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

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 quasilinearly 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.

Conditional nonlinear optimal perturbation: Applications to stability, sensitivity, and predictability

“…Considering that the linearity of SVs does not take into account some mechanisms presented during the nonlinear development (such as wave-mean flow interactions), Riviere et al [26] applied a CNOP-like approach, a modified version of nonlinear singular vector (NLSV) approach developed by Mu [17] , in a two-layer quasigeostrophic model to investigate the effect of nonlinearities on the behavior of baroclinic unstable flows. In that study, the authors defined the objective function related to the so-called NLSV with perturbation growth rate, and confined the initial perturbations on the boundary of a constraint.…”

“…For example, Liu [25] proved theoretically that CNOPs locate at the boundary of a given constraint, which coincides with the numerical results demonstrated by other papers [5,18] . Riviere et al [26] applied CNOP-like approach in a two-layer quasi-geostrophic model of baroclinic instability to investigate the effect of nonlinearities on the behavior of baroclinic unstable flows. Terwisscha van Scheltinga [27] computed the CNOPs of the double-gyre ocean circulation by an implicit 4D-Var methodology and studied the finite amplitude stability of the double-gyre flow.…”

The time and regime dependencies of sensitive areas for tropical cyclone prediction using the CNOP method

“…The CNOP method represents an extension of the linear SV method to nonlinear regimes (Mu and Duan, 2003). It has been used for studying several pertinent areas: ENSO predictability (Duan and Mu, 2006;Mu et al, 2007b;Yu et al, 2009;Duan and Zhang, 2010), the sensitivity and passive variability of the ocean thermohaline circulation (THC) (Mu et al, 2004;Sun et al, 2005), double-gyre ocean circulations (Terwisscha van Scheltinga and Dijkstra, 2008), the nonlinear behavior of baroclinically unstable flows (Rivier et al, 2008), and ensemble forecasting (Mu and Jiang, 2008). In the study of Mu et al (2007aMu et al ( , 2009 for targeted observations, they found that the forecasts benefit more from reductions of CNOP-type initial errors than from reductions of SV-type initial errors.…”

An extension of conditional nonlinear optimal perturbation approach and its applications

“…The former has been largely investigated, and many theories and methods have been proposed or introduced (Lorenz, 1965;Toth and Kalnay, 1997;Mu et al, 2003;Mu and Zhang, 2006;Riviere et al, 2008), in which optimal methods are important for estimating the limit of the predictability of weather and climate events. The application of a singular vector (SV; Lorenz, 1965;Farell, 1989) in meteorology is pioneer in this scenario.…”