Potential Enstrophy Research Articles

Interactions between Geostrophic Eddies and the Mean Circulation over Large-Scale Bottom Topography

Abstract The presence of intense recirculations in the abyssal oceans has been revealed by both observations and modeling studies. A suggested mechanism is the interactions between geostrophic eddies and the mean circulation in the presence of variable bottom topography. Here such interactions are studied using an idealized numerical model, consisting of a single active abyssal layer overlying variable bottom topography. An initial ensemble of eddies quickly organize themselves to generate a mean anticyclonic circulation around seamounts. During this rearrangement energy is approximately conserved while potential enstrophy is dissipated, consistent with previous studies of geostrophic turbulence. A parameterization of geostrophic eddies is formulated in terms of an eddy-induced transport, U*. Based on results from the eddy-resolving experiments it is hypothesized that U* should dissipate potential enstrophy while conserving energy. Using variational methods, a solution for U* is found that dissipates pote...

Eddy Fluxes and Second-Order Moment Balances for Nonhomogeneous Quasigeostrophic Turbulence in Wind-Driven Zonal Flows*

Abstract The balances of momentum and second-order moments (potential enstrophy, energies, and potential vorticity flux) of wind-driven zonal flow, using a suite of numerical eddy resolving experiments in a two-layer channel, governed by quasigeostrophic dynamics, are investigated. The flow regime in these experiments does not satisfy the usual scaling of quasigeostrophic large-scale dynamics: relative vorticity is a significant contribution to the quasigeostrophic potential vorticity (QPV) in the deep layer and the lateral Reynolds stress divergence is comparable to the interfacial form stress in the top layer. The balances of second-order moments confirm that the eddy-induced fluxes of QPV and layer thickness are downgradient but significant contributions of triple moments occur. Existing parameterizations and scaling laws of the eddy fluxes of QPV and layer thickness are tested against data from the numerical experiments and it is shown that the usual downgradient forms of parameterization with diffusi...

Stochastic Models of Shear-Flow Turbulence with Enstrophy Transfer to Subgrid Scales

Abstract A stochastic model for shear-flow turbulence is constructed under the constraint that the parameterized nonlinear eddy–eddy interactions conserve energy but dissipate potential enstrophy. This parameterization is appropriate for truncated models of quasigeostrophic turbulence that cascade potential enstrophy to subgrid scales. The parameterization is not closed but constitutes a rigorous starting point for more thorough parameterizations. A major simplification arises from the fact that independently forced spatial structures produce covariances that can be superposed linearly. The constrained stochastic model cannot sustain turbulence when dissipation is strong or when the mean shear is weak because the prescribed forcing structures extract potential enstrophy from the mean flow at a rate too slow to sustain a transfer to subgrid scales. The constraint therefore defines a transition shear separating states in which turbulence is possible from those in which it is impossible. The transition shear...

Evolution of Mean-Flow Fofonoff Gyres in Barotropic Quasigeostrophic Turbulence

Abstract Numerical experiments are performed over a wide range of parameters to show that mean flows in the form of Fofonoff gyres, characterized by a linear relationship between streamfunction and potential vorticity, are universally produced in the statistically steady state of inviscid unforced barotropic quasigeostrophic turbulence, provided that the initial state is sufficiently well resolved. Further, as the resolution is increased, the mean-flow energy approaches the total energy, and the mean-flow potential enstrophy reaches a minimum value, which is lower than the value with no flow. This is in agreement with the predictions of the theory of equilibrium statistical mechanics. The timescale for the appearance of these flows is on the order of 5–10τϵ, where τϵ is a mean eddy turnover time. When viscosity is turned on, the mean-flow Fofonoff gyres become internally homogenized and eventually disappear entirely as the flow decays to zero. This evolution of the gyres can be universally scaled with a t...

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.

A discussion of non-linear mechanism on subtropical anticyclone's extending/shrinking over East-Asia in summer

Based on the actual circulation structure as well as weather characters over East-Asia subtropical region in summer, by using three-dimension non-linear forced/dissipated dynamic model, the activities of subtropical anticyclone over East-Asia have been studied and discussed. The potential enstrophy criteria of system stability have been derived and also been analysed. The criterion can provide useful reference for analysing and predicting subtropical anticyclone's extending/shrinking as well as corresponding weather over East-Asia in summer.

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.

The Maintenance of Extratropical Intraseasonal Transient Eddy Activity in the GEOS-1 Assimilated Dataset

Abstract Diagnostic potential enstrophy (PE) analyses are formulated for studying the dynamics of transient eddies within discrete frequency bands. The PE analyses are derived for nonconservative quasigeostrophic motion on a sphere and relate local PE tendencies to 1) conversions from the climatological-mean flow, 2) nonlinear dynamical sources, and (3) nonconservative physical sources. Local conversions are associated with cross-gradient residual eddy fluxes of pseudopotential vorticity (PV) while the nonlinear and nonconservative sources of PE result from correlations between PV anomalies and anomalous PV source terms. The PE analyses provide a useful local diagnostic approach for studying fundamental aspects of transient wave dynamics within a PV-based framework. Wintertime-average PE analyses are constructed for both bandpass (BP) and low-pass (LP) intraseasonal timescales. These analyses are applied to the multiyear GEOS-1 assimilated dataset to ascertain the primary sources and sinks of BP and LP tr...

On the evolution of finite-amplitude disturbance to the barotropic and baroclinic quasigeostrophic flows

Some new results on the evolution of finite-amplitude disturbances to the nonlinearly stable and unstable quasigeostrophic flows are presented. Both barotropic and multilayer baroclinic problems are investigated. The upper (lower) bounds on the energy and potential enstrophy of disturbances to the nonlinearly stable(unstable) basic flows are established.

A comparative study of implicit and explicit initialization for a tropical barotropic model

The explicit nonlinear normal mode initialization (ENMI) scheme is applied to a tropical barotropic limited area shallow water model in spherical coordinates. The model is formulated by considering potential enstrophy conserving finite difference scheme. It is seen from the results of this study that the ENMI scheme is fully capable of filtering out the spurious gravity wave oscillations. The results are compared with those using an implicit nonlinear normal mode initialization (INMI). The latter scheme gives equally satisfactory results, requiring less computational time than the explicit scheme.

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.

Dissipation and Cascades to Small Scales in Numerical Models Using a Shape-Preserving Advection Scheme

Abstract A nondivergent barotropic mode1 and a shallow-water model are presented that exploit a high-order shape-preserving scheme for the advection of vorticity or potential vorticity as well as tracers. The dissipation associated with the advection scheme is found to be due to the spreading of features as they are advected across a finite-resolution grid. The strength and scale selectivity of this dissipation are quantified using some simple tests. The resolved tracer variance and (potential) enstrophy are not conserved by the advection scheme; this can be interpreted as a cascade to unresolved scales. In simulations of turbulent mixing, satisfactory cascade of tracer variance, energy, and (potential) enstrophy are obtained without the need for any special parameterization of the cascade to unresolved scales.

Pseudoinviscid Wake Formation by Mountains in Shallow-Water Flow with a Drifting Vortex

Numerical solutions to the shallow-water equations are used to examine the generation of wake vorticity as a cyclone drifts past a mountain. In cases with sufficient vortex strength. mountain height, and vortex-mountain proximity, the flow becomes supercritical over the mountain and hydraulic jumps generate wake vorticity. The dissipative vorticity transport in jumps modifies the usual vorticity integral constraints for inviscid shallow-water flow regarding potential enstrophy, vorticity centroid, and vortex size. The increase in vortex size during wake formation represents a weakening of the vortex. These changes, and the macroscopic flow patterns, are independent of the viscosity coefficient. The generation of vertical vorticity within a viscous jump, and the associated Bernoulli loss, arise from a shear stress induced at the sloping upper interface of the layer and transmitted down through the layer by a secondary flow. Applied to the problem of a typhoon drifting past Taiwan, the shallow-water equations capture many of the observed phenomena such as upstream blocking, downstream sheltering, corner winds, and foehn and secondary vortex formation.

Linear Modes and Storm Tracks in a Two-Level Primitive Equation Model

Linear stability analyses of three-dimensional time-mean flows in a two-level primitive equation model are presented. The model's stationary wave is generated by implementing idealized mountains at the lower boundary. Two sets of experiments are performed: the first with one mountain and the second with three mountains at equal distance from each other. Structures of streamfunctions and heat fluxes from the linearly unstable normal mode are compared with their bandpass transient counterparts in the nonlinear model simulation. The three-dimensional time-mean flow, about which the equations are linearized, is convectively unstable for both the one and three mountain cases. For the three mountain case, there is reasonable agreement between the linear mode and the bandpass transient eddies in terms of both the longitudinal location of the largest eddy amplitudes and the potential enstrophy budget, suggesting that the global mode can capture the correct structure of the climatological storm tracks for different reasons. For the one mountain case, however, the largest eddy amplitude of the linear mode extends farther downstream than that of the bandpass transient eddies. The reasonable correspondence between the linear modes and the bandpass eddies for the three mountain case appears to be due to the relative proximity of successive unstable regions. Between these successive unstable regions are diffluent flows, resulting in increased deformation and enhanced horizontal diffusion, which plays an important role in dissipating transient eddy enstrophy. It is suspected that this locally enhanced dissipation represents strained eddies by the deformation field.

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.

On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow

Abstract Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-geostrophic flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews...

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.

Arnol′d′s second nonlinear stability theorem for general multilayer quasi-geostrophic model

Arnol′d′s second nonlinear stability criterion for motions governed by a general multilayer quasi-geostrophic model is established. The model allows arbitrary density jumps and layer thickness, and at the top and the bottom of the fluid, the boundary condition is either free or rigid. The criterion is obtained by the establishment of the upper bounds of disturbance energy and potential enstrophy in terms of the initial disturbance field.

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.