The Use of Linear Filtering as a Parameterization of Atmospheric Diffusion

A simple linear filter is adapted for use in numerical models of the large-scale circulation to act in place of an explicit horizontal diffusion term in the equations. The filter can be shown to be ideally suited for this purpose in the sense that it can be made increasingly scale-dependent as the order of the filter is increased. The one-dimensional filter of order n is constructed from n three-point symmetrical operators and involves 2n/1 grid points. It is capable of eliminating two-grid-interval waves completely, yet allowing little or no damping of longer waves. In one space dimension, the use of the n = 1 order filter can be shown to be equivalent to the incorporation of a one-dimensional Fickian diffusion term in the differential equation. For any order n, the use of the one-dimensional filter is equivalent to the incorporation of a one-dimensional linear diffusion of order 2n. It is therefore apparent that as n increases, the ability of the filter to discriminate in its response to short- and long-wavelength components becomes increasingly sensitive. The damping properties of linear diffusion are examined by means of the two-dimensional, horizontal Fickian equation and compared with the response of the order n = 8 filter in two dimensions. With typical values for the space and time increments and with K=4×109 cm2 sec−1, K−2> reduces two-grid-length waves by a factor e in 15 hr, four-grid-length waves in 30 hr, and ten-grid-length waves in 147 hr (6.1 days). The filter, applied each time step, removes the two-grid-length waves immediately and reduces three-grid-length waves by a factor e in 0.8 hr. This compares with ∼20 hr for K∇2. For four-grid-length waves, the effect of the filter is nearly the same as that of K∇2, but for longer waves the damping effect is drastically reduced. The filter requires 5 × 105 days to damp 10-grid-interval waves by a factor e and 3 × 1010 days for 20-grid-interval waves. Thus, waves shorter than four grid lengths are effectively eliminated, while longer waves are essentially unaffected, except in so far as nonlinear interactions among the longer waves produce a cascade to shorter wavelengths. If the grid length is properly chosen for the physical system being studied and the equations adequately model the large-scale dynamics, the filter will automatically handle the cascade to smaller scale and should represent the effect of viscous damping that takes place in the atmosphere. Some desirable properties of the filter are demonstrated by a one-dimensional example with no dynamics in which the order n = 8 filter is applied successively ten thousand times to grid-point data of the sea level pressure field.