Basic Zonal Flow Research Articles

Application of Vertical Normal Mode Expansion to Problems of Baroclinic Instability

As an alternative to the finite difference method, we explore the use of the spectral method with normalmodes as the basis functions for discretizing dependent variables in the vertical direction in order to obtainnumerical solutions to time dependent atmospheric equations. The normal modes are free solutions to the timedependent perturbation equations linearized around the atmosphere at rest. To demonstrate the feasibility ofnormal mode representation in the spectral vertical discretization, the vertical normal mode expansion is appliedto the quasi-geostrophic potential vorticity equation to investigate the traditional baroclinic instability of Charneyand Green types on a zonal flow with a constant vertical shear. The convergence of the numerical solutions isexamined in detail in relation to the spectral resolution of expansion functions.We then extend the method of vertical normal mode expansion to solve the problem of baroclinic instabilityon the sphere. Two aspects are different from the earlier example. One is use of the primitive equations insteadof the quasi-geostrophic system and the other is application of normal mode expansions in the horizontal, aswell as vertical direction. First, we derive the evolution equations for the spectral coefficients of truncated seriesin three-dimensional normal mode functions by application of the Galerkin procedure to the global primitiveequations linearized around a basic zonal flow with vertical and meridional shear. Then, an eigenvalue-eigenfunction problem is solved to investigate the stability of perturbation motions superimposed on the 30' jetexamined earlier by Simmons, Hoskins and Frederiksen. From these two examples, it is concluded that thenormal mode spectral method is a viable numerical technique for discretizing model variables in the vertical.

On rossby waves modified by weak sinusoidal shear

The spatial structure of beta-plane Rossby waves in a sinusoidal basic zonal flow U 0cos(γ,y) is determined analytically in the (stable) asymptotic limit of weak shear, U 0γ2 0/β≈1. The propagating neutral normal modes are found to take their greatest amplitude in the region of maximum westerly flow, while their most rapid phase variation is achieved in the region of maximum easterly flow. These results are shown to be consistent with what is obtained by ray-tracing methods in the limit of small meridional disturbance wavelength.

The domain of validity of the KdV-type solitary rossby waves in the shallow water β-plane model

Abstract The Domain, where the necessary and sufficient conditions for the existence of the KdV-type solitary Rossby waves are satisfied is defined in the shallow water β-plane model. The KdV-type solitary Rossby waves are the Rossby waves whose time-longitude dependence is determined by the KdV equation. As far as an appropriate amplitude and an appropriate ratio of the scales of the east-west and north-south directions are given, the KdV-type solitary Rossby waves can exist for every basic zonal flow. This result suggests the large validity of the soliton model in geophysical fluid dynamics. The KdV-type solitary Rossby waves are classified into four categories: (1) shear solitons studied by Long, Larsen, Benny, Redekop, and Hukuda, (2) β-divergent solitons studied by Clarke, Yamagata, and Nogami, (3) β-solitons found in the case of the strong stratification, and (4) divergent solitons which exist in the planetary-geostrophic-scale zonal flow. A remarkable result is that, in addition to the conventional east-west elongated solitons, the north-south elongated solitons can also exist for the case of the divergent solitons.

THE SPECTRA OF BAROTROPIC QUASIGEOSTROPHIC MODEL AND THE EVOLUTION OF DISTURBANCES

The spectra of linearized barotropic quasigeostrophic model is one of classical but not yet completelysolved problems in dynamic meteorology. This paper is devoted to a detailed investigation of this problem.It is obtained that in the case of zonal basic flow μ(y), there exist two types of spectra, the discrete andcontinuous. Each discrete spectrum (eigenvalue) gives an eigenfunction (normal mode); and the real num-bers in the interval [μ_(min), μ_(max)] are the continuous spectra; every point c ? [μ_(min), μ_(max)] gives a spectralfunction which is bounded and whose derivative is unbounded but integrable. Every disturbance can berepresented as a linear combination of two parts, ψ'(x, y, t) and ψ'(x, y, t), the former is expressed bythe discrete spectra, and the latter by the continuum. ψ'_d can be easily derived from the initial conditionby the use of the generalized "weighted orthogonality" of normal modes. It is also proved that ψ'_c as wellas its energy E'_c and mean y-direction scale l_c all approach zero as i→∞. A finite difference method for calculating the spectra and spectral functions is suggested. This numer-ical scheme keeps the same global characteristics and the criteria of instability as the original differentialoperator has, although the continuous spectra are distorted as computational discrete spectra due to the fi-nite numbers of grid points. However, the evolutionary characteristics of ψ'c can be correctly representedby this method for a long time from the initial. Of course, the computational spectra in the intervalμ_(min)≤c≤μ_(max) become denser and denser as the grid size becomes smaller and smaller, and cover the wholeinterval as the grid size approaches zero.

Response of Planetary Waves to Stationary Tropical Heating in a Global Atmosphere with Meridional and Vertical Shear

Abstract The response of planetary waves to stationary tropical heating in a stratified global atmosphere linearized with respect to a basic zonal mean flow is investigated. The basic zonal wind has meridional and vertical shear. The basic equations are solved by using the method of three-dimensional normal-mode expansion. Forced solutions to a prescribed tropospheric equatorial heating distribution with a specific wavenumber in longitude are examined. Without the basic zonal flow, the internal vertical modes whose equivalent depths are on the order of a few hundred meters are favorably excited, but the response of the external mode (“barotropic” mode) is relatively small. With the inclusion of a zonal flow, the vertical shear of the zonal wind permits the coupling of the external mode with the internal vertical modes. As a result of the coupling, a significant response occurs in the external mode due to the excitation of the “baroclinic” internal modes by tropical heating. The meridional structures of in...

Das Ansprechen der Lösungen der barotropischen Stabilitätsanalyse der niedertroposphärischen, zonalen Monsunströmung auf den meridionalen Netzabstand des Modells

The sensitivity of the solutions of a numerical barotropic stability analysis of low level monsoonal zonal flow to the meridional grid set up was studied using a variable meridional grid in the channel 5°N to 25°N. It was found that the errors in the solutions arising due to the replacement of the continuous variability of the basic zonal flow with the discrete values at the grid points can be attributed to two different factors associated with the grid. Firstly the solutions (growth rates and wave lengths of unstable waves) are found to be quite sensitive to the grid length and the second and more serious problem was the apparent sensitivity of the solutions to the grid point locations. This sensitivity seems to yield significantly different sets of solutions for different grid setups (in respect of their grid point locations), but for basically the same basic flow profile and grid sizes. The above errors, which are significant for large grid sizes, tend to disappear when the grid size is decreased to a critical or less than a critical value, which was found to be about 0.8° latitude for the low level monsoonal zonal flow. The critical values for different types of zonal flows may be different and should be taken care of before performing barotropic stability analysis.

Baroclinic Instability with Cumulus Heating

Abstract We have investigated baroclinic instability with cumulus heating using a vertically discrete, linearized, quasi-geostrophic model on a β-plane. Two formulations of cumulus heating were used. The first formulation (η-model) rests on the assumption that heating at all levels is proportional to the vertical p-velocity at the top of the lowest model layer. The second formulation (AS-model) follows the cumulus parameterization proposed by Arakawa and Schubert. We present results for basic states with a constant temperature lapse rate and zonal flows linear in pressure. With both formulations, we found the Green modes for easterly shears destabilized by cumulus heating. We discuss the mechanism of this destabilization along with the vertical structure and energetics of the perturbations. We extended the analyses for basic zonal flows similar to those observed during the Indian summer monsoon season, with the AS-model. The wavelength, phase speed, growth rate and vertical structure corresponding to a pe...

Nonlinear meridional scale interaction in baroclinic instability

Baroclinic instability of a zonal flow with latitudinal structure is examined using a nonlinear quasi-geostrophic, two-level β-plane model. An initially small perturbation with the structure of the linearly most unstable mode is allowed to grow to finite amplitude through nonlinear interaction. Because of latitudinal asymmetries of the basic zonal flow, a spectrum of meridional modes is generated in the perturbation. The time evolution of zonal wind and perturbation meridional structures, and their Fourier meridional mode spectra are examined. The radius of deformation is an important meridional scale in both the zonal flow and perturbation. This is especially true during the barotropic decay phase of the baroclinic wave. Time series of energy conversion terms show there is no energy accumulation.

準共鳴ロスビー波について

Non-linear resonance of external Rossby wave in a continuously stratified atmosphere under cyclic condition in the meridional direction is studied by the perturbation expansion method. The feed-back effects of the mean zonal flow induced by the wave-topography (and/or heating) and wave-self interaction are included. As the forcings, a sinusoidal mountain and/or heating are assumed, while the Ekman damping, Ray Leigh friction and Newtonian cooling are included as the dissipations.In the case of topographic forcing alone, it is shown that for long (short) waves the amplitude response curves incline toward negative (positive) side of the detuning parameter δ (deviation of the basic zonal flow from the linear resonant condition), to give the socalled multiple (three) equilibrium solutions for a certain range of δ for small damping coefficients (e.g., Charney and DeVore, 1979; hereafter referred to as CD). The positions of maximum response on δ-axis shift in proportion to 2nd power of the magnitude of forcing and -2nd power of the damping coefficients. At a 'critical' wave number, the amplitude response curve is similar to that of linear resonance as a result of the cancellation between non-linear terms; the multiplicity of equilibrium vanishes.The stabilities of equilibrium solutions are examined by perturbation method, to show that the upper and the lower branches are stable, while the middle branch is unstable as has been pointed out by CD. All the equilibrium states outside the multiple equilibrium region are stable.If thermal forcing is incorporated, the response becomes complicated, depending upon its magnitude and its phase relative to the topography; for example, the amplitude response curve forms a loop if plotted against δ and the largest amplitude state is shown to be unstable. This loop becomes larger and wider as the heating increases. It is noted that such a loop can be found also in CD's model if the mountain becomes high, though it seems to have been overlooked so far. Finally, we show the time-dependent behaviors of the solutions by numerical integrations of the equation obtained by the present method.

遠心力バランスした傾圧流の安定性に関する予備的研究

We investigate the stability of a baroclinic flow in which a centrifugal force acting on the zonal flow cannot be neglected compared to the Coriolis force. It is found that in the governing equations of this problem, the coefficient which corresponds to the Coriolis parameter in the usual baroclinic instability theory becomes a function of height, and the wave number in the longitudinal direction (k) becomes an independent external parameter. When the value of the Rossby number (R0=k/_??_ ΔH, here Δ is a vertical shear of the basic zonal flow and H is a depth of the layer) is small, the dynamic balance of the basic flow is always in the vicinity of pure geostrophic balance, so that the departure of eigenvalues obtained from pure geostrophic case remains small. For R0 larger than about 3.0, with an increase of ΔH, the dynamic balance of the basic flow is continuously changed from geostrophic balance to cyclostrophic one. It is found that in the balance close to the latter, there appear unstable waves which have fairly larger growth rate than unstable waves appearing in the usual baroclinic instability.

A Combined Barotropic-Baroclinic Instability Study of the Upper Tropospheric Tropical Easterly Jet

A combined barotropic-baroclinic stability analysis is performed for an upper tropospheric tropical easterly jet representing the observed mean monsoon zonal flow during summer. Numerical solutions are obtained by time integration of a 20-layer linear spectral quasi-geostrophic model, which is based on truncated Fourier series representations in y. It is seen from the growth rate and phase speed spectra that the asymmetric barotropic-baroclinic preferred wave has a wavelength of 6500 km, an e-folding time of 3.3 days, a westward phase speed of 20.5 m s− and a period of 3.8 days. The geopotential, vertical velocity and temperature fields associated with the most unstable barotropic-baroclinic wave are computed. The most unstable wave has a vertical scale of 125 mb, a meridional scale of 1650 km and a zonal scale of 2135 km. The relationship between the vertical and meridional scales of the wave with the corresponding basic zonal flow scales is discussed. The large southward easterly momentum transports associated with the unstable wave are essentially due to the antisymmetric components of the jet. The computed sensible heat transports are found to be down the basic state meridional temperature gradient. The energetics of the unstable wave is computed and it is inferred that the energy sources for the wave growth lie in a narrow vertical layer around the jet level. It is also found that the contribution of baroclinic process is larger than the contribution of barotropic in the wave growth. The contribution of different processes in the movement of the unstable wave are also investigated. The beta effect is identified as the most important physical factor responsible for the westward propagation of the wave.

表面地形がある場合の傾圧帯状流の不安定性

Stability properties of a baroclinic zonal flow in the presence of surface topography are investigated by using 2-layer and 20-layer, quasi-geostrophic, low-order models. The effect of the topography is approximately incorporated into the model by using a rather simple lower boundary condition. Compared with the conventional baroclinic instability analysis, the present analysis has two differences: Inclusion of (1) effect of surface topo-graphy and (2) perturbation of zonal components.Two kinds of instabilities are obtained in both models. One is the modified baroclinic instability, which is essentially the conventional baroclinic instability. However, the growth rate, the phase speed and the vertical structure of the wave perturbation vary in time depending on the phase relation between the wave and the topography. The other is the topographic instability, in which the wave perturbation grows exponentially in time without phase propagation. In the 20-layer model the second modes of both instabilities, in which the perturbation has a nodal structure in vertical, are obtained in addition to the first modes similar to those obtained in the 2-layer model.Except for the time-dependency, however, the structure and energetics are similar in both instabilities. The eddy energy is supplied by the baroclinic conversion from the available potential energy of the basic zonal flow. The conversion rate of kinetic energy by the surface topography is small compared with the baroclinic conversion.

Internal hydromagnetic planetary-gravity waves in zonal wind-magnetic shears. Two-variables technique

The technique of slow and fast variables is used to investigate the relation between the amplitude of hydromagnetic-planetary-gravity waves and the strength of the basic state zonal flow and zonal magnetic field.

Hydromagnetic planetary-gravity waves in vertically sheared zonal flow and zonal magnetic field

Hydromagnetic planetary-gravity waves propagating in a basic state zonal flow and zonal magnetic field both of which are sheared in the vertical direction are studied. It is found that the waves possess four critical levels at each of which they exhibit the valve effect behaviour, an effect due to the variation of density with latitude. It is also shown that on crossing a critical level the waves are either attenuated or escalated by an exponential factor which depends on the shear in the flow and the latitudinal variation in density.

Baroclinic Waves and Frontogenesis. Part II: Uniform Potential Vorticity Jet Flows-Cold and Warm Fronts

Abstract The semi-geostrophic equations are used to study the baroclinic instability of jet flows having zero potential vorticity gradient. Normal modes of the linear problems are obtained and it is shown that the most unstable modes tend to tilt in the horizontal with the ambient shears of the basic zonal flows. These most unstable modes are used as initial conditions for integrations of the nonlinear equations. When an upper westerly jet is present, the slightly unrealistic frontogenesis in the Eady mode independent of y (Hoskins and Bretherton, 1972) becomes a realistic cold front development. As the wave grows further a warm front is produced. The same solution may be used for baroclinic growth on a flow with positive vorticity at the surface on the equatorial side of the mode. This exhibits the development of a very different sort of warm front. No cold front is produced. The three kinds of frontogenesis found in these models are analyzed, with emphasis being placed on 1) the ageostrophic horizontal ...

Growth Rates and Phase Speeds of Baroclinic Waves in Multi-Level Models on a Sphere

Abstract The complete growth rate and phase spectra of unstable baroclinic disturbances are studied in multi-level quasi-geostrophic spherical models and in a two-level non-geostrophic model, for two basic zonal jet-profile flows. The eigenvalue method employed has the advantages over initial value approaches of determining all the disturbance modes, being more efficient computationally and avoiding problems of slow convergence or lack of convergence when there are two modes of similar growth rates. The method is used to re-examine the spectra for Simmons and Hoskins' 30° jet and Gall's 42.1° jet. It is found that the disrepancies between our two-level and multi-level results are not as great as found by Simmons and Hoskins using primitive equation models, particularly with respect to phase speeds and momentum fluxes. Further, the short-wave maximum at zonal wavenumber 16 in their five-level primitive equation model does not occur here. Instead, our growth rates for the fastest growing modes are very simi...

On the Wavelength of Maximum Baroclinic Instability

Abstract A four-level quasigeostrophic model of a baroclinic atmosphere is used to examine the instability of short (∼2000 km) baroclinic waves. It is determined that only a slight decrease in the low-level static stability or increase in the low-level wind shear relative to the stale stability and wind shear in the middle and upper troposphere can mean the difference between the maximum growth rate occurring at a wavelength of 4000 km (∼wavenumber 7) or 2000 km (∼wavenumber 15). Similar changes of static stability in the upper troposphere relative to the middle and lower troposphere have very little effect on the growth-rate spectrum. This effect of vertical variations in the static stability and wind shear on the growth-rate spectrum is consistent with the structure of the short wavelengths. Wavelengths 4000 km extend through the depth of the troposphere. Therefore, changes in the static stability of the basic zonal flow near the earth's...

Baroclinic Instability of a Slowly Varying Zonal Flow. Part 2

Further results are presented using Eady's model of baroclinic instability with a zonal flow varying slowly in the meridional direction. Now, however, the basic flow is more in keeping with observations and this means the WKB method used to solve the ensuing linear eigenvalue problem has to be considered in the complex plane. The overall effect of meridional shear is a slight stabilization and the meridional scale of the growing waves is governed by the scale of the jet in the basic zonal flow and by the radius of deformation. Further results are also presented for the two-layer model including the β-effect, and far from neutral stability they complement the findings of the continuous model. Close to neutral stability, however, the method can be used to explain the differences between the previous solutions of Simmons and Stone.

強制プラネタリー波の準共振的発達に対する大気の応答特性

The forcing of the second-order zonally-averaged wind and temperature fields due to the quasiresonant growth of planetary waves in a two-layer model is considered. Dissipation in the form of linear drag and Newtonian cooling is allowed for. Calculations are made in a mid-latitude β-plane channel bounded by two vertical walls. The linear or first order solution is constrained so as to not transport momentum meridionally. Three neutral modes in the adiabatic two layer model are brought to resonance separately by adjusting the basic zonal flow configuration. For a dissipation time scale of 2 days, the quasi-resonant enhancement of the response is negligible. A time scale of 4 days is used.For resonance involving the two layer equivalent of the Rossby wave, the strong westerly zonal flow configuration causes most of the second order effects to be confined to the lowest few scale heights, where a slow easterly acceleration of the zonal flow occurs.A second resonance associated with weak westerlies surmounted by moderate westerlies leads to a very rapid high level easterly acceleration lasting about two weeks. A final resonant wind configuration with easterlies above westerlies is associated with very rapid temperature rises near the pole.Calculations suggest that the quasi-resonant enhancement of the planetary wave-zone flow interaction resulting from certain basic flow configurations might be the explanation of the stratospheric warming.It is hypothesized that the timing of warmings is determined by how well tuned the zonal flow structure is in the sense that appreciable secondorder effects will result in response to small influxes of planetary-wave energy from below. Furthermore, it is suggested that the subsequent evolution of a warming into a full scale warming depends on the ability of the zonal flow configuration to evolve through a series of tuned or resonant states.

Numerical and Analytical Studies on the Energy Conversions in a Baroclinic Model

Abstract The conversions of both zonal to eddy kinetic energy C(KzKr) and eddy potential to eddy kinetic energy C(AzKz) are studied in a β-plane channel confined within latitudinal walls. The initial flow field consists of zonal waves depending on the meridional direction Y, which are superimposed on a zonal basic flow. The latter is a function of y and pressure height P. The tendencies, ∂[C(KZKE)]/∂t and ∂[C(AEKE)/∂t, at time t=0, are evaluated by solving analytically the quasi-geostrophic vorticity equation and the ω equation for t=0. In order to test the above results and to investigate how far in time they can he used to determine the behavior of the wave patterns, respective numerical computations with a five-layer primitive equation model have been performed. The comparisons show a fairly good agreement between the two calculations within about one day of model time.