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Chantal Rivest and Brian F. Farrell


In a preceding paper a simple dynamical model for the maintenance of upper-tropospheric waves was proposed: the upper-level Eady normal modes. In this paper it is shown that these modes have counterparts in basic states with positive tropospheric gradients of potential vorticity, and that these counterparts can be maintained and excited on time scales consistent with observations.

In the presence of infinitesimal positive tropospheric gradients of potential vorticity, the upper-level normal-mode solutions no longer exist. That the normal-mode solution disappears when gradients are infinitesimal represents an apparent singularity and challenges the interpretation of upper-level synoptic-scale waves as related to the upper-level Eady normal modes. What happens to the upper-level modal solution in the presence of tropospheric gradients of potential vorticity is examined in a series of initial-value experiments. Our results show that they become slowly decaying quasi modes. Mathematically the quasi modes consist of a superposition of singular modes sharply peaked in the phase speed domain, and their decay proceeds as the modes interfere with one another. We repeat these experiments in basic states with a smooth tropopause in the presence of tropospheric and stratospheric gradients, and similar results are obtained.

Following a previous study by Farrell, a class of near-optimal initial conditions for the excitation of upper-level waves is identified. The initial conditions consist of upper-tropospheric disturbances that lean against the shear. They strongly excite upper-level waves not only in the absence of tropospheric potential vorticity gradients, but also in their presence. This result is important mathematically since it suggests that quasi modes are as likely to emerge from favorably configured initial disturbances as true normal modes, although the excitation is followed by a slow decay.

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Chantal Rivest, Andrew Staniforth, and André Robert


Semi-Lagrangian semi-implicit techniques are now well established and used by an increasing number of meteorological centers. However, it is demonstrated by both analysis and numerical integration that there is a serious problem incorporating orographic forcing into semi-Lagrangian models, since spurious resonance can develop in mountainous regions for Courant numbers larger than unity. A solution, consisting of two classes of schemes, is proposed, analyzed, and then evaluated using a global shallow-water model. Simply off-centering the semi-implicit scheme eliminates the spurious resonances. Although this can be achieved with a first-order scheme, it is at the expense of decreased accuracy, and therefore a second-order scheme is recommended.

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Chantal Rivest, Christopher A. Davis, and Brian F. Farrell


Upper-tropospheric waves are important in midlatitude synoptic-scale dynamics. Their life duration is typically longer than time scales for disruption by the ambient shaer and they often induce surface cyclogenesis. In this paper a dynamical model is proposed for the maintenance of these upper-level waves based on the upper-level Eady normal modes.

An analytical model of waves is developed on Eady basic states that have uniform tropospheric and stratospheric potential vorlicity. While the standard Eady basic state represents the limiting case of infinite stratospheric static stability, it is found that the standard Eady normal-mode characteristics hold in the presence of realistic tropopause and stratosphere. In particular, the basic states studied support upper-level normal modes of synoptic scale, which are also nonlinear solutions of the quasigeostrophic equations.

It was demonstrated earlier that the upper-level modal solutions do not exist in the presence of even infinitesimal tropospheric gradients of potential vorticity. However, it is shown here that the limit of the Eady neutral mode is approached smoothly by solutions that decay even more slowly as the gradients of potential vorticity vanish.

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