# A Heuristic Theory of Large-Scale Turbulence and Long-Period Velocity Variations in Barotropic Flow

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- Tellus A
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## Abstract

By combining an equation for the conservation of zonal momentum with the eddy vorticity equation for nondivergent barotropic flow through a channel, it has been found possible to express the time variations of the longitude-averaged zonal wind in terms of the mean zonal wind itself and certain statistics of the eddy velocity field. In large-scale turbulence-where the characteristic dimension of the eddies is comparable with the entire “width” of the flow— these eddy statistics take on the properties of autocorrelation functions; thus, they depend primarily on the overall scale of the eddies and intensity of turbulence, rather than on details of the flow pattern. Changes in the scale and intensity of large-scale turbulence are related to changes in the mean zonal flow through the principles of conservation of total kinetic energy and variance of vorticity. Under a general hypothesis regarding the “stability” of the eddy statistics, the equations for the zonally-averaged motion comprise a complete system of nonlinear equations, which can be integrated by finite-sum and -difference methods. Accordingly, these equations constitute a mathematical theory of large-scale “free” turbulence, and provide the basis for a numerical method of predicting the zonally-averaged motion. With the assumption that the eddy scale is invariant, the equation governing the mean zonal wind takes a form related to that of the classical wave equation. Numerical integrations of this equation verify that extremes of mean zonal wind (e.g., pronounced jets) maintain their identity over long periods of time, and migrate either northward or southward (or in both directions at once) in a very regular way—a phenomenon that has been observed by Riehl bt al. (1950). The results also show that a strong jet in the mean westerlies tends to split into two weaker jets, which then move in opposite directions to produce the separated double jet structure typical of widespread blocking. The latter has been described as a characteristic denouement from strong zonal flow by Cressman (1950). The theoretical speed of migration of extremes in the mean zonal wind is found to agree closely with the observed speed in a case where published data are available. Since these features of long-period velocity variations—the most striking and regular discovered to date—have a characteristic time scale of a week or two, numerical prediction methods based on the present theory show some promise of aiding in the preparation of extendedrange forecasts.DOI: 10.1111/j.2153-3490.1957.tb01854.x

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