Generalized energetics for linear and nonlinear symmetric instabilities

The generalized energy conservation integral is applied to linear and nonlinear symmetric instabilities (SIs) and conditional symmetric instabilities (CSIs) in uniform and nonuniform basic states. The generalized energy conservation integral for linear SI in nonuniform basic state is refined to include initial thermal-inertial perturbations. A generalized conservation integral is derived for a nonlinear SI with uniform and nonuniform basic states. The integral reveals that: (1) the stability conditions of nonlinear and linear SI are not the same; (2) when the basic state is symmetrically stable the nonlinear evolution of a symmetric disturbance is energetically bounded by its initial energy; and (3) when the basic state is symmetrically unstable the growth of the symmetric perturbation is approximated by the linear SI theory. Precipitative heating is incorporated into the generalized energy integral for nonlinear CSI; the relation between nonlinear evolution of a CSI circulation and precipitative heating is examined.