Potential Vorticity Equation Research Articles

A new method to determine the mean sea surface dynamic topography from satellite altimeter observations

A method is presented to calculate mean sea surface dynamic topography from satellite altimeter observations of its temporal variability. Time averaging of a simplified version of the quasi‐geostrophic potential vorticity equation for the upper ocean layer results in a differential equation for the averaged relative vorticity in which the mean divergence of the eddy vorticity fluxes acts as a source or sink. The essential part is that these eddy fluxes can be determined from the altimeter observations. Consequently, no parameterisations appear in the averaged vorticity equation. From the average vorticity field, surface geostrophic velocities and related mean dynamic sea surface topography can then simply be derived. The usefulness of the method is established using “perfect” data, namely numerical output from the United Kingdom Fine Resolution Antarctic Model. The method appears applicable to areas of the ocean with strong enough mesoscale variability such as the major western boundary currents and their extensions and to frontal regions of the Antarctic Circumpolar Current. Quite realistic results are presented for active regions of the world ocean. Results are compared with hydrographic observations for the major western boundary current extensions of the Southern Ocean. An important application is to combine the newly derived averaged flow field with the observed eddy field to derive the total time‐varying geostrophic surface velocity field. As a striking example this is applied to the Agulhas Current retroflection, where the repeated shedding of large rings can now be synoptically reconstructed as a continuous process.

Maintaining the inshore shear of continental boundary currents

Abstract It is suggested that the large inshore shear of Western Boundary Currents, such as is observed on the continental slope of the Florida Current, can be maintained by instability waves which produce a downgradient potential vorticity flux and an upgradient flux of relative vorticity. This is shown for a non-inflected barotropic shear layer above continuously varying bottom topography, and also for a ‘ 1 1 2 layer’ quasi-geostrophic current bounded below by a density interface. The linear eigenfunctions are substituted into the nonlinear potential vorticity equation to obtain the mean vorticity tendency, and then the quantitative increase (in the case of the barotropic model) is computed by a pseudo-spectral calculation of the nonlinear equation.

Derivation of quasi-geostrophic potential vorticity equations

In this paper we derive the quasi-geostrophic potential vorticity equations, starting from primitive type equations, as the Rossby number goes to zero, and prove the convergence as long as the limit system has strong enough solutions. We in particular take account of the boundary layers and investigate the existence of global weak solutions of the limit system with physical boundary conditions.

Quantifying transport in numerically generated velocity fields

Geometric methods from dynamical systems are used to study Lagrangian transport in numerically generated, time-dependent, two-dimensional (2D) vector fields. The flows analyzed here are numerical solutions to the barotropic, β-plane, potential vorticity equation with viscosity, where the partial differential equation (PDE) parameters have been chosen so that the solution evolves to a meandering jet. Numerical methods for approximating invariant manifolds of hyperbolic fixed points for maps are successfully applied to the aperiodic vector field where regions of strong hyperbolicity persist for long times relative to the dominant time period in the flow. Cross sections of these 2D “stable” and “unstable” manifolds show the characteristic transverse intersections identified with chaotic transport in 2D maps, with the lobe geometry approximately recurring on a time scale equal to the dominant time period in the vector field. The resulting lobe structures provide time-dependent estimates for the transport between different flow regimes. Additional numerical experiments show that the computation of such lobe geometries are very robust relative to variations in interpolation, integration and differentiation schemes.

Vanishing Viscosity in the Barotropic β-Plane

The initial boundary value problems associated with the inviscid barotropic potential vorticity equation in the β-plane and its viscous analogue are considered. It is shown that the solution velocity to the viscous equation converges to the inviscid solution in aC1sense for finite times and that, under additional smoothness assumptions on the inviscid flow, this convergence can be extended toC3. Moreover, this convergence occurs as O(ε), where ε is the viscous parameter. This particular form of vanishing viscosity is of relevance in analysing viscosity induced advection for barotropic models.

Coherence Maps for Wind-Forced Quasigeostrophic Flows*

Abstract Coherence maps are a useful tool to study the oceanic response to atmospheric forcing. For a specific frequency band these maps display the coherence between the oceanic current (or pressure) at a single mooring location and the atmospheric forcing field at other locations as a function of separation. This paper calculates such coherence maps from a simple linear quasigeostrophic model forced by a statistically stationary and homogeneous wind field. The calculated coherence maps show values less than one. Such values are not due to the presence of noise but are a consequence of the ocean being forced at many locations. The maps also show characteristic patterns with maxima either at the mooring location or away from it. The locations of the maxima do not indicate the locations of the forcing but instead reflect the scales of the atmospheric forcing spectrum and of the Green’s function of the potential vorticity equation. Coherence maps can be used to estimate the Green’s function in a multiple re...

Hamiltonian moment reduction for describing vortices in shear

This paper discusses a general method for approximating two-dimensional and quasigeostrophic three-dimensional fluid flows that are dominated by coherent lumps of vorticity. The method is based upon the noncanonical Hamiltonian structure of the ideal fluid and uses special functionals of the vorticity as dynamical variables. It permits the extraction of exact or approximate finite degree-of-freedom Hamiltonian systems from the partial differential equations that describe vortex dynamics. We give examples in which the functionals are chosen to be spatial moments of the vorticity. The method gives rise to constants of motion known as Casimir invariants and provides a classification scheme for the global phase space structure of the reduced finite systems, based upon Lie algebra theory. The method is illustrated by application to the Kida vortex [S. Kida, J. Phys. Soc. Jpn. 50, 3517 (1981)] and to the problem of the quasigeostrophic evolution of an ellipsoid of uniform vorticity, embedded in a background flow containing horizontal and vertical shear [Meacham et al., Dyn. Atmos. Oceans 14, 333 (1994)]. The approach provides a simple way of visualizing the structure of the phase space of the Kida problem that allows one to easily classify the types of physical behavior that the vortex may undergo. The dynamics of the ellipsoidal vortex in shear are shown to be Hamiltonian and are represented, without further approximation beyond the assumption of quasigeostrophy, by a finite degree-of-freedom system in canonical variables. The derivation presented here is simpler and more complete than the previous derivation which led to a finite degree-of-freedom system that governs the semi-axes and orientation of the ellipsoid. Using the reduced Hamiltonian description, it is shown that one of the possible modes of evolution of the ellipsoidal vortex is chaotic. These chaotic solutions are noteworthy in that they are exact chaotic solutions of a continuum fluid governing equation, the quasigeostrophic potential vorticity equation.

Low-frequency finite-amplitude oscillations in a near resonant topographically forced barotropic flow

Abstract In this paper, an inviscid, barotropic, nondivergent potential vorticity equation with topography in a β-plane channel is used to investigate nonlinear interaction between a travelling Rossby wave in a uniform basic flow and a single-wave topography with the same zonal wavenumber. It is found that for the CDV-type monopole near resonance, when the uniform zonal basic westerly wind is near the resonant basic westerly wind, 30–60 day low-frequency oscillations (LFO) can arise from the wave-topography interaction. It has both travelling and standing components. In mid-latitudes (30°–45°N), westward-travelling 30–60 day LFO are dominated by Zonal Wavenumbers 1–3, while eastward-travelling 30–60 day LFO are dominated by Zonal Wavenumbers 1 and 3. In high latitudes (near 60°N), 30–60 day LEO are observed to travel westward, and are dominated by Zonal Wavenumber 1. However, for the LG-type (Legras and Ghil, 1985) dipole near resonance it requires a weaker, more realistic basic flow. When the uniform zonal basic westerly wind is near the resonant basic westerly wind, 30–60 day LFO can be excited by the wave-topography interaction. In the mid-bigh latitudes(30°–60°N), 30–60 day LFO are observed to slowly travel either eastward or westward, depending on the intensity of the near resonant westerly wind, and are dominated by Zonal Wavenumbers 1–3. This demonstrates that nonlinear wave-topography interaction appears to be able to induce low-frequency finite-amplitude oscillations with periods of 30–60 days in the Northern Hemisphere (NH) extratropics.

Low-Frequency Variability in a GCM: Three-Dimensional Flow Regimes and Their Dynamics

Abstract General circulation models (GCMs) can be used to develop diagnostics for identifying weather regimes. The author has looked for three-dimensional (3D) weather regimes associated with a 10-yr run of the U.K. UGAMP GCM with perpetual January boundary conditions; 3D low-pass empirical orthogonal functions (EOFs), using both the 500- and 250-mb streamfunctions (ψ) have been computed. These EOFs provide a low-order phase space in which weather regimes are studied. The technique here is an extension to 3D of that of Haines and Hannachi. They found, within the 500-mb ψ EOF phase space, two local minima of area-averaged ψ-tendency (based on barotropic vorticity dynamics), which were identified as ±Pacific–North America (PNA). In this work, the author demands that both the flow and its tendency be within the phase space spanned by the 3D EOFs. The streamfunction tendency is computed from the two-level quasigeostrophic potential vorticity equation and projected onto the EOF phase space. This projection pro...

A Qualitatively Realistic Low-Order Model of the Extratropical Low-Frequency Variability Built from Long Records of Potential Vorticity

Abstract A low-order deterministic qualitative model is formulated in order to simulate extratropical low-frequency variability. This deterministic model is based on a filtering of the potential vorticity equation on the 315-K isentrope and a projection onto its leading empirical orthogonal functions. The model has an empirical formulation, and the feedback of unresolved scales is taken into account. The model building procedure is novel, since it is not based on a severe truncation of the physical evolution equations but on an empirical analog averaging of each relevant dynamical process. It can be applied to any geophysical system for which long observational data series are available. The model is used to diagnose weather regimes with its multiple equilibria and intraseasonal oscillations as periodic orbits. These equilibria result from the balance between large-scale advection, transient feedback, and residual forcing. The authors analyze their forcing budgets and show in particular that transient fee...

Simulation of Stratospheric Vortex Erosion Using Three Different Global Shallow Water Numerical Models

ABSTRACT Simulation of planetary wave breaking and polar vortex erosion in the stratosphere has been carried out using three different global shallow water numerical models. The models are an Eulerian spectral model, a semi-Lagrangian finite difference model based on a vector discretization of the momentum equation and a semi-Lagrangian finite difference model based on the potential vorticity and divergence equations. The results of the integrations indicate advantages of the potential vorticity-based model in the vortex erosion problem.

A note on 1-dimensional calculations of baroclinic and barotropic instabilities

Linear instability of zonal jets are calculated within the framework of the quasi-nondivergent potential vorticity equation. The vertical structure of the flow is treated using vertical normal mode expansion. Three simple 1-dimensional cases are investigated. Firstly, the baroclinic flow on the β-plane, where instability is seen as function of the shape and strength of a jet mimicking the vertical structure of a typical zonal jet in the atmosphere. The study shows that the jet is most unstable for a low-lying jetmaximum. Secondly, a purely barotropic flow on a β-plane, with fixed boundaries in north and south and with periodic boundaries in east—west, mimicking the west wind belt is investigated. In this simple case, the linearized instability equation is solved both spectrally and using finite differences in order to illustrate the dependence on resolution. The stability of the jet is independent on the position within the channel. Finally, for each vertical normal mode, equivalent to a barotropic flow, the stability of a typical west-wind jet on the sphere is calculated. In this case, the jet is more stable the further away from the equator, thus the Coriolis force stabilizes the flow. These very simple examples are in general agreement with findings from full 3-dimensional stability analysis.

Eliassen–Palm Fluxes and the Momentum Equation in Non-Eddy-Resolving Ocean Circulation Models

Abstract The concepts of residual-mean circulation, transformed Eulerian-mean equations, and Eliassen–Palm fluxes are generalized when the averaging is a low-pass operator in time and space rather than a zonal average. Thus, the eddy motions being considered are ocean eddies on short time and small space scales rather than either purely transient eddies or steady, zonally averaged, standing eddies as commonly considered for the atmosphere. The generalized Eliassen–Palm fluxes are then parameterized as downgradient momentum diffusion plus the appropriate Coriolis term. This gives a momentum equation for use in non-eddy-resolving ocean circulation models. The resulting potential vorticity equation is then analyzed and the quasigeostrophic limit taken. When the adiabatic tracer parameterization of Gent and McWilliams is also used, this equation is close to showing that quasigeostrophic potential vorticity is advected by the geostrophic velocity and diffused by a Laplacian operator. A discussion of the Antarc...

The Fargo Snowstorm of 6–8 January 1989

This paper documents the snowstorm that hit eastern North Dakota and western Minnesota during 6–8 January 1989. Fargo, North Dakota, set records for both the greatest 24-h snowfall and the greatest single-storm snowfall with a total of 62 cm. Investigation of regional surface observation time series showed that the snowfall occurred during the early stages of a cyclone event, but the entire snowfall ended before primary cyclogenesis began. Instead, an arctic front, not directly associated with the incipient cyclone, was a focus for increasing frontogenesis and ascent over the Dakotas and Minnesota during the event. Upper-level two-dimensional frontogenesis calculations identified a sloping maximum in the lower troposphere associated with an 850-mb wave, which progressed through North Dakota during the period of heaviest snowfall. Snowfall ceased as the cold side of the sloping maximum reached the Fargo area. Small values of equivalent potential vorticity indicate that weak conditional symmetric stability existed, possibly helping the organization of the snowfall in a narrow band. This study points out the gross deficiencies in our knowledge of Northern Plains snowstorms and calls attention to the pitfalls of analyses based on quasigeostrophic flow.

Free and Rigid Boundary Quasigeostrophic Models in Pressure Coordinates

Starting from the hydrostatic primitive equations in pressure coordinates, a quasigeostrophic (QG) model is derived with temperature and ground surface pressure (GSP) as the governing prognostic fields. In this model two different tendency equations for the GSP exist: one expressing mass conservation, and the other the condition of zero vertical velocity at the ground in common physical space. Comparison of them equations loads to a diagnostic relationship that provides a boundary condition for the omega equation. Equivalence of the system of equations to the QG model that employs the potential vorticity equation is established. Introduction of the GSP as one of the main prognostic fields highlights the problem of the lower boundary condition in pressure coordinates. Two models are introduced and studied, and their main features are compared–the free surface (FS) and the rigid boundary (RB) models, the latter being most common in pressure coordinate QG studies. The choice has an effect on the boundary conditions for the omega equation and affects the physical qualities of the model. The model with the FS has additional energy, similar to the energy of an elastic membrane, which the RB model lacks. Both models give similar results for synoptic-scale processes but differ essentially for scales larger than the external Rossby deformation radius (∼3000 km). As the scale analysis shows, the FS model is accurate in this case, while the RB model, which filters mass fluctuations of vertical unit columns of the atmosphere, seriously distorts the phase speeds.

Radiating Instability of Nonzonal Ocean Currents

Abstract The linear stability of nonzonal flow, uniform in the alongjet direction on a beta plane, is studied. The flow is balanced by a forcing term in the potential vorticity equation. The problem is analyzed in both barotropic and two-layer models. The stability computations are performed for piecewise constant and continuous velocity profiles. New stability properties of nonzonal jets are discussed. The destabilizing effect of the meridional tilt of the jet is demonstrated. The study focuses on the ability of the current to support radiating instabilities capable of significant penetration into the far field. The radiating properties of nonzonal currents are found to be very different from those of zonal currents. In particular, purely zonal flows do not support radiating instabilities, whereas flows with a meridional component are capable of radiating long waves. The new mechanism for radiation is related to the nonzonality of a current.

An objective determination of the polar vortex using Ertel's potential vorticity

We have developed objective criteria for choosing the location of the northern hemisphere polar vortex boundary region and the onset and breakup dates of the vortex. By determining the distribution of Ertel's potential vorticity (Epv) on equivalent latitudes, we define the vortex edge as the location of maximum gradient of Epv constrained by the location of the maximum wind jet calculated along Epv isolines. We define the vortex boundary region to be at the local maximum convex and concave curvature in the Epv distribution surrounding the edge. We have determined that the onset and breakup dates of the vortex on the 450 K isentropic surface occur when the maximum wind speed calculated along Epv isolines rises above and falls below approximately 15.2 m s−1. We use 1992–1993 as a test case to study the onset and breakup periods, and we find that the increase of polar vortex Epv values is associated with the dominance of the term in the potential vorticity equation involving the movement of air through the surface due to the diabatic circulation. We also find that the decrease is associated with the dominance of the term involving radiatively induced changes in the stability of the atmosphere.

The Potential Vorticity Budget of an Atmospheric General Circulation Model

Abstract The flux form of the potential vorticity (PV) equation in isentropic coordinates is used to examine the effects of advection and the effects of parameterized mechanical and thermal forcing on the PV budget of the second generation Canadian Climate Centre general circulation model (CCC GCM). Model data corresponding to Northern Hemisphere winter are used. The simulated PV flux contains significant nonadvective contributions in the planetary boundary layer, in the gravity wave drag regions of the Northern Hemisphere, and in the tropical midtroposphere in regions of intense latent heat release associated with persistent moist convection. The model advective PV flux is compared to the advective PV flux calculated from a National Centers for Environmental Prediction observational dataset. Large discrepancies are seen where parameterized gravity wave drag dominates the mechanical forcing field in the model. The zonally averaged model PV flux in the upper troposphere and lower stratosphere is characteri...

球面上の停滞性波動の準地衡風3次元E-Pフラックスについて

We formulate the quasi-geostrophic 3-dimensional E-P flux of stationary waves by means of a quasi-geostrophic potential vorticity equation on a sphere with a divergent component introduced by Matsuno (1970, 1971). It is found that the expression for the E-P flux has the same form as that of Plumb (1985), even when the wave is represented by a geostrophic wind with a divergent component.

Orthogonal expansion functions for meridional profiles

In order to find an efficient spectral representation for the variables in a non-linear partial differential equation, one may require that the expanding functions should satisfy the equation, not exactly as in the linear case, but instead as well as possible in the least square sense. This variational approach is here applied to the non-linear potential vorticity equation for a zonal flow and finite amplitude perturbations in a beta-plane channel. The solutions are required to be neutral. For zonal wave numbers 2–10 primary modes are found for which the zonal wind has a typical jet-like profile. Its form varies very little with the zonal scale and it is shown that a short wave will have the same meridional profile as a much longer wave. The longest waves are quasi-stationary as is usually observed in the atmosphere. Shorter waves move east with phase velocities which are higher than those obtained with the Rossby formula. They agree, however, well with observations. For higher orthogonal modes a mathematical modification is made in the equation for the primary modes so that a complete Sturm-Liouville orthogonal system is obtained for all modes. These higher modes also show little dependence on zonal wave number and one may therefore question if it is necessary to have separate expansion functions in the meridional direction for each zonal wave number. For the beta-plane model this necessity was not confirmed in an expansion of the meridional profile of the perturbation Jacobian, where a comparison also was made with Fourier expansions. In this case the functions determined here are found to give a much faster convergence. DOI: 10.1034/j.1600-0870.1996.00003.x