# Presentation: 490071

## Formulation of radar reflectivity in two moment cloud parameterizations 

** ****Comparison of predicted radar reflectivity (*****Z*****) with observations, such as e.g. from the W-band ARM Cloud Radar (WACR), is an essential part of verifying the fidelity of cloud bulk microphysics parameterizations.  **
** ****The majority of cloud models use two-moment cloud parameterizations which do not predict radar reflectivity directly. The commonly used method to obtain the *****Z***** from the *****Z-R***** relationship is rather inaccurate. **
** ****The radar reflectivity may be precisely calculated from the drop size distributions (DSD); they however are unavailable in bulk models where microphysics is represented only by four moments of the DSD, such as cloud and rain mixing ratios and drop concentrations.**** **
** ****Recently we showed that approximation of DSD by a combination of two Gamma-type functions may rather accurately represent the higher moments of the drop spectra, including radar reflectivity. **
**The question is how to define the two-mode Gamma type function through the predictive variables available in the bulk model formulation?**
**  ****We simulated several cases of stratocumulus clouds observed during the ASTEX field experiment. **
**  ****The simulations represent cloud layers with different intensities of drizzle in the cloud (****Fig. 1****) and provided over 19,200 DSDs for  each case. **
**Fig 1. Mean and standard deviation of drop spectra parameters for light (LD), moderate (MD) and heavy (HD) drizzling cases.**** ***** ***

**Formulation of radar reflectivity in two moment cloud parameterizations**** **
** **
***Zena N. Kogan and Yefim L. Kogan, ***
-  Cooperative Institute for Mesoscale Meteorological Studies, The University of Oklahoma, Norman, OK

**Comparison of analytical fits**

**Motivation and Objective**

**Description of Data**

**Conclusions**

** ****The Gamma-type bimodal fits represent rain rates and radar reflectivities much more accurately than lognormal fits.**
** ****Radar reflectivity field can be represented reasonably well in the 2-moment bulk schemes, but most accurately in the 3-moment bulk microphysical parameterizations.**

**In the LD case the rain rate is rather well approximated by a unimodal *****L-fit*****, although the reflectivity is overestimated. For the HD case (****Fig 2****)**** the unimodal fits fail to capture contribution from the tail of the spectrum; thus, rain rate and radar reflectivity are significantly underestimated by either an *****L-fit***** (left), or a *****G-fit***** (right panels).**
**Fig 3**** ****shows that a) bimodal fits result in significantly smaller bias ****relative to the unimodal fits****, and b) the bimodal *****G-fits***** have a ****substantially reduced scatter and**** a much smaller error envelope (mean ± standard deviation) than *****L-fits*****.**

**Results**

**Fig. 2. Rain rate and radar reflectivity approximated by a unimodal fit.**

***Acknowledgments.***** ****This research was supported by the Office of Science (BER), U.S. Department of Energy, Grant No. DE-FG02-05ER64062 and by the Office of Naval Research, DOD.**

**where *****N*****,  ********** , **********. are parameters; ********(*****x*****) is the gamma function.**
**  **
**The 3 parameters defining each fit are expressed through the 0****th****, 1****st**** and 2****nd****  moments of the LES derived DSDs. The 4****th**** and 6****th**** moments of the fit are then compared with corresponding moments of the DSD from the LES dataset. Note that in Sc these moments represent drizzle flux and reflectivity.**
**  **
**Depending on drizzle intensity, drop spectra in Sc may exhibit one or two modes; the 1****st**** mode for cloud and the 2****nd**** for drizzle drops (*****r*****>25********). We thus consider two fits. The unimodal fit is defined by 3 parameters ****expressed through moments of the DSD integrated *****over the whole drop size range*****. **
**The bimodal fit is a sum of two fits, each defined by partial moments integrated in the 1****st**** fit ****over the***** cloud drop *****sizes****, and in the 2****nd**** ****over the***** drizzle drop *****sizes.***** ***

**where *****r******m***** is the modal radius, *****N***** concentration, and ********** logarithmic drop spectrum width.**
**The three parameter Gamma fit (*****G-fit*****) is:**

**Method**

**The three parameter lognormal fit (*****L-fit*****) is:**

***heavy drizzle case***

**Fig. 3. As Fig. 2, but using bimodal *****L-*****(*****left*****) and *****G-fits (right)*****. **

***L-fit***

***G-fit***

***L-fit***

***G-fit***

**Reflectivity in 2-moment**** ****parameterizations**

**Most CRM and NWP models use two-moment microphysics parameterizations and, therefore, only two variables are available for definition of a G-fit for. We show that the drizzle drop dispersion, can be approximated as a function of drizzle concentration and mixing ratio (Fig. 4).**
**Assuming a fixed cloud drop dispersion and approximating drizzle drop dispersion as in Fig. 4, we can reduce the number of parameters defining a *****G-fit***** to four microphysical variables predicted in the two-moment schemes.**
**Fig. 5 compares the radar reflectivity field *****Z***** in a simulation of a drizzling Sc: a) benchmark field defined by the LES predicted DSDs, b) *****Z***** parameterized by 3-parameter  *****G-fit*****; c)**** *****Z***** parameterized by the 2-parameter Gamma fit with approximated drizzle drop dispersion**. **Panels *****d-e***** show fields of total and drizzle liquid water content illustrating the extent of correlation between them and *****Z*****. Panel *****f***** shows much better, but still inferior relation of *****Z***** to the drizzle rate.**

**Fig. 4****. D*****rizzle***** drops dispersion ******** as a function of drizzle drops concentration stratified by the value of *****Q******r***

**Fig. 5****. ****The isolines of Sc cloud parameters: a) benchmark reflectivity; b)  *****Z***** defined by the 3-parameter *****G-fit*****; c)  *****Z***** defined by the 2-parameter *****G-fit*** **with parameterized dispersion d) total liquid water content *****Q******t*****; e) drizzle water content *****Q******r*****; f) drizzle rate *****R*****.**** **