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<br />28 <br /> <br />than the second order Crowley technique used for state parameters. This up- <br /> <br />stream difference technique can be represented as <br /> <br />N At <br />fN+1 = fN _ ~U - IU l)(f~l - fz) 2Az <br />z z ~ z z <br /> <br /> + (UZ + IUzP(f~ - f~_l) t:Z] (eq. 35) <br />where: fN = the number density of particles of size class <br /> z f at height z at time step N <br /> U = V -v = updraft - terminal velocity of <br /> z z z particles of size f at height z <br /> at time N <br /> At = integration time step <br /> Az = vertical spatial grid spacing <br /> <br />The sum and difference terms involving absolute values insure that a one <br /> <br />sided spatial derivative of fN is taken only from the lIupstreamll direction. The <br />z <br /> <br />equation can thus be conceptualized as requiring the advective change of particle <br /> <br />density at a size class f and vertical location Z during a finite time step At to be the <br /> <br />sum of first order estimates of the flux into location Z from upstream, less the flux <br /> <br />out from Z to downstream locations during time step At. The scheme requin!s that <br /> <br />~At < 1 so that material cannot be advected further than the grid spacing ~Z in a <br />"-J.z - <br /> <br />single time step b"t 0 So long as this criteria is enforced, the scheme will never <br /> <br />generate negative number densities of hydrometeors at any node 0 The upstrE!am <br /> <br />difference scheme is also used for particle mass advection (i.e., condensation, sub- <br />