Disturbance Potential Enstrophy Research Articles

Disturbance evolution and the nonlinear stability to the basic flows for two-dimensional quasi-geostrophic motion

The status of disturbances of both initial values and parameters in the models is further investigated, the exact explicit estimates on the disturbance energy and disturbance potential enstrophy are given; and while the initial disturbance fields rely only on the initial disturbance potential enstrophy, initial disturbance velocity circulation along the boundary, disturbance parameters, and the nonlinear stability criteria paralleling to Arnold’s second theorem are obtained, and the main results of Mu are generalized.

On disturbance evolution and nonlinear stability of the maltilayer quasi-geostrophic basic flows

The status of disturbances of both initial values and parameters in the model is further investigated, and the exact explicit estimates on the disturbance energy and disturbance potential enstrophy are given, while the initial disturbance fields depend only on the initial disturbance potential enstrophy, initial disturbance velocity circulation along the boundary, and disturbance parameters. And based on the generalized nonlinear stability concept in the sense of Lyapunov, the nonlinear stability criteria paralleling to Arnold’s second theorem are obtained.

Nonlinear stability of continuously stratified quasi-geostrophic flow

Nonlinear stability theorems analogous to Arnol'd's second stability theorem are established for continuously stratified quasi-geostrophic flow with general nonlinear boundary conditions in a vertically and horizontally confined domain. Both the standard quasi-geostrophic model and the modified quasi-geostrophic model (incorporating effects of hydrostatic compressibility) are treated. The results establish explicit upper bounds on the disturbance energy, the disturbance potential enstrophy, and the disturbance available potential energy on the horizontal boundaries, in terms of the initial disturbance fields. Nonlinear stability in the sense of Liapunov is also established.

Nonlinear Stability Theorem for Eady's Model of Quasigeostrophic Baroclinic Flow

Abstract In this paper, the linear stability criterion for Eady's problem is proved to be also a nonlinear stability one. Upper bounds for the disturbance potential enstrophy, the disturbance energy, and the disturbance available potential energy on the rigid lids are also established.

Nonlinear Stability of Eady's Model

A nonlinear stability theorem is established for Eady's model of baroclinic flow. In particular, the Eady basic state is shown to be nonlinearly stable (for arbitrary shear) provided (Δz)(Δy)1/2f(πN) ,where Δz is the height of the domain, Δy the channel width, f the Coriolis parameter, and N the buoyancy frequency. When this criterion is satisfied, explicit bounds can be derived on the disturbance potential enstrophy, the disturbance energy, and the disturbance available potential energy on the rigid lids, which are expressed in terms of the initial disturbance fields. The disturbances are completely general (with nonzero potential vorticity) and are not assumed to be of small amplitude. The results may be regarded as an extension of Arnol'd's second nonlinear stability theorem to continuously stratified quasigeostrophic baroclinic flow.

Nonlinear stability of multilayer quasi-geostrophic flow

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

Nonlinear stability of intermediate baroclinic flow on a sloping bottom

The governing equations describing the dynamics of mesoscale gravity currents or coupled density fronts and steadily-travelling coherent cold eddies on a sloping bottom are shown to possess a non-canonical hamiltonian structure. We exploit the hamiltonian formalism to obtain a variational principle that describes arbitrary steady solutions in terms of a suitably constrained hamiltonian. Two Arnol’d-like stability theorems are obtained which can establish the linear stability in the sense of Liapunov of these steady solutions. Based on this analyses two a priori estimates are derived which bound the disturbance energy and the Liapunov norm with respect to the initial disturbance potential enstrophy and energy. In the limit of parallel shear flow solutions corresponding to a current flowing along isobaths, the first formal stability theorem reduces to a previously established normal-mode stability result. Based on the formal stability analysis, convexity conditions are given for the constrained hamiltonian that can rigorously establish nonlinear stability in the sense of Liapunov for the steady current solutions. A variational principle is also presented which can describe steadily-travelling isolated cold eddy solutions of the model. The principle is based on constraining the hamiltonian with appropriately chosen Casimir and momentum invariants. It is shown that a suitably extended form of Andrews’ theorem holds for our model equations. Therefore, the stability of the steadily travelling isolated eddy solutions cannot be established using the energy-Casimir analysis developed here.