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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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Piotr K. Smolarkiewicz

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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

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No abstract available.

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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

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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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Richard Rotunno and Piotr K. Smolarkiewicz

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Vanda Grubišić and Piotr K. Smolarkiewicz

Abstract

The effect of a critical level on airflow past an isolated axially symmetric obstacle is investigated in the small-amplitude hydrostatic limit for mean flows with linear negative shear. Only flows with mean Richardson numbers (Ri) greater or equal to ¼ are considered. The authors examine the problem using the linear, steady-state, inviscid, dynamic equations, which are well known to exhibit a singular behavior at critical levels, as well as a numerical model that has the capability of capturing both nonlinear and dissipative effects where these are significant.

Linear theory predicts the 3D wave pattern with individual waves that are confined to paraboloidal envelopes below the critical level and strongly attenuated and directionally filtered above it. Asymptotic solutions for the wave field far from the mountain and below the critical level show large shear-induced modifications in the proximity of the critical level, where wave envelopes quickly widen with height. Above the critical level, the perturbation field consists mainly of waves with wavefronts perpendicular to the mean flow direction. A closed-form analytic formula for the mountain-wave drag, which is equally valid for mean flows with positive and negative shear, predicts a drag that is smaller than in the uniform wind case. In the limit of Ri d⃗ ¼, in which linear theory predicts zero drag for an infinite ridge, drag on an axisymmetric mountain is nonzero.

Numerical simulations with an anelastic, nonhydrostatic model confirm and qualify the analytic results. They indicate that the linear regime, in which analytic solutions are valid everywhere except in the vicinity of the critical level, exists for a range of mountain heights given Ri > 1. For Ri d⃗ ¼ this same regime is difficult to achieve, as the flow is extremely sensitive to nonlinearities introduced through the lower boundary forcing that induce strong nonlinear effects near the critical level. Even well within the linear regime, flow in the vicinity of a critical level is dissipative in nature as evidenced by the development of a potential vorticity doublet.

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Piotr K. Smolarkiewicz and Richard Rotunno

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Piotr K. Smolarkiewicz and Richard Rotunno

Abstract

The present paper contains a continuation of our study of the flow of a density-stratified fluid past three-dimensional obstacles for Froude number ∼O(1). Linear theory (large Froude number) and potential-flow-type theory (small Froude number) are both invalid in this range, which is of particular relevance to natural, atmospheric flows past large mesoscale mountains. The present study was conceived to provide a systematic investigation of the basic aspects of this flow. Thus, we have excluded the effects of friction, rotation, nonuniform ambient flow, and the complexity of realistic terrain. In Part I of this study we focused on the pair of vertically oriented vortices forming on the lee side when the Froude number decreases below 0.5 (approximately), and argued that their formation may be understood in terms of nonlinear aspects of inviscid gravity waves, i.e., without invoking traditional arguments on the separation of the friction boundary layer. Herein we examine the zone of flow reversal on the windward side of the obstacle, which is also a characteristic feature of the low-Froude number flow. We find that flow stagnation and a tendency for flow reversal upwind of a symmetric bell-shaped obstacle is well predicted by linear inviscid gravity-wave theory. This finding stands in contrast with “horseshoe-vortex” arguments (attributed to frictional boundary layer separation) often invoked in the literature. We also perform experiments on obstacles of varying aspect ratio, β (across-stream length/along-stream length). Here the utility of the linear theory is less clear: considering cases with Fr = 0.33 and, β → ∞, we find that for β ≤ 1 upstream-propagating columnar modes are essentially absent, however, for β increasing beyond unity, they appear with increasing strength. It has been argued in the literature that having a sufficiently strong columnar mode is the means by which the upwind flow is brought to stagnation. The absence of this effect for β ≤ 1 (even though there is upwind-flow stagnation) and its appearance for β > 1 indicate the coexistence of two distinct gravity-wave effects that decelerate the flow upwind of an obstacle at low Froude number.

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