Eady Model Research Articles

The Kelvin-Helmholtz instability in a non-geostrophic baroclinic unstable flow

In an influential paper [1], P.H. Stone has included non-geostrophic effects in an asymptotic study of Eady's model of baroclinic flow. A careful analysis of his paper leads to the revision of one of his conclusions: the case of a large longitudinal wave number does not exhibit non-zero growth rates. Therefore, the symmetric instability has the largest growth rates for the Richardson number in the whole interval 0 ⩽ Ri ⩽ 0.95.

Resonating Neutral Modes of the Eady Model

Abstract The resonance between a neutral Eady mode and a continuous mode with the same phase speed is used to interpret the growth of neutral modes discussed by Farrell. It is shown that the existence of such a resonance may lead to spurious linear growth of error in a discretized quasigeostrophic (QG) model if one of the levels lies very close to the steering level of the neutral Eady normal mode. For a primitive equation model, this spurious resonance manifests itself as an exponentially growing-decaying pair of short waves. Such waves are similar to those found by Arakawa and Moorthi for the Lorenz grid for the QG equations. However, here it is found that for the primitive equation model, these spurious short waves exist both for the Lorenz grid and the Charney–Phillips (CP) grid, although the growth rate is somewhat smaller for the CP grid. It is also shown that the growth rate of the nongeostrophic short waves discussed by Stone is erroneously overestimated in vertically discretized models.

A Semigeostrophic Eady-Wave Frontal Model Incorporating Momentum Diffusion. Part III: Wave Dispersion and Dissipation

Abstract The two-dimensional, semigeostrophic and uniform potential vorticity Eady model is considered. An unstable baroclinic wave develops large velocity and temperature gradients in a narrow zone. Momentum diffusion and wave dispersion are incorporated into the model to prevent the ultimate development of a discontinuity in the alongfront geostrophic velocity υ(υx = ∞). Diffusion and dispersion act to reduce the amplitude of the growing baroclinic wave, and these processes also act to expand the width of the frontal zone, where the maximum velocity gradient is located. Explicit relationships are derived that reveal how these processes are dependent on two parameters: ϵ, the nondimensional eddy diffusion coefficient, and λ the ratio of a dispersion coefficient μ to ϵ2. The total dissipation of kinetic energy D is separated into two parts,D1andD2:D1 provides the dissipation that is largely confined to the relatively narrow frontal zone, and D2 = D − D1 provides the dissipation that is associated with the...

Supercritical Baroclinic Disturbances under the Influence of Topography

Nonlinear evolution of baroclinic waves in the presence of surface topography is investigated in an Eady model modified to include Ekman dissipation and sloping horizontal boundaries. The topographic form drag competes with baroclinicity for the control of amplitude evolution and propagation characteristics of the various disturbance modes. The effectiveness of the topography to phase lock and equilibrate a given mode versus that of baroclinicity to propagate and vacillate that mode depends on the topographic height, its zonal structure, and the level of supercriticality. When topography is sinusoidal and of the same wavelength as the baroclinically most unstable mode, it induces a short-period amplitude modulation whose envelope represents the baroclinic evolution. The dynamics of this modulation is explored analytically. When the sinusoidal topography is of a longer zonal wavelength, a “mixed” wave state persists with amplitude dominance shifting between the topographic mode and the baroclinic mode, depending on the relative strength of form drag and baroclinicity. When topography is a superposition of the most unstable wavelength and a longer one, amplitude dominance and phase locking tend to shift to the mode associated with the “taller” mountain harmonic. When the harmonic structure of topography corresponds to that observed at 35°N, amplitude dominance is concentrated in the longest seven wavelengths. Model results are also compared, in appropriate parameter limits, with near-resonance asymptotic results and annulus experiments.

Moist Surface Frontogenesis Associated with Interior Potential Vorticity Anomalies in a Semigeostrophic Model

Abstract We previously examined surface frontogenesis as an initial value problem in a dry inviscid semigeostrophic Eady model focusing on the frontal dynamics associated with interior potential vorticity anomalies. This work explores frontogenesis associated with potential vorticity anomalies in the presence of latent heat release using a generalization of the two-dimensional) semigeostrophic Eady model that insures that the heating term gives correct asymptotic behavior at high and low temperatures. A variety of initial conditions are considered in which upper-level potential vorticity disturbances induce strong positive potential vorticity anomalies near the lower surface. With uniform interior potential vorticity, baroclinic coupling between the upper and surface induced disturbance is weak despite a near neutral moist stability. On the other hand, examples of initial disturbances with interior potential vorticity show greatly enhanced baroclinic coupling between upper and lower disturbances. Comparis...

Dry Surface Frontogenesis Arising from Interior Potential Vorticity Perturbations in a Semigeostrophic Model

We examine the role of interior potential vorticity perturbations in surface frontogenesis using the two-dimensional semigeostrophic Eady model. Fronts form rapidly for properly configured small disturbances even at zonal wavenumbers for which no exponentially unstable modes exist. While lack of exponential growth does not preclude frontal development, the strength of the disturbance must exceed an amplitude threshold for frontal formation. In practice, the finite amplitude constraint is not severe requiring meridional winds of a few meters per second. Model integrations illustrate the variety of frontogenesis phenomena that arise from interior potential vorticity perturbations including formation of near surface vertical velocity maxima.

Modified Eady Waves and Frontogenesis Part I: Linear Stability Analysis

Baroclinic instability in an inviscid fluid with parabolic potential temperature profiles is investigated. Unlike the classical Eady model, there is no short wave cutoff. In addition to the longwave disturbances, which are similar to the Eady waves, shortwave disturbances can also develop in the lower atmosphere, where the stratification is weaker. The growth rate of the short waves increases with increasing stratification aloft. The results show that shortwave disturbances can penetrate into the upper stable layer. The growth rate and disturbances of those waves may be associated with an effective Burger number, which is defined as(The equation is abbreviated)where h(superscript *)is the height of the maximum vertical heat flux(w(superscript ')θ(superscript '))and λ is the horizontal wavelength. Numerical simulations obtained from a nonlinear mesoscale model in Part Ⅱ also confirm that the short waves can develop into a surface front within a few days. Those short waves may correspond to the medium-scale disturbances observed over the AMTEX (Air Mass Transformation EXperiment) region.

Nonlinear Equilibration of Two-Dimensional Eady Waves

The initial-value problem for Eady's model is reexamined using a two-dimensional (x–z) primitive equation model. It is generally accepted that a finite amplitude instability of Eady's basic state will produce a frontal discontinuity in a finite time. When diffusion prevents the frontal discontinuity from forming, the wave amplitude eventually stops growing and begins to oscillate. We analyze this equilibration and suggest that it is a result of enhanced potential vorticity in the frontal region that is mixed into the interior from the boundaries. The dynamics of equilibration is crudely captured in a modified quasi-geostrophic model in which the zonal-mean static stability is allowed to vary. The magnitude of the meridional wind speed of the equilibrated wave is O(N0H), where N0 is the initial buoyancy frequency and H is the depth of the fluid. This is of the same order as the amplitude of the wave predicted by semigeostrophic theory at the point of frontal collapse. Scaling arguments are presented to determine the three-dimensional flows for which this equilibration mechanism should be important. It is argued that this mechanism is likely to be of some importance for shallow cyclones forming in regions of weak low-level static stability.

Linear Baroclinic instability with the Geostrophic Momentum Approximation

Abstract The linear Eady model of baroclinic instability with the geostrophic momentum (GM) approximation is solved analytically in physical space and shown to be identical to linear three-dimensional semigeostrophic theory. Both the growth rates and the wavenumber of the short-wave cutoff are larger than those predicted by quasi-geostrophic (QG) theory. This behavior arises because the effective static stability is reduced in the GM case. These results are opposite to those using standard nongeostrophic (NG) theory, and the discrepancy increases with decreasing Richardson number. Energetically, the unstable GM normal modes enhance the conversion of available potential energy compared to the QG modes and also convert available kinetic energy to eddy kinetic energy. With regards to the structure of the unstable modes, the northward tilt with height in the GM case is more consistent with NG theory than is the QG solution which displays no meridional tilt. Additional analysis addresses the effect of assuming...

Petterssen's “Type B”Cyclogenesis in Terms of Discrete, Neutral Eady Modes

Abstract We investigate B. Farrell's hypothesis that the development of a surface cyclone with the passage of an upper trough, as observed by S. Pettessen and coworkers, may be understood in terms of an initial-value problem on the Eady model. We consider the response of the Eady model to perturbations whose horizontal wavelengths are short enough to ensure their stability, and whose perturbation potential vorticity is zero. We depart from Farrell with the latter condition as it eliminates the continuous spectrum and allows the evolution of the perturbation to be understood solely in terms of the two normal modes of the Eady model—one with maximum amplitude at the upper lid, which propagates eastward with respect to the midlevel flow, and one westward propagating, with maximum amplitude at the lower surface. Imagine an initial upper-level disturbance with no surface perturbation; this is represented by the two Eady modes in combination such that the initial surface perturbation pressure is zero. As the fl...

Cumulus Heating and CISK in the Extratropical Atmosphere. Part I: Polar Lows and Comma Clouds

Abstract The effects of cumulus heating and the possibility of CISK in extratropical latitudes are explored by combining the Eady model of baroclinic instability with wave-CISK. For perturbations about a state of rest, there is no intrinsic time scale, however the equations can be solved for geopotential tendency. It was found that CISK disturbances could not grow unless the heating exceeded a threshold value. This value is reduced when the boundary layer moisture content is increased, the static stability decreased, or the heating distributed lower in the troposphere. Perturbations about a baroclinic basic state, for small heating rates resemble a baroclinic wave with heating acting mainly to reduce the effective static stability, resulting in faster growth and shorter wavelengths. However, as the heating is increased, the instability gradually takes on the characteristics of the pure CISK disturbance. The transition level shows the same dependencies on model parameters as for the threshold in the CISK c...

Baroclinic waves and frontogenesis with an embedded zone of small moist symmetric stability

AbstractThree‐dimensional semigeostrophic moist baroclinic waves and frontogenesis are studied both analytically and numerically. It is found that a slantwise zone of small (either unconditional or quasi‐conditional) moist symmetric stability embedded in a dry and baroclinic basic state traps the energy of baroclinic waves, forming a strong meridional structure with maximum wave amplitude in the moist zone and exponential decay of the wave amplitude away from the moist zone. The unstable waves absorb energy from both the moist and dry baroclinic basic state, but the strongest energy conversion is in the moist zone, so growth rates decrease with the width of the moist zone. As the moist zone becomes infinitely narrow (wide) the model degenerates into a dry (moist) Eady model. The smaller the moist symmetric stability, the larger the growth rate and the wavenumber of the most unstable wave. The valid range of the model parameters depends on the semigeostrophic approximation, which sets a limit to the smallness of the moist symmetric stability.In semigeostrophic space, the moist zone changes shape due to geostrophic deformation. As the wave develops in real space, the surface pressure low (high) shifts northward (southward) due to geostrophic deformation, tightens (expands) due to ageostrophic convergence (divergence), and goes on to form a coupled warm‐cold frontal system.A numerical model of the semi‐Lagrangian finite element method is developed. The method is shown to be accurate in computing wave growths and efficient in resolving frontal structures, although the semigeostrophic approximation is found to become poor or even invalid locally in the narrow (meso‐β‐scale) cold frontal region.

On the scale of baroclinic instability in deep, compressible atmospheres

A simple model of linearized, inviscid baroclinic instability in an adiabatic, hydrostatic, compressible atmosphere of arbitrary (though finite) depth, based on the well-known Eady model, is used to investigate the variation of growth rates and favoured horizontal length scales as functions of δ, the ratio of the model depth D to the density scale height Hs. Both geometric height coordinates (with w = 0 horizontal boundary conditions) and log-pressure (with ω = 0 boundary conditions) are considered. For δ > 3 and a given zonal velocity scale, growth rates are significantly reduced relative to comparable instabilities in an incompressible fluid (δ = 0), and may be suppressed altogether in a laterally-bounded channel for large enough δ at a given value of static stability. Where instability does occur, the length scales favoured are longer than for an incompressible fluid, and are generally comparable to a deformation radius based on D (rather than Hs). The relationship between these results and those obtained in comparable recent studies (which have found scales comparable to a deformation radius based on Hs to be important) is examined. Some implications for the role of baroclinic instability in the atmospheres of the major planets are also discussed.

On the scale of baroclinic instability in deep, compressible atmospheres

AbstractA simple model of linearized, inviscid baroclinic instability in an adiabatic, hydrostatic, compressible atmosphere of arbitrary (though finite) depth, based on the well‐known Eady model, is used to investigate the variation of growth rates and favoured horizontal length scales as functions of δ, the ratio of the model depth D to the density scale height Hs. Both geometric height coordinates (with w = 0 horizontal boundary conditions) and log‐pressure (with ω = 0 boundary conditions) are considered. For δ > 3 and a given zonal velocity scale, growth rates are significantly reduced relative to comparable instabilities in an incompressible fluid (δ = 0), and may be suppressed altogether in a laterally‐bounded channel for large enough δ at a given value of static stability. Where instability does occur, the length scales favoured are longer than for an incompressible fluid, and are generally comparable to a deformation radius based on D (rather than Hs). The relationship between these results and those obtained in comparable recent studies (which have found scales comparable to a deformation radius based on Hs to be important) is examined. Some implications for the role of baroclinic instability in the atmospheres of the major planets are also discussed.

A study of absolute and convective instabilities with an application to the Eady model

Abstract The formalism for linear absolute and convective instabilities developed in plasma physics (Briggs, 1964; Bers, 1973) is extended. The conclusion is that any order saddle point as well as any singular point of a frequency as a function of a wavenumber Ω(k) may contribute to instability. Moreover, contributions may come from k-independent branches of the dispersion relation and from regular nonsaddle points of Ω(k). Accordingly, a variety of algebraic-exponential asymptotic behaviors, in particular, purely algebraic growths and sinusoidal oscillations, of a disturbance is possible. In shear flows and stratified flows instability may also be caused by the singularities associated with critical layers. It is shown that in such flows the asymptotic pattern of a growing disturbance may vary in the direction of the shear and/or stratification. Such variability of pattern may be present only in the presence of instabilities related to the interactions between critical layers and incident amplifying wave...

Favorable environments for explosive cyclogenesis in a modified two-layer Eady model

Favorable environments for explosive cyclogenesis in a modified two-layer Eady model

Favorable environments for explosive cyclogenesis in a modified two-layer Eady model

A modified two-layer Eady model is used to find favorable environments for explosive cyclogenesis by examining baroclinic instability of both short and long waves. The model includes Ekman dissipation and low-level sensible heating, and allows both static stabilities and vertical shears to be different but constant in each layer. It is found that an enhanced low-level shear, a reduced low-level static stability, a relatively large upper-level static stability, and an appropriate height of the interface are favorable for the short-wave development, and that an enhanced upper-level shear and a relatively small upper-level static stability are favorable for the long-wave development, while having opposite effects on the short wave. A small Ekman dissipation is favorable for the development of both waves. Newtonian sensible heating does not directly enhance the development of both waves. A favorable environment for explosive cyclogenesis may not be expressed by a single parameter, but rather by a combination of baroclinic instability with several parameters which contribute collectively to the development of both short and long waves. The relative phase of these waves also plays an important part: the development is enhanced when the surface cyclone is ahead of the upper level wave trough by 1/4π. DOI: 10.1111/j.1600-0870.1987.tb00301.x

87:6195 Favorable environments for explosive cyclogenesis in a modified two-layer Eady model: Weng, H.Y. and A. Barcilon, 1987. Tellus, 39A(3): 202–214

87:6195 Favorable environments for explosive cyclogenesis in a modified two-layer Eady model: Weng, H.Y. and A. Barcilon, 1987. Tellus, 39A(3): 202–214

Linear Development of Quasi-Geostrophic Baroclinic Disturbances with Condensational Heating

Abstract This paper presents the linear solution to the initial value problem for the Eady model of baroclinic instability including condensational heating using a wave–CISK formulation with a uniform heating profile in the vertical. As in the dry case, the continuous spectrum completes the class of free mode solutions but is asymptotically stable. In the moist case, both the dry and the moist normal modes contribute to the solution to the initial value problem. Analysis of the moist Eady dispersion relation indicates that the heating increases the growth rate and the wavenumber of the most unstable mode and of the short-wave cutoff. For all values of the heating amplitude, the growth rate is bounded, both wavenumbers are finite, and the very short waves are always stable. Shallow clouds, however, increase both wavenumbers more than deep clouds. For sufficiently large values of the heating amplitude, the free modes display unphysical behavior with steering levels either above the rigid-lid tropopause or b...

A comparison between frontogenesis in the two‐dimensional Eady model of baroclinic instability and summertime cold fronts in the Australian region

AbstractA two‐dimensional, anelastic, numerical model with a simple turbulence parametrization is integrated with Eady's linear normal mode solution for an unstable baroclinic wave as initial condition. The structure of the surface front which develops after five days of model integration is compared with data on summertime cold fronts, or ‘cool changes’, in south‐eastern Australia obtained during recent field experiments of the Australian Cold Fronts Research Programme. It is shown that the ridge‐trough structure of an amplifying baroclinic wave and its attendant surface front captures many of the important features of ‘cool changes’, together with the broad‐scale flow in which they develop. Indeed, it is argued that the model provides a useful theoretical framework in which the dynamics of the Australian summertime ‘cool change’ may be understood.