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Alain Joly

Abstract

The adjoint method for finding optimal or singular modes is employed for studying the finite time stability of steady, tw0-dimensional atmospheric fronts as represented by the uniform potential vorticity semigeostrophic model.

The most unstable singular models over a given period of time are computed for a wide range of scalar products. The reference scalar products are relevant to physical space and include total, kinetic, or potential energy; geopotential variance; and enstrophy.

A front inspired by observations from FRONTS 87 and including a surface potential temperature anomaly is examined first through the usual linear results. The validity of the linear approximation is considered as a function of amplitude. The modes are also integrated in nonlinear simulations and their life cycles am shown.

Results indicate that each norm and wave has its own preferred spatial scale. This severely restricts the concept of scale selection. Energy and geopotential variance modes increase mostly by improving the energy collection by barotropic processes. Enstrophy modes favor baroclinic processes. The linear approximation is more restrictive for the former than for the latter. In the nonlinear regime, the enstrophy mode exhibits faster deepening rates and larger vertical velocities.

Similar conclusions arise for the Hoskins-Bretherton deformation front in the same range of wavelengths, although this front is stable in the sense of Charney and Stern. The discussion examines the scale selection process inherent to the different scalar products.

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Alain Joly
and
Alan J. Thorpe

Abstract

A methodology suitable for assessing the stability of any time-dependent basic state is presented. The equivalent of the normal modes for steady basic states are the eigenvectors of the resolvent matrix; this matrix incorporates the evolution of the large-scale flow, and growth rates are replaced by amplification rates. This method is applied to the three-dimensional stability of two-dimensional fronts undergoing frontogenesis in the presence of latent heat release in a semigeostrophic model. Disturbances developing in this flow are therefore geostrophically balanced. The concepts are first illustrated in a dry time-dependent uniform shear and potential vorticity flow. At any time during the evolution of the basic flow the stability can be compared to that obtained by assuming that the frontogenesis has, at that instant, ceased. Although differences between the results from the two methods exist, general conclusions as to the scales and structure of the modes are not altered; only large-scale waves are unstable. The situation in moist baroclinic waves is dramatically different. Growth rates are enhanced compared to the steady state analysis, but the possibility for frontal waves on the 1000-km scale to amplify most rapidly depends on the rate of development of the parent wave. Such waves dominate the spectrum only when that rate is slow and then only when the frontal ascent takes on a small cross-frontal width and the vorticity maximum penetrates over a deep layer. The short-wave growth is mostly due to latent heat release in the wave. This heating is shown, in a simplified case, to modify the necessary conditions for instability. It is concluded that shearing deformation does not intrinsically inhibit frontal instability, but paradoxically it greatly favors two-dimensional growth in the early stages due to the more rapid frontogenesis in the presence of latent heating. The role of stretching deformation may be substantially different but is not considered here.

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Alain Joly
and
Alan J. Thorpe

Abstract

The stability of the steady two-dimensional horizontal shear front to geostrophic disturbances in the along-front direction is examined within the framework of semi-geostrophic theory. The basic state corresponds to the geostrophic along-front flow at any time during the nonlinear evolution of a two-dimensional Eady wave. The matrix resulting from the stability analysis can be transformed into a weakly nondiagonal form. Its structure shows that the selection of the most unstable along-front wavenumber is independent of the “intensity” of the front. The growth rate is a linear function of this amplitude. The most unstable along-front mode is a modified Eady mode stationary with respect to the front. It draws a fraction of its energy from the shear. For smaller along-front wavelengths, the solution is dominated by propagating modes near the boundaries. These are also baroclinic, with a larger contribution from the basic kinetic energy and much smaller growth rates. It is apparent that the existence of a vorticity maximum at fronts, however large, is not sufficient to produce the observed small scale of frontal waves. Anomalous potential vorticity at the front is necessary to provide a deep zone of large horizontal shear and hence the reduced horizontal scale of waves.

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Matthieu Plu
,
Philippe Arbogast
, and
Alain Joly

Abstract

Midlatitude cyclogenesis as interpreted in the framework of either baroclinic development or potential vorticity thinking heavily relies on the concept of synoptic-scale anomaly. Given the existence of potential vorticity inversion and attribution, what is at stake to provide a mathematical definition for this concept is a complete finite-amplitude alternative to the linear-based theory of cyclogenesis. The existence of a reasonably objective way to represent anomalies in both real and idealized flows would not only help understanding cyclogenesis, it would also have many other applications for both theory and in practical forecasts. Inspired by the recent theory of wavelet representation of coherent structures in two-dimensional fluid mechanics, a wavelet representation of three-dimensional potential vorticity anomalies is built. This algorithm relies on the selection of the appropriate two-dimensional wavelet coefficients from the stationary wavelet transform in order to guarantee the critical translation-invariance property. The sensitivity of the algorithm to the position, size, and shape of the structures is assessed.

The wavelet extraction is then applied to the upper-level precursor of a real-case storm of December 1999 and is compared to a basic monopolar extraction. Using potential vorticity inversion and forecasts with a primitive-equation model, it is found that both anomalies have similar implications on the development of the surface cyclone. However, the coherence in time of the extracted wavelet structure in the forecast and analysis sequence is more satisfactory than the extracted monopole: this suggests that the underlying mathematical description of an anomaly proposed here does, indeed, point toward the direction of an actual physical reality of the concept.

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