Quasi-geostrophic Potential Vorticity Equation Research Articles

Experiments to study the beta-effect in atmospheric dynamics

In atmospheric dynamics, it is well known that the beta-effect, namely, the variation of Coriolis force with latitude causes the mid-latitude large-scale tropospheric waves such as the westerlies to propagate vertically into the stratosphere. Although this has been well proved in the theoretical works and atmospheric observations, to test this also in a laboratory we demonstrate here rotating fluid annulus experiments setting a conical bottom in a tank to create an equivalent beta-effect. Three clear evidences of the beta-effect were observed in the wavenumber-2 flow: the wave flow propagated vertically, its critical level rose, and its drift velocity reduced. Then the quasi-geostrophic potential vorticity equation succeeded in describing this phenomenon below a height slightly lower than the critical level, but failed near that level.

A high-latitude quasi-geostrophic delta plane model derived from spherical geometry

For quasi-geostrophic models, the beta plane approximation is well established and can be derived from spherical geographic coordinates. It has been argued that such a connection does not exist for a higher-order approximation, the so-called delta plane. Here it will be demonstrated that a quasi-geostrophic potential vorticity equation on the delta plane can formally be derived using rotated geographic instead of geographic coordinates. The rigorous derivation of such a model from the shallow-water equations leads to a correction of previous more intuitive-based formulations of the delta plane model. Some applications of the corrected delta plane model are given. It is shown that the delta plane model describes well the low-frequency basin modes of a polar plane shallow-water model. Moreover, it is found that the westward phase speed of the delta plane model shows a dependency on latitude comparable to a model on the sphere. The ratio of delta to beta plane zonal phase speed decreases monotonically with increasing latitude, in qualitative agreement with the phase speed ratio obtained by comparing a spherical to a beta plane model. Finally, it is demonstrated analytically that Rossby wave energy rays are curved on the delta plane, in contrast to the beta plane. Ray curvature is important for a realistic description of energy dispersion at high latitudes. The results suggest that the quasi-geostrophic delta plane model is a suitable tool for conceptual studies on Rossby wave dynamics at high latitudes.

A high-latitude quasi-geostrophic delta plane model derived from spherical geometry

For quasi-geostrophic models, the beta plane approximation is well established and can be derived from spherical geographic coordinates. It has been argued that such a connection does not exist for a higher-order approximation, the so-called delta plane. Here it will be demonstrated that a quasi-geostrophic potential vorticity equation on the delta plane can formally be derived using rotated geographic instead of geographic coordinates. The rigorous derivation of such a model from the shallow-water equations leads to a correction of previous more intuitive-based formulations of the delta plane model. Some applications of the corrected delta plane model are given. It is shown that the delta plane model describes well the low-frequency basin modes of a polar plane shallow-water model. Moreover, it is found that the westward phase speed of the delta plane model shows a dependency on latitude comparable to a model on the sphere. The ratio of delta to beta plane zonal phase speed decreases monotonically with increasing latitude, in qualitative agreement with the phase speed ratio obtained by comparing a spherical to a beta plane model. Finally, it is demonstrated analytically that Rossby wave energy rays are curved on the delta plane, in contrast to the beta plane. Ray curvature is important for a realistic description of energy dispersion at high latitudes. The results suggest that the quasi-geostrophic delta plane model is a suitable tool for conceptual studies on Rossby wave dynamics at high latitudes.

Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation**The project supported by National Natural Science Foundation of China under Grant No. 40305009

The (2+1)-dimensional nonlinear barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a beta-plane channel is investigated by using the classical Lie symmetry approach. Some types of group-invariant wave solutions are expressed by means of the lower-dimensional similarity reduction equations. In addition to the known periodic Rossby wave solutions, some new types of exact solutions such as the ring solitary waves and the breaking soliton type of vorticity solutions with nonlinear and nonconstant shears are also obtained.

The Response of the North Pacific Ocean to Decadal Variability in Atmospheric Forcing: Wind versus Buoyancy Forcing*

Abstract Both wind and buoyancy forcing result in variability in the North Pacific Ocean thermocline. A vertical modal analysis of the density deviations in a 30-yr run of an ocean general circulation model of the North Pacific forced by atmospheric variability is used to identify the spatial and temporal patterns of the different baroclinic modes. The different dynamic vertical modes show distinct propagation characteristics, with the first baroclinic mode exhibiting consistent westward propagation at all latitudes. The higher baroclinic modes show westward phase propagation at low latitudes but propagate eastward at higher latitudes. The propagation characteristics of each mode can be understood by the inclusion of the zonal mean flow in the vertical structure equation. Evaluation of the Ekman pumping and diapycnal fluxes in the quasigeostrophic potential vorticity equation for each dynamic vertical mode distinguishes their effects on the thermocline variability. Wind variability dominantly forces the f...

Experiments to study baroclinic waves penetrating into a stratified layer by a quasi-geostrophic potential vorticity equation

The upward penetration of the mid-latitude baroclinic waves has been well known to influence the air motion in the stratosphere. To study this in the laboratory, we present a new type of rotating fluid annulus experiment where a radial temperature difference is imposed on water near the flat bottom to produce a baroclinic wave in a lower layer and the water surface is heated to create a stratified upper layer. To test this model, we carried out experiments to study baroclinic waves penetrating into the stratified layer by a quasi-geostrophic potential vorticity equation in the atmospheric dynamics. The baroclinic waves produced in the lower fluid layer were observed to penetrate into the stratified upper layer. As expected from the equation, the vorticities of the flows for wavenumbers 3, 4 and 5 decayed exponentially with height and showed a weaker decay for lower wavenumbers.

The Scaling Law of Quasi-Geostrophic Turbulence with Weak Energy Dissipation

When a problem without viscosity is considered as an ideal case, a small dissipation term is necessary in many numerical calculations. Quasi-geostrophic turbulence, which is described by the barotropic version of quasi-geostrophic potential vorticity equation, (also called Charney-Hasegawa-Mima equation as an equivalent) is one of these situations: a dissipation term must be added in order to prevent the energy from piling up in the large wavenumber region. Consequently, the total energy decreases gradually, which should be conserved in an ideal situation. In this article, the total energy dissipation rate in quasi-geostrophic turbulence is estimated, based on the assumption that the energy dissipated in this system is equal to the energy transported to the large wavenumber region. This estimation complements the dynamic scaling laws for the quasi-geostrophic turbulence developed by Watanabe et al. (1998): the parameter expressing the total energy dissipation, which was determined from the result of the numerical calculation in their study, is derived from the parameters of the setting of the numerical calculation. This estimation also suggests the means to determine the appropriate artificial hyperviscosity coefficient used in numerical simulations.

Nonlinear geostrophic adjustment in the presence of a boundary

The process of nonlinear geostrophic adjustment in the presence of a boundary (i.e. in a half-plane bounded by a rigid wall) is examined in the framework of a rotating shallow water model, using an asymptotic multiple-time-scale theory based on the assumed smallness of the Rossby number ε. The spatial scale is of the order of the Rossby scale. Different initial states are considered: periodic, ‘step’-like, and localized. In all cases the initial perturbation is split in a unique way into slow and fast components evolving with characteristic time scales f−1 and (εf)−1, respectively. The slow component is not influenced by the fast one, at least for times t [les ] (fε)−1, and remains close to geostrophic balance. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance and Kelvin waves confined near the boundary.The theory provides simple formulae allowing us to construct the initial profile of the Kelvin wave, given arbitrary initial conditions. With increasing time, the Kelvin wave profile gradually distorts due to nonlinear-wave self-interaction, the distortion being described by the equation of a simple wave. The presence of Kelvin waves does not prevent the fast–slow splitting, in spite of the fact that the frequency gap between the Kelvin waves and slow motion is absent. The possibility of such splitting is explained by the special structure of the Kelvin waves in each case considered.The slow motion on time scales t [les ] (εf)−1 is governed by the well-known quasigeostrophic potential vorticity equation for the elevation. The theory provides an algorithm to determine initial slow and fast fields, and the boundary conditions to any order in ε. For the periodic and step-like initial conditions, the slow component behaves in the usual way, conserving mass, energy and enstrophy. In the case of a localized initial disturbance the total mass of the lowest-order slow component is not conserved, and conservation of the total mass is provided by the first-order slow correction and the Kelvin wave.On longer time scales t [les ] (ε2f)−1 the slow motion obeys the so-called modified quasi-geostrophic potential vorticity (QGPV) equation. The theory provides initial and boundary conditions for this equation. This modified equation coincides exactly with the ‘improved’ QGPV equation, derived by Reznik, Zeitlin & Ben Jelloul (2001), in the step-like and localized cases. In the periodic case this equation contains an additional term due to the Kelvin-wave self-interaction, this term depending on the initial Kelvin wave profile.

Instability of time-dependent wind-driven ocean gyres

The wind-driven ocean circulation at midlatitudes is susceptible to several types of instabilities. One of the simplest models of these flows is the quasigeostrophic barotropic potential vorticity equation in an idealized ocean basin. In this model, the route to complex spatio/temporal flows is through successive bifurcations. The aim of this study is to describe the physics of the destabilization process of a periodic wind-driven flow associated with a secondary bifurcation. Although bifurcation theory has proven to be a valuable tool to determine the physical mechanisms of destabilization of fluid flows, the analysis of the stability of time-dependent (for example, periodic) flows, using this methodology, is computationally unpractical, due to the large number of degrees-of-freedom involved. The approach followed here is to construct a low-order model using numerical Galerkin projection of the full model equations onto the dynamically active eigenmodes. The resulting reduced model is shown to capture the local dynamics of the full model. The physical mechanism of the destabilization of the periodic wind-driven flow is deduced from the reduced model. While there are several stabilizing processes, notably rectification, the destabilization occurs due to time-dependent increase of the background horizontal shear in the flow.

Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model

We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f−10 and (f0Ro)−1 respectively, where f0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction.The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H0, where ΔH and H0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t [les ] (f0Ro)−1. We find modifications to this equation for longer times t [les ] (f0Ro2)−1. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance.For large relative elevations (ΔH/H0 [Gt ] Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.

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.

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 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...

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.

Three-dimensional global scale permanent-wave solutions of the nonlinear quasigeostrophic potential vorticity equation and energy dispersion

The three-dimensional nonlinear quasi-geostrophic potential vorticity equation is reduced to a linear form in the stream function in spherical coordinates for the permanent wave solutions consisting of zonal wavenumbers from 0 ton andr n vertical components with a given degreen. This equation is solved by treating the coefficient of the Coriolis parameter square in the equation as the eigenvalue both for sinusoidal and hyperbolic variations in vertical direction. It is found that these solutions can represent the observed long term flow patterns at the surface and aloft over the globe closely. In addition, the sinusoidal vertical solutions with large eigenvalueG are trapped in low latitude, and the scales of these trapped modes are longer than 10 deg. lat. even for the top layer of the ocean and hence they are much larger than that given by the equatorialΒ-plane solutions. Therefore such baroclinic disturbances in the ocean can easily interact with those in the atmosphere.

Baroclinic–Barotropic Instabilities of the Gulf Stream Extension

Abstract In this paper, the baroclinic-barotropic instability of the Gulf Stream is studied numerically. The quasigeo-strophic potential vorticity equation is linearized around the mean flow, which is modeled using data from field measurements in the Gulf Stream off Cape Hatteras. The perturbation around the mean flow is decomposed into waves along the streamwise direction using a Fourier transformation in space and a Laplace transformation in time. For each wave, an eigenvalue problem is obtained, which is solved numerically to yield the frequency as a function of the wavenumber. The instability is of the convective type, in the sense that any localized perturbation will propagate out of any fixed region in space in finite time. External noise can, however, drive the instability creating spatially growing waves. The spatial growth rates of the waves are computed as a function of their frequency using an iterative procedure and are found to be in good agreement with growth rates from field measurements. V...

Climate variability in a low-order coupled atmosphere-ocean model

The dynamical behavior of the climate system is investigated through the use of a low-order coupled atmosphere-ocean general circulation model. The goal is to gain some qualitative understanding of how non–linear interactions between the individual system components may affect the climate. Both the atmosphere and ocean models are fully dynamic: the former is defined by 3 ordinary differential equations derived from a truncated Fourier series expansion of the mean and perturbation components of the quasi-geostrophic potential vorticity equation, while the latter is specified by 6 ordinary differential equations representing the time-dependent variations of ocean temperature and salinity in a 3-box model of the North Atlantic. Despite the existence of 2 basic equilibrium ocean model responses to perpetual atmospheric conditions, equilibrium states are never attained in the coupled system within 10000 years of integration; the deep ocean flow continually adjusts to the atmospheric regime changes associated with particular ocean circulations, which leads to new circulations and new atmospheric regimes. Low-frequency quasi-periodic oscillations about a single state of the thermohaline circulation result from an advective-diffusive process, modulated by the correlation of the atmospheric behavior with the phase of the ocean cycle. The climate is strongly effected by interactions with the ocean, leading to distinct atmospheric patterns for different phases in the oscillations, and a conversion of some of the high-frequency atmospheric signal to lower frequencies. This conversion also results in a measurable ocean response at high frequencies. Furthermore, owing to the richness of the atmospheric response to small modifications in the meridional and zonal gradients in diabatic heating, even modest adjustments in the ocean circulation resulting from interactions with the high-frequency atmospheric component can also lead to climate change over relatively short time periods. The results of the model are applied to recent deductions of climate variability in the North Atlantic, obtained from Greenland ice-cores. DOI: 10.1034/j.1600-0870.1995.t01-3-00006.x

The Acceleration Field Associated with Vorticity Flux Convergence

It is shown that each term that represents vorticity flux convergence or divergence in the quasigeostrophic potential vorticity equation can be inverted to obtain fluid acceleration associated with that term, and the same can be obtained for the vortex stretching term. The computed acceleration in the form of a “force potential” shows precisely how the process given by each vorticity term changes the flow field and contributes to its evolution, such as formation and propagation of current features. This complements the standard vorticity diagnostic, which evaluates the magnitudes of the terms only. Application to some simple unstable flows is presented.

Organization of Rainfall by an Unstable Jet with an Application to African Waves

African waves are believed to originate as shear instabilities, although in certain cases rainfall is organized so that latent heating contributes to wave growth. What determines whether the shear instability can organize rainfall is considered here; in particular, why African waves organize rainfall mainly during the late summer, despite the regular occurrence of shear instability and rainfall throughout the season.During GATE, moisture convergence by the waves was also largest toward the late summer. It is assumed that an African wave will organize rainfall if it converges moisture—as measured by the ascent at the top of the moist layer—with sufficient amplitude. The wave amplitude is specified at some level beneath the 600-mb African jet, whose instability is a plausible source of the wave. The ascent is calculated using the quasigeostrophic potential vorticity and thermodynamic equations, and depends on the zonal wind separating the unstable jet from the top of the moist layer.Before turning to the example of the African jet, the more general behavior of the model is considered. In the absence of shear, a wave can arrive at the moist layer with undiminished amplitude. However, the ascent corresponding to this wave is small—less than the estimated ascent for Phase I of GATE when rainfall remained unorganized. For larger values of the shear, this threshold can be exceeded, although the ascent decays beneath the jet. Thus, the question arises whether a wave source can organize rainfall from an arbitrarily large distance above the moist layer. It is suggested that organization can only occur if the unstable jet is within a few kilometers of the moist layer and separated by large shear, although exceptions are noted.The calculation is applied to a wind profile resembling the observed 600-mb African jet. The wave amplitude decays beneath the jet so that the ascent at the top of the moist layer increases as the separation of the jet and moist layer decreases. Evidence is presented that the waves are closer to the moist layer during the late summer, resulting in larger ascent at this time.Large variations in the ascent can also occur even if the separation of the jet and moist layer remains constant. It is shown that the ascent can vary greatly as a result of small changes in the jet that are within its observed summer variability.

Acoustic ray chaos induced by mesoscale ocean structure

It has previously been shown that some acoustic ray trajectories in ocean models with periodic range dependence exhibit chaotic behavior, thereby imposing a limit on one’s ability to make deterministic predictions using ray theory. The objective of the work reported here is to quantify the limitations of ray theory to make predictions of underwater sound fields in the presence of realistic mesoscale structure. This is done by numerically investigating sound ray propagation in an analytically prescribed sound speed model consisting of Munk’s canonical profile perturbed by a randomly phased superposition of several baroclinic modes of the linearized quasigeostrophic potential vorticity equation. The ray equations used are consistent with the parabolic wave equation. To investigate ray chaos, power spectra are calculated and Lyapunov exponents are estimated. For realistic strengths of the mesoscale field, near-axial ray trajectories are found to be chaotic with characteristic e-folding distances (inverse Lyapunov exponents) of several hundred km. Steep rays, on the other hand, are found to be stable. This suggests that, in the presence of mesoscale structure, deterministic predictions of near-axial underwater acoustic energy using ray theory are limited to ranges of one to two thousand km, while steep rays should be predictable over longer ranges.