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Peter Lynch

Abstract

A filtering integration scheme based on a modification of the inversion integral for the Laplace transform (LT) is developed and implemented in a barotropic limited-area model. The LT scheme is compared to a conventional scheme and shown to simulate faithfully the low-frequency evolution of the atmosphere while eliminating high-frequency oscillations. The scheme is combined with a Lagrangian treatment of advection giving stable integrations for long time steps.

Simple perturbation experiments show that the LT model can absorb an imposed disturbance without data shock. It is superior in this respect to more conventional schemes and may prove useful for asynoptic data assimilation.

An alternative formulation of the filtering scheme using the Z transform is described. This techniques applied to a system of equations that have been discretized with respect to time. The Z-transform scheme is shown to behave in a manner similar to the Laplace-transform scheme.

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Peter Lynch

The numerical forecasts made in 1950 using the Electronic Numerical Integrator and Computer (ENIAC) paved the way for the remarkable advances that have been made over the past half-century in weather prediction and climate modeling. We review the circumstances in which the forecasts were made, the nature of the ENIAC machine, and the roles of the people involved. The basis for the forecasts was the barotropic vorticity equation, and the initial data were prepared manually from standard weather charts. Now that the NCEP-NCAR reanalysis extends back to 1948, the initial height fields for the forecasts are readily available in digital form. We describe the reconstruction of the forecasts using reanalyzed data.

Were the ENIAC forecasts any good? To date, no objective verification of the four integrations has been available. A comparison of the original and reconstructed forecasts shows them to be in good agreement. Quantitative verification of the forecasts yields surprising results. On the basis of root-mean-square errors, persistence beats the forecast in three of the four cases. The mean error, or bias, is smaller for persistence in all four cases. However, when SI scores are compared, all four forecasts show skill and three are substantially better than persistence.

A small modification of the prediction equation, in which the streamfunction replaces the geopotential height as the prognostic variable, can be implemented without any computational penalty. The modified equation yields slightly improved forecasts in all four cases.

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Peter Lynch

The wave solutions discovered by Rossby are of fundamental importance for atmospheric dynamics. The nonlinear interactions between these waves determine the primary characteristics of the energy spectrum. These interactions take place between triplets of waves known as “resonant triads” and, for a small amplitude, they are described by the three-wave equations. These same equations also govern the dynamics of a simple mechanical system, the elastic pendulum or swinging spring. This equivalence allows us to deduce properties, not otherwise evident, of resonant triads from the behavior of the mechanical system. In particular, the characteristic stepwise precession of the swing plane, so obvious from observation of the physical spring pendulum, is also found for the Rossby triads. This phenomenon has not been previously noted and is an example of the insight coming from the mathematical equivalence of the two systems. The implications of the precession for predictability of atmospheric motions are considered. The pattern of breakdown of unstable Rossby waves is very sensitive to unobservable details of the perturbations, making accurate prediction very difficult.

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Peter Lynch

Abstract

Analyzed data for numerical prediction can be effectively initialized by means of a digital filter. Computation time is reduced by using an optimal filter. The construction of optimal filters involves the solution of a nonlinear minimization problem using an iterative procedure. In this paper a simple filter based on the Dolph–Chebyshev window, which has properties similar to those of an optimal filter, is described. It is shown to be optimal for an appropriate choice of parameters. It has an explicit analytical expression and is easily implemented. Its effectiveness is demonstrated by application to Richardson’s forecast: the initial pressure tendency is reduced from 145 hPa per 6 h to −0.9 hPa per 6 h. Use of the filter is not restricted to initialization; it may also be applied as a weak constraint in four-dimensional data assimilation.

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Peter Lynch

Abstract

The horizontal wind field may be deduced from the vorticity and divergence by solving Poisson equations for the velocity potential and streamfunction or, more directly, by the solution of a single Poisson equally accurate.

If the domain is of limited extent, boundary conditions must be specified. It is sufficient to prescribe a single component of the boundary velocity. Methods which use both components overdetermine the solution and may not converge in general.

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Peter Lynch

Abstract

Tne partitioning of a global windfield into rotational and divergent components is unique. These components are orthogonal and imply a corresponding partitioning of the kinetic energy. For a limited domain the partitioning is neither unique nor (necessarily) orthogonal and depends on the boundary conditions. Several simple boundary conditions are examined and the resulting wind components derived. A natural partitioning into three mutually orthogonal components, the rotational, divergent and harmonic components is proposed. For a global domain the harmonic component vanishes reducing the partitioning to the usual form.

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Peter Lynch

Abstract

The Laplace transform technique of initialization is used to initialize the data for a barotropic forecasting model over a limited area. The initialization is successful in suppressing high-frequency oscillations during early forecast hours. It has negligible effect upon the resulting 24-hour forecast.

A variation of the linearization, wherein the Coriolis parameter is held constant, is investigated. It is found that the fields which result after a single nonlinear iteration of the modified scheme are almost identical to those resulting from the more general scheme.

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Peter Lynch

To elucidate his numerical technique and to examine the effectiveness of geostrophic initial winds, Lewis Fry Richardson carried out an idealized forecast using the linear shallow-water equations and simple analytical pressure and velocity fields. This barotropic forecast has been repeated and extended using a global numerical model, and the results are presented in this paper. Richardson's conclusions regarding the use of geostrophic winds as initial data are reconsidered.

An analysis of Richardson's data into normal modes shows that almost 85% of the energy is accounted for by a single eigenmode, the gravest symmetric rotational Hough mode, which travels westward with a period of about five days. This five-day wave has been detected in analyses of stratospheric data. It is striking that the fields chosen by Richardson on considerations of smoothness should so closely resemble a natural oscillation of the atmosphere.

The numerical model employed in this study uses an implicit differencing technique, which is stable for large time steps. The numerical instability that would have destroyed Richardson's barotropic forecast, had it been extended, is thereby circumvented. It is sometimes said that computational instability was the cause of the failure of Richardson's baroclinic forecast, for which he obtained a pressure tendency value two orders of magnitude too large. However, the initial tendency is independent of the time step (at least for the explicit scheme used by Richardson). In fact, the spurious tendency resulted from the presence of unrealistically large high-frequency gravity-wave components in the initial fields.

High-frequency oscillations are also found in the evolution starting from the idealized data in the barotropic forecast. They are shown to be due to the gravity-wave components of the initial data. These oscillations may be removed by a slight modification of the initial fields. This initialization is effected by means of a simple digital filtering technique, which is applicable not only to the linear equations used here but also to a general nonlinear system.

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Xiang-Yu Huang and Peter Lynch

Abstract

A digital-filtering initialization scheme, which includes the effects of diabatic processes, has been formulated. This scheme gives a lower noise level in the forecast and a better organized initial pressure-tendency field than for the corresponding adiabatic initialization. The implementation of the scheme is very easy, requiring only the calculation of the filter coefficients and minor adjustments to the model code.

The computational expense of the digital-filtering initialization is directly proportional to the length of the time span over which the filter is applied. By a careful choice of filter weights, based on optimal filter theory, the span of the filter can be reduced by a factor of 2 or 3, with a corresponding increase in efficiency.

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David M. Schultz and Peter Lynch
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