Transilient Turbulence Theory. Part III: Bulk Dispersion Rate and Numerical Stability

Roland B. Stull Boundary Layer Research Team, Department of Meteorology, University of Wisconsin, Madison, WI 53706

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Abstract

Even though a continuum of mixing parameters γ(t, z, ζ) is used in transilient turbulence theory to describe the effects of many superimposed eddy sizes (ζ) on the mean field at height z, the overall bulk dispersive rate at any height can be measured by one number, N2(z). By utilizing second moment measures of dispersion, it is shown theoretically, for various special cases that the variance of tract position, σs2, is given by σz2(z, t = N2(z)t, where N2(z) = ∫ ζ2γ(z, ζ)dζ. An analogous expression for N2(z) is derived for the discrete version of transilient theory, as can be used for grid point models. These bulk dispersive rates can easily be compared to eddy diffusivity, K(z), because the variance of tracer position for K theory is known to be σs2(z, t) = 2K(z)t.

The discrete version of transilient turbulence theory is shown to be absolutely numerically stable, regardless of the timestep size or the grid point spacing. In addition, it is shown how the values of the discrete transilient coefficients are determined by two factors: 1) the physics governing the turbulence mixing, and 2) the nature of the discretization (i.e., size of timestep and grid spacing. Thus, it is possible to employ transilient turbulence theory for both the diffusive and boundary layer parameterizations in a large-scale numerical forecast model that, by operational necessity, has coarse grid spacing and large timesteps.

Abstract

Even though a continuum of mixing parameters γ(t, z, ζ) is used in transilient turbulence theory to describe the effects of many superimposed eddy sizes (ζ) on the mean field at height z, the overall bulk dispersive rate at any height can be measured by one number, N2(z). By utilizing second moment measures of dispersion, it is shown theoretically, for various special cases that the variance of tract position, σs2, is given by σz2(z, t = N2(z)t, where N2(z) = ∫ ζ2γ(z, ζ)dζ. An analogous expression for N2(z) is derived for the discrete version of transilient theory, as can be used for grid point models. These bulk dispersive rates can easily be compared to eddy diffusivity, K(z), because the variance of tracer position for K theory is known to be σs2(z, t) = 2K(z)t.

The discrete version of transilient turbulence theory is shown to be absolutely numerically stable, regardless of the timestep size or the grid point spacing. In addition, it is shown how the values of the discrete transilient coefficients are determined by two factors: 1) the physics governing the turbulence mixing, and 2) the nature of the discretization (i.e., size of timestep and grid spacing. Thus, it is possible to employ transilient turbulence theory for both the diffusive and boundary layer parameterizations in a large-scale numerical forecast model that, by operational necessity, has coarse grid spacing and large timesteps.

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