Eady Problem Research Articles

Eady Instability of Isolated Baroclinic Jets with Meridionally Varying Tropopause Height

Abstract An efficient solution procedure is developed for a nonseparable semigeostrophic Eady problem on a semi-infinite f plane. A model is designed to study the effects of meridionally isolated jets and tropopause morphology on baroclinic instability. Potential vorticity (PV) is assumed to be piecewise constant and discontinuous at the tropopause, whose height is allowed to vary with latitude. The computational domain is discretized in a stretched coordinate to maintain adequate resolution in the vicinity of the tropopause and axis of the jet. The basic state is numerically inverted from the two PV values with the specification of only the meridional profiles of potential temperature (PT) at the surface and (unspecified) tropopause height. The nonseparable eigenvalue problem about this basic state is solved for the discrete (zero-PV) normal modes. Since continuum modes are not sought, the size of the problem is reduced greatly, keeping the storage and CPU requirements moderate even at relatively high sp...

Linear Amplification and Error Growth in the 2D Eady Problem with Uniform Potential Vorticity

The concept of a singular mode underlies optimal linear amplification theories. This concept is studied in the frame of the two-dimensional, quasigeostrophic Eady problem with uniform potential vorticity. Analytical solutions are produced for the relevant physical norms. Exact relations are also derived for the amplifications, which give the lower and upper bounds to any linear development. Results show significant differences in the structure of the singular modes, as well as in the associated amplifications, when the horizontal wavenumber is varied or the inner product is changed. It is found that the singular modes can depart significantly from the normal modes, though the dynamics of the problem are very simple. Comparisons with previous works are also performed. Finally, the derived equations are used to present the linear evolution of error growth within the Eady problem, as predicted by a Kalman filter. Considerations on the spectral space error covariance matrix are made, and a particular case of error dynamics in the 2D physical space is shown. The derivation of the general algebraic solutions is included in the appendix.

The Effect of Basic-State Potential Vorticity Gradients on the Growth of Baroclinic Waves and the Height of the Tropopause

The stability characteristics of normal mode perturbations on idealized basic states that have meridional potential vorticity (PV) gradients that are zero in the troposphere, very large at the tropopause, and order b in the stratosphere are checked. The results are compared to the corresponding models that have a lid at the tropopause. The dispersion relations and the vertical structures of the modes are similar in the two models, thus confirming the relevance of the Eady problem to unbounded atmospheres. The effect of replacing the lid with a more realistic tropopause is to complicate the interaction of tropopause and surface waves, such as to inhibit phase locking for a range of wavenumbers. This causes the short-wave cutoff of the Eady model to move to longer waves. Also, there is a slight destabilization of the long waves, which have large amplitudes in the stratosphere. The effect of gradually changing the tropospheric PV gradients from zero (Eady-type profile) to b (Greentype profile) on the stability of normal modes is checked. The dispersion relations show a smooth transition from the Green profiles to the Eady profiles, and a short-wave cutoff is gradually formed. Finally, the possibility of neutralizing the atmosphere through the short-wave cutoff of the Eady model by lifting the tropopause while keeping PV gradients zero in the troposphere is examined. It is found that instability depends on some minimal amount of tunneling of waves between the surface and the tropopause. The amount of tunneling depends on the vertical integral of N in the troposphere. It is necessary for ∫ Nd zto increase for

Free boundary effects on baroclinic instability

The effects of a free boundary on the stability of a baroclinic parallel flow are investigated using a reduced-gravity model. The basic state has uniform density stratification and a parallel flow with uniform vertical shear in thermal-wind balance with the horizontal buoyancy gradient. A finite value of the velocity at the free (lower) boundary requires the interface to have a uniform slope in the direction transversal to that of the flow. Normal-mode perturbations with arbitrary vertical structure are studied in the limit of small Rossby number. This solution is restricted to neither a horizontal lower boundary nor a weak stratification in the basic state.In the limit of a very weak stratification and bottom slope there is a large separation between the first two deformation radii and hence short or long perturbations may be identified:(a) The short-perturbation limit corresponds to the well-known Eady problem in which case the layer bottom is effectively rigid and its slope in the basic state is immaterial.(b) In the long-perturbation limit the bottom is free to deform and the unstable wave solutions, which appear for any value of the Richardson number Ri, are sensible to its slope in the basic state. In fact, a sloped bottom is found to stabilize the basic flow.At stronger stratifications there is no distinction between short and long perturbations, and the bottom always behaves as a free boundary. Unstable wave solutions are found for Ri→∞ (unlike the case of long perturbations). The increase in stratification is found to stabilize the basic flow. At the maximum stratification compatible with static stability, the perturbation has a vanishing growth rate at all wavenumbers.Results in the long-perturbation limit corroborate those predicted by an approximate layer model that restricts the buoyancy perturbations to have a linear vertical structure. The approximate model is less successful in the short-perturbation limit since the constraint to a linear density profile does not allow the correct representation of the exponential trapping of the exact eigensolutions. With strong stratification, only the growth rate of long enough perturbations superimposed on basic states with gently sloped lower boundaries behaves similarly to that of the exact model. However, the stabilizing tendency on the basic flow as the stratification reaches its maximum is also found in the approximate model. Its partial success in this case is also attributed to the limited vertical structure allowed by the model.

Nonlinear Stability Theorem for Eady's Model of Quasigeostrophic Baroclinic Flow

Abstract In this paper, the linear stability criterion for Eady's problem is proved to be also a nonlinear stability one. Upper bounds for the disturbance potential enstrophy, the disturbance energy, and the disturbance available potential energy on the rigid lids are also established.

On improving a one-layer ocean model with thermodynamics

A popular method used to incorporate thermodynamic processes in a shallow water model (e.g. one used to study the upper layer of the ocean) is to allow for density variations in time and horizontal position, but keep all dynamical fields as depth independent. This is achieved by replacing the horizontal pressure gradient by its vertical average. These models have limitations, for instance they cannot represent the ‘thermal wind’ balance (between the horizontal density gradient and the vertical shear of the velocity) which dominates at low frequencies. A new model is now proposed which uses velocity and density fields varying linearly with depth, with coefficients that are functions of horizontal position and time. This model can explicitly represent the thermal wind balance, but its use is not restricted to low- frequency dynamics.Volume, mass, buoyancy variance, energy and momentum are conserved in the new model. Furthermore, these integrals of motion have the same dependence on the dynamical fields as the exact (continuously stratified) case. The evolution of the three components of the absolute vorticity field are correctly represented. Conservation of density–potential vorticity is not fulfilled, though, owing to artificial removal of the vertical curvature of the velocity field.The integrals of motion are used to construct a ‘free energy’ [Escr ]f, which is quadratic to the lowest order in the deviation from a steady state with (at most) a uniform velocity field. [Escr ]fis positive definite, and therefore the free evolution of the system cannot lead to an ‘explosion’ of the dynamical fields. (This is not the case if the velocity shear and/or the density vertical gradient is excluded in the model, which results in a non-negative definite free energy.)In a model with one active layer, linear waves on top of a steady state with no currents are, to a very good approximation, those of the first two vertical modes of the continuously stratified model. These are the familiar geophysical gravity and vortical waves (e.g. Poincaré, Rossby, and coastal Kelvin waves at mid-latitudes, equatorial waves, etc.).Finally, baroclinic instability is well represented in the new model. For long perturbations (wavelengths of the order of the deformation radius of the first mode) the agreement with more precise calculations is excellent. On the other hand, the comparison with the eigenvalues of Eady's problem (which corresponds to wavelengths of the order of the deformation radius of the second mode) shows differences of the order of 40%. Nevertheless, the new model does have a high-wavenumber cutoff, even though it is constrained to linear profiles in depth and therefore cannot reproduce the exponential trapping of Eady's problem eigensolutions.In sum, the integrals of motion, vorticity dynamics, free waves and baroclinic instability results all give confidence in the new model. Its main novelty, however, lies in the ability to incorporate thermodynamic processes.

The Eady problem with linear horizontal shear

Abstract We consider the quasi-geostrophic linear shear stability analysis of an inviscid, channel flow with linear horizontal and vertical velocities. This modified Eady basic state yields a variety of flow behaviors as the horizontal shear is increased. The presence of y -shear allows for unstable modes with the same growth rate, different phase speeds and different amplitude structures. We speculate that these modes may contribute to amplitude and structural vaccilation. We find that the two-mode truncation in y provides both quantitative, for small horizontal shears, and qualitative, for larger y -shears, results which compare favorably with the 25 y -mode numerical results. The instability in this model is baroclinic modified by the projection of the horizontal basic flow onto the y -modes. The destabilization of relatively short waves, associated with the horizontal shear, is interpreted physically using the constraints provided by energy conservation. It is demonstrated that the presence of a horizontal shear may be interpreted as a reduction in the ‘effective’ vertical channel depth in a baroclinic Eady model. In other words, waves with smaller vertical penetration depth, with wavenumbers larger than the wavenumber of the short-wave cut-off in the Eady model, can maintain the necessary phase relationship to provide instability.

The Eady Problem for a Basic State with Zero PV Gradient butβ≠ 0

The classic Eady problem is modified to include β ≠ 0, but with the basic distributions of temperature and zonal flow adjusted to preserve zero meridional gradients of basic-state potential vorticity in the fluid interior. Much of the mathematical simplicity of the classic problem is retained; however, the results differ in important ways. Specifically, the instability now has a long-wave cutoff in addition to the traditional short-wave cutoff. The former is associated with the fact that the phase speeds of the edge waves begin to differ so much as wavenumber is reduced that the two edge waves can no longer interact in order to form unstable modes. For the unstable modes, this manifests itself in that the steering level for unstable modes is always below the middle of the fluid and approaches the lower boundary near the long-wave cutoff. Relatedly, the amplitude of the unstable geopotential perturbations is larger at the upper boundary than at the lower boundary. Finally, below the long-wave cutoff, one of the neutral waves has a phase speed that becomes increasingly easterly as wavenumber decreases. This allows a resonant response to planetary-scale stationary forcing.

Cyclogenesis in a Saturated Environment

The dynamics of baroclinic wave growth in a saturated environment is examined using linear and nonlinear models employing a parameterization of latent heat release that assumes all rising air is saturated, and saturation equivalent potential temperature is conserved on ascent. Piecewise potential vorticity (PV) diagnostics are used to interpret the results. When the stability to vertical displacements in saturated air is allowed to increase with height, as it must in an atmosphere with a constant, positive lapse rate of potential temperature, the growth rates of the most unstable modes of the Eady problem grow only marginally faster than the modes of the dry problem. The vertical variation of moist static stability produces a gradient of moist potential vorticity in the rising air, eliminating the short wave cutoff present in the dry Eady problem. The destabilization of the short waves is shown to be associated with the interaction between surface potential temperature anomalies and diabatically generated lower-tropospheric potential vorticity anomalies. Nonlinear primitive equation simulations, starting from normal-mode initial conditions, show that while the dry wave grows at nearly the linear growth rate until maximum amplitude is reached, the moist wave grows significantly faster than the linear growth rate at finite amplitude. This enhanced growth is associated with the rapid amplification of a mesoscale PV anomaly generated by latent heat release at the warm front. The rapid amplification of the surface cyclone results from the superposition of the circulation associated with this mesoscale PV anomaly upon the circulation associated with the surface and upper boundary potential temperature anomalies. Additional integrations with finite-amplitude initial conditions more typical of atmospheric conditions exhibit similar behavior. It is suggested that many of the rapid cyclogenesis events that occur as upper-tropospheric PV anomalies cross the east coasts of continents may arise from the rapid generation of PV anomalies by condensational heating in the moist maritime lower troposphere.

A Coordinate Transformation for Objective Frontal Analysis

The analysis of a frontal discontinuity is difficult as the hypothesis of isotropy commonly assumed by inter- polation schemes is obviously erroneous in that case. Starting from the semigeostrophic theory, the authors propose a kind of flow-dependent analysis based on the use of geostrophic coordinates. The idea behind this approach is that the discontinuity should appear much more regular in geostrophic space, and then it should better fulfill the above-mentioned hypothesis of isotropy. The validity of such a scheme is first checked in the Hoskins and Bretherton dry and inviscid shear model of frontogenesis. In order to treat more realistic cases, the authors introduce a filtered geostrophic advection coordinate (FGAC) using real wind instead of geostrophic wind. Applied to simulations of the wet and viscous Eady's problem, this procedure brings a clear gain with a maximum positive impact when data spacing is in the 100–200-km range. Finally, the authors apply this method to a high-resolution dropsonde dataset collected during the FRONTS 87 experiment. Again, the FGAC transformation is shown to greatly improve the analysis, producing more consistent wind and mass fields.

Baroclinic Neutrality and the Tropopause

It is noted that the atmosphere has a baroclinically neutral state that permits strong temperature gradients at the ground. In this state, the concentrated potential vorticity gradient at the ground is separated from the concentrated potential vorticity gradient at the tropopause by a region where the potential vorticity gradient is zero. The resulting basic state is analogous to the basic state in the Eady problem. A minimum meridional wavenumber is established by the width of the subtropical jet, and neutrality is established by moving the tropopause to a sufficient height so that the total wavenumber corresponds to the short-wave cutoff in the Eady problem. The resulting height is approximately the observed tropopause height.

Eady問題における種々のじょう乱発達メカニズムの波運動量による解析

To understand various development mechanisms of disturbances in the Eady problem, the quasigeostrophic equation system is extended to include the horizontal boundary conditions. First, we distinguish continuous-spectrum waves from the modal waves by using potential vorticity. Next, we derive the "wave momentum" and clarify its relationship with the Eliassen-Palm flux. Finally, the necessary condition for the development of waves in the baroclinic flow is expressed by evaluating the EP-flux divergence. In Farrell (1984) or Thorncroft and Hoskins (1990), shallow vortices develop algebraically, as the result of wave-wave interaction of a boundary modal wave and internal continuous-spectrum waves. These interactions are clearly indicated by the EP-flux convergence around the critical level.

Nonhydrostatic Simulation of Frontogenesis in a Moist Atmosphere. Part I: General Description and Narrow Rainbands

Abstract A series of experiments using a two-dimensional, nonhydrostatic, numerical cloud model with fine horizontal and vertical resolution is performed with the Hoskins-Bretherton solution to the Eady problem as initial condition. Dry and wet simulations are presented with 5-, 10-, and 40-km horizontal resolutions and vertical resolution from 160 m at the ground to 330 m at the domain top. Sensitivity experiments on the initial Brunt-Vaisala frequency and vertical shear are also discussed. Two classes of narrow bands are identified: 1) A narrow cold-frontal rainband at the surface cold front, consisting of a line of shallow convection triggered by the frictionally induced instability in the boundary layer at the surface front. The associated precipitation is organized in a narrow line with a large rainfall rate. Latent heating due to condensation contributes in large part to the tilting of isentropes and to the increasing of the vertical jet strength. Sensitivity experiments show that both friction and ...

Baroclinic instability in a two-layer, vertically semi-infinite domain

A quasi-geostrophic two-layer system with uniform, but different, shears and stratifications in two layers is examined for the occurrence of baroclinic instability. A derived necessary condition for instability requires that the shear in the upper, semi-infinite layer does not exceed a certain value (related to the stratification). For each configuration of the basic flow, there appear two short and two long neutral modes as well as a decaying and a growing mode at intermediate wavelengths. The short neutral modes are bound to either the surface or the interface, the long neutral modes to one or other of the layers. Instability appears if the foregoing modes propagate at a similar velocity. The depth of the waves is shown to be of less importance in the framework of potential vorticity dynamics. For some configurations, the growth rates surpass those of the classical Eady problem. Finally some consideration is given to the linkage between these results and the formation of polar lows. DOI: 10.1034/j.1600-0870.1991.t01-4-00003.x

The semi-geostrophic Eady problem as a testbed for numerical simulations of frontogenesis

The semigeostrophic Eady problem is used as a testbed for frontogenetical calculations using a specific numerical model. Over 3 days of integration, the numerical solutions agree very well with their analytical counterparts. The increasing deviations in the vertical field of motion during the last day before the analytical model breaks down are attributed to the inability of a grid-point model to handle evergrowing contrasts correctly. The dependence of the time-scales on the external parameters is addressed. DOI: 10.1034/j.1600-0870.1990.00017.x

The semi-geostrophic Eady problem as a testbed for numerical simulations of frontogenesis

The semigeostrophic Eady problem is used as a testbed for frontogenetical calculations using a specific numerical model. Over 3 days of integration, the numerical solutions agree very well with their analytical counterparts. The increasing deviations in the vertical field of motion during the last day before the analytical model breaks down are attributed to the inability of a grid-point model to handle evergrowing contrasts correctly. The dependence of the time-scales on the external parameters is addressed. DOI: 10.1034/j.1600-0870.1990.00017.x

A Comparison between the Hoskins-Bretherton Model of Frontogenesis and the Analysis of an Intense Surface Frontal Zone

Abstract The Hoskins-Bretherton model of frontogenesis employed here represents the counterpart of the two-dimensional Eady problem expressed in geostrophic coordinate space. The fundamental characteristics of the model solution are shown to be derivable from the properties of the nonlinear one-dimensional advection equation and the linearized Eady problem. Detailed comparisons are made between the predictions of this model and the analysis of an intense frontal zone presented by Sanders. Qualitative agreement is found in details of the horizontal wind field and potential temperature distributions. The major discrepancy occurs in the vertical velocity field: the most intense vertical velocities occur at midlevel in the model and are significantly smaller in magnitude than the rising narrow jet above the analyzed zone of maximum cyclonic relative vorticity. The presence of this jet is responsible for the most significant frontogenetical properties of the front associated with vertical tilting of potential ...

A Note on Horizontal Boundary Conditions and Stability of Quasi-Geostrophic Flow

sufficient conditions are determined for the stability of flow governed by the modified quasi-geostrophic system of equations, which contain a non-Doppler term in the boundary conditions. Some instability characteristics of the modified Eady problem are determined and compared with calculations made by White.

Uniform Potential Vorticity Flow: Part I. Theory of Wave Interactions and Two-Dimensional Turbulence

Uniform potential vorticity flows are examined. In the quasi-geostrophic system, conservation of total energy and conservation of available potential energy on plane rigid horizontal boundaries imply a restriction on energy exchanges as a result of scale interactions. It is shown that for the Eady problem instability is always associated with energy transfer both up and down the vertical wavenumber spectrum although energy transfer from small to large three-dimensional wavenumbers may occur over a finite range of the spectrum. An inertial theory of two-dimensional turbulence is also presented. The formal analysis, based on Leith's diffusion approximation, predicts two inertial subranges: −5/3 and −1 power dependences on the horizontal wavenumber for available potential energy on horizontal boundaries. In the former range, available potential energy on horizontal boundaries cascades at a constant rate toward higher wavenumbers; in the latter range, the depth-integrated total energy cascades at a constant rate toward lower wavenumbers. Analysis of the semi-geostrophic equations, in the form presented by Hoskins, shows that a formal analogy exists between energy exchanges in this system and energy exchanges in the quasi-geostrophic system. The transformation back to physical space reveals that the mean strain rate, due to vertical wind shear, affects the complete spectrum of interacting waves. This latter result brings the concept of a local inertial energy transfer theory of turbulence for synoptic- and subsynoptic-scale motions into question, although it is concluded that further analysis and observational evidence would be required to resolve the problem.

A Note on the Baroclinic Instability of General Time-Dependent Basic Fields of the Eady Type

Abstract The Eady problem concerning the stability of quasi-geostrophic baroclinic flows which have uniform vertical shear is generalized to include time-dependent basic flows. By means of a simple transformation an exact solution can be obtained. The instability of the flow, in the sense of boundedness as t → ∞, is determined by the usual Eady criterion derived for steady basic flows, but the growth rates can be significantly enhanced or diminished. In the special case of purely periodic time dependence, the flow is always stable in the above sense.