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William Blumen

Abstract

A simple theoretical approach is formulated, based on part I of this study, in which inadequate modelling of physical processes and the use of numerical algorithms are assumed to introduce random phase errors between model predictions of large-scale atmospheric disturbances and the true state of the atmosphere. An attempt is made to prevent excessive growth of these errors by updating the model predictions with error-contaminated observations, available either periodically or aperiodically. It is demonstrated that the root mean square prediction error can be controlled by updating,, when the technique of weighted assimilation is employed. This well-known technique uses both predicted and observed values of an atmospheric variable to form an estimate of the true state. In general, this estimate is a better data source than direct replacement by an observed value. However, the results show that when observation errors are relatively small, weighted assimilation is essentially equivalent to replacement by the observed variable. When the model prediction errors are relatively small, significant improvement over replacement by the observed value is attained. These results are displayed for various model and observation errors and for different length scales of the wave disturbance.

A critique of the present results and inherent difficulties that are met in application to numerical weather prediction are discussed.

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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

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

A dynamical/statistical approach to initialization that is compatible with the dynamics of potential vorticity conservation is proposed. This approach consists of combining weighted assimilation, which minimizes the analysis error by means of linear regression, with a dynamical constraint imposed by this conservation principle. As a consequence, the initial analysis is shown to be optimal and dynamically compatible with the forecast model used in the present study.

Two situations that contribute to error growth in numerical prediction models are considered. 1) differences in the phase propagation speed of the model disturbance relative to that of the true or control state and 2) distortion of the initial error field by nonlinear wave interactions. In each case results obtained with the proposed dynamical/statistical initialization are compared with results from weighted assimilation using uncorrelated observational errors and from initialization by direct use of error-contaminated observations. These comparisons demonstrate the theoretical advantage of using an initialization scheme that is compatible with the model dynamics. However, it is pointed out that practical aspects involving additional computations and use of data from mixed observing systems have not been taken into account.

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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 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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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

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.

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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

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.

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