution of features such as the tropopause. Negative static stability and isentropic overturning frequently occur in finescale atmospheric motion. This presents a challenge to nonhydrostatic modeling with the isentropic vertical coordinate. This paper presents a new nonhydrostatic atmospheric model based on a generalized vertical coordinate. The coordinate is specified in a manner similar to that of Konor and Arakawa, but “arbitrary Eulerian–Lagrangian” (ALE) methods are used to maintain coordinate monotonicity in regions of negative static stability and to return the coordinate surfaces to their isentropic “targets” in statically stable regions. The model is mass conserving and implements a vertical differencing scheme that satisfies two additional integral constraints for the limiting case of z coordinates. The hybrid vertical coordinate model is tested with mountain-wave experiments including a downslope windstorm with breaking gravity waves. The results show that the advantages of the isentropic coordinate are realized in the model with regard to vertical tracer and momentum transport. Also, the isentropic overturning associated with the wave breaking is successfully handled by the coordinate formulation.

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