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William Blumen

Abstract

The geostrophic coordinate transformation is a contact transformation that preserves the correspondence between the slopes of geopotential surfaces ϕ in physical space and the slopes of the surfaces ϕ, the maps of ϕ, in transformed space. The transformation back to physical space may be accomplished by integration along characteristic surfaces. This technique may be used to determine the time and place where a discontinuity would form as a function of the initial conditions. A model solution is used to illustrate properties of the transformation.

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William Blumen

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 ε 2. The total dissipation of kinetic energy D is separated into two parts,D 1andD 2:D 1 provides the dissipation that is largely confined to the relatively narrow frontal zone, and D 2 = DD 1 provides the dissipation that is associated with the decaying waves that trail behind the front. These evaluations are carried out for a range of parameter values (ε, λ). Results show that the dissipation is not confined exclusively to the frontal zone but that D 2D 1 when λ is large. Limitations of the present model development are associated with the excessive growth of the unstable Eady wave in the absence of dissipation and the lack of fine-scale measurements that may be used to design a dynamical model of the frontal zone.

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William Blumen

Abstract

A model of quasi-geostrophic uniform potential vorticity flow, previously examined by Blumen (1978a,b), is considered. The total depth-integrated energy and the available potential energy on level boundaries are conserved by the motion. Nonlinear interactions between three different scales of motion are examined. The linear system is first analyzed to determine the normal modes of the model. There are two sets of normal modes, corresponding to two different unstable growth rates. It is then shown that if normal mode initial conditions are specified for the nonlinear initial-value problem, the two conservation principles may be combined to yield a single constraint on the nonlinear interactions that occur between three scales of motion. The properties of normal mode initial conditions are also used to cast this constraint into a relatively simple form that is appropriate during the initial stages of the finite amplitude motion.

Numerical integrations of the basic set of equations reveal that the solutions are quite sensitive to the initial conditions. When normal mode initial conditions corresponding to the largest unstable growth rate are used, the simpler constraint continues to apply past the initial stages of growth. Analytical confirmation of this result is also provided. Nonlinear motions, associated with the other set of normal mode initial conditions, are also examined. The initial stages of the motion are similar to those above, but then the solutions tend to become aperiodic and the simpler form of the constraint on scale interactions does not apply. Extension of the range of integration over a broader range of initial conditions is suggested by these results.

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William Blumen

Abstract

A model of inertial oscillations that may occur with, and be modulated by, deformation frontogenesis is formulated. The deformation parameter is α ∼ 10−5 s−1 and the Coriolis parameter is f ∼ 10−4 s−1. This timescale separation, distinguished by the ratio α/f ∼ 10−1, provides the basis for application of a two-timescale analysis that separates the frontal evolution from the inertial frequency oscillations. To lowest order, the inertial oscillations do not influence frontogenesis, described by the classical Hoskins and Bretherton model. The frontal evolution, characterized by the alongfront geostrophic wind, does, however, provide an amplitude modulation of the inertial wind oscillation and of the temperature that also undergoes an oscillation at the inertial frequency. Parameter values are chosen to illustrate frontal contraction and translation characteristics that can distort the wind hodograph from circular motion. Ground-level temperature traces also exhibit unusual attributes, such as an initial temperature increase with a cold frontal passage, that can be associated with the relative phase of the oscillation compared to the leading edge of the front. Lack of adequate observations for verification purposes and neglect of the boundary layer provide two important limitations.

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William Blumen

Abstract

Predictability experiments are carried out with a divergent barotropic model that describes the evolution of quasi-geostrophic planetary waves and high-frequency gravity-inertia waves. Error growth, relative to a model-determined control state, is initiated by an initialization procedure that is not compatible with the model equations. An analysis of error growth due to improper representation of the physics incorporated in prediction models is also carried out with the present model. The error growth rate and the range of predictability determined from these experiments, based on a simple triad solution of the nonlinear forecast equation, compare very well with the results from experiments carried out with multi-level numerical models. The mechanism of predictability decay by nonlinear energy exchange is shown to differ from the corresponding mechanism discussed by Lorenz and Leith, which is based on a model of two-dimensional turbulence.

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William Blumen

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 isotherms and isopleths of the horizontal velocity component parallel to the frontal zone. In contrast, ageostrophic convergence and horizontal distortion of potential isotherms make the largest contribution to frontogenesis in the model.

Ekman-layer pumping is introduced into the model to simulate the vertical velocity jet. Yet this feature is not sufficient to increase the contribution of vertical tilting to frontogenesis because the vertical gradients of potential temperature and geostrophic velocity are weaker in this case.

Trajectories of the air motion tend to show the pattern of upgliding warm air ahead of the frontal zone with relatively stagnant cold air to the rear. In general, the model is able to provide qualitative agreement with gross features of this frontal situation. Discrepancies seem to be associated with the absence of a realistic boundary layer formulation and mesoscale mixing processes in the model.

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William Blumen

Abstract

Continuous partial reflection of linear hydrostatic gravity waves that propagate through a stratified shear flow is examined. The complex reflection coefficient R satisfies a Riccati equation, which is a first-order nonlinear differential equation. It is shown that |R|<1 since critical levels and overreflection are not considered. In this case the conservation of wave action flux may be expressed as a relationship between |R| and E l −1, where E is the wave energy and l a characteristic inverse vertical length scale of the background state.

It is demonstrated that R for a layered model represents a limiting solution of the Riccati equation. A general solution is also derived, under the assumption that the characteristic woe l is directly proportional to the inverse scale height of the characteristic impedance associated with a stratified shear flow. It is shown that the vanishing of |R| at a specific level is analogous to the vanishing of |R| in a three layer model, when the characteristic impedances in the top and the bottom layers satisfy a matching condition. Finally, various properties of the reflection coefficient are displayed for a particular background state. The extension of the theory to encompass other types of wave motion is indicated.

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William Blumen

Abstract

The nonlinear evolution of unstable two-dimensional Eady waves is examined by means of a two-layer version of the Hoskins and Bretherton (1972) model. The upper layer is characterized by a higher static stability than the lower layer. Two types of unstable solutions are realized: the relatively long-wave solution has a vertical structure that extends throughout the vertical depth of the fluid and is the counter-part of the solution for a single layer system, while the shorter wave is essentially confined to the lower fluid layer. Model parameters, lower layer depth and static stability difference are chosen such that the two waves have comparable growth rates. The solution is determined by means of a Stokes expansion and terminated at second-order in the amplitude. The nonlinear interaction process between these growing baroclinic waves is then related to the wave interaction process described by the one-dimensional advection equation. Finally, an interpretation is proposed to explain disparate observations of cyclogenesis in polar air streams.

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William Blumen

Abstract

Momentum diffusion has been introduced into a semigeostrophic Eady-wave frontal model by Blumen (Part I). This model is used to determine the kinetic energy and enstrophy dissipations within a frontal zone that extends from the ground to a midtropospheric level. The largest amount of kinetic energy dissipation occurs when a relatively small nondimensional eddy viscosity coefficient is used, and the front attains an equilibrated state—a balance between unstable growth and momentum diffusion. The magnitude of kinetic energy dissipation ranges from about 50 to 250 W m−2 for parameter values that characterize surface-based fronts. These values are comparable to the 75 W m−2 determined by Kennedy and Shapiro from observations in an upper-level front, but are about one to two orders of magnitude larger than previous estimates of kinetic energy dissipated locally in clear-air turbulence zones and in the planetary boundary layer. An estimate of global kinetic energy dissipation in the planetary boundary layer is provided. A comparison establishes that fronts may make a relatively large contribution to the dissipation occurring during the life cycle of a cyclone, but the global contribution is less than that associated with the planetary boundary layer.

Frontal equilibration is characterized by a balance between production and dissipation of enstrophy. However, as frontolysis sets in, the dissipation of enstrophy becomes the dominant feature. Finally, it is noted that the physical process associated with the cascade of energy and enstrophy to dissipative scales differs from the cascade process described by the theory of homogeneous turbulence, and a different spectral decay law is realized.

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William Blumen

Abstract

The Hoskins-Bretherton (HB) model is adopted to study two-dimensional frontogenesis in unsteady basic shear flows. The solutions exhibit the nonlinear evolution of an unstable Eady wave up to the formation of a frontal discontinuity. This development is described by the HR solution for a steady shear flow with time reinterpreted by means of a coordinate transformation. The computations are carried out by means of a relatively simple but hightly accurate approximation to the exact solution. The results show that typical wintertime variability of midlatitude zonal flows may either retard or accelerate frontal development compared to frontogenesis in a steady basic flow. Rapid frontal development is also associated with relatively rapid frontal movement and a more intense ageostrophic circulation. In contrast, a prolonged period of development is associated with relatively slow movement and a weaker ageostrophic circulation. The effect of different time-varying basic flows is examined and the results interpreted in relation to atmospheric frontogenesis.

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