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Brian J. Hoskins

Abstract

Consideration of flows in which the rate of change of momentum is much smaller than the Coriolis force suggests that the advected quantity momentum may be approximated by its geostrophic value but that trajectories cannot be so approximated. The resulting set of equations imply full forms of the equations for potential temperature, three-dimensional vorticity, potential vorticity and energy. A transformation of horizontal coordinates products the “semi-geostrophic” system in which conservation of potential vorticity and potential temperature suffice to determine the motion. The system is capable of describing the formation of fronts, jets, and the growth of baroclinic waves into the nonlinear regime. It sheds some light on the success and failure of the quasi-geostrophic equations.

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Brian J. Hoskins

Abstract

From spectral analyses of the 1° terrain heights of Gates and Nelson (1975), the representation of the earth topography by truncated series of spherical harmonics is obtained. Rhomboidal and triangular truncations at various wavenumbers are exhibited.

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Paul J. Valdes
and
Brian J. Hoskins

Abstract

In this paper we show that the observed zonally averaged atmosphere is unstable to normal mode type perturbations even when the effects of surface friction as modeled by an Ekman layer or Rayleigh damping are included. Growth rates are reduced and short wavelengths stabilized. Further, we show that thermal damping generally has a weaker effect and in some circumstances can destabilize the flow.

The atmosphere is baroclinically unstable, and normal mode baroclinic instability theory provides an invaluable guide to the mechanisms responsible for and the structure of middle latitude weather systems. However there is no doubt that the initial value problem and the growth of nonnormal mode disturbances is crucial to the development of individual systems. Linear calculations are presented that exhibit large initial transient growth. The growth rate can be more than twice that of the normal mode rates, but only for one or two days. Thereafter the normal mode dominates the evolution.

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Michael J. Revell
and
Brian J. Hoskins

Abstract

Recently, the effects of nonlinearity on waves forced by sinusoidal orography in severely truncated barotropic and baroclinic models have been explored. Multiple equilibria were found for fixed forcing and these have been associated with zonal and blocked states of the global circulation, although the contrast between states was less marked in the baroclinic model.

The presence of multiple equilibria is dependent on instability of the basic forced solution. This instability in barotropic and baroclinic models is the subject of this study. In the barotropic case, the instability seems to be new but the baroclinic counterpart is shown to be a variation of the dynamics exhibited by Simmons in his study of planetary-scale waves in the polar winter stratosphere. These two instabilities are shown to play important, but different, roles in determining the behavior of simple models in the presence of forcing and dissipation.

An extension is made to a five-layer, σcoordinate, primitive equation model on the sphere, using more degrees of freedom. Taking as the basic state the Northern Hemisphere winter zonal mean flow, orographically unstable modes are found. In all but one case, the associated growth rates are much smaller than those corresponding baroclinic instability, even for rather large mountains. This is in contrast with the results from simple, highly truncated β-plane models and suggests that in more realistic situations, orographically induced instabilities may not be so important. However, the work has deepened our understanding of some of the possible interactions of Rossby waves with mountains.

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Brian J. Hoskins
and
Paul J. Valdes

Abstract

Given that middle latitude weather systems transport heat in a manner such as to weaken the baroclinicity that is thought to be crucial to their growth, it is perhaps surprising that concentrated regions of such eddy activity, i.e. storm-tracks, are found in the Northern Hemisphere winter. The existence and possible self-maintenance of storm-tracks is investigated using a linear, stationary wave model with storm-track region forcings taken from data averaged over a number of winters. It is found that the direct thermal effect of the eddies does indeed act against the existence of the storm-track. Their vorticity fluxes lead to some reduction of this effect. It is argued that the mean diabatic heating in the storm-track region is an indirect eddy effect. This heating is found to maintain the mean maximum in baroclinicity in the region. Further, the mean low-level flow induced by the eddy effects is such as to enhance the warm western oceanic boundary currents that are crucial to the existence of the storm-tracks. The extent to which the Northern Hemisphere storm-tracks can be considered self-maintaining is discussed.

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Paul J. Valdes
and
Brian J. Hoskins

Abstract

A spectral, σ-coordinate model linearized about a zonal mean flow is used to investigate the maintenance of the December–February climatological stationary waves. The model is generally run with 15 levels in the vertical and a T31 truncation. The sum of the response to orography, diabatic heating and transient eddy flux convergences specified from data gives a generally good Northern Hemisphere simulation at 200 mb with some deterioration in the lower troposphere. The same is true for the Southern Hemisphere, but only when the forcing terms from recent years are used. The response is broken down into that associated with particular orographic features, tropical and midlatitude heating and transient vorticity and heat flux convergence. All of these are shown to be significant. In particular, orography accounts for approximately 30% of the total 200 mb planetary wave response. In midlatitudes the diabatic heating maxima in the two storm tracks are important. This heating is related to latent and sensible heating associated with transient disturbances. The explicit transient eddy terms in the vorticity and thermodynamic equations force a complicated upper-level flow pattern. The effects of truncation, dissipative parameterization, tropical heating profile and nonlinearity are discussed.

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Paul J. Valdes
and
Brian J. Hoskins

Abstract

Traditionally, stationary wave models have been linearized about a zonal-mean flow and the response calculated to various fixed orographic and thermal forcings. In this paper it is shown that the inclusion of nonlinear interactions can significantly modify the solution for orographic forcing. This arises primarily from the improved boundary condition that allows the flow to be deflected around the mountain as well as over it. In this context a useful conceptual model is obtained by linearization based on the smallness of the latitudinal extent of a mountain. The nonlinear model is considerably less sensitive to the zonal-mean surface flow and, in some instances, the perturbation amplitude decreases with increasing surface flow. The nonlinear, hemispheric solutions for full orography and wintertime basic state are shown for both the Northern and Southern hemispheres. They suggest that the direct effect of orographic forcing alone accounts for less than one-half of the observed time mean asymmetries.

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Brian J. Hoskins
and
Michael J. Revell

Abstract

There has been an apparent inconsistency between the comparatively large growth rates given for long-wavelength baroclinic instability modes for jet flows on the sphere and the small values given by the Charney and Green models. It is shown that this discrepancy is due to the consideration of a fixed meridional structure in the quasi-geostrophic theories. When this restriction is removed there is good agreement. For reasonable parameters, the stabilizing effect of the β-parameter is no more important for the most unstable mode at wavenumber 1 than it is at wavenumber 5. The most unstable wavenumber 1 still has essentially the structure of an Eady mode.

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Brian J. Hoskins
and
David J. Karoly

Abstract

Motivated by some results from barotropic models, a linearized steady-state five-layer baroclinic model is used to study the response of a spherical atmosphere to thermal and orographic forcing. At low levels the significant perturbations are confined to the neighborhood of the source and for midlatitude thermal forcing these perturbations are crucially dependent on the vertical distribution of the source. In the upper troposphere the sources generate wavetrains which are very similar to those given by barotropic models. For a low-latitude source, long wavelengths propagate strongly polewards as well as eastwards. Shorter wavelengths are trapped equatorward of the poleward flank of the jet, resulting in a split of the wave-trains at this latitude. Using reasonable dissipation magnitudes, the easiest way to produce an appreciable response in middle and high latitudes is by subtropical forcing. These results suggest an explanation for the shapes of patterns described in observational studies.

The theory for waves propagating in a slowly varying medium is applied to Rossby waves propagating in a barotropic atmosphere. The slow variation of the medium is associated with the sphericity of the domain and the latitudinal structure of the zonal wind. Rays along which wave activity propagates, the speeds of propagation, and the amplitudes and phases along these rays are determined for a constant angular velocity basic flow as well as a more realistic jet flow. They agree well with the observational and numerical model results and give a simple interpretation of them.

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Adrian J. Simmons
and
Brian J. Hoskins

Abstract

Some aspects of the nonlinear behavior of mid-latitude baroclinic waves are investigated by means of a series of integrations of the primitive equations with spherical geometry. Each integration has as initial conditions a balanced zonal flow perturbed by a small-amplitude disturbance of normal-mode form. Results are presented in detail for several zonal flows and perturbations which are confined initially to either zonal wavenumber 6 or zonal wavenumber 9.

In each case a disturbance grows by baroclinic instability and develops a structure in some agreement with the usual synoptic picture of an occluding system. Its growth rate at low levels decreases more rapidly than that at higher levels, as found by Gall using a more severely truncated model, and upper-level amplitudes become larger relative to surface values than in the initial linear mode. This is more marked for wavenumber 6 than for wavenumber 9, and differences in linear structure are thus enhanced in the nonlinear regime.

Barotropic processes become important during the occlusion of the disturbance as the forcing of vertical motion by thermal advection decreases in importance, although the vorticity actually changes at about half the rate that would occur in a barotropic fluid. In these examples the barotropic effects bring about a decay of the wave at a rate similar to that of its earlier baroclinic growth, and a well-defined life cycle exists.

Large-scale eddy fluxes of heat and momentum averaged over this life cycle have a structure that is substantially different from that given by linear stability analyses, and agreement with observation is improved. Net changes to the zonal-mean temperature gradient are largely confined to the lower troposphere and, to a lesser extent, the lower stratosphere. The change in surface zonal-mean flow is much as suggested by linear theory but at upper levels the westerly jet is strengthened as the disturbance decays.

Additional barotropic integrations have been performed to examine the changes in structure of longer wavelength disturbances at upper levels. Predominantly poleward momentum fluxes result from latitudinal variations in phase speed, and movement at a particular latitude is found to be governed largely by the zonal-mean velocity and vorticity gradient at that latitude. Additional baroclinic experiments provide an example of interactions involving a slower growing, longer wavelength component, and examples of some truncation errors that may result from use of lower resolution models. The sensitivity of results to the inclusion of dissipative processes is also examined.

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