Search Results

You are looking at 11 - 20 of 41 items for :

  • Author or Editor: William Blumen x
  • Refine by Access: All Content x
Clear All Modify Search
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.

Full access
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.

Full access
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.

Full access
William Blumen

Abstract

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.

Full access
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.

Full access
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.

Full access
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.

Full access
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.

Full access
William Blumen

Abstract

Nonlinear features of the geostrophic adjustment process in a one-dimensional barotropic atmosphere are investigated by means of a perturbation expansion in the Froude number. The initial unbalanced velocity field is a continuous (nonconstant) even function of the spatial coordinate. The steady-state solution shows the southward shift of the axes of maximum geostrophic velocity and zero pressure, first found by Rossby. In addition, the geostrophic fields are asymmetric about their respective axes.

The nonlinear oscillation of the whole current system approaches the inertial period and decays like t −½ as time t→∞. However, this oscillation continues for a significantly longer time, before approximate geostrophic balance is reached, than the “adjustment time” determined from a linear analysis. A possible shortcoming in the quasi-geostrophic approximation, used in some large-scale dynamical models, is indicated by this result.

Full access
William Blumen

Abstract

The momentum flux by small-amplitude gravity waves produced by steady-state flow over a three- dimensional circular mountain in an isothermal plane rotating atmosphere is investigated. There is an upward transfer of momentum normal to the basic current by external-type gravity-inertia waves. This momentum transfer yields a flux convergence of momentum primarily in the lowest kilometer of the atmosphere. In contrast, the component of momentum parallel to the basic current is transported downward by internal-type gravity waves. This flux is independent of height and is essentially independent of the earth's rotation. Computed values of this surface drag are comparable with estimates of the frictional drag over ordinary terrain. The dependence of the various drag coefficients on atmospheric and mountain-shape parameters is also presented.

Full access