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Nonlinear Generalization of Singular Vectors: Behavior in a Baroclinic Unstable Flow

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  • 1 Laboratoire de Météorologie Dynamique/IPSL, Ecole Normale Supérieure/CNRS, Paris, and Ecole Nationale des Ponts et Chaussées, Marne la Vallée, France
  • 2 Laboratoire de Météorologie Dynamique/IPSL, Ecole Normale Supérieure/CNRS, Paris, France
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Abstract

Singular vector (SV) analysis has proved to be helpful in understanding the linear instability properties of various types of flows. SVs are the perturbations with the largest amplification rate over a given time interval when linearizing the equations of a model along a particular solution. However, the linear approximation necessary to derive SVs has strong limitations and does not take into account several mechanisms present during the nonlinear development (such as wave–mean flow interactions). A new technique has been recently proposed that allows the generalization of SVs in terms of optimal perturbations with the largest amplification rate in the fully nonlinear regime. In the context of a two-layer quasigeostrophic model of baroclinic instability, the effect of nonlinearities on these nonlinear optimal perturbations [herein, nonlinear singular vectors (NLSVs)] is examined in terms of structure and dynamics. NLSVs essentially differ from SVs in the presence of a positive zonal-mean shear at initial time and in a broader meridional extension. As a result, NLSVs sustain a significant amplification in the nonlinear model while SVs exhibit a reduction of amplification in the nonlinear model. The presence of an initial zonal-mean shear in the NLSV increases the initial extraction of energy from the total shear (basic plus zonal-mean flows) and opposes wave–mean flow interactions that decrease the shear through the nonlinear evolution. The spatial shape of the NLSVs (and especially their meridional elongation) allows them to limit wave–wave interactions. These wave–wave interactions are responsible for the formation of vortices and for a smaller extraction of energy from the basic flow. Therefore, NLSVs are able to modify their shape in order to evolve quasi linearly to preserve a large nonlinear growth. Results are generalized for different norms and optimization times. When the streamfunction variance norm is used, the NLSV technique fails to converge because this norm selects very small scales at initial time. This indicates that this technique may be inadequate for problems for which the length scale of instability is not properly defined. For other norms (such as the potential enstrophy norm) and for different optimization times, the mechanisms of the NLSV amplification can still be viewed through wave–wave and wave–mean flow interactions.

Corresponding author address: G. Lapeyre, Laboratoire de Météorologie Dynamique, Ecole Normale Supérieure, 24 Rue Lhomond, 75005 Paris, France. Email: glapeyre@lmd.ens.fr

Abstract

Singular vector (SV) analysis has proved to be helpful in understanding the linear instability properties of various types of flows. SVs are the perturbations with the largest amplification rate over a given time interval when linearizing the equations of a model along a particular solution. However, the linear approximation necessary to derive SVs has strong limitations and does not take into account several mechanisms present during the nonlinear development (such as wave–mean flow interactions). A new technique has been recently proposed that allows the generalization of SVs in terms of optimal perturbations with the largest amplification rate in the fully nonlinear regime. In the context of a two-layer quasigeostrophic model of baroclinic instability, the effect of nonlinearities on these nonlinear optimal perturbations [herein, nonlinear singular vectors (NLSVs)] is examined in terms of structure and dynamics. NLSVs essentially differ from SVs in the presence of a positive zonal-mean shear at initial time and in a broader meridional extension. As a result, NLSVs sustain a significant amplification in the nonlinear model while SVs exhibit a reduction of amplification in the nonlinear model. The presence of an initial zonal-mean shear in the NLSV increases the initial extraction of energy from the total shear (basic plus zonal-mean flows) and opposes wave–mean flow interactions that decrease the shear through the nonlinear evolution. The spatial shape of the NLSVs (and especially their meridional elongation) allows them to limit wave–wave interactions. These wave–wave interactions are responsible for the formation of vortices and for a smaller extraction of energy from the basic flow. Therefore, NLSVs are able to modify their shape in order to evolve quasi linearly to preserve a large nonlinear growth. Results are generalized for different norms and optimization times. When the streamfunction variance norm is used, the NLSV technique fails to converge because this norm selects very small scales at initial time. This indicates that this technique may be inadequate for problems for which the length scale of instability is not properly defined. For other norms (such as the potential enstrophy norm) and for different optimization times, the mechanisms of the NLSV amplification can still be viewed through wave–wave and wave–mean flow interactions.

Corresponding author address: G. Lapeyre, Laboratoire de Météorologie Dynamique, Ecole Normale Supérieure, 24 Rue Lhomond, 75005 Paris, France. Email: glapeyre@lmd.ens.fr

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