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Joseph Egger

Abstract

The numerical solution to the linear advection equation with oscillating forcing is derived in analytic form for two-level schemes and for the leapfrog scheme with an Asselin filter. The numerical solutions are compared to the analytic solution of the advection equation with emphasis on the forced part. A detailed analysis is presented for the trapeze, the backward, and the leapfrog scheme with Asselin filter. Large deviations are found in quasi-resonant situations where the period of forcing and advection are close. Damping schemes fail completely to capture the resonant case. As for amplitude errors, the backward scheme is generally better than the trapeze scheme outside the quasi-resonant domain. However, the backward scheme produces large phase errors while the trapeze solutions are free of such errors. The leapfrog scheme has a resonant solution but generates large-amplitude errors near resonance. On the other hand, phase errors are particularly small in that case. The amplitude of the numerical mode tends to be large if either the forcing period or the advective period are but coarsely resolved. Addition of the Asselin filter removes the numerical high-frequency oscillations but destroys the resonant solution.

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Joseph Egger

Abstract

It is proposed to monitor the conservation of “material” volumes in the phase space of a numerical model. In principle, strict volume conservation is required for reversible flow problems. In particular, the phase space volume occupied by the initial distribution of an ensemble forecast should not change in time for such flows. It is not obvious to what extent numerical schemes satisfy this requirement. A corresponding test for volume conservation is designed. Euler, upstream, Lax–Wendroff and Runge–Kutta schemes are exposed to this test as well as a time centered implicit scheme and the leapfrog method. Members of the former group are numerically irreversible, while those of the latter are reversible in time. Tests are performed for gridpoint representations of advection equations and of one-dimensional inviscid shallow-water flow. The numerically irreversible schemes exhibit spurious contraction or expansion. The error is O(C) for the upstream scheme and O(C 2) for all the other irreversible schemes except fourth-order Runge–Kutta (C is the Courant number). The implicit scheme conserves volume for reversible flow problems. It is a surprising result that the leapfrog scheme conserves volume for any type of equations. Applications to Monte Carlo forecasts are presented. It is shown that clouds of initial states contract or expand indeed as predicted by the test in the early stages of an integration. This affects the quality of the Monte Carlo forecast.

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Joseph Egger

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The stochastic model of Weickmann et al. for the global angular momentum budget is modified to become applicable to latitude belts. In particular, a Langevin equation is added for the flux divergence of angular momentum in a belt. The friction torque Tf is assumed to be purely damping with respect to angular momentum M. The mountain torque To is generated by red noise but also damps angular momentum directly as suggested by recent stochastic models. The model parameters are tuned such that the variances of all model variables come close to the observations. The corresponding equations for the covariance functions of all variables are solved analytically. The results are compared to observations for selected belts. It is found that the model captures the observed decay rates of all covariance functions. The covariance of the flux divergence and the angular momentum is simulated successfully for positive lags but rarely for negative ones. The covariance of friction torque and angular momentum is reproduced reasonably well. The model is also successful with respect to the covariance of mountain torque and M in the Tropics, but there are large discrepancies at midlatitudes because the observed mountain torque events are accompanied by flux divergences in these belts.

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Joseph Egger
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Joseph Egger
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Joseph Egger

Abstract

A shallow inversion layer with southeasterly outflow and a cyclonic vortex in the troposphere are the basic characteristics of the Antarctic mean circulation. An attempt is made to model this pattern in a two-layer representation of the atmosphere where all equations are averaged horizontally over the Antarctic domain. Cooling at the slope drives a direct circulation that acts as a source of westerly angular momentum. This momentum is transferred out of Antarctica by topographically modified large-scale waves, enforced at the northern boundary of the model. Two types of steady states are found for fixed frequency and zonal wavenumber: one where the wave is quite effective in performing the required momentum transport so that a qualitatively realistic circulation results and another one with strong upper-level westerlies but virtually no surface easterlies. A model climatology can be derived if stochastic forcing is added to the equations. It turns out that the distribution of the flow states is centered near a “realistic” equilibrium if a wave spectrum is prescribed at the northern boundary according to observations.

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Joseph Egger

Abstract

The spurious numerical generation and/or destruction of various types of entropies in models is investigated. It is shown that entropy s θ of dry matter tends to be generated if potential temperature is advected by a damping scheme. There is no mean tendency of entropy if the reversible leapfrog scheme is used. Generalized entropies can be assigned to conserved quantities. In particular, the generalized entropy s ζ of the vorticity of two-dimensional nondivergent flow is shown to grow in presence of irreversible diffusive processes. This entropy increases numerically if the vorticity equation is integrated with an upstream scheme. There are weak oscillations of s ζ if a leapfrog time step is combined with the Arakawa scheme. Similar results are obtained for an entropy s p related to potential vorticity. Information entropy provides a gross measure of the information contained in ensemble forecasts. It is shown that information entropy decreases spuriously if schemes are used that are contracting in phase space. It is argued that the evaluation of entropies provides a useful check of the quality of numerical schemes.

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Joseph Egger

Abstract

Lateral momentum transport by gravity waves is investigated within the framework of a linear steady-state model of inviscid, nonrotating flow over and around orography. Obstacles protruding laterally from massifs are considered as well as more conventional shallow mountains. For flow around obstacles, an analytic solution to the three-dimensional gravity wave equations with a lateral radiation condition is derived for constant zonal mean flow and stability. Laterally propagating gravity waves transport momentum toward the mountain if a rigid lid is imposed. Downward transports are important as well if the lid is removed. These results are compared to those obtained on the basis of the shallow mountain approximation.

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Joseph Egger

Abstract

Although mountains are generally thought to exert forces on the atmosphere, the related transfers of energy between earth and atmosphere are not represented in standard energy equations of the atmosphere. It is shown that the axial rotation of the atmosphere must be included in the energy budget in order to resolve this issue. The energy transfer resulting from mountains turns out to be closely related to mountain torques. The energetic effects of a changing rotation of the earth are discussed, as well as those of friction torques and those of the nonspherical shape of the earth.

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Joseph Egger

Abstract

Planetary-scale orography exerts a substantial pressure drag on the atmosphere. This drag appears to be partially balanced by the convergence of momentum transports by Rossby waves induced by these mountains. Simple models of this process are presented within the framework of inviscid barotropic zonally periodic β-plane flow. Lateral radiation conditions are imposed for steady flow so as to allow for meridional wave radiation. Solutions are presented to the nonlinear vorticity equation and to the linear shallow-water equations. The relation between drag and flow configuration is discussed. An attempt is made to relate the results to the situation in the atmosphere and also in the ocean. In particular, it is demonstrated that coastlines with capes may exert a drag on ocean currents.

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