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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

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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

Interactions between waves that satisfy uniform potential vorticity are considered. The formal analysis is restricted to a triad of eigenfunctions and the reduced system is constrained to satisfy conservation of total energy and conservation of available potential energy on plane rigid horizontal boundaries. A linear stability analysis is used to establish the properties of unstable waves in two cases: basic flows with anti-symmetry, and basic flows with symmetry in the vertical direction. A necessary condition for instability is that the vertical wavenumber of the basic flow must fall between the vertical wavenumbers associated with the perturbation waves. The properties of unstable waves in both cases are compared and analogies with the stability properties of the two-layer model are pointed out.

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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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William Blumen

Abstract

The stability of a parallel flow, exhibiting both vertical and horizontal shear in a Boussinesq fluid, is investigated by a linear analysis. The effect of the earth's rotation is not considered and the disturbances are assumed to be hydrostatic, adiabatic and inviscid. A theorem, restricting the range of the complex phase speed c=cr+ici is exhibited. Then neutral wave solutions are found when the basic flow is represented by ū(y,z) = (a + z) tanhy and cr = ci = 0. It is concluded that these neutral waves are adjacent to unstable waves (cr = 0, ci > 0), which owe their existence to inertial instability associated with the inflection point in the profile of tanhy. The effect of the vertical shear is passive, only modifying the characteristics of the instability but not being responsible for its occurrence.

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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

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William Blumen

Abstract

The Physical process responsible for short-wave baroclinic instability (zonal wavenumbers>10) is examined by means of a linearized two-layer Eady model. The static stability is uniform but different in each layer and the wind shear is uniform throughout both layers. Analysis of the unstable growth rates reveals that the instability is associated with the delta function distribution of potential vorticity at one boundary and at the interface between the two layers. This interpretation complements the interpretation of the unstable modes of a multi-layer model by Staley and Gall (1977). However, the present analysis also demonstrates how the short- and long-wave baroclinic instabilities depend on the relative layer depths as well as on the jump in static stability between the two layers. The effect of a jump in zonal wind shear is shown to be analogous to a jump in static stability in the present model. Finally, some implications of modeling atmospheric flows by multi-layered models, exhibiting discontinuities in potential vorticity, are pointed out.

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William Blumen

Abstract

The divergent barotropic model presented in Part I is used to investigate reduction of rms forecast errors by periodic updating with model-produced observations. Results show that an asymptotic error level is reached in about 2 days. This rapid adaptation reflects the initial balancing provided to the data at each update. Asymptotic rms forecast errors are increasing functions of both the updating period and the observation error, but the asymptotic error level is shown to be independent of the initial error. These results are in basic agreement with experiments carried out with various numerical models. Error reduction by statistically optimal assimilation of data is expected to yield results similar to those obtained in a previous study by the author.

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