Quasi-geostrophic Approximation Research Articles

Cyclogenesis and frontogenesis

Detailed linear analysis of the stability of realistic atmospheric frontal structures using the full hydrostatic primitive equations has demonstrated that such localized baroclinic zones support instability at two distinct and well-separated length scales, even when the frontal mean states have uniform potential vorticity. The first scale is associated with the well-known Charney-Eady mode of baroclinic instability with a wavelength of 3000–4000 km that is responsible for the generation of observed mid-tropospheric long waves. The second scale is defined by an apparently new mode of baroclinic instability, that we have previously called the “cyclone mode”, which has a wavelength near 1000 km and is boundary confined rather than being troposphere filling as is the case of the Chamey-Eady mode. The most important characteristic of the cyclone mode is that it is completely filtered by both the quasi-geostrophic and semi-geostrophic approximations to the equations of motion. In this paper, we demonstrate explicitly through the analysis of a specific case of polar low development that this new mode is an important actor in at least one recurrent atmospheric dynamical process. Besides reviewing and extending these linear stability analyses, we describe a number of rather interesting results that we have obtained from a new series of f-plane analyses of the nonlinear life cycles of frontal baroclinic waves. These have already provided considerable new insight into the detailed dynamics of the processes through which frontal waves “break”. DOI: 10.1034/j.1600-0870.1990.00002.x

Vertical propagation of stationary waves

In this paper we study the vertical propagation of stationary Rossby waves in the troposphere. We compute the refraction index of the troposphere, using the quasi-geostrophic approximation. We find that the approximations used by many authors in the stratosphere cannot be extended to the troposphere. Our results show a refraction index totally independent of horizontal wave numbers that becomes imaginary close to the tropopause. These features follow from the use of the experimental profile of the vertical stability, that in the troposphere cannot be assumed constant.

The ventilated thermocline in quasigeostrophic approximation

The ventilated thermocline in quasigeostrophic approximation

Nonseparable Baroclinic Instability. Part I: Quasi-geostroPhic Dynamics

Abstract We address the problem of the stability of nonseparable baroclinic mean states against small-amplitude quasi-geostrophic perturbations. A general numerical methodology is developed that allows us to construct solutions to such problems without requiring any simplifying assumptions. A previous investigation of frontal stability (Moore and Peltier) which was based upon the use of the primitive equations, demonstrated that frontal zones are unstable to a cyclone-scale mode of baroclinic instability. This mode had not been detected in any previous stability analysis. No evidence of this new mode is found in the results for the quasi-geostrophic problems reported here. In fact, it is shown that the dynamical constraints implied by the quasi-geostrophic approximation forbid the existence of such modes. We also show that recently reported quasi-geostrophic stability analyses that make simplifying assumptions regarding the meridional structure of the basic state have a markedly limited validity. The pres...

Balanced and Unbalanced Upper-Level Frontogenesis

The dynamics of frontogenesis at upper levels are investigated using a hierarchy of three numerical models. They are, in order of decreasing sophistication, the anelastic (AN), the geostrophic momentum (GM), and the quasi-geostrophic (QG) approximations to the full equations of motion. Each model is two-dimensional and assumes the same basic-state, which incorporates the frontogenetical mechanisms of confluence and horizontal shear. The dependence of the numerical solutions on the initial vertical shear of the cross-front component of the geostrophic wind, λ, and its associated along-front temperature gradient is examined in detail. For the values of λ chosen, the along-front temperature gradient is either zero (λ = 0) or such that cold air is advected along the upper front (λ < 0). Intercomparison of the broad-scale structure of the upper-level jet–fronts as described by the AN and GM models shows close agreement. For zero or weak shears (λ = 0 s−1 or λ = −2 × 10−3 s−1), the solutions are essentially identical. Vertical shear in the cross-front geostrophic wind serves to increase the amplitude of the cross-front circulation and displace the subsiding branch toward the warmer air. In the cases of weak or zero shear, the dominant mechanism for generating vertical vorticity at upper levels is the stretching of preexisting vertical vorticity, whereas for stronger shear (λ = −5.741 × 10−3 s−1) the key process becomes the tilting of horizontal vorticity into the vertical by differential vertical motion. In contrast, the QG model exhibits marked differences with its AN and GM counterparts, which become even more pronounced as |λ| is increased. These differences are related largely to the neglect of vortex tilting in generating vertical vorticity in the OG model. The GM and QG models assume cross-front thermal wind balance at all time. A posteriors examination of the numerical solutions shows this to be an excellent approximation when the vertical shear in the cross-front geostrophic wind is weak. For strong vertical shear of the cross-front geostrophic wind, the unbalanced along-front ageostrophic wind is proportional to the vertical advection of the cross-front velocity. Diagnoses of these simulations reveal thermal wind balance to be less well satisfied. It is shown that in contrast to the GM and QG models, wherein the along-front ageostrophic velocity is passive and thus cannot contribute to the evolution of the jet–front system, the unbalanced along-front flow contributes significantly to the dynamics as described by the AN model.

Singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation

A short-time existence theorem is proven for the initial-boundary-value problem for a class of quasilinear symmetric hyperbolic systems containing a constant-coefficient spatial operator multiplied by a large parameter λ. Solutions u are shown to remain bounded independently of λ for a time independent of λ provided that u(0) and u l (0) are bounded independently of λ and certain structural conditions are satisfied. This result is applied to the quasigeostrophic approximation of the shallow water equations and the constant-pressure approximation in combustion, and solutions for these cases are shown to converge to solutions of limiting equations as λ → ∞.

Free Planetary Waves in Finite-Difference Numerical Models

The effects of spatial finite-differencing, viscosity and diffusion on unbounded planetary waves in numerical models are investigated using a quasi-geostrophic approximation to the midlatitude, β-plane, shallow-water equations. The two-dimensional Arakawa B- and C-grid numerical schemes are used to illustrate finite-difference effects, which are found to depend on two nondimensional resolution parameters in each spatial direction: wave resolution = 2 × grid-spacing/wavelength and grid resolution = grid-spacing/2 × Rossby radius. The study is particularly relevant to baroclinic waves, for which the latter parameter can he very large. The properties of the finite-difference frequency dispersion relationships as functions of wave and grid resolutions are studied in detail. If the wave resolution is fine, then both grids perform well, with the C-grid giving a slightly better approximation to the continuum behavior. Neither grid performs well, though, when the wave resolution is poor. In particular, for poor zonal wave resolution, the group velocity is eastward for all meridional wave resolutions and at all grid resolutions on both grids. This latter property explains, though, the dynamics of the western boundary layer in a coarse grid-resolution model in which short waves (in a continuum sense) are not resolved. The properties of the finite-difference frequency dispersion relationships and their derivatives are tabulated for the planetary wave and the inertia-gravity wave (the other clan of wave of primary importance in interior oceanic adjustment) for ready reference. Unbounded planetary waves propagating in a continuum are damped by viscosity and diffusion—long waves are most severely affected by vertical and lateral diffusion of heat, short waves by lateral viscosity. For fine wave resolution (“long” waves) on the B- and C-grids, vertical and lateral diffusion of heat are the main damping mechanisms. For coarse wave resolution (“short” waves), lateral and vertical viscosity are the dominant damping mechanisms on the c-grid, but on the B-grid, lateral viscosity is only important at fine grid resolutions. The theory also gives an estimate of the damping required to effectively trap the energy of poorly resolved waves with eastward group velocity on both grids, and hence the scale of the viscous-diffusive western boundary layer.

A primitive equation barotropic instability study of the monsoon onset vortex, 1979

A primitive equation barotropic stability analysis of the 700 mb zonal wind profile over the Arabian Sea was performed for each day between 10 and 14 June 1979, during the onset phase of the monsoon. the flow was found to become progressively more unstable with the passage of time. the most unstable inviscid primitive equation mode has an e-folding time of 3.1 days, a wavelength of 3500 km and a westward phase speed of 1.9ms-1. The non-geostrophic effect reduces the growth rate and changes the structure of the unstable mode significantly. It is noticed that when the normalized unstable mode is superposed on the basic state, the total field is closer to the observations. Momentum transports by the primitive equation mode and by the quasi-geostrophic approximation are in opposite directions in equatorial regions. the energetics of the primitive equation unstable mode are computed and the physical processes responsible for the westward propagation of the wave are identified.

Low-order model of channelled flow

Results concerning the middle-latitude atmospheric flow in barotropic and quasi-geostrophic approximations are presented. With no dissipation, two semi-infinite regions of stability bound a finite region of instability, ensuring that the solutions are limited. In the presence of dissipation, above a critical value of forcing below which there is one stationary stable state, there are three stationary states, one unstable and two stable. Numerically we find that the two basins of attraction in phase space are strongly asymmetric, the one, corresponding to the solution resonant with the mountain, being much larger than the other. The model used for this study is the Charney-DeVore one.

On nonlinear planetary waves: a class of solutions missed by the traditional quasi-geostrophic approximation

On nonlinear planetary waves: a class of solutions missed by the traditional quasi-geostrophic approximation

General stability conditions for zonal flows in a one-layer model on the β-plane or the sphere

Sufficient stability conditions are derived for a zonal flow on the β-plane or the sphere. Two conditions guarantee both shear stability (to perturbations with vanishing zonal average) and inertial stability (to longitude-independent perturbations). These conditions are not restricted to normal-mode disturbances, and are derived without making use of the quasi-geostrophic approximation. The main limitation of the model is to have only one layer.On the β-plane, the conditions are: (i) that the product of the meridional gradient of potential vorticity and the difference between an arbitrary constant and the zonal velocity be everywhere non-negative; and (ii) that the absolute value of this difference be nowhere larger than the local phase speed of long gravity waves. Inertial stability is independently assured if the Cariolis parameter and the potential vorticity are everywhere of the same sign (this well-known condition can be easily violated near the equator, but the flow may nonetheless be stable).If the meridional gradient of potential vorticity has everywhere the same sign, then conditions (i) and (ii) can be shown to be consequences of the conservation of a total pseudo-energy E0 and pseudomomentum P0, defined so that their lowest-order contribution is quadratic in the deviation from the fundamental state (even in the case that the perturbation is longitude-independent). Thus, if there exists a value of α such that the integral of E0 − αP0 is positive-definite, then the flow is stable. In this case, the stability conditions are valid for small, rather than infinitesimal, perturbations.The parameters of stable flows, as guaranteed by these conditions, are investigated for the family of Gaussian jets centred at the equator; both the cases of an unbounded ocean and a semi-infinite ocean, poleward from a zonal wall, are considered. Easterlies with the width of a Kelvin wave and westerlies with that width or wider may be unstable, even though the gradient of potential vorticity is positive for any strength of the jet.

On nonlinear planetary waves: A class of solutions missed by the traditional quasi-geostrophic approximation

Weakly nonlinear quasi-geostrophic planetary waves on a beta-plane and topographic waves over a linearly inclined bottom are examined by use of shallow water equations for a small beta parameter. Long solitary wave solutions missed by the use of the traditional quasi-geostrophic approximation are found in a channel ocean with neither a sheared current nor a curved (non-linearly inclined) bottom topography. The solutions are missed in the traditional approach because the irrotational motion associated with the geostrophic divergence is neglected by the quasi-geostrophic approximation. Another example which calls attention to the limitation of the traditional quasi-geostrophic approximation is the nonlinear evolution of divergent planetary eddies whose scale is much larger than the Rossby's radius of deformation. Some aspects of a new evolution equation are briefly discussed.

A note on the adjustment to hydrostatic balance

This note investigates the problem of the relation between the initial conditions for the primitive equations and the full set of initial conditions for the Euler equations of adiabatic motion. We formulate for the primitive equations a problem analogous to the one that was considered by Rossby (1938) concerning the quasi geostrophic approximation: a problem which is now well known as the adjustment to geostrophy. The major conclusion is that the initial conditions for the primitive equations may be derived from a full set of initial conditions by solving a one-dimensional unsteady problem of vertical motion. DOI: 10.1111/j.2153-3490.1982.tb01791.x

A note on the adjustment to hydrostatic balance

This note investigates the problem of the relation between the initial conditions for the primitive equations and the full set of initial conditions for the Euler equations of adiabatic motion. We formulate for the primitive equations a problem analogous to the one that was considered by Rossby (1938) concerning the quasi geostrophic approximation: a problem which is now well known as the adjustment to geostrophy. The major conclusion is that the initial conditions for the primitive equations may be derived from a full set of initial conditions by solving a one-dimensional unsteady problem of vertical motion. DOI: 10.1111/j.2153-3490.1982.tb01791.x

On the dynamics of a solitary vortex

The dynamic equilibrium of a solitary vortex of permanent form but arbitrary shape is investigated on a β plane. The vortex is assumed to propagate in the east-west direction. Forces and moments acting on the vortex are computed and the balances between them obtained. The dynamic balances are required for the equilibrium of the vortex and represent the conditions necessary for the existence of a solitary vortex of permanent form. These conditions have two useful expressions, one as integral constraints on the vortex structure and the other as formulas for the propagation speed of the vortex. For the latter, the speed is shown to be proportional to the averaged velocity moment of the vortex and inversely proportional to its averaged relative vorticity. The propagation speed also depends on the size of the vortex; it increases with size for westward propagation and decreases with size for eastward propagation. The effect of current shear on the vortex is also investigated. The results of this study are applicable to the nonlinear barotropic vortex in a nondivergent fluid as well as to the baroclinic vortex, provided that the latter satisfies the quasigeostrophic approximation.

Homogeneous intrusions in a rotating stratified fluid

A study is made of intrusions of fixed volumes of fluid of constant density into a uniformly stratified and uniformly rotating environment. In particular, the eventual steady-state configuration is sought for the ideal fluid case. Exact two-dimensional solutions can be found using a co-ordinate transformation which converts the equations satisfied outside the intrusion into the Cauchy-Riemann equations. The same technique does not, however, yield linear equations in the axisymmetric case.Effects of friction are also considered for cases where the quasi-geostrophic approximation can be made. Vertical advection by the viscosity-induced motion is shown to have the same effect on the temperature field as a greatly enhanced lateral diffusion, and therefore tends to spread the intrusion out laterally.

Pressure wind adjustment relationships during dynamic initialization process with tropical wind data

In an attempt to understand the kind of balance the initially imbalanced low latitude mass and motion fields tend to adjust to during an iterative dynamic initialization process we carry out a systematic investigation of the pressure-wind adjustment laws. This is achieved by comparing, through root mean square differences, the mass field (z) from the initialization process with each of the three other mass fields (z) obtained, using motion fields from initialization process, from diagnostic expressions which relate mass and motion fields. The diagnostic relationships used in this study are the nonlinear, linear and quasi-geostrophic balances equations. The results from these investigations show thatz fields from nonlinear balance approximation are closets to thez fields accruing from the initialization process. Next in closeness is the linear balancez fields which appear to be not very different from the nonlinear balance approximation. The last in closeness is the quasi-geostrophic balance approximation which appears to be 2–3 times, in terms of rms values, as far away as the nonlinear approximation. In fact, the root mean square differences betweenz fields from the initialization process and those from nonlinear balance equation are about 10 metres and the root mean square differences involving thez fields from the quasi-geostrophic balance equation are 30 metres.

A Statistical-Dynamical Model for Stationary Perturbations in the Atmosphere

Abstract The governing equations for stationary perturbations are derived by subtracting the equations for the mean zonally averaged flow from the equations for the time-averaged flow. The averaging period for the ensemble is chosen such that the terms involving the time derivatives can be neglected compared to the other terms in the equations. These equations contain the first moments (means and gradients of means) of stationary perturbations and the zonally averaged flow, second moments such as eddy transports of momentum and heat, heating due to diabatic processes, and frictional forces. The closure for the equations is sought by relating the second moments and frictional forces to the first moments, parameterizing the diabatic heating in terms of first moments and radiation-convection parameters, and evaluating the stationary perturbation from a quasi-geostrophic approximation. The equations are linearized by neglecting the products of the perturbation quantities and the products of perturbations and ...

Response of a two-layer ocean with a baroclinic current to a moving storm, part I

A storm moves with a constant speed parallel to a stationary geostrophic current which flows only in the upper layer of a two-layer, infinite ocean. It is assumed that the lower layer is motionless. The quasi-geostrophic approximation is valid for a moving speed less than 4 ms−1 for a storm radius of 100 km. The primary change of the upper layer thickness is caused by the wind stress divergence and the time integral of the wind stress curl. A cyclonic storm generates upwelling in its wake. The effect of the stationary flow similar to a western boundary current is minor by an order of magnitude and noticeable only on the left edge of the flow. Scaling of equations of motion and continuity for a more general upper geostrophic flow leads to expansion with a parametera 2=gεH m(fL)−2, wheregε is reduced gravity,H m is the maximum thickness of the upper layer,f is Coriolis' parameter andL is the storm radius. The zeroth order perturbations of transport and thickness do not include the stationary flow which appears only in the first order perturbations ina 2. When there is a coast, the change of the interface near the coast is dependent on the time integral of the wind stress component parallel to the coast, thus leading to upwelling or downwelling according to the center being to the left or right of the coastline.

A Numerical Circulation Model with Topography for the Martian Southern Hemisphere

A quasi-geostrophic numerical model, including friction, radiation and the observed planetary topography is applied to the general circulation of the Martian atmosphere in the Southern Hemisphere at latitudes south of about −35°. Near-equilibrium weather systems developed after about 5 model days. To avoid violating the quasi-geostrophic approximation, only 0.8 of the already smoothed relief was employed. Weather systems and velocity fields are strikingly tied to topography. A 2 mb middle-latitude jet stream is found of remarkably terrestrial aspect. Highest surface velocities, both horizontal and vertical, are predicted in western Hellas Planitia and eastern Argyre Planitia, which are common sites of orgin of major Martian dust storms. Mean horizontal velocities >30 m s−1 and mean vertical velocities >0.2 m s− are sometimes found just above the surface velocity boundary layer, apart from eddy velocity contributions. When consideration is taken of scaling to full topography and the probable gustiness of Martian winds, it seems very likely that the general circulation is adequate, at certain times and places, to transport dust from the surface of Mars, as observed. Certain sources and sinks of vertical dust streaming are suggested; the entire south circumpolar zone appears to be a dust sink in winter.