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Piotr K. Smolarkiewicz

Abstract

The conservation form of the second-order Crowley advection scheme, as some authors have pointed out, may lead to numerical instability for multi-dimensional blows. Replacing the original second-order-accurate approximation of the first spatial partial derivative, with one that includes information about the dimensionality of the field, and also considering the cross-space difference, eliminates this instability without destroying the level of conservation or the accuracy of the original Crowley scheme. The paper also presents some flux correction solutions, the use of which avoids the development of negative values in the solution for positive definite scalars. The paper also discusses the solution of the advection equation in the em of strong deformational flow.

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Piotr K. Smolarkiewicz

Abstract

No abstract available.

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Piotr K. Smolarkiewicz

Abstract

No abstract available.

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Piotr K. Smolarkiewicz

Abstract

The development of negative values for positive definite scalars in the solution of the advection equation is an important difficulty in numerical modeling. This paper proposes a new positively definite advection scheme which has a simple form, small implicit diffusion and low computational cost. Comparisons of the present scheme with some other known positive definite schemes are also presented.

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Piotr K. Smolarkiewicz

Abstract

The purpose of this note is to show that the popular Crowley advection scheme, which is usually considered to have first-order accuracy in time for a nonuniform velocity field, can be constructed so as to have second-order accuracy both in time and space for an arbitrary flow field.

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Piotr K. Smolarkiewicz

Abstract

This note discusses the extension of the dissipative advection schemes, often referred to in meteorological literature as Crowley-type schemes, on advection equations with arbitrary forcing and/or source terms included. Since such equations constitute a prototype of prognostic equations for fluids, the considerations herein are relevant to a variety of atmospheric problems. The thesis of this note is that, no matter how accurate the advection scheme employed, the entire equation is approximated to, at most, Ot), which is a consequence of disregarding forcing terms in the derivation of Crowley-type schemes. The consequences of this truncation error may be quite severe depending on the particular problem at hand. The remedy proposed is simple and easy to implement in any numerical model using forward-in-time differencing. Theoretical considerations are illustrated with an example of a flow of the density-stratified fluid past a two-dimensional mountain.

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Wojciech W. Grabowski
and
Piotr K. Smolarkiewicz

Abstract

This paper discusses two-time-level semi-Lagrangian approximations for the bulk warm-rain microphysics embedded in the framework of an anelastic cloud model. The central theoretical issue is a semi-Lagrangian integration of the rain-evolution equation. Because departure points of rain trajectories differ from those of flow trajectories and the terminal velocity of the precipitation depends on the concentration of the precipitation itself, effective semi-Lagrangian approximations are not necessarily straightforward. Some simplifying assumptions are adopted that compromise formal accuracy and computational efficiency of the method. Theoretical considerations are illustrated with idealized simulations of precipitating thermal convection and orographically forced clouds. Comparisons with corresponding results obtained using a more traditional, flux-form Eulerian cloud model document highly competitive performance of the semi-Lagrangian approach. Although derived for the warm-rain parameterization, the method presented in this paper is universal; that is, it may be easily extended to any standard microphysical parameterization, including ice physics or detailed microphysics.

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Piotr K. Smolarkiewicz
and
Len O. Margolin

Abstract

This paper extends the discussion of fully second-order-accurate, forward-in-time, finite-difference schemes for the advection equation with arbitrary forcing (which is viewed as a prototype for the prognostic equations of fluid dynamics) to an arbitrary curvilinear system of coordinates. Since forward-in-time schemes derive ultimately from Taylor series analysis of the uncentered-in-time differencing, it is important to include the appropriate metric terms explicitly into the algorithm's design. A rigorous truncation-error analysis leads to a compact scheme that preserves (to second-order accuracy) the consistency of Eulerian and Lagrangian formulations for fluids. Alternative approximations to the advective velocity in the transport flux are also discussed. In order to achieve second-order accuracy of the forward-in-time approximation, the advective velocity must be evaluated to at least first-order accuracy at the intermediate time level. Such a temporal staggering is usually simulated by means of either linear interpolation or linear extrapolation. The alternative considered in this paper employs an extrapolation consistent with the governing equations of motion. This approximation is derived from concepts inherent in Runge-Kutta methods for ordinary differential equations. It allows at least twice the usual time step in simulations of elastic systems, where high-speed propagating modes dominate the computational stability. These theoretical considerations are illustrated with idealized tests and examples of shallow-water flows on the sphere.

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Wojciech W. Grabowski
and
Piotr K. Smolarkiewicz

Abstract

We discuss herein numerical difficulties with finite-difference approximations to the thermodynamic conservation laws near sharp, cloud-environment interfaces. The Conservation laws for entropy and water substance variables are coupled through the phase change processes. This coupling of the thermodynamic equations may lead to spurious numerical oscillations that, in general, are not prevented by direct application of traditional monotone methods developed for the uncoupled equations. In order to suppress false oscillations in the solutions, we consider special techniques which derive from the flux-corrected-transport (FCT) methodology. In these, we incorporate physical information about condensation-evaporation processes directly into the limiters constraining the antidiffusive fluxes of the FCT methods. We elaborate upon two different advection-condensation schemes relevant to the two formulations of the advection-condensation problem, commonly used in cloud modeling. For the fractional-time-steps formulation, we develop an FCT advevtion scheme which preserves the monotone character of transported thermodynamic variables and also ensures monotonicity of the relative humidity field diagnosed after the advection step. This results in monotone solutions after the entire advection-condensation cycle. For the conservative variables formulation, we derive a simple and efficient FCT scheme that ensures nonoscillatory thermodynamic fields diagnosed from the advected variables. Theoretical considerations are illustrated with idealized one-dimensional kinematic tests and with examples of two-dimensional simulations of a small cumulus cloud. Dynamical calculations show that morphology of the cloud-environment interface strongly depends on the numerical scheme applied.

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Wojciech W. Grabowski
and
Piotr K. Smolarkiewicz

Abstract

The recently reported nonhydrostatic anelastic numerical model for simulating a range of atmospheric processes on scales from micro to planetary is extended to moist processes. A theoretical formulation of moist precipitating thermodynamics follows the standard cloud models; that is, it explicitly treats the formation of cloud condensate and the subsequent development and fallout of precipitation. In order to accommodate a broad range of temporal scales, the customized numerical algorithm merges the explicit scheme for the thermodynamics with the semi-implicit scheme for the dynamics, where the latter is essential for the computational efficiency of the global model. The coarse spatial resolutions used in present global models result in a disparity between the timescales of the fluid flow and the much shorter timescales associated with phase-change processes and precipitation fallout. To overcome this difficulty the approach based on the method of averages is employed, where fast processes are evaluated with adequately small time steps (and lower accuracy) over the large time step of the model, to provide an accurate approximation to the large time step integral of fast forcings in the stiff system. This approach allows for stable integrations when cloud processes are poorly resolved and it converges to the formulation standard in cloud models as the resolution increases. The theoretical developments are tested in simulations of small-, meso-, and planetary-scale idealized moist atmospheric flows. Results from the small-scale simulations demonstrate that the proposed approach compares favorably with traditional explicit techniques used in cloud models. Planetary simulations, on the other hand, illustrate an ability to capture moist processes in low-resolution large-scale flows.

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