By Jerry M. Straka

This e-book focuses in particular on bin and bulk parameterizations for the prediction of cloud and precipitation at a variety of scales - the cloud scale, mesoscale, synoptic scale, and the worldwide weather scale. It presents a history to the elemental ideas of parameterization physics, together with techniques eager about the construction of clouds, ice debris, liquid water, snow mixture, graupel and hail. It offers complete derivations of the parameterizations, permitting readers to construct parameterization programs, with various degrees of complexity in response to details within the ebook. Architectures for a number of dynamical types are given, during which parameterizations shape an important device for investigating huge non-linear numerical platforms. version codes can be found on-line at www.cambridge.org/9780521883382. Written for researchers and complex scholars of cloud and precipitation microphysics, this publication can be a necessary reference for all atmospheric scientists inquisitive about types of numerical climate prediction.

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3 Number-concentration-weighted mean diameter The number-concentration-weighted mean diameter is a simple measure of diameter. 27), DNx Tx ¼ Dnx Àðvx þ 1Þ ; Àðvx Þ ð2:55Þ and lastly for the negative exponential distribution, DNx Tx ¼ Dnx Àð2Þ ¼ Dnx : Àð1Þ ð2:56Þ Next, there is a need to find a relation to diagnose Dn. As will be shown later, Dn can be predicted from the first and second moment in terms of mass. The method to diagnose Dn is derived later. 5 Mean-volume diameter The mean-volume diameter can be shown to be equal to the following for any distribution, DMVx ¼ 6rQx prx NTx 1=3 ; ð2:64Þ where r is the density of air, Qx is the mixing ratio of the hydrometeor species, rx is the density of the hydrometeor species, and NTx is the number concentration of the hydrometeor species.

LnðDx =Dnx Þ 2 Dx p : d Dnx 2sx ð2:138Þ ! 6 Log-normal distribution 43 By letting y ¼ ln(u), u ¼ exp(y), du/u ¼ dy, so DNx T Dnx ¼p 2psx 1 ð expðyÞ exp 1 ! y 2 p dy; 2sx ð2:140Þ where the limits of the integral change as u approaches zero from positive values, and ln(u) approaches negative infinity. Likewise, for the upper limit, as u approaches positive infinity, ln(u) approaches positive infinity. 143) results in 1 Ð Deff ¼ D2x exp 0 1 Ð Dx exp 0 ! ½lnðDx =Dnx Þ2 dDx 2s2x ! : ½lnðDx =Dnx Þ2 dDx 2s2x ð2:144Þ Dividing by Dnx gives 1 Ð Dx Dnx Deff ¼ Dnx 0 1 Ð Dx 0 2 exp exp Dnx ½lnðDx =Dnx Þ2 2s2x ½lnðDx =Dnx Þ2 2s2x d DDnxx : d DDnxx ð2:145Þ 44 Foundations of microphysical parameterizations Now letting u ¼ Dx/Dnx, 1 Ð Deff ¼ u2 exp Dnx 01 Ð u exp 0 ðln uÞ2 2s2x ðln uÞ2 2s2x du : ð2:146Þ du By letting y ¼ ln(u), u ¼ exp(y), du/u ¼ dy, so 1 Ð 1 Deff ¼ Dnx 1 Ð expð3yÞ exp expð2yÞ exp 1 y2 2s2x y2 2s2x dy ; ð2:147Þ dy where the limits of the integral change as stated above.

Distribution of number versus diameter for raindrops recorded at Ottawa, summer 1946. 0 mm hr 1. NDdD is the number of drops per cubic meter, of diameter between D and D þ dD. g. Feng and Beard 1991; Beard et al. 1991; Tokay and Beard 1996; Bringi et al. 1998; and Andsager et al. 1999). Nevertheless, for diameters greater than 1 mm, drops become increasingly oblate with size. Raindrops generally fall with their minor axis oriented in the vertical, though a rare few drops might be temporarily elongated vertically, possibly because of oscillations, collisions, or both.