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Miodrag Rančić

Abstract

A conservative and monotonic remapping algorithm is developed, which could be used as a component of a semi-Lagrangian transport scheme for numerical models of the atmosphere. The algorithm is a version of the monotonic piecewise parabolic interpolations combined with a “cascade” approach, where only one-dimensional operators are used for formulation of the multidimensional scheme.

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Miodrag Ranc̆ić

Abstract

Horizontal advection schemes on the semi-staggered grid E are presented, which in their linearized versions have a fourth-order space accuracy. The first is a scheme for evaluation of the horizontal nonlinear terms in the momentum equation. It prevents false energy cascade in the manner of the Janjić scheme, i.e., by conserving rational C grid energy and enstrophy. However, it is derived by transformation of the fourth-order rather than of the second-order Arakawa Jacobian. In an extension of the case which includes the divergent part of the flow, care is taken to achieve the conservation of the physically important integral properties of the general flow. The other is a scheme for advection of a passive scalar variable. It is derived by a modification of the former scheme.

The momentum equation scheme is tested in a long-term integration of the shallow water equations. As a consequence of the imposed conservation constraints, it is able to simulate general characteristics of the flow up to roughly 100 days of integration. The scalar advection scheme is tested in a traditional experiment with a rotating cone-shaped perturbation of the variable. The experiment demonstrates its advantages over a similar scheme, which is a modification of the original Janjić scheme.

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Miodrag Ranc̆ić

Abstract

A conservative semi-Lagrangian algorithm with a snall computational diffusion is presented that may be applied to advection of passive scalars in numerical models of the atmosphere. The technique is preferable for the horizontal semistaggered grids where a scalar point is surrounded by four velocity points. It is a coupling of the semi-Lagrangian approach and the piecewise parabolic method (PPM). Unlike the original PPM when applied to the advection of passive scalars, the new scheme is a fully two-dimensional algorithm. Also, it is not restricted by the linear stability condition. This paper describes the two steps comprising the two-dimensional algorithm. The first one is the interpolation pressure for getting the piecewise biparabolic function, and the second is a conservative remapping from the original grid to the grid made by the departure domains. Several test integrations are presented in which the described scheme performs very successfully.

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Miodrag Rančić and Gordana Sindjić

Abstract

A simple noninterpolating semi-Lagrangian advection algorithm is presented, which uses a two-time-level scheme with minimized dissipation and dispersion errors in the Eulerian step. Advection experiments with uniform Bow demonstrate the stability of the algorithm. Moreover, the increased time step even improves the obtained results. The technique could be applied as a fast and economical method for computing the advection of scalar fields in numerical models of the atmosphere.

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Miodrag Rančić, Hai Zhang, and Verica Savic-Jovcic

Abstract

Successful treatment of nonlinear momentum advection is one of the outstanding challenges for the application of rectangular quasi-uniform spherical grids in global circulation models. Quasi-uniform grids (e.g., cubic and octagonal), which are virtually assembled by connecting a set of regional domains along their boundaries, appear to be an excellent choice for the expansion of regional atmospheric models to global coverage. However, because of an unavoidable lack of orthogonality of these grids in the proximity of the singular points (i.e., the corner points connecting three neighboring rectangular tiles), a common-sense approach is to first generalize underlying numerical schemes to the general curvilinear coordinates, and then to apply globalization. In this procedure, assuming that a “weak conservative formulation” for the generalization is applied, the advective formalism of the Arakawa-type momentum schemes and some of their properties, especially those important for the long-term “climate type” simulations, may be lost. This paper discusses challenges faced in the application of Arakawa-type nonlinear advection schemes on the quasi-uniform semistaggered grids and suggests a solution that is based on discretization of the momentum equation in the vector form. Both the second- and the fourth-order energy-conserving nonlinear advection schemes are considered. The potential merits of this approach are demonstrated in a series of benchmark test integrations of a shallow-water model on the octagonal quasi-uniform grid.

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Miodrag Rančić, R. James Purser, Dušan Jović, Ratko Vasic, and Thomas Black

Abstract

The rapid expansion of contemporary computers is expected to enable operational integrations of global models of the atmosphere at resolutions close to 1 km, using tens of thousands of processors in the foreseeable future. Consequently, the algorithmic approach to global modeling of the atmosphere will need to change in order to better adjust to the new computing environment. One simple and convenient solution is to use low-order finite-differencing models, which generally require only local exchange of messages between processing elements, and thus are more compatible with the new computing environment. These models have already been tested with physics and are well established at high resolutions over regional domains. A global nonhydrostatic model, the Nonhydrostatic Multiscale Model on the B grid (NMMB), developed at the Environmental Modeling Center of the National Centers for Environmental Prediction during the first decade of this century is one such model. A drawback of the original version of global NMMB is that it is discretized on the standard longitude–latitude grid and requires application of Fourier polar filtering, which is relatively inefficient on massively parallel computers. This paper describes a reformulation of the NMMB on the grid geometry of a novel cubed sphere featuring a uniform Jacobian of the horizontal mapping, which provides a uniform resolution close to that of the equiangular gnomonic cubed sphere, but with a smooth transition of coordinates across the edges. The modeling approach and encountered challenges are discussed and several results are shown that demonstrate the viability of the approach.

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