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Relationship between Precipitation Rates at the Ground and AloftÐA Modeling Study

NIKOLAI DOTZEK AND THORSTEN FEHR Institut fuÈr Physik der AtmosphaÈre, DLR, Oberpfaffenhofen, Germany

(Manuscript received 17 June 2002, in ®nal form 1 January 2003)

ABSTRACT Two cloud-resolving mesoscale models, the Karlsruhe Atmospheric Mesoscale Model (KAMM) and the ®fth- generation Pennsylvania State University±National Center for Atmospheric Research Mesoscale Model (MM5), were used to study re¯ectivity factor±rain-rate (Z±R) relationships and instantaneous and horizontally averaged pro®les of precipitation rate for convective storms of varying intensity. Simulations were conducted for idealized terrain. KAMM modeled a single cloud; MM5 was used to study split-storm development. Both models consistently con®rm analytical results from earlier studies: convective drafts and strati®cation of air density signi®cantly alter the local rain rate and, therefore, also any Z±R relation relying on conditions of stagnant air and sea level air density. Air density effects can be almost completely corrected for by a recently proposed algorithm, but effects of convective drafts remain. They can lead to upward precipitation mass ¯uxes of signi®cant magnitude and subsequent horizontal displacement of precipitation. Applicability of simple Z±R relations over complex terrain with distinct watershed boundaries will be strongly degraded by such convection effects on precipitation mass ¯uxes.

1. Introduction ␳ ␣ R ϭ w ϩ w 00 ␳q, (3) As recently quanti®ed by Dotzek and Beheng (2001) []t,00 ΂΃␳ and brie¯y summarized here in the appendix, standard re¯ectivity±rainfall (Z±R) relations of the form Z ϭ aRb with ␳q denoting hydrometeor mass per unit volume are signi®cantly altered in deep convective clouds by and w t,00 hydrometeor fall speed. The term f(␳) from convection effects on precipitation rate R, de®ned as Eq. (2) was chosen in its most widely used form with vertical mass ¯ux density of hydrometeors with diam- its best-®t value of ␣ ϭ 0.45 for terminal velocities, eter D and bulk density ␳ , that is, after Beard (1985). Even though Beard proposed ␣ ϭ h 0.42 for direct rain-rate computations, we have to stay ϱ ␲ with ␣ ϭ 0.45, because it enters the equations from the R ϭ ␳ n(D)D3 w dD. (1) 6 h ͵ s fall speed term. 0 By convention in radar , R is taken to be Not only is R subject to variations in particle number positive for precipitation falling to the ground, despite spectra n(D), but there is a much stronger dependence a negative w s. We will conform with that convention on the effective sedimentation velocity (Dotzek and Be- and show the negative of R de®ned in Eq. (3) in our heng 2001): graphs and Z±R and R±␳q relations. wst(D, w, ␳) ϭ w ϩ w (D, ␳) ϭ w ϩ f (␳)w t,00(D). (2) Lower air density aloft increases R because hydro- meteor fall speed depends on ␳ in a way that lower- Here, w denotes ambient vertical air velocity, w (D) t,00 density air exerts a smaller friction force on the falling is terminal fall speed of hydrometeors at sea level con- hydrometeors, allowing them to fall faster than at sea ditions, and effects of air density ␳ are covered by f(␳). level air density (e.g., Foote and du Toit 1969; Battan Therefore, aside from n(D), variations of R may result 1973). Dotzek et al. (2002) were able to show that, from up- and downdrafts and, secondarily, from vertical although being a secondary effect on the precipitation density strati®cation from its sea level value to small- ␳ 00 rate pro®le alone, this effect can have signi®cant con- er values aloft. After integration of Eq. (1), the precip- sequences for radar-based quantitative precipitation es- itation rate in bulk variables reads timation (QPE) at large radar ranges. Also, downdrafts increase R, whereas updrafts de- crease the precipitation rate. In the limit of updrafts Corresponding author address: Dr. Nikolai Dotzek, DLRÐInstitut fuÈr Physik der AtmosphaÈre, Oberpfaffenhofen, D-82234 Wessling, greater than the hydrometeor fall speed, R can become Germany. highly negative even for arbitrarily high hydrometeor E-mail: [email protected] contents by the action of strong updrafts. As noted by

᭧ 2003 American Meteorological Society

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FIG. 1. Conceptual view of precipitation with and without mean windV (z) plus convection. For a rain cell with R ϭ 0 at the ground, application of a Z±R relation implies accumulated

precipitation Pac right below the cloud (graph ``a''). In reality, hydrometeor trajectories at later

times will lead to Pac (graph ``b'').

Dotzek and Beheng (2001) and many other researchers 2002). Here, with R at the ground (where w ϭ 0) being before (e.g., Battan 1976; Aniol et al. 1980; Wilson and derived from Z measured by low-elevation radar scans, Brandes 1979; Zawadzki 1984; Austin 1987; Joss and the problem arises as to how to relate Z at a few kilo- Waldvogel 1990; Atlas et al. 1995, 1999; DoÈlling et al. meters above radar to an instantaneous rain rate at (or 1998; Jordan et al. 2000), this effect makes application below) the radar level. Even in the case of stratiform of standard Z±R relations for deep convective clouds clouds, vertical drafts, density strati®cation, and the pro- dif®cult or even questionable. Various Z±R relations, ®le of the horizontal wind may introduce considerable however, have been publishedÐfor instance, by Battan error to this nonlocal approach. At least, the effect of (1973) and Sauvageot (1992). air density strati®cation on terminal fall velocity and, Besides, deep moist convection also induces hori- hence, on Z±R relations of the form Z ϭ aRb can nearly zontal components and is usually superposed to strong be eliminated (Dotzek et al. 2002). vertically sheared mean horizontal air¯ow, as schemat- So, to supplement Dotzek and Beheng (2001) and ically outlined in Fig. 1. For a developing rain cell with- Dotzek et al. (2002), in our paper Z±R relations and in a radar scan volume but with no rain at the ground vertical pro®les of instantaneous (R) and horizontally yet (i.e., R 00 ϭ 0), application of a Z±R relation would averaged (͗R͘) precipitation rate are studied in more erroneously diagnose R 00 Ͼ 0 and also a precipitation detail with a cloud-resolving-model analysis. Despite accumulation Pac on the ground. In reality, rain will only the cloud microphysical simpli®cations inherent to Kes- reach the ground at later times and with a horizontal sler-type bulk cloud models (Kessler 1969), they offer displacement due to a combination of convection and the unique opportunity to compare consistent ®elds of transport by the mean wind pro®le. Although it is a hydrometeor content, vertical drafts, and air density minor effect over large areas with homogeneous orog- strati®cation. From these ®elds, R and Z can be com- raphy, this horizontal shift can become important over puted at every grid point to conduct a study that re¯ects complex terrain with small distances between neigh- effects of deep convective motions in a strati®ed at- boring river catchments. It is evident that the rain rate mosphere alone, unbiased by microphysical processes for a given value of Z in convective clouds changes in hardly accessible for either models or observations. space and time and that no unique Z±R relation exists The paper is organized as follows. Section 2 brie¯y in general [see the discussion given by Battan (1976), presents the two model simulations applied for our study Atlas et al. (1995), and Dotzek and Beheng (2001)]. with their initialization and procedure. In section 3, Practical problems of this kind in the context of QPE Karlsruhe Atmospheric Mesoscale Model (KAMM) re- are usually addressed by combining data from one sults for a single-cell cumulonimbus are discussed; sec- (Dotzek et al. 2002) or several radars (Gourley et al. tion 4 addresses the ®fth-generation Pennsylvania State

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University±National Center for Atmospheric Research TABLE 1. Vertical pro®les of virtual potential temperature ⌰␷ and Mesoscale Model (MM5) supercell and multicell storm RH up to 10 km AGL used for the KAMM simulation. simulations, respectively. Sections 5 and 6 present dis- z (km AGL) ⌰␷ (K) RH (%) cussion and conclusions. 10.0 333.19 4.89 9.5 326.34 6.60 9.0 320.47 8.23 2. Model setup 8.5 316.26 9.68 Two different three-dimensional nonhydrostatic me- 8.0 314.51 11.25 7.5 313.20 12.90 soscale models, KAMM (e.g., Adrian and Fiedler 1991; 7.0 311.81 13.51 Dotzek 1999) and MM5 (e.g., Grell et al. 1994), were 6.5 309.78 8.35 used to model individual storms over idealized orog- 6.0 307.58 23.28 raphy with 1-km horizontal grid size. Note that both 5.5 306.28 20.97 5.0 304.68 43.13 simulations were initially conducted to clarify scienti®c 4.5 303.96 39.57 problems other than the one considered here. Only after 4.5 302.43 57.73 the ®ndings by Dotzek and Beheng (2001), did it appear 3.5 300.11 73.75 fruitful to exploit the two simulations to determine the 3.0 298.11 93.06 in¯uence of deep convection on vertical pro®les of pre- 2.5 296.20 98.45 2.0 294.41 86.88 cipitation rate. In both case studies, model initialization 1.5 293.46 68.51 was performed with a single atmospheric sounding and 1.0 292.93 65.12 convection was arti®cially triggered by specifying local 0.5 293.03 63.48 0.0 294.30 61.87 perturbations of equivalent potential temperature ⌰e (Klemp and Wilhelmson 1978). The basic differences in both simulations are that (i) MM5 used a ¯at terrain while the KAMM orography had a 500-m-high bell-shaped mountain in the center of The model domain size was 64 km in both horizontal the model domain, (ii) KAMM modeled a single-shower directions x and y and 18 km in the vertical direction. cell whereas MM5 was used for a split-storm supercell A bell-shaped, 500-m-high mountain with a half-width simulation, and (iii) both model runs used separate ini- of 2 km was positioned in the center of the model do- tialization soundings. main. In comparison with an otherwise identical sim- ulation for ¯at terrain, this mountain led to an intensi- ®cation of the storm as it moved over it, but for eval- a. KAMM uation of the KAMM run in the current context this KAMM was applied in a substantially revised and result does not play a major role. Grid spacing of the extended version to simulate deep convection, including model was 1 km horizontally and from 10 m at ground a bulk microphysical cloud module predicting rain water level to about 100 m near the model top vertically. The basic state was a barotropic ¯ow of 10 m sϪ1 from west- ␳qr, cloud water ␳qc, and cloud ice ␳qi concentrations (Dotzek 1998, 1999). Aside from the prognostic quan- southwest, and the pro®les of temperature and humidity tities ␳q, the model also provides the ®elds of w, ␳, and were chosen to be similar to the synoptic setting de- scribed by Hannesen et al. (1998) in their radar analysis the terminal fall speed of hydrometeors w t,00. All hy- drometeor fall speeds in the model are subject to a var- of a small-supercell F1 (on the Fujita scale) in iation caused by density strati®cation according to Eq. southern Germany on 9 September 1995. Pro®les of Ϫ3 (3) with ␳ 00 ϭ 1.225 kg m and ␣ ϭ 0.45 (Beard 1985). virtual potential temperature ⌰␷ and relative humidity As outlined by Dotzek and Beheng (2001), for pre- RH up to 10-km altitude are shown in Table 1. Note cipitation KAMM has only one prognostic quantitiy, the continuous decrease of RH toward the ground below

␳qr. However, effects of mixed- and ice-phase precip- 2.5 km above ground level (AGL). With the barotropic itation could be included in KAMM according to the ¯ow in the current simulation, however, supercell for- simple approach by Tartaglione et al. (1996): below mation could not be expected (Weisman and Klemp freezing-level height, a terminal velocity law for water 1982). The composite sounding from Hannesen et al. drops is applied. Above the freezing level, between 0Њ (1998) revealed a convective available potential energy and Ϫ35ЊC, hydrometeor fall speeds follow from linear (CAPE) of 440 J kgϪ1 with a level of free convection interpolation with temperature between raindrop fall at about 1.2 km and a neutral buoyancy level at ap- speed and a constant terminal velocity w t,00 ϭϪ2.5 m proximately 7.3 km AGL, allowing for overshooting sϪ1, representative of a mixture of snow and lump grau- cloud tops of at most 8±9 km AGL. pel. For temperatures less than Ϫ35ЊC, only this as- Convection was initiated by a local 3-K boundary ymptotic ®xed value is assumed. As Tartaglione et al. layer ⌰e perturbation to the basic state with a moist and (1996) showed, storm dynamics are improved signi®- warm air mass centered at (x ϭ 10, y ϭ 25, z ϭ 1.5) cantly by this more realistic description of hydrometeor km, similar to the procedure of Klemp and Wilhelmson sedimentation. (1978). The perturbation was introduced to the system

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TABLE 2. Maximum total hydrometeor mixing ratios, highest values of up- and downdrafts, and storm strength S (Weisman and Klemp 1984) for the KAMM and MM5 simulations.

␳qtot,max wmax wmin Stage t (LST) (g mϪ3) (m sϪ1) (m sϪ1) S

FIG. 2. Projection of maximum re¯ectivities derived from the KAMM KAMM single-cell storm simulation onto the x, z plane in different (a) 1220 12.34 11.10 Ϫ1.97 0.37 stages of cumulonimbus development: (a) 1220 LST growth, (b) 1230 (b) 1230 15.67 5.78 Ϫ5.10 0.19 LST maturity, and (c) 1245 LST decay. Light gray shading shows (c) 1245 8.52 3.16 Ϫ9.75 0.11 Ն Ն cloud dimensions (Z 0dBZ); regions with Z 40 dBZ are dark MM5 gray. IC 1630 11.05 27.43 Ϫ12.27 0.53 SC 1730 10.18 37.73 Ϫ19.29 0.73 SC 1930 4.02 16.37 Ϫ11.38 0.32 after 1 h of spinup simulation, corresponding to 1200 MC 1730 8.90 28.33 Ϫ10.19 0.55 local standard time (LST). MC 1930 9.60 36.67 Ϫ20.49 0.71 Soon after initiation, a rapidly developing cumulus cloud appeared, moving east-northeastward with the mean ¯ow. Figure 2 shows a synthetic radar composite mulus Ensemble microphysics scheme (Tao and Simp- from the model study, a projection of maximum re¯ec- son 1993), with cloud water ␳qc, rain ␳qr, cloud ice ␳qi, tivity values Z in the volume data to a vertical x, z plane. snow ␳qs, and graupel ␳qg as prognostic variables, was Re¯ectivity factors were computed from the prognostic applied. bulk variables as outlined by Dotzek and Beheng (2001) The vertical pro®le for model initialization was a and the appendix, with identical formulas for KAMM composition of radiosonde, aircraft, and dropsonde data and MM5. Light gray shading denotes re¯ectivities representative of a supercell storm situation in southern above 0 dBZ, representative of cloud shape, and dark Germany on 21 July 1998. Resulting initial conditions gray areas correspond to high re¯ectivities of more than for the case were given by Fehr (2000) and are sum- 40 dBZ. The location of cloud initialization was chosen marized in Fig. 3, which shows the model initialization such that the storm would be in its mature stage when sounding. Large CAPE of 1343 J kgϪ1, a convective passing over the bell-shaped mountain in the center of inhibition value of Ϫ37 J kgϪ1, and a bulk Richardson the domain. At 1220 LST, Fig. 2 shows that the cloud top was at about 7 km AGL (location ``a'') and a core of high re¯ectivity has developed. Only 10 min later, heavy precipitation fell out of the now-mature cumu- lonimbus cloud with its top at 9 km AGL (location ``b''). The decaying stage with weakening precipitation and cloud transition to an ice-®lled anvil can be seen at location ``c'' from 1245 LST. The highest computed re¯ectivities in this storm were above 60 dBZ, initial instantaneous rain rate at the ground peaked at 420 mm hϪ1, and the largest precipitation accumulation at a sin- gle point was 34 mm. As given in Table 2, updrafts in this cloud hardly exceeded 11 m sϪ1 (at 1220 LST), downdrafts reached their peak of about Ϫ10msϪ1 at 1245 LST, and a weak gust front developed at later times. However, note that gust-front formation was mainly suppressed by the mountain orography: an iden- tical simulation run over ¯at terrain showed a very dis- tinct gust front with a boundary layer arc cloud moving away from the storm (Dotzek 1999). In general, ther- modynamic strati®cation, as shown in Table 1, was re- sponsible for enhancing precipitation-induced down- drafts. This cumulonimbus cell represents in many ways a typical central European heavy rain shower during the warm season. b. MM5

Adapted by Fehr (2000), MM5 was used for the split- FIG. 3. Model initialization sounding used for the MM5 split-storm storm simulation over ¯at terrain. The Goddard Cu- simulation (Fehr 2000).

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ed Ϫ25 m sϪ1. Storm strength (Weisman and Klemp 1984) w S ϭ max , (4) ͙2 ϫ CAPE also given in Table 2, shows that the supercell life cycle undergoes a maximum in the middle, whereas the mul- ticell storm is continuously growing in strength and size. However, the vertical drafts in the multicell are gen- erally shorter lived than those in the supercell storm. After 2 h of quasi steady state, the supercell rapidly decayed. Another example of a more vigorous left-¯ank- mover development is the simulation presented by Wil- helmson and Klemp (1981). Storm anvils reached up to 12 km AGL with a max- imum overshoot of roughly 2 km, so cloud tops peaked at approximately 14 km. In its most vigorous phase, from about 60 to 120 min simulation time, strong ver- FIG. 4. Evolution of MM5 multicell/supercell system shown by the total hydrometeor concentration at an altitude of 4.4 km AGL. Lo- tical vorticity on the order of 0.01 sϪ1 was built up and cation of the gust front is also shown; labels are minutes of simulation maintained within a mesocyclone and in the boundary time, starting at 1600 LST. layer as low as 1.0 km AGL, a characteristic of supercell storms. The simultaneous development of an S-shaped cloud geometry shown in the horizontal section of Fig. 4 was caused by the formation of a strong rear-¯ank number Rib of 25.5 support split-supercell development (Weisman and Klemp 1984). and a weaker forward-¯ank downdraft. This is also in The simulation was run with nested grids. The coarse accordance with the concept of supercell thunderstorms. grid covered 363 km ϫ 246 km with 3-km horizontal grid spacing, and the inner domain was 130 km ϫ 130 3. KAMM results: Single-cell storm km wide, with 1-km horizontal grid size. The high- resolution inner domain tracked the developing clouds. Figure 2 shows three characteristic stages of cloud There were 50 vertical levels between the ground and development for the single-cell storm. The radar re¯ec- approximately 20 kmÐ11 below 2 km and 12 above tivity factor computed for Fig. 2 represents the total the tropopause. After a short spinup time of 10 min, sum of Z values from any present hydrometeors, that convection was initialized by a large 8-K ⌰e pertur- is, Ztot ϭ Zc ϩ Zi ϩ Zr (appendix; Smith 1984; Dotzek bation at 1600 LST. Smaller values for the ⌰e pertur- and Beheng 2001). bation did not lead to a storm splitting in the early stages In Fig. 2 stage a, storm growth, a larger volume of of cumulonimbus development, probably caused by the cloud and precipitation experiences updrafts (see also short spinup time. The model was then run for 3.5 h, Table 2) and reduced rain rates than in stage b, cloud that is, up to 1930 LST. At this time storm anvils reached maturity, because here maximum up- and downdrafts Ϫ1 the boundary of the inner model domain. were nearly in equilibrium (wmax ϭ 5.78 m s , wmin ϭ The complete storm development at midlevels can be Ϫ5.10 m sϪ1). In stage c, storm decay, precipitation seen in Fig. 4, showing total hydrometeor mixing ratio rates tend to be enhanced because of more dominant at 4.4 km AGL and near-surface gust-front location, downdrafts. identi®ed by a 0.5-K drop in temperature. At 1630 LST, In analyzing all three stages (a±c), we will ®rst present 30 min after convective initialization, the storm con- Z±R and R±␳q relations with and without convective sisted of a single cell that split into a northern left-¯ank drafts. Then, scatterplots that give information on in- and a southern right-¯ank storm during the following dividual precipitation pro®les R will be shown before 30 min. The two cloud systems evolved as individual horizontally averaged vertical pro®les of ͗R͘ provide entities: the left-¯ank mover developed into a multicell information on mean properties of rain rates under the storm and the right-¯ank mover developed into a su- in¯uence of convection. percell storm (cf. Wilhelmson and Klemp 1981; Weis- man and Klemp 1984). a. Z±R and R±␳q relations As Table 2 and Fig. 4 show, storm development was supercell dominated in the ®rst 2 h. The maximum up- Analytical Z±R and R±␳q relations were given by draft velocity remained above 30 m sϪ1 during this pe- Dotzek and Beheng (2001), focusing mainly on stage b riod, reaching values close to 50 m sϪ1 (Fehr 2000) in Fig. 2. From the same regression analysis, we ®nd while maximum supercell downdraft velocities exceed- the following Z±R relations for KAMM's precipitation

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bulk hydrometeor density ␳qr given both without con- lations should approach a form dependent on the most vection and with modeled vertical drafts. dominant hydrometeor form. As a consequence, the ex- The equations give Z±R and R±␳q relations with their ponents for Z±R relations should be between the ana- coef®cients split into mean value and standard devia- lytical values for rain (1.54) and for snow/lump graupel tion, as derived from a least squares ®t to a scatterplot (1.75), and for the R±␳q relation between 1.14 and 1.00, from data at all model grid points with hydrometeors. respectively (appendix; Dotzek and Beheng 2001). By setting w ϭ 0 in computing R from Eq. (3) and Equations (6a)±(6c) indeed show such a tendency, again assuming ␳qr only consists of rain water, we obtain strongest for 1230 LST when the cloud top is highest. At the same time, mean values of Z±R prefactors de- Z ϭ (163 Ϯ 12)R1.54Ϯ0.01 and crease while their standard deviation slightly increases. R ϭ (23.02 Ϯ 1.06)(␳q)1.13Ϯ0.01 for 1220 LST, (5a) Including mixed-phase precipitation and convection, w ± 0 in Eq. (3) leads to Z ϭ (170 Ϯ 26)R1.55Ϯ0.02 and 1.75Ϯ0.24 R ϭ (22.38 Ϯ 2.21)(␳q)1.13Ϯ0.01 for 1230 LST, (5b) Z ϭ (715 Ϯ 495)R and and R ϭ (7.36 Ϯ 3.36)(␳q)0.89Ϯ0.12 for 1220 LST, (7a) Z ϭ (173 Ϯ 20)R1.56Ϯ0.01 and Z ϭ (197 Ϯ 94)R1.55Ϯ0.11 and R ϭ (21.63 Ϯ 1.61)(␳q)1.12Ϯ0.01 for 1245 LST. (5c) R ϭ (18.55 Ϯ 7.29)(␳q)1.12Ϯ0.10 for 1230 LST, (7b) Equations (5a)±(5c) only show small scatter stemming and from the density strati®cation of the atmosphere. Var- iation of mean values for prefactors and exponents of Z ϭ (112 Ϯ 49)R1.55Ϯ0.10 and Z±R and R±␳q relations is not signi®cant. Temporal var- R ϭ (28.17 Ϯ 5.80)(␳q)1.17Ϯ0.05 for 1245 LST. (7c) iations of the standard deviation, however, can be in- terpreted as being in¯uenced by the density dependence In¯uences from density strati®cation or mixed-phase of R according to Eq. (3); Fig. 2 shows that cloud top hydrometeor ensembles are indeed minor when com- and vertical extent of in-cloud precipitation are highest pared with the effect of vertical convective drafts, as is at 1230, lowest at 1220, and somewhere in between at obvious from Eqs. (7a)±(7c). Now, just by including the 1245 LST. In accord, density-induced scatter also peaks modeled vertical drafts, large scatter occurs. Also the at 1230 and is smallest at 1220 LST. mean values of prefactors and exponents undergo sig- For completeness, we stress that simple inversion of ni®cant changes. As already found by Dotzek and Be- the Z±R relations given in Eq. (5a) would lead to an heng (2001) for the data from 1230 LST, our present erroneous diagnosis of rainfall below the developing evaluation of all three cloud stages substantiates the shower cloudÐFig. 2 shows that at 1220 LST all pre- dominant in¯uence of convection on Z±R relations even cipitation is contained within the cloud or still very close more. Storm maturity at 1230 LST is closest to the to cloud base. This is an example illustrating the sche- relations with w ϭ 0. Up- and downdrafts, as well as matic of Fig. 1. their variances, have similar magnitudes there. We focus The same data as before, but including the Tartaglione here on the two other times, because they show the most et al. (1996) parameterization to treat ␳qr as mixed- drastic changes in their average Z±R and R±␳q relations. phase precipitation yield At 1220 LST, updrafts are strongest, cloud top is well above the freezing level (at about 3.0 km AGL), and Z ϭ (138 Ϯ 20)R1.59Ϯ0.03 and the high precipitation content only begins to fall out R ϭ (20.80 Ϯ 1.07)(␳q)1.12Ϯ0.01 for 1220 LST, (6a) from cloud base (Fig. 2). We can expect that precipi- tation rates are strongly reduced because of the updrafts, Z ϭ (121 Ϯ 40)R1.64Ϯ0.08 and which are most prominent during cloud growth. As soon R ϭ (18.98 Ϯ 2.60)(␳q)1.11Ϯ0.03 for 1230 LST, (6b) as precipitation-cooled downdrafts dominate cloud dy- namics, rain rates should be enhanced. Equation (7a) and shows very strong variation in the prefactors of both Z± Z ϭ (133 Ϯ 32)R1.63Ϯ0.06 and R and R±␳q relations. For the latter, the prefactor is reduced by almost 65% and the exponent's variation R ϭ (20.37 Ϯ 1.21)(␳q)1.12Ϯ0.02 for 1245 LST. (6c) encompasses 1.00Ðthe value appropriate for hydro- We can expect that, according to the Tartaglione et al. meteors falling at constant terminal velocity. The fact (1996) parameterization, cloud regions above the freez- that the mean value of the exponent is below 0.9 may ing level ␳qr will now represent a mixture of super- be attributed to the way that mixed-phase precipitation cooled liquid and frozen hydrometeors. Because this is represented by the Tartaglione et al. (1996) param- leads to a modi®cation of the hydrometeor fall speeds eterization. In a similar way, the Z±R prefactor attains between the extremes of rain and snow/lump graupel a very high value, consistent with observed Z±R rela- with constant terminal velocity, also Z±R and R±␳q re- tions reported for thunderstorms, and the exponent of

Unauthenticated | Downloaded 09/29/21 02:43 AM UTC SEPTEMBER 2003 DOTZEK AND FEHR 1291 about 1.75 again points at particles falling at constant itation rates R. Even slowly subsiding hydrometeors like speed (Dotzek and Beheng 2001). cloud ice and, more so, cloud droplets can yield high The reason for this behavior cannot be a larger pro- upward mass ¯uxes that exceed 100 mm hϪ1 because portion of ice-phase precipitation, as comparison with of strong updrafts. Eq. (6a) reveals. The amount of hydrometeors in the ice Most obvious from Figs. 5c±f is a distinct spatial phase is the same in both cases. Instead, a probable coherence of the downdrafts enhancing R. The data explanation is that during storm growth a large amount points reveal several discrete pro®les corresponding to of water-phase precipitation falls at constant speed, those model grid columns containing the main down- namely, the speed ws Ӎ 0. This accumulation of hy- draft. Although Figs. 5c and 5e show development to- drometeors ¯oating at constant altitude in the updraft ward vertically homogeneous rain-rate pro®les below leads to the observed exponents and prefactors. the freezing level at about 3 km AGL, the corresponding At 1245 LST on the other hand, updrafts no longer Figs. 5d and 5f reveal negative mass ¯uxes above 5 km play a relevant role in the decaying cloud and its rain AGL and strongly enhanced precipitation rates below 4 shaft. As Eq. (7c) shows, both Z±R and R±␳q relations km AGL (Fig. 5d). The largest instantaneous rain rate do show enhanced rain rates, but prefactors and expo- at the ground for this modeled cumulonimbus was 420 nents are relatively close to their values from Eq. (6c) mm hϪ1, and the absolute maximum is R Ӎ 675 mm and their analytical ranges. Again, the mean value of hϪ1 at z Ӎ 900 m. In Fig. 5f, the largest precipitation the R±␳q exponent slightly exceeds its analytical bound rate is located at 3 km AGL, and the remnants of weak of 1.14. Table 2 gives a maximum downdraft speed of updrafts in the anvil region above 6.5-km altitude can Ϫ9.75 m sϪ1, which can roughly double typical terminal only induce small negative mass ¯uxes. velocities in heavy rain. c. Horizontally averaged rain-rate pro®les b. Individual rain-rate pro®les For horizontally averaged precipitation rate pro®les,

Information on individual horizontal gridpoint ver- at each vertical grid column in the model, Rtot was av- tical pro®les R(z) and the total vertical mass-¯ux budget eraged over 500-m vertical intervals provided that the

Rtot ϭ Rc ϩ Ri ϩ Rr can be gathered from the scatterplots total hydrometeor content at grid points was larger than of R and z, shown in Figs. 5a±f for the modeled cu- 10Ϫ3 gmϪ3. The resulting vertical pro®les with 500-m mulonimbus cloud of Fig. 2. For all grid points within spatial resolution were horizontally averaged to yield the cloud and the rain shaft, R(x, y, z) ϭ R(r, z) was ͗R͘. evaluated from Eq. (3) using w, ␳, and the hydrometeor Figure 6a gives the pro®les without convective drafts. concentrations ␳q. Individual hydrometeor types are dis- Again, for 1220 LST a triangular shape of ͗R͘, indicative tinguished by symbols for precipitation (rain or snow/ of pure sedimentation, is obvious (Wacker and Seifert lump graupel: ϩ), cloud ice (*), and cloud water (⅙), 2001), whereas precipitation near the ground peaks only respectively. Whenever Rtot exceeds the contribution of later at 1230 LST. Figure 6b with convection identi®es any single hydrometeor type, it is plotted by a cross those regions in which even in horizontal averages con- (ϫ). siderable upward hydrometeor mass ¯ux occurs and also Figures 5a, 5c, and 5e exclude vertical air motions, shows the downdraft-induced shift of maximum rain thereby providing the traditional view of precipitation rate at 1245 LST near 1 km AGL. falling through stagnant air. Figures 5b, 5d, and 5f in- Note the sharp drop in precipitation rate at about 1.75 clude vertical air motions and give the real hydrometeor km AGL, which is evident from both Figs. 6a and 6b. mass-¯ux pro®les. The reason for this phenomenon, not visible in the in- At 1220 LST, the scatterplot in Fig. 5a gives a tri- dividual pro®les of Fig. 5, could be identi®ed by careful angular shape of the hydrometeor mass-¯ux distribution. inspection of the KAMM data. It is not caused by up- Note the similarity between our Fig. 5a and the rain- drafts or diminished downdrafts between 1.5 and 2.25 rate pro®les presented in the analytical study by Wacker km AGLÐthis would also be visible in the individual and Seifert (2001). At this point in time, we clearly gridpoint data of Fig. 5. Instead, it is caused by a larger observe sedimentation of primary precipitation inside horizontal extent of the cloud droplet region at this level the cloud; there is no secondary rain formation going close to cloud base: The shower cloud itself is very on yet. Hardly any symbols other than those for pre- small in horizontal extent. When there is lateral cloud cipitation or the total rain rate can be seen; that is, mass growth near cloud base, the relative change in the hor- ¯uxes of cloud ice and cloud water are negligible for izontal cloud section area is large while the total hy- w ϭ 0. This changes when w ± 0 is taken into account. drometeor mass ¯ux over this area remains nearly un- In Fig. 5b, a completely different view is presented: altered. So the observed drop in ͗R͘ is a purely geo- Updraft speeds in the at stage a metrical averaging effect. in Fig. 2 were large, and so some portions of the hy- This explanation is substantiated by Fig. 6c, which drometeors were no longer descending to the ground gives the mass-¯ux pro®les for cloud water only. We but rising upward. These give rise to negative precip- see that at 1230 and 1245 LST the average cloud water

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FIG. 5. Scattergrams between rain rate R(r, z) and height z for rain, snow (ϩ), cloud ice (*), and cloud water (⅙) (a), (c), (e) without and (b), (d), (f) with vertical convective motions for KAMM. Crosses (ϫ) denote total precipitation rate.

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FIG. 6. Horizontally averaged rain-rate pro®les for the KAMM single-cell storm, showing cloud growth (1220 LST, solid), cloud maturity

(1230 LST, dashed), and cloud decay (1245 LST, dotted): (a) w is set to zero; (b) the general case w ± 0. For the latter case: (c) ͗Rc͘ only and (d) ͗Rr ϩ Ri͘ only. mass ¯ux is zero. Figure 6d correspondingly, depicts led to gust-front formation driven by evaporatively the mass ¯ux of solely cloud ice and precipitation. As cooled downdrafts. So, in this case, rain falling through compared with Fig. 6b, there is no drop in rain rate any cloud base has led to a shallow layer with condensation more. of cloud droplets, which can explain the lateral cloud A physical interpretation of this effect is readily at growth near cloud base. hand. During their lifetime, liquid precipitation particles evaporate as soon as the local relative humidity is less 4. MM5 results: Initial cell, supercell, and than 100%. KAMM's bulk microphysical scheme in- multicell storms cludes this effect. As mentioned earlier, the thermody- namic strati®cation chosen for this model run supported Similar to snapshots taken from the KAMM simu- relatively strong evaporation below cloud base: The de- lation shown in Figs. 2a±c, the chosen MM5 volume crease in relative humidity toward the ground below 2.5 datasets contain the initial cloud stage before storm km AGL, visible in Table 1, is evidence for strong down- splitting (t ϭ 30 min, 1630 LST), cloud maturity after draft potential. An identical simulation for ¯at terrain splitting (t ϭ 90 min, 1730 LST), and storm decay (t

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FIG. 7. Scattergrams between rain rate R(r, z) and height z for rain and graupel (ϩ), cloud ice and snow (*), and cloud water (⅙) (a) without and (b) with vertical convective motions at the MM5 initial cell stage t ϭ 30 min. Crosses (ϫ) denote total precipitation rate.

ϭ 210 min, 1930 LST) for both the multicell and su- Z ϭ (264 Ϯ 119)R1.41Ϯ0.11 and percell storms. This is illustrated by Fig. 4, providing R ϭ (23.95 Ϯ 7.70)(␳q)1.20Ϯ0.08 for 1630 LST. (9) a plan view of total hydrometeor ®elds at 4.4 km AGL and gust-front location during3hofsimulated storm Large scatter occurs, which encompasses again well- development. Our analysis will follow the same lines known separate prefactors for rain and thunderstorm as those for the KAMM evaluation. In particular, the cloud bursts. formulas to determine Z from the modeled hydrometeor The scatterplot of Figs. 7a and 7b reveals drastic ®elds ␳q were identical to those used with KAMM, in changes from the vertical distribution of rain rate de- order not to introduce any bias that might complicate pending on whether w is set to zero. There is a slight comparison of the two simulations. similarity between KAMM's Fig. 5c and Fig. 7a for the shape of the distribution. The effects of vertical drafts in Fig. 7b, however, are much more prominent in this a. Initial cell larger storm cell than for the rain shower modeled by In focusing ®rst on the initial single-cell MM5 storm, KAMM. there is one big difference from the KAMM storm in A large maximum in precipitation rate is found at about 500 m AGL because of the rain-induced down- its stage a from Fig. 2, aside from the larger storm Ϫ1 strength S (cf. Table 2): This initial cell is already strong- drafts, nearly 2 times the value of 300 mm h at the ly precipitating with a rain rate at the ground of about ground. Above 2 km AGL, the mass ¯ux becomes strongly negative, with R ( ) peaking at nearly 500 300 mm hϪ1. tot ϫ Ϫ mm hϪ1 at 5.5-km altitude. This total mass ¯ux consists For Z±R and R±␳q relations of the initial cell without of several contributions, rain and graupel (ϩ) peaking convection, relations very similar to that for the KAMM Ϫ1 model occur: at Ϫ300 mm h at 6 km AGL, cloud ice and snow (*) peaking at Ϫ100 mm hϪ1 at 10 km AGL, and cloud Z ϭ (169 Ϯ 10)R1.55Ϯ0.01 and water peaking at Ϫ200 mm hϪ1 at 2 km AGL. R ϭ (22.26 Ϯ 0.81)(␳q)1.13Ϯ0.004 for 1630 LST. (8) b. Supercell This was to be expected because for stagnant air these relations are fully determined by the cloud microphys- Without convection, radar meteorological relations ical parameterizations chosen in the particular model are again mainly determined by model parameteriza- under consideration. Because of the principal similarity tions. Only the density effect plays a minor role: of KAMM and MM5 as bulk cloud models, the relations Z ϭ (170 Ϯ 10)R1.54Ϯ0.01 and for air at rest must also be similar. R ϭ (22.37 Ϯ 0.84)(␳q)1.13Ϯ0.004 for 1730 LST (10a) If convection is allowed for, then the following re- lations appear: and

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FIG. 8. As in Fig. 7, but for the right-¯ank-moving MM5 supercell storm at times (a), (b) t ϭ 90 min and (c), (d) t ϭ 210 min.

Z ϭ (170 ϩ 9)R1.55Ϯ0.01 and any signi®cant way. This corresponds nicely to the no- tion that should indeed be in a quasi-station- R ϭ (22.29 Ϯ 0.78)(␳q)1.13Ϯ0.01 for 1930 LST. (10b) ary dynamical mode up to their decay. With convection, the relations are of course affected, This homogeneity is not so obvious from the scat- yet with little time dependence: terplots in Figs. 8a±d. At 1730 LST we see from Fig. 8a a strong increase in precipitation near and below the Z ϭ (158 Ϯ 38)R1.47Ϯ0.05 and freezing level at 3.9 km AGL. This freezing-level al- R ϭ (26.23 Ϯ 4.20)(␳q)1.16Ϯ0.03 for 1730 LST (11a) titude is valid for the reference environmental sounding from Fig. 3, but not necessarily so for in-cloud regions and where lifting of warm boundary layer air is likely to Z ϭ (157 Ϯ 35)R1.46Ϯ0.06 and deform the T ϭ 0ЊC plane and to raise it by a few hundred meters. Above and below T ϭ 0ЊC, rain rates R ϭ (26.02 Ϯ 3.88)(␳q)1.17Ϯ0.04 for 1930 LST. (11b) are relatively homogeneous with height, indicating that Even though at 1730 and 1930 LST the supercell is in the warm rain parameterization in MM5 is simply very different stages of development, Z±R and R±␳q switched off above the freezing level. This is a subject relations do not re¯ect variations in cloud evolution in inviting future investigation.

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FIG. 9. Horizontally averaged rain-rate pro®les ͗R͘(z) for the MM5 supercell, showing cloud growth (30 min, solid), maturity (90 min, dashed), and decay (210 min, dotted). In (a), w ϭ 0.

Figure 8b shows the largest values of total precipi- c. Multicell tation rate, including convection for all MM5 data under Switching off convection in the analysis of the mul- consideration. Total mass-¯ux peaks at 650 mm hϪ1 at ticell storm again yields the by-now already familiar Z± z ϭ 1.3 km and Ϫ750 mm hϪ1 at 8 km AGL. Individual R and R±␳q relations: upward contributions to total mass ¯ux at this level mainly stem from graupel (ϩ) and snow (*). Z ϭ (171 ϩ 10)R1.54Ϯ0.01 and Figures 8c and 8d for 1930 LST give similarly shaped R ϭ (22.29 Ϯ 0.86)(␳q)1.13Ϯ0.01 for 1730 LST (12a) distributions, yet with much smaller values for the mass ¯ux. Without convection there is still a small discon- and tinuity in rainfall intensity at 4.0 km AGL. For the case Z ϭ (169 Ϯ 10)R1.54Ϯ0.01 and of w ± 0 both peak levels of rain rate have moved downward, coupled to the sedimentation of precipitation R ϭ (22.36 Ϯ 0.82)(␳q)1.13Ϯ0.004 for 1930 LST. (12b) cores during storm decay. With convection, the resulting changes are discernible Horizontally averaged precipitation rate pro®les but not as large as for the supercell's most active stage ͗R͘(z) in Figs. 9a and 9b yield much smaller values than or even the KAMM cell: those from KAMM in Fig. 6. This is not surprising: The longer-lived and larger storms from the MM5 simulation Z ϭ (172 ϩ 51)R1.46Ϯ0.07 and have well-developed ``passive'' cloud regions with low R ϭ (25.66 Ϯ 5.17)(␳q)1.17Ϯ0.05 for 1730 LST (13a) precipitation around the main rain shafts, whereas the small and short-lived KAMM shower cell had practi- and cally no weak intensity fringes in its active life stages. Z ϭ (181 Ϯ 54)R1.49Ϯ0.07 and Furthermore, it should be kept in mind that the two initialization soundings were signi®cantly different. Fig- R ϭ (22.37 Ϯ 4.38)(␳q)1.13Ϯ0.05 for 1930 LST. (13b) ure 9a gives a smooth decrease of ͗R͘ with height. Figure It is likely that this is due to the fact that, by de®nition, 9b, on the other hand, gives negative mean rain rates in multicells there is always a mixture of individual above 3.9 km for the initial cell at 1630 LST and for precipitation cores at different stages of evolution. the supercell at 1730 LST. During storm decay at 1930 Treating them all as a single large entity surely leads to LST, average mass ¯ux is negligible above 3.9 km, the some sort of intrinsic averaging. Therefore, we ®nd pre- freezing level. factors in the Z±R and R±␳q relations with mean values All pro®les intersect at ͗R͘ϭ0 in the freezing-layer close to normal, that is, the case of stagnant air, but with region. Below that, the increase toward the ground is a higher standard deviation as for the supercell storm. nearly linear for all three storm stages chosen. Figures 10a±d indeed show that the shape of the scat-

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FIG. 10. Same scattergrams of R(r, z) and z as in Fig. 8, but for the left-¯ank-moving MM5 multicell storms. terplots for the multicell storm is invariant of cloud 11a and 11b reveal the vital dynamical difference be- evolution. Without convection, the jump in mass ¯ux tween the initial single cell at 1630 LST and the left- below the freezing level again becomes apparent from ¯ank-moving multicell. The latter leads to pro®les sim- Figs. 10a and 10c. Below 3.9 km AGL, precipitation ilar to each other, both with and without convection. rates are constant with height at about 190 mm hϪ1. Again, all three mean pro®les intersect around the freez- With convection, the S shape of the distribution is char- ing level when w ± 0. The biggest difference between acterized by a maximum rain rate of 450±480 mm hϪ1 multicell and supercell with w ± 0, as given in Fig. 9b, at 1.6 km AGL and a minimum of Ϫ520 to Ϫ420 mm is that, for the multicell, the shape of the ͗R͘(z) pro®les hϪ1 at 7.5±8.5 km AGL. From the higher density of above the freezing level is preserved over timeÐhow- symbols in Fig. 10d when compared with Fig. 10b (as ever, with smaller amplitude. This was not the case for well as from Fig. 4), we can infer that the multicell the supercell. storm complex has continuously grown in size and also has lightly intensi®ed during its life cycle. That left- 5. Discussion ¯ank movers need not decay quickly in principle was already shown by Wilhelmson and Klemp (1981). In our modeling study, we have compared the in¯u- Horizontally averaged precipitation pro®les in Figs. ence of convective drafts on vertical pro®les of precip-

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FIG. 11. Horizontally averaged R pro®les as in Fig. 9, but for the left-¯ank-moving MM5 multicell storms. itation rate for the whole spectrum of storm intensities, large scatter occurs in the Z±R relation at individual ranging from a shower cell to a mature supercell thun- locations in convective clouds and in the rain-®lled derstorm. For both KAMM and MM5, Z±R relations downdraft. Precipitation rate R is easily increased by a and R (r, z) scatterplots for individual grid points, as factor of 2 in these regions. Maximum values of rain well as horizontally averaged ͗R͘(z) pro®les, consis- rate consistently occur for both mesoscale models at a tently show the limitations of the nonlocal approach to height of roughly 1 km AGL. Here, the precipitation- derive R at the ground from Z aloft. driven downdrafts reach their largest magnitude; at low- If convective drafts were set to zero, resembling the er levels, increasing interaction of the downward-mov- case of purely stratiform rain clouds, Z±R relations ing air with the ground leads to a locally high dynamic showed limited scatter because of increasing rain rate perturbation pressure under the rain shaft, decelerating caused by air density decreasing with height. This effect the downward air¯ow. on Z±R relations, however, can be completely compen- The horizontally averaged ͗R͘(z) pro®les from sated by the correction term for measured re¯ectivity KAMM attain large negative values of more than Ϫ20 as given by Dotzek and Beheng (2001). Application of mm hϪ1 in upper cloud regions during cloud growth and this physically based Z correction to observational radar maturity. Because the KAMM cloud is a single and network data from a stratiform rain event in central short-lived shower cell, there is not much time to de- Oklahoma (Dotzek et al. 2002) showed an increase in velop a signi®cant passive, low-re¯ectivity, fringe precipitation accumulation of 10%±50% at radar ranges around the core of the storm. Instead, the cloud is almost from 100 to 300 km, at which even the base-level 0.5Њ- in its whole volume also precipitation ®lled. Besides tilt-angle radar beam is at several kilometers in altitude. higher instantaneous rain rates in the MM5 simulation, Note that the correction term given here by Eq. (A13) in the appendix also applies for convective clouds of this effect leads to the larger magnitude of the averaged any intensity. Therefore, after the removal of artifacts KAMM rain rates when compared with the MM5 storms in radar-observed re¯ectivity ®elds, such as the melting- and also to the signi®cant drop in average precipitation layer signature, it should be routinely applied to any rate as soon as the cloud-droplet region grows in size radar observation intended to serve for quantitative pre- near cloud base. cipitation estimation. Our MM5 results for large and strong cumulonimbus Nevertheless, for horizontally averaged vertical pro- clouds con®rm the KAMM data. One has to keep in ®les of rain rate, both KAMM and MM5 showed that, mind, though, that for the more intense thunderstorms in general, the correlation of precipitation mass ¯ux aloft from the MM5 study a completely different sounding with rain intensity at the ground decreases rapidly over was used for initialization. In addition, the MM5 storms the lowest kilometers above ground, in accordance with develop considerably larger regions with low Z values the observed radar re¯ectivity correlation pro®le pre- around their main precipitation cores during their longer sented by Dotzek et al. (2002). life cycles. This leads to the somewhat smaller mag- When vertical air motion is included, however, very nitude of the horizontally averaged ͗R͘(z) pro®les. Nev-

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ertheless, these also become negative and reach their to eliminate the in¯uence of air-density pro®le on R minimum values of Ϫ18 mm hϪ1 at 5.5-km altitude. by the physical correction procedure outlined by In the split-storm MM5 simulation, there is a fun- Dotzek and Beheng (2001) and tested using radar damental difference between the right-¯ank- and left- data by Dotzek et al. (2002). ¯ank-moving storms. The supercell showed stationary 7) The same is true for model-derived precipitation, Z±R relations at 1730 and 1930 LST but more-time- should it be computed from re¯ectivity instead of by dependent R(r, z)±z scatterplots at these times. The con- directly computing R from cloud microphysical mod- trary was true for the left-¯ank-moving multicell storms. el variables. Here, Z±R relations showed stronger time dependence Over complex terrain, horizontal displacement of pre- than the R(r, z)±z scatterplots. A reason for this might cipitation plays an additional critical role. be that the supercell, being quasi stationary dynami- cally, always maintains very similar convection-in¯u- Acknowledgments. The authors thank J. J. Gourley at enced Z±R relations throughout its whole life cycle. The NOAA NSSL for many stimulating talks. Insightful scatterplots, however, better represent the development comments by our referees greatly helped to ®nalize the of the storm from maturity to decay from 1730 to 1930 paper. This work was partly funded by the Commission LST, leading to smaller magnitudes of rain rate while of the European Communities under contract ENV4- still following the average Z±R relation. CT97-0409 and by the German Ministry for Education The multicell, on the other hand, almost always con- and Research BMBF under contract 07ATF 45 in the tains a similar spectrum of cell intensities, therefore project Vertikaler Austausch und Orographie (vertical inducing more or less stationary R±z scatterplots. The transport and orography; VERTIKATOR) within the at- Z±R relation is more affected by the actual number of mospheric research program AFO 2000. grid points contained within cells of a particular inten- sity, however. Thus, we observe a larger variation here. APPENDIX We did not analyze in detail the horizontal transport of hydrometeors as a consequence of upward in-cloud Derivation of Z±R Relations mass ¯uxes as schematically outlined in Fig. 1. This would be a more fruitful task for thunderstorm simu- In this section we give a brief outline of the analytical lations over highly resolved complex terrain. However, procedure described by Dotzek and Beheng (2001). for the growing shower cloud at 1220 LST, KAMM was Normalized ⌫-type number distribution functions are able to reproduce the sedimentation-dominated rain-rate assumed for any hydrometeor type: pro®les described by Wacker and Seifert (2001) in their ⌫(4) D ␥Ϫ1 ϪD/D0 analytical study. Both in the R(r, z)±z scatterplot and n(D) ϭ Ne0 . (A1) the ͗R͘(z) pro®le, the typical triangular shape of the ⌫(␥ ϩ 3)΂΃D0 pro®le became evident. Here D denotes particle diameter, ␥ is a shape parameter,

N0 presents a ``particle load'' of the distribution, and D0 is a formal scaling diameter [in fact, D0 ϵ 1/␭ in 6. Conclusions the notation of Marshall and Palmer (1948)] that can From our modeling study the following conclusions easily be related to any speci®c measure of particle size, can be drawn: such as the volume median diameter. The normalization in Eq. (A1) ensures that hydro- 1) Individual vertical pro®les of R can attain large neg- meteor content ␳q does not depend on the shape param- ative values in the presence of strong updrafts. The eter ␥ [cf. Eq. (A3)] and that, for ␥ ϭ 1, the exponential same also holds for horizontally averaged pro®les of Marshall and Palmer (1948) spectrum is reproduced. ͗R͘. With increasing ␥, the spectra broaden and conserve ␳q

2) Rapid decorrelation of ͗R͘(z) from R 00 signi®cantly by lowering nmax. We de®ne the moment Mm of order limits the applicability of Z aloft to determine R at m by

the ground. ϱ 3) Using Z±R relations merely in subregions of con- m Mm ϭ n(D)DdD vective clouds will most likely not yield meaningful ͵ 0 results. ⌫(4) 4) Maximum rain-rate enhancement from downdrafts ϭ⌫(␥ ϩ m)NDmϩ1. (A2) occurs at about 1 km AGL. Below this level, vertical ⌫(␥ ϩ 3) 00 perturbation pressure gradient forces decelerate the Most bulk microphysics cloud models, following the downdrafts. work of Kessler (1969), use hydrometeor content ␳q as 5) Temporal variability of Z±R relations, and instan- the main prognostic variable: taneous rain rates versus height, were signi®cantly ␲ different for the supercell and multicell storms. ␳q ϭ ␳ M ϭ ␲␳ ND4. (A3) 6) In QPE, radar-observed Z values should be corrected 6 h 3 h 00

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Spectral parameters can be substituted by the moments which is, after multiplication by ␳q, identical to the M, and so the quantity ␳q from Eq. (A3) will be intro- mean mass ¯ux density or precipitation rate R. In Eq.

duced to any following equation to eliminate either N 0 (A7), again either N0 or D0 can be eliminated by intro- or D 0, depending on which spectral parameter aside ducing the hydrometeor content from Eq. (A3): from ␳q is chosen to describe the particle spectrum. Note w ⌫(␥ ϩ 3 ϩ ␤) that this does not imply any loss of generality for the w ϭ 0 DNÃ Ϫ␤ Ϫ␤/4(␳q)␤/4 (A8a) t,00(␲␳ )␤/4 ⌫(␥ ϩ 3) 0 spectra. None of N 0, D 0, ␥, or ␳q are assumed to be h constant, and all of these quantities can be functions of and time and space. ⌫(␥ ϩ 3 ϩ ␤) D ␤ w ϭ w 0 . (A8b) a. Radar re¯ectivity factor t,00 0 ⌫(␥ ϩ 3) ΂΃DÃ The radar re¯ectivity factor Z for spherical particles Note that Eq. (A8b) is independent of ␳q. under the assumption of Rayleigh's approximation (radar In the general case in which an external vertical ve- wavelength much larger than particle size) is given by locity ®eld w is imposed so that with Eq. (2) the pre- ⌫(␥ ϩ 6) cipitation rate reads 7 Z ϭ M600ϭ⌫(4) ND ⌫(␥ ϩ 3) ␣ ␳ 00 ϭ⌫(4)(␥ ϩ 5)(␥ ϩ 4)(␥ ϩ 3)ND7. (A4) R ϭϪw st,␳q ϭϪw ϩ w 00 ␳q. (A9) 00 []΂΃␳ From Eq. (A3) we arrive at the following two desired Without loss of generality we may set relationships between Z and ␳q by eliminating either D0 ␣ or N 0: ␳ w ϭ Xw 00 . (A10) ⌫(4) t,00 ␳ Z ϭ (␥ ϩ 5)(␥ ϩ 4)(␥ ϩ 3)N Ϫ3/4(␳q) 7/4 (A5a) ΂΃ (␲␳ )7/4 0 h Here, negative values of X correspond to updrafts. The and local value of w is a superposition of an imposed vertical wind and an additional downdraft component induced ⌫(4) 3 by the hydrometeor drag. Introducing Eq. (A10) into Z ϭ (␥ ϩ 5)(␥ ϩ 4)(␥ ϩ 3)D0 ␳q. (A5b) ␲␳h Eq. (A9) the expression for the rain rate becomes

Note that the Z±R relation of Eq. (A5b), being linear in ␣ ␳q, is related to equilibrium or statistically stationary ␳ 00 R ϭϪ(X ϩ 1)w t,00 ␳q. (A11) rain (Jameson and Kostinski 2002, and references there- ΂΃␳ in). However, most published Z±␳q relations show the power 7/4 (Kessler 1969) or similar empirical values such as 1.82 (Douglas 1964; Smith et al. 1975) for rain. c. Z±R relations It is clear that Z does not depend on either vertical As shown by Dotzek and Beheng (2001), the ®nal Z± velocity or on the variation of fall speed due to the R relations take on the form vertical air density gradient. 7/(4ϩ␤) Z ϭ af00,NN(␳)g N(w)R , with

b. Terminal fall velocity and precipitation rate Ϫ␣[7/(4ϩ␤)] ␳ 00 Specifying the index ``00'' for all quantities at the f N(␳) ϭ , and (A12a) ␳ chosen reference level, that is, sea level conditions with ΂΃ vertical air velocity w ϭ 0 and air density ␳ ϭ ␳ 00, ®rst Z ϭ af(␳)g (w)R, with the terminal fall velocity w of the hydrometeors as a 00,DD D t,00 Ϫ␣ function of ␳q will be computed by setting ␳ 00 f (␳) ϭ , (A12b) D ␳ D ␤ ΂΃ w (D) ϭ w . (A6) t,00 0 Dà in which the f(␳) represent the density terms that can ΂΃ be corrected for. Dotzek et al. (2002) have used the term à Here D is the unit diameter, usually 1 mm, and w 0 is f (␳) in their radar data correction algorithm for Z ϭ à N the terminal fall velocity of hydrometeors with D ϵ D. aRb and the fall-speed-law exponent ␣ from Eq. (A9): Ϫ1 For rain, Kessler (1969) proposed w 0 ϭϪ4.11 m s ␣b and ␤ ϭ 1/2. By using Eq. (A6), a volume-weighted ␳ 00 mean fall velocity w can be calculated, Zcorr ϭ Z . (A13) t,00 []␳(z) ␲␳ ϱ w ϭ h n(D)w (D)DdD,3 (A7) For precipitation aloft, the measured radar re¯ectivity t,00 6␳q ͵ t,00 0 factor Z has to be increased according to this relation

Unauthenticated | Downloaded 09/29/21 02:43 AM UTC SEPTEMBER 2003 DOTZEK AND FEHR 1301 before any standard sea level Z±R relation can be applied on Radio Meteorology and Proc. 11th Radar Meteorology Conf., to yield a representative rain rate R. That is important Boulder, CO, Amer. Meteor. Soc., 146±149. Fehr, T., 2000: Mesoskalige Modellierung der Produktion und des drei- both for R pro®les in high clouds and for base-level dimensionalen Transports von Stickoxiden in Gewittern (Meso- re¯ectivities of convective and stratiform clouds at large scale modeling of the production and three-dimensional transport range. The latter was quanti®ed by Dotzek et al. (2002) of nitrogen oxides in thunderstorms). Ph.D. dissertation, Univer- in greater detail. For the case of the Oklahoma City, sity of Munich, 116 pp. [Available online at http://www.op.dlr. de/ϳpa10/.] Oklahoma, Weather Surveillance Radar-1988 Doppler Foote, G. B., and P. S. du Toit, 1969: Terminal velocity of raindrops (WSR-88D) radar site (KTLX), the authors showed that aloft. J. Appl. Meteor., 8, 249±253. the precipitation accumulation from the corrected Z ®eld Gourley, J. J., R. A. Maddox, K. W. Howard, and D. W. Burgess, can be 10%±50% higher for radar ranges from 100 to 2002: An exploratory multisensor technique for quantitative es- timation of stratiform rainfall. J. Hydrometeor., 3, 166±180. 300 km. Grell, G. A., J. Dudhia, and D. R. Stauffer, 1994: A description of the ®fth-generation Penn State/NCAR mesoscale model (MM5). NCAR Tech. Rep. NCAR/TN-398 ϩ STR, 122 pp. REFERENCES Hannesen, R., N. Dotzek, H. Gysi, and K. D. Beheng, 1998: Case study of a tornado in the Upper Rhine valley. Meteor. Z., 7, 163±170. [Available online at http://www.op.dlr.de/ϳpa4p/.] Adrian, G., and F. Fiedler, 1991: Simulation of unstationary wind and Jameson, A. R., and A. B. Kostinski, 2002: When is rain steady? 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