ON THE PARAMETERIZATION OF EVAPORATION OF RAINDROPS

BELOW CLOUD BASE

AXEL SEIFERT

German Weather Service, Offenbach, Germany

1. INTRODUCTION

Evaporation of raindrops can lead to a significant re-duction of the surface precipitation compared to theprecipitation flux at cloud base. A precise parame-terization of this process is therefore an importantissue in quantitative precipitation forecasts. Evapo-ration of raindrops provides also an important linkbetween cloud microphysics and cloud dynamics.In mesoscale convective systems the evaporation ofraindrops determines the strength of the cold pooland subsequently the organization and life time ofconvective systems. For boundary layer clouds ob-servations show that often more than 80 % of thedrizzle drops evaporate below cloud base and theassociated cooling of the boundary layer has an im-portant impact on the macroscopic cloud structure.In cloud-resolving numerical models the evapora-tion of raindrops received surprisingly little atten-tion up to now. Usually the parameterizations fol-low Kessler’s assumptions, e.g. using an exponentialdrop size distribution combined with a power lawrelation for the fall speed. For convection-resolvingmodels these assumptions might be insufficient asthe variability of the size distribution in convectivesituations is much larger than in stratiform rain. Us-ing a multi-moment approach, as it is done in somecurrent research models, does not a-priori solve thisproblem but poses additional ones, like the questionof size effects of evaporation. Does evaporation in-crease or decrease the mean size of the raindrops?To shed some light on theses issues the process ofevaporation of raindrops below cloud base is in-vestigated by numerical simulations using a ide-alized one-dimensional rainshaft model with high-

Corresponding author’s address: Dr. Axel Seifert,Deutscher Wetterdienst, GB Forschung und Entwick-lung, Kaiserleistr. 42, 63067 Offenbach, Germany. E-mail:axel.seifert@dwd.de

resolution bin microphysics. The simulations reveala high variability of the shape of the raindrop sizedistributions which has important implications forthe efficiency of evaporation below cloud base. Anew parameterization of the shape of the raindropsize distribution as a function of the mean vol-ume diameter is suggested and applied in a two-moment microphysical scheme. In addition, the ef-fect of evaporation on the number concentration ofraindrops is parameterized.

2. THE RAINDROP SIZE DISTRIBUTION

A crucial step for all parameterizations of evapora-tion of raindrops is the choice of an appropriate rain-drop size distribution (RSD). Here a gamma distri-bution given by

n(D) = N0Dµ exp(−λD) (1)

is assumed wheren(D) is the drop size distributionin m−4, D is the drop diameter in m,N0 the inter-cept parameter with units m−(µ+4), λ the slope inm−1 andµ the dimensionless shape parameter.The variability of the shape parameterµ and its pa-rameterization has been the focus of many investiga-tions (Ulbrich 1983; Testud et al. 2001, and others).Recently Zhang et al. (2001) suggested the empiri-cal relationship

λ = 0.0365µ2 + 0.735µ + 1.935. (2)

between the shape parameterµ and the slopeλbased on disdrometer measurements in Florida (seealso Brandes et al. 2003; Zhang et al. 2003; Brandeset al. 2007). Using a simple rainshaft model Seifert(2005, S05 hereafter) showed that thisµ-λ-relationis probably a result of gravitational sorting, colli-sion/coalescence and collisional breakup.

Figure 1: Shape parameterµ as a function of the slopeλ (left) or mean volume diameterDm (right). Thedashed line is theµ-λ-relation of Zhang et al. (2001), the dotted line is theµ-Dm-relation of Milbrandt andYau (2005a). The dashed-dotted line is Eq. (3).

3. NUMERICAL EXPERIMENTS USING A1D RAINSHAFT MODEL

To investigate the variability of the shape of the RSDa simplified model of a non-stationary precipitationevent is used. As in S05 a homogeneous initial cloudis assumed between a cloud base heightzbase and acloud topztop with a initial cloud droplet distribu-tion given by

f0(x, z) =

{

Ae−Bx, ztop ≥ z > zbase

0, else.

The parameters A and B are calculated from theinitial liquid water contentL0 and the initial meanvolume radiusr0. Below cloud base a constanttemperatureTpbl and relative humidityRHpbl isassumed. In the following all simulations assumeTpbl = 20

◦C. This 1D rainshaft model is numeri-cally solved using 130 spectral bins with a verticalgrid spacing of50 m and a timestep of 1 s.

Diagnostic relations for µ

From the bin microphysics simulation the shape ofthe RSD can be derived. Fig. 1 shows a scatter plotof the shape parameterµ for various initial condi-tions defined byL0, r0, zbase, ztop andRHpbl. Thesame data is also shown as a function of the mean

volume diameterDm. The mean volume diame-ter has several advantages. First, it makes it easierto identify the breakup equilibrium regime aroundDm = Deq, second, it makes it possible to dis-tinguish RSDs with smaller mean diameters fromRSDs with larger mean diameters that have the sameλ. For an individual ’convective’ rain event the pre-cipitation at the ground starts with large drops andhighµ, thenµ reaches a minimum during the precip-itation maximum, maybe being close to equilibriumin strong precipitation (see Fig. 1 of S05) and thenthe raindrops become smaller andµ might becomelarger again or not depending on the relative hu-midity/evaporation and the rainwater content. Thisbehavior can be roughly seen in Fig. 1. Especiallywhen evaporation is taken into account the scatterin the µ-λ- or µ-Dm-relation becomes very large.Therefore any diagnostic parameterization ofµ canonly be a very crude approximation of the com-plicated time evolution of the RSD. A formulationwhich proved to be useful is

µ =

{

6 tanh[c1 ∆D]2 + 1, Dm ≤ Deq

30 tanh[c2 ∆D]2 + 1, Dm > Deq

(3)

where∆D = Dm −Deq, c1 = 4 mm−1 andc2 = 1mm−1. For Dm ≈ Deq = 1.1 mm this parame-terization will give low values ofµ assuming that

the RSD is close to the equilibrium distribution, forsmall mean diameters larger values ofµ will occurbut the parameterization is arbitrarily constrained toa maximum value of 7. For large mean volume di-ameters gravitational sorting dominates which canproduce very narrow size distribution and thereforevery high values ofµ.

Parameterization of evaporation in a two-momentbulk model

Assuming a gamma distribution the parameteriza-tion of the bulk evaporation rate of the rainwatercontent is quite straightforward, unfortunately thisis not the case for the number concentration. A prag-matic approach to the problem is

∂Nr

∂t

∣

∣

∣

∣

eva= γ

Nr

Lr

∂Lr

∂t

∣

∣

∣

∣

eva(4)

where the coefficientγ nicely hides all unknown de-tails. Khairoutdinov and Kogan (2000), Milbrandtand Yau (2005) as well as Morrison and Grabowski(2007) assumeγ = 1, i.e. that the mean vol-ume diameter does not change during evaporation.Khairoutdinov and Kogan (2000) show some re-sults for drizzling stratocumulus that support this as-sumption in their case (their Fig. 2), but in generala compelling physical explanation ofγ ≈ 1 can-not easily be found. ForDm ≫ 80 µm andµ ≫ 1,for example, one would expectγ = 0, since within asmall time interval evaporation would only make theraindrops smaller without evaporating any of them.Therefore the assumption ofγ = 1, which is equiv-alent to a constantDm during evaporation, is proba-bly only a good one for broad DSDs and/or drizzle,but not for strong convective rain.Using Eq. (3) for the shape parameter, the evaluationof the 1D bin model suggests a parameterization forγ as

γ =Deq

Dmexp(−0.2µ) (5)

with Deq = 1.1 mm (see Seifert 2008, for more de-tails). Although this parameterization points towarda delicate dependency of the size effect of evapora-tion on the shape parameter of the DSD, it is only acrude first attempt to model this complicated behav-ior.

Results of the two-moment bulk model

The parameterizations of the shape parameterµ andthe evaporation coefficientγ which have been in-troduced in the previous sections can be combinedwith the warm rain scheme of Seifert and Beheng(2001) and Seifert and Beheng (2006). This schemecan now be compared with the results of the binscheme for the idealized rain event simulated by the1D rainshaft model. Fig. 2 shows the time evolu-tion of the surface rainrate, the mean volume di-ameter and the shape parameter for an initial cloudwith L0 = 7 g/m3, r0 = 13 µm, zbase = 3 km,ztop = 8 km (this case differs from Fig. 1 of S05only in the cloud base height). Again, we can seethe time evolution a typical strong ’convective’ rainevent in three stages: During the first stage only thelargest drops arrive at the surface, the second stage ischaracterized by strong precipitation with the DSDbeing close to breakup equilibrium, i.e. a broad sizedistribution, during the last stage the smaller dropsdominate like in the stratiform region of a convec-tive system. These three stages can be distinguishedby the mean volume diameter withDm > Deq dur-ing stage one,Dm ≈ Deq during the equilibriumstage andDm < Deq during the final stratiform-likeperiod. Over the complete event the mean volumediameter decreases monotonically, although duringthe equilibrium stageDm is almost constant. Theshape parameterµ reaches its minimum in the equi-librium stage, and decreases (increases) during thefirst (last) stage. The two-moment parameterizationis able to reproduce this behavior qualitatively, andfor this individual event also the quantitative agree-ment is very good, except for the fact theµ startsto increase again too early and reaches only a valueof 7 while the bin model simulates much larger val-ues at the end of the event. In the evaporating caseFig. 2b with a relative humidity below cloud base of70 % the maximum rainrate is reduced from about150 mm/h to 80 mm/h, and overall about 55 % of theprecipitation evaporates before reaching the ground.Compared to the simulation without evaporation themean volume diameter drops off more rapidly dur-ing the decaying third stage of the event and theshape parameter does not increase to high valuesbut remains low reaching only a value of 3 at theend of the event. The two-moment scheme captures

a)RHpbl = 100 % b) RHpbl = 70 %

Figure 2: Time evolution of the rainrateR (blue), the shape parameterµ (red, plotted is0.1µ) and the meanvolume diameterDm (green) for a strong rain event withL0 = 7 g/m3, r̄c = 13 µm, zbase = 3 km, ztop = 8km andRHpbl = 100 %, i.e. no evaporation, (a) as well asRHpbl = 70 % (b). Solid lines are the results ofthe bin model, dashed lines represent the bulk model.

these differences to the non-evaporating case quitewell, although the increase ofµ in the final stage ofthe event is now too strong. This could only be im-proved by makingµ a function of relative humidity.As it is now, theµ-Dm-relation Eq. (3) tries to makea compromise between non-evaporating and heavilyevaporating situations. More details and a compari-son with other assumptions can be found elsewhere(Seifert 2008).

4. SUMMARY AND CONCLUSIONS

An improved two-moment parameterization of rain-drop evaporation below cloud base has been sug-gested and tested against a spectral bin referencemodel. It has been shown that an accurate param-eterization of evaporation is a challenging prob-lem and the suggested relations can only be afirst step towards a better understanding of thiscomplicated process. The complications arise dueto the high variability of the raindrop size distri-bution, especially of the shape parameterµ, andthe non-linear feedbacks between evaporation andbreakup/coalescence as already shown by Hu andSrivastava (1995).

The suggested diagnostic relations for the shape pa-rameterµ and the evaporation parameterγ are stillvery uncertain for several reasons:

• The idealized rainfall simulation is probablynot realistic enough, although is reproducesthe observations of Zhang et al. (2001) in astatistical sense.

• Both, theµ-Dm-relation as well as the pa-rameterization of the evaporation coefficientγ, rely heavily on the ability of the spectralbin microphysics model to simulate the co-alescence/breakup process in a realistic way.Especially for collisional breakup the uncer-tainties in the kernels are still significant. It isquite possible that this leads to an overestima-tion of evaporation in the spectral bin model,especially in moderate to heavy rain.

• In light precipitation the bin model allowshigh µ-values atDm=Deq, i.e. the systemis not in coalescence/breakup equilibrium al-though the mean volume diameter is identi-cal to the equilibrium diameter. The parame-terization always predictsµ=1 for Dm=Deq,

since it would assume coalescence/breakupequilibrium.

• The dependency of the DSD on relative hu-midity is poorly understood and has, to theauthor’s knowledge, not yet been investigatedbased on observations.

Overall this study shows once more that the key toan improved understanding and parameterization ofthe warm rain processes are reliable measurementof the drop size distribution which would be neces-sary to validate - of falsify - the theoretical models.

Acknowledgments The author thanks Bjorn Stevens,Klaus D. Beheng, Ed Brandes and Uli Blahak forinteresting and helpful discussions.

References

Brandes, E. A., K. Ikeda, G. Zhang, M. Schönhuber, andR. M. Rasmussen, 2007: A statistical and physicaldescription of hydrometeor distributions in Coloradosnowstorms using a video disdrometer.J. Appl. Met.,46, 634–650.

Brandes, E. A., G. Zhang, and J. Vivekanandan, 2003:An evaluation of a drop distribution-based polarimet-ric radar rainfall estimator.J. Appl. Met., 42, 652–660.

Hu, Z. and R. C. Srivastava, 1995: Evolution of raindropsize distribution by coalescence, breakup, and evap-oration: Theory and observations.J. Atmos. Sci., 52,1761–1783.

Khairoutdinov, M. and Y. Kogan, 2000: A new cloudphysics parameterization in a large-eddy simulationmodel of marine stratocumulus.Mon. Wea. Rev., 128,229–243.

Milbrandt, J. and M. Yau, 2005: A multimoment bulkmicrophysics parameterization. Part II: A proposedthree-moment closure and scheme description.J. At-mos. Sci., 62, 3065–3081.

Morrison, H. and W. Grabowski, 2007: Comparison ofbulk and bin warm rain microphysics models using akinematic framework.J. Atmos. Sci., accepted.

Seifert, A., 2005: A note on the shape-slope rela-tion of the drop size distribution in convective rain.J. Appl. Met., 44, 1146–1151.

— 2008: On the parameterization of evaporation of rain-drops as simulated by a one-dimensional rainshaftmodel.J. Atmos. Sci., accepted.

Seifert, A. and K. Beheng, 2006: A two-momentcloud microphysics parameterization for mixed-phaseclouds. Part I: Model description.Meteorol. At-mos. Phys., 92, 45–66.

Seifert, A. and K. D. Beheng, 2001: A double-momentparameterization for simulating autoconversion, ac-cretion and selfcollection.Atmos. Res., 59-60, 265–281.

Testud, J., S. Oury, R. A. Black, P. Amayenc, and X. Dou,2001: The concept of ”normalized” distribution to de-scribe raindrop spectra: A tool for cloud physics andcloud remote sensing.J. Appl. Met., 40, 1118–1140.

Ulbrich, C. W., 1983: Natural variations in the an-alytical form of the raindrop size distribution.J. Clim. Appl. Met., 22, 1764–1775.

Zhang, G., J. Vivekanandan, and E. Brandes, 2003: Theshape-slope relation in observed gamma raindrop sizedistributions: Statistical error or useful information?J. Atmos. Ocean. Tech., 20, 1106–1119.

Zhang, G., J. Vivekanandan, and E. A. Brandes, 2001: Amethod for estimating rain rate and drop size distri-bution from polarimetric radar measurements.IEEETrans. Geosci. Remote Sens., 39, 830–841.