FILTERED EQUATIONS AND FILTERING INTEGRATION SCHEMES

Peter Lynch

Meteorological Service

Dublin, Ireland

I still consider the elimination or dampening of noise to be the crucial problem in weather analysis and prediction. K.H. Hinkelmann, WMO Bulletin, 34(4), 279, Oct., 1985

1.   INTRODUCTION

The spectrum of atmospheric motions is vast, encompassing phenomena having periods ranging from seconds to millennia. The motions of interest to the forecaster have timescales of the order of a day, but the mathematical models used for numerical prediction are quite general, and describe a broader span of dynamical features than those of direct concern. For many purposes these higher frequency components can be regarded as noise contaminating the motions of meteorological interest. The elimination of this noise has been achieved by adjustment of the initial fields (a process called initialization) or by modification of the governing equations (called filtering the equations). There is a close relationship between these two approaches: the diagnostic constraints imposed to initialize the fields may also be used to replace prognostic components of the prediction system, and thus the constraints may be applied throughout the forecast. Filtered equation systems are discussed in § 2 below, and their relationship to normal mode initialization is considered.

As an alternative to modification of the equations, a numerical integration scheme may be employed having the property that it selectively eliminates or dampens elements of the solution which are considered to be noise, while simulating the meteorologically significant components accurately. A number of such filtering integration schemes are examined in § 3. Following that, § 4 introduces the theory of digital filters. Two applications of these filters are described in § 5, one to initialization and one to integration, and it is argued there that an integration scheme having a specified frequency response may be constructed using filter theory. The final section (§ 6) attempts to synthesize the ideas discussed in the preceeding parts, and raises some important unsolved problems.

1.1    Signals and Noise: the Spectrum of Atmospheric Motions

The natural oscillations of the atmosphere fall into two groups (see, for example, Holton, 1975, § 2.4). The solutions of meteorological interest have low frequencies and are close to geostrophic balance. They are called rotational modes since their vorticity is greater than their divergence; if divergence is ignored, these modes reduce to the Rossby-Haurwitz waves. There are also very fast gravity-inertia wave solutions, with phase speeds of hundreds of metres per second and large divergence. For typical conditions of large scale atmospheric flow (when the Rossby and Froude numbers are small) the two types of motion are clearly separated and interactions between them are weak. The high frequency gravity-inertia waves may be locally significant in the vicinity of steep orography, where there is strong thermal forcing or where very rapid changes are occurring; but overall they are of minor importance and may be regarded as undesirable noise.

Observations show that the atmospheric pressure and wind fields in regions not too close to the equator are close to a state of geostrophic balance and the flow is quasi-nondivergent. The bulk of the energy is contained in the slow rotational motions and the amplitude of the high frequency components is small. The existence of this geostrophic balance is a perennial source of interest; it is a consequence of the forcing mechanisms and dominant modes of hydrodynamic instability and of the manner in which energy is dispersed and dissipated in the atmosphere. The gravity-inertia waves are instrumental in the process by which the balance is maintained, but the nature of the sources of energy ensures that the low frequency components predominate in the large scale flow. The atmospheric balance is subtle, and difficult to specify precisely. It is delicate in that minor perturbations may disrupt it but robust in that local imbalance tends to be rapidly removed through radiation of gravity-inertia waves in a process known as geostrophic adjustment.

When the primitive equations are used for numerical weather prediction the forecast usually contains spurious large amplitude high frequency oscillations. These result from anomalously large gravity-inertia waves which occur because the balance between the mass and velocity fields is not reflected faithfully in the numerically analysed fields. The problem is that small errors in the initial fields of pressure and wind can lead to large deviations from a balanced state. As a result, high frequency oscillations of large amplitude are engendered, and these may persist for a considerable time unless strong dissipative processes are incorporated in the forecast model. It was the presence of such imbalance in the initial fields which gave rise to the totally unrealistic pressure tendency of 145 hPa/6h obtained by Lewis Fry Richardson in the first-ever objective numerical weather forecast.

1.2   The Primitive Notion of Filtering

The concept of filtering has a rôle in virtually every field of study, from topology to theology, seismology to sociology. The process of filtering involves the selection of those components of an assemblage having some particular property, and the removal or elimination of those components which lack it. A filter is any device or contrivance designed to carry out such a selection. It may be represented by a simple system diagram, having an input with both desired and undesired components, and an output comprising only the former: