Quasi-geostrophic Dynamics Research Articles

Quasigeostrophic Wire-Vortex Model

A wire-vortex model is developed in order to investigate the dynamics of vortex interactions in a quasigeostrophic turbulence. Each coherent vortex is modeled by a straight wire of finite length. A tilted wire-vortex rotates around the vertical axis steadily by its self-induction in an otherwise quiescent fluid, whereas its inclination angle changes if it is embedded in a local strain field induced by other vortices. The dynamics of wire-vortices are shown to be a Hamiltonian system of finite degree of freedom extracted from the partial differential equations that describe quasigeostrophic vortex dynamics. The center of vorticity and the angular momentum are conserved, besides the total energy and Casimirs of the system, such as the vortex height and the vortex volume. Chaotic motions are observed even in a two-body system.

The Total Energy Norm in a Quasigeostrophic Model

Abstract Total energy E as the sum of kinetic and available potential energies is considered here for quasigeostrophic (QG) dynamics. The discrete expression for E is derived for the QG model formulation of Marshall and Molteni. While E is conserved by the nonlinear unforced model equations, an analogous expression in terms of perturbed fields is, in general, not conserved for tangent-linearized versions of the model, thereby allowing for growth (or decay) in this total energy norm. Examples for structures linearly growing optimally (i.e., the so-called singular vectors) in terms of either the total energy or just the kinetic energy norm are briefly illustrated and contrasted. It is argued that E might preferably be used (rather than kinetic energy) in predictability and data assimilation studies that are based on the QG model considered here.

Role of Nonmodal Growth and Nonlinearity in Cyclogenesis Initial-Value Problems

Observationally motivated idealized initial-value problems of cyclogenesis are studied for quasigeostrophic dynamics. The goals of this investigation are to assess the contributions of normal-mode and nonmodal growth mechanisms and the influence of nonlinearity during incipient cyclogenesis. The initial condition is represented by a coherent vortex superposed on a zero potential vorticity parallel flow. Nonlinear solutions are qualitatively in accord with observations, producing typical deepening of the surface cyclone, an asymmetry in the strength of the cyclone and anticyclone, and the formation of an upper-level front downstream from the cyclogenesis. The growth rate for the projection of the model state vector onto the most unstable mode closely approximates the linear value during the early stages of surface development. Nonlinear dynamics become important after approximately 30 h, beyond which the modal-projection growth rate declines approximately 30%. Linear solutions accurately approximate the intensity and zonal location of the surface cyclone, as well as the asymmetry between the cyclone and upstream anticyclone. The development of the surface cyclone is explained, almost entirely, by the projection onto the growing normal modes. The growing normal modes also account for the development of a prominent ridge of high pressure that forms on the tropopause downstream from the vortex. Nonmodal processes (the complementary subset to the growing normal modes) capture the dispersion of the upper vortex but do not contribute to deepening the surface cyclone. The upper-level front is captured by the linear solutions, and results from a favorable superposition between the growing normal modes and the neutral modes. Tests reveal that surface development declines markedly for vortex length scales smaller than those of observed precursor disturbances. This effect is attributed to a reduction in the vortex projection onto the unstable normal-mode spectrum.

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...

The interaction of two baroclinic geostrophic vortices on the β–plane

The interaction of two anticyclonic baroclinic vortices on the beta-plane is studied, taking into account the effects of vortex-vortex advection and the radiation of barotropic Rossby waves. Quasi-geostrophic dynamics is assumed in each of the two layers and the vortices are represented by delta-function anomalies in the upper-layer potential vorticity. The vortices are strong (i.e, their swirl velocities are large compared with the long baroclinic Rossby wave speed), but are sufficiently widely separated so that the advection of one vortex due to another is of the same order as the meridional drift caused by Rossby wave radiation. Centre of mass arguments are used to demonstrate that the velocity of a vortex is due to the sum of (a) advection due to the other vortex and (b) drift due to wave radiation.The far-field structure of the Rossby wave wake of two zonally aligned vortices (i.e. they have zero meridional displacement relative to each other) is found as a function of their relative position enabling the total drag on the centre of mass to be found. The drag and subsequent meridional drift of each vortex due to the wave radiation alone is then deduced.Equilibria are found in which the vortices have the same southward and westward drift, thus preserving their zonal alignment. Depending on the vortex separation distance, the equilibrium configuration may have meridional drift significantly different from that of a single vortex by itself. For example, when the ratio of the upper-layer depth to bottom-layer depth is 0.4, the meridional drift may be as little as 35% or as much as 117% of the single-vortex drift speed. The stability of the equilibria as a function of the vortex separation distance is found.

On the wave‐activity invariants and stability of the two‐layer shallow‐water semi‐geostrophic model

AbstractAlthough the two‐layer shallow‐water semi‐geostrophic (TLSWSG) model has been proven to be an important tool for the investigation of the dynamics of anisotropic and baroclinic flow in frontal zones, its general instability mechanism, however, has not been well understood. In light of recent work on the stability of the one‐layer SWSG model, this issue is re‐examined in this paper by extending a body of well developed theorems in the two‐layer quasi‐geostrophic (QG) model to the TLSWSG model. The generalized linearized TLSWSG model equations with respect to a parallel basic state are presented first. Conservation equations for two small‐amplitude wave‐activity invariants analogous to pseudo‐momentum and pseudo‐energy densities in QG dynamics are derived and then used to examine the stability of a parallel basic state. It is found that the TLSWSG model possesses two kinds of stability conditions: Arnold's first stability condition in the interior, plus conditions on lateral boundaries, and Arnold's second stability condition. In contrast to previous work, Ripa's ‘subsonic’ condition \documentclass{article}\pagestyle{empty}\begin{document}$ \[\left({\left({U_{G1},U_{G2} } \right)\max < \sum\nolimits_{j = 1}^2 {\left({{1 \mathord{\left/ {\vphantom {1 {cj}}} \right. \kern-\nulldelimiterspace} {cj}}} \right)^{ - 1} } } \right) \] $\end{document}, where UG1 and UG2 are basic flows at two levels and Cj is the phase speed of gravity waves at the jth level) is found to be unnecessary, and the underlying physical reasons, associated with the coastal Kelvin waves in the model, are given. It is also found that the phase speed of unstable normal‐mode disturbances of slow motion satisfies Howard's semi‐circle theorem.The energy—Casimir method is applied to derive small‐ and finite‐amplitude wave‐activity invariants and corresponding stability criteria for non‐parallel basic states. It turns out that the stability criteria for non‐parallel basic states include not only those for parallel basic states but also conditions associated with the anisotropy of basic flow. The nonlinear stability analyses in this work help us to understand the failure of the energy—Casimir method in deriving finite‐amplitude stability criteria in the continuously stratified semi‐geostrophic model.

Onset of Edge Wave Breaking in an Idealized Model of the Polar Stratospheric Vortex

Abstract A mechanism for the breakdown of vertically propagating edge waves in a Boussinesq fluid is investigated within the context of the destruction of the polar stratospheric vortex. Under inviscid, quasi-linear, and slowly varying conditions in a three-dimensional, quasigeostrophic contour dynamics model it is analytically predicted that planetary-scale edge wave breaking will occur if the zonal mean flow is decelerated by more than approximately one-half its initial value via a positive group-velocity–mean-flow feedback mechanism. Fully nonlinear model simulations confirm this “one-half rule” and detail the sequence of events leading to the breaking.

An Unconditionally Stable Scheme for the Shallow Water Equations*

Abstract A finite-difference scheme for solving the linear shallow water equations in a bounded domain is described. Its time step is not restricted by a Courant–Friedrichs–Levy (CFL) condition. The scheme, known as Israeli–Naik–Cane (INC), is the offspring of semi-Lagrangian (SL) schemes and the Cane–Patton (CP) algorithm. In common with the latter it treats the shallow water equations implicitly in y and with attention to wave propagation in x. Unlike CP, it uses an SL-like approach to the zonal variations, which allows the scheme to apply to the full primitive equations. The great advantage, even in problems where quasigeostrophic dynamics are appropriate in the interior, is that the INC scheme accommodates complete boundary conditions.

A remark on the three dimensional baroclinic quasi-geostrophic dynamics

We consider time-periodic patterns of the dissipative three dimensional baroclinic quasi-geostrophic model in spherical coordinates, under time-dependent forcing. We show that when the forcing is time-periodic and the spatial square-integral of the forcing is bounded in time, the model has time-periodic solutions.

Local Lyapunov exponents of the quasi-geostrophic ocean dynamics

The predictability of the quasi-geostrophic ocean model is considered in this paper. This is a simple dynamical model which assumes the ocean depth to be divided into n layers of different water density. This model is driven by the wind stress on the surface, it includes effects of lateral and bottom friction, of the Earth rotation and the nonlinear interaction between adjacent layers. Internal instability of the system leads to the rapid divergence of its trajectories limiting time of deterministic prediction of the system. To estimate the divergence rate of trajectories we compute Lyapunov exponents of this system. Along with Lyapunov exponents, their generalization, local Lyapunov exponents, are computed as a measure of divergence rate on finite time scales. These exponents provide us with the information about principal mechanisms of local instability. They can be useful in the forecasting of the forecast skill in meteorology and oceanology. Numerous studies are performed in order to understand the influence of physical (form of the basin, number of layers) and numerical (grid resolution) parameters on the predictability of the system. To study the geographical structure of the perturbation, the distribution of initial error with the fastest norm growth is distinguished. This distribution shows us the initial data structure the most sensible to errors and points out the regions where the most accuracy should be required. The structure of this function depends much on the time scale of prediction.

Sub-inertial dynamics of density-driven flows in a continuously stratified fluid on a sloping bottom. I. Model derivation and stability characteristics

A theory is developed describing the sub–inertial baroclinic dynamics of bottom–intensified density–driven flows within a continuously stratified fluid of finite depth with variable bottom topography. The evolution of the density–driven current is modelled as a geostrophically balanced homogeneous flow, which allows for finite–amplitude dynamic thickness variations and for which the pressure fields in each layer are strongly coupled together. The evolution of the overlying fluid is governed by baroclinic quasi–geostrophic dynamics describing a balance between the production of relative vorticity, the vortex–tube stretching/compression associated with a deforming gravity current height, and the rectifying influence of a background topographic vorticity gradient. The model is derived as a systematic asymptotic reduction of the two-fluid system in which the upper fluid is described by the Boussinesq adiabatic equations for a continuously stratified fluid and a lower homogeneous layer described by shallow–water theory appropriate for an f -plane with variable bottom topography. The model is shown to possess a non–canonical Hamiltonian formulation. This structure is exploited to give a variational principle for arbitrary steady solutions and stability conditions in the sense of Liapunov. The general linear stability problem associated with parallel shear flow solutions is examined in some detail. Necessary conditions for instability are derived. The instability is convective in the sense that it proceeds by extracting the available gravitational potential energy associated with the lower–layer water mass sliding down the sloping bottom. For the normal–mode instability problem, a semicircle theorem is derived. The linear stability characteristics are illustrated by solving the normal–mode equations for a simple linearly varying lower–layer height profile. In the overlying fluid, the unstable normal modes correspond to amplifying bottom–intensified topographic Rossby waves.

Analytical Prototypes for Ocean–Atmosphere Interaction at Midlatitudes. Part I: Coupled Feedbacks as a Sea Surface Temperature Dependent Stochastic Process*

Effects of ocean‐atmosphere feedback processes and large-scale atmospheric stochastic forcing on the interdecadal climate variability in the North Atlantic and North Pacific Oceans are examined in a simple midlatitude ocean‐atmosphere model. In the ocean, the authors consider a linearized perturbation system with quasigeostrophic shallow-water ocean dynamics and a sea surface temperature (SST) equation for a surface mixed layer. The atmosphere is represented as stochastic wind stress and heat flux forcing. This includes a noise component that depends on SST, as well as an additive component that is independent of SST. Coupling is represented by the SST dependent stochastic process, in which SST influences the probability density function of the atmospheric noise both in shifting the mean and affecting the variance. It thus includes a multiplicative noise component. The model results in both oceans indicate that large-scale additive atmospheric stochastic forcing alone (the uncoupled case) can give coherent spatial patterns in the ocean and sometimes even a weak power spectral peak at interdecadal periods. Coupling due to the SST dependent stochastic process can produce a more distinct power-spectral peak relative to the uncoupled ocean. Moreover, the time and spatial scales of the interdecadal mode are insensitive to the standard deviation of the multiplicative noise. Thus a deterministic feedback limit can be used to simplify the coupled model for further investigation of the physical mechanisms of the interdecadal mode. In both uncoupled and coupled cases, the period of the interdecadal oscillation is determined by the zonal length scale of atmospheric wind stress and oceanic Rossby wave dynamics. The atmospheric spatial pattern sets the length scale of large-scale wave motion in the ocean. This wave propagates to the west due to oceanic Rossby wave dynamics and is dissipated at the western boundary. However, in the coupled case, the SST anomalies generated by geostrophic current can feed back to the atmosphere, which in turn brings some information back to the east and reexcites oceanic waves there. Although the magnitude of the feedback of SST on the atmosphere is much smaller than atmospheric internal variability, its effects are significant.

Further Results on the Stability of Rapidly Rotating Vortices in the Asymmetric Balance Formulation

Two kinds of wave-activity invariants, analogous to the pseudomomentum and pseudoenergy densities in the quasigeostrophic theory, are derived for both three-dimensional baroclinic and two-dimensional barotropic asymmetric balance models, and applied to examine the stability of asymmetric circular basic-state vortices under general disturbances. Stability theorems obtained in this paper extend the results of Montgomery and Shapiro, which are valid only for normal mode disturbances. The general properties of normal mode disturbances in quasigeostrophic dynamics are generalized to the asymmetric balance models, which include an orthogonality principle, bounds on the angular phase speeds of neutrally stable regular normal modes, and a semicircle theorem bounding the angular phase speeds and growth rates of unstable normal modes.

Quasigeostrophic Dynamics of the Western Boundary Current

A local model is used to investigate the dynamics of the western boundary current in a midlatitude, wind-driven gyre. This current is important for the gyre as a whole, and its local instability is correlated with structural changes of the separated eastward jet and the interior gyres. In particular, the eastward jet can be disrupted and broadened in the regime with a strong, local instability in the western boundary current. Such a regime occurs with a no-slip lateral boundary condition. Alternatively, in the absence of local instability, the eastward jet is narrow and penetrates farther in the basin interior. This behavior is typical with free-slip boundary condition. Both the linear stability and nonlinear time-dependent behavior of the western boundary current are analyzed for a wide range of parameters. The current loses stability at moderate Reynolds numbers, and the stability threshold strongly depends upon the vertical stratification profile. The nonlinear time-dependent flow contains well-defined mesoscale eddies with adjacent meanders. The finite amplitude dynamics is fundamentally different in the no-slip and free-slip situations, because the free-slip boundary substantially stabilizes the flow. It is shown that fluctuations in nonlinear regime are rather different from the linearly unstable modes. Multiple stable equilibria are also found.

Linear Stability of the Three-Dimensional Semigeostrophic Model in Geometric Coordinates

Abstract Motivated by the recent work on the stability properties of balanced dynamics, the author investigates in this paper the stability of parallel basic flow in the baroclinic semigeostrophic (SG) model in geometric coordinates. The linearized baroclinic SG equation with a nonconstant Coriolis parameter is presented. Conservation equations for two wave-activity invariants, analogous to the pseudomomentum and pseudoenergy in baroclinic quasigeostrophic dynamics but with the contributions from lateral boundaries, are derived and then used to examine the stability properties of the SG model. It is found that the lateral boundary contributions to the invariants, which were often ignored, are important to the stability mechanism of the SG model. The general properties of normal mode disturbances in the SG model are investigated. Results obtained in this work include an orthogonality relation, the semicircle theorem to bound the phase speed of normal mode disturbances, and an estimate of upper bound on uns...

A Comparison of Extended and Quasigeostrophic Dynamics for a Case of Small–Rossby Number Extratropical Cyclone Development

Abstract This note compares the extended (EXT) and quasigeostrophic (QG) dynamics of a small–Rossby number extratropical cyclone using the Zwack–Okossi (ZO) equation. Applied to a cyclone that occurred on 8–9 November 1985 over the North Atlantic Ocean, results show that although differences exist, both the EXT and QG forms of the ZO equation provide very adequate estimates of the large-scale forcing processes associated with this case.

Circulation and Dynamics of the Western North Atlantic. Part II: Dynamics of Meanders and Rings

Abstract A multiparameter kinematic synthesis of multiscale feature models for the circulation of the western North Atlantic was developed in Part I of this study. The dispersion characteristics and dynamical behavior of this circulation model are presented here in detail. Insertion of the kinematically synthesized features into a numerical dynamical model dynamically adjust the features and provide the basis for a multiparameter sensitivity study with respect to the reasonable range of parameter variation consistent with observations. Two primary characteristics of the Gulf Stream meandering behavior, namely, the wave-growth characteristics and the ring formation and absorption statistics are studied via both quasigeostrophic and primitive equation dynamics. In achieving realistic dispersion characteristics, a comprehensive methodology for dynamical model tuning and validation in this limited region ocean is developed. The realistic regimes of parameter variation are identified on the basis of observatio...

Parameterization of mesoscale eddy fluxes in zonal ocean flows

Quasi-geostrophic dynamics in an eddy-resolving zonal re-entrant channel in the Southern Hemisphere have been studied for east- and westward wind forcing scenarios. The main difference is seen in the zonally averaged velocity profiles. In the case of eastward forcing, transient eddies strongly intensify the flow in the channel center into a jet, a feature totally absent in the westward forcing cases. The free jet is associated with a five times higher available potential energy compared to the westward flow. We have used these two distinctly different flow regimes to investigate possible parameterizations of the eddy fluxes in both situations. Parameterizations using a diffusion concept for the quasi-geostrophic potential vorticity (QPV) fluxes, based on earlier work by Green and Welander, have raised a number of questions concerning the transfer (or diffusion) coefficients. These coefficients must satisfy three basic integral constraints, the balances of momentum, energy and enstrophy. It is shown that the constraints related to momentum and energy conservation are associated with Pedlosky's instability conditions. An analytical solution developed in this paper shows that the transfer coefficients have a theoretical upper limit in the eastward-forcing scenarios, resulting in a greater Reynolds number than a theoretically derived critical Reynolds number. A general parameterization scheme, based on the quasi-geostrophic eddy enstrophy balance, is presented, which accounts for both scenarios, a weakly baroclinic westward flow and a strongly baroclinic eastward flow. This new parameterization reproduces the main difference in the east- and westward flows; i.e., a strong jet in the eastward-forcing case and a broad smooth flow in the westward-forcing case, in agreement with the numerical results of the eddy-resolving experiments.

Ocean circulation variations associated with the Antarctic Circumpolar Wave

Altimeter data analysis indicates persistent sea surface height (SSH) anomalies propagating eastward about the Antarctic continent. The spatial and temporal characteristics match variations in the atmosphere and sea surface temperature (SST) that are associated with the Antarctic Circumpolar Wave (ACW). During the observation time period, the SSH appears quasiperiodic with a dominant 4 year period and 180° longitude wavelength. Thus, the SSH signature of the ACW appears as two anomalies on opposite sides of the Antarctic continent propagating eastward at 10 cm/s. The SSH response to observed wind forcing agrees in terms of amplitude and phase with simple quasigeostrophic dynamics.

Recirculation Gyres Forced by a Beta-Plane Jet

Abstract A numerical model, with quasigeostrophic and barotropic dynamics, is used to study the forcing of mean flows by an unstable jet. The initially zonal jet has specified shape and transport at the western inflow boundary and is sufficiently intense and narrow that the potential vorticity gradient changes sign, giving rise to barotropic instabilities. The resulting eddies act to smooth the potential vorticity anomalies transported into the domain and produce homogenized regions in which recirculations develop to the north and south of the jet. The intensity of these recirculations, as a function of nondimensional beta, is investigated and a simple, kinematic interpretation offered.