Microphysical and Geometrical Controls on the Pattern of Orographic Precipitation

MARCH 2006 R O E A N D B A K E R 861 Microphysical and Geometrical Controls on the Pattern of Orographic Precipitation GERARD H. ROE AND MARCIA B. BAKER Department of Earth and Space Sciences, University of Washington, Seattle, Washington (Manscript received 25 May 2004, in final form 2 May 2005) ABSTRACT Patterns of orographic precipitation can vary significantly both in time and space, and sch variations mst ltimately be related to montain geometry, clod microphysics, and synoptic conditions. Here an extension of the classic pslope model is presented, which incorporates an explicit representation in the vertical dimension, represents the finite growth time of hydrometeors, their downwind advection by the prevailing wind, and also allows for evaporation. For a simple montain geometry the athors derive an analytical soltion for the precipitation rate, which can be nderstood in terms of for nondimensional parameters. The finite growth time and slanting hydrometeor trajectories give rise to some interesting possibilities: a precipitation rate that maximizes at intermediate vales of the horizontal wind speed, localized precipitation efficiencies in excess of 100%, and a reverse rain shadow with more precipitation falling on the leeward flank than on the windward flank. 1. Introdction The inflence of srface topography on patterns of precipitation leads to some of the most prononced climate gradients on earth, reflected in sharp transitions in flora and fana across many montain ranges. Not only is orographic precipitation important for natral ecosystems and for the management of hman water resorces, bt it also has significant interactions with other physical components of the earth system on a wide range of time scales. Over millions of years, for example, patterns of erosion, rock exhmation, and the form of montain ranges themselves reflect patterns of precipitation (Beamont et al. 1992; Willett 1999; Montgomery et al. 2001; Reiners et al. 2003; Anders et al. 2004, npblished manscript hereafter A04). Dring the Pleistocene the location of, and ice flx within, the great continental-scale ice sheets of the ice ages were controlled in part by patterns of snow accmlation (Sanberg and Oerlemans 1983; Kageyama et al. 1999; Roe and Lindzen 2001; Roe 2002). On shorter time scales, natral hazards sch as avalanches and landslides are impacted by precipitation intensity in montainos regions (e.g., Caine 1980; Conway and Corresponding athor address: Dr. Gerard Roe, Dept. of Earth and Space Sciences, University of Washington, Seattle, WA 98195. E-mail: gerard@ess.washington.ed Raymond 1993). So, while the existence of a rain shadow across a montain range whose axis is perpendiclar to the prevailing wind direction is one of the most confident expectations in atmospheric science, it is worthwhile to ask how, in different climates, patterns of orographic precipitation might change. In these contexts, it is to be hoped that simple representations of orographic precipitation can yield an nderstanding of the sensitivity of patterns to conceivable changes in forcing and that they can be implemented in investigations of these copled systems. Research efforts to categorize and model orographic precipitation have prsed an hierarchy of approaches, from statistical regressions (e.g., Nordø and Hjortnæs 1966; Daly et al. 1994; Wratt et al. 2000), to sophisticated mesoscale modeling, inclding complex microphysical schemes and in some instances resolving clods (e.g., Katzfey 1995a,b; Colle et al. 1999; Rotnno and Ferretti 2001; Lang and Barros 2004; Smith et al. 2003). Intermediate approaches developing redced models of orographic precipitation have also been prsed. These models have generally focsed on the precipitation that oght to reslt from stable ascent, althogh it is important to note that observations also indicate that convection can modify patterns (e.g., Browning et al. 1974; Medina and Hoze 2003). The simplest models (often referred to as pslope models) are essentially two-dimensional with streamlines everywhere parallel to the montain (e.g., Sawyer 1956; 2006 American Meteorological Society

862 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63 Smith 1979; Alpert 1986) and assme that the condensation rate is proportional to the rate of ascent of the air mltiplied by the vertical gradient of the satrated moistre content. Therefore condensation, and ths precipitation, maximize over the steepest windward slopes. Another set of models aim to solve for the airflow in a more realistic way (e.g., Myers 1962; Smith 1979; Sarker 1966; Fraser et al. 1973; Colton 1976; Smith and Barstad 2004). It is commonly fond in these models that some downwind smoothing of the condensation rate needs to be applied in order to get precipitation rates consistent with observations. It is arged that this smoothing reflects the growth and downwind advection of hydrometeors (precipitation particles large enogh to fall). Demonstrating the potential of this mechanism, Hobbs et al. (1973) sed comptations of the airflow across a two-dimensional ridge and incorporated explicit calclations of hydrometeor trajectories. Their reslts showed that the concentration of the condensation ice nclei and the growth process greatly affected the descent trajectories. A final category of diagnostic nmerical models exist that, to varying degrees of complexity, diagnose the topographically modified atmospheric flow from the largescale circlation and inclde formlations of precipitation formation (e.g., Colton 1976; Rhea 1978; Sinclair 1994; Barros and Lettenmeier 1993). All of these models are proposed to some degree as generalized models of orographic precipitation. Data are hard to come by in montainos terrains, so sch models are often evalated (or more typically calibrated) for a limited dataset, sometimes for a particlar storm and sometimes for the climatological distribtion. Indeed, Barros and Lettenmeier (1994a) arge for the necessity of a range-by-range calibration of any redced precipitation model, asserting that the myriad factors controlling precipitation formation make the prevailing synoptic and microphysical conditions for each montain range essentially niqe. The prpose of this paper is to develop a simple framework for the precipitation pattern in terms of a few model parameters, where the vale of those parameters can be related to different precipitation sitations. Or goal is to interpret precipitation patterns in terms of the prevailing wind, several clod microphysics properties, and the dimensions of the montain range. We concentrate on three main defining aspects, which together might be said to characterize the precipitation pattern: the maximm precipitation rate, the location of the maximm precipitation rate, and the strength of the rain shadow. The model can be regarded as an extension of the classic pslope model, with an explicit representation in the vertical that incorporates the conversion of condensate into hydrometeors, their terminal velocity and advection by the horizontal wind field, and leeside evaporation. It can also be modified to incorporate vertical wind shear. Within this redced framework, an analytical soltion is derived for the precipitation pattern, the dependency of which on for nondimensional parameters can be clearly ascertained. The chief costs of deriving this soltion are a physically realizable atmospheric flow simple mechanical lifting is assmed, which omits any gravity wave response or blocking of the flow and a simplified montain geometry. The reslts of this stdy can be contrasted with those of Smith and Barstad (2004; also Jiang and Smith 2003; Smith 2003) in which a soltion is presented that acconts for linear atmospheric response to flow over topography, bt in which moistre is represented in terms of a vertically integrated colmn and hydrometeor growth and fallot occrs on a characteristic time scale. Many of the aspects of the patterns are similar bt in some cases the physical reasons are different. These are highlighted. The discssion identifies the aspects left ot of or model formlation that likely reslt in the largest departre from natre. The model simplifications probably preclde it from being applied to simlate a set of observations, nor is that intended. It is a framework within which the effects of slanting hydrometeor trajectories and montain geometry on the pattern of precipitation can be explored. In this regard, the model contines from the work of Hobbs et al. (1973), who demonstrated the importance of these effects. It is also in contrast to many other simple models of orographic precipitation, which integrate the condensation rate in a vertical colmn above the montain slopes. The condensation rate in these models is ths distribted on the landscape in a fndamentally different way from that in the model presented here. 2. The model a. Montain geometry The model framework is illstrated in Fig. 1. Atmospheric flow impinges on a wedge-shaped montain described by the fnction z s (x s ), z s H L 1 L 1 x s, L 1 x s 0 z s H L 2 L 2 x s, 0 x s L 2 z s 0, elsewhere. 1 If L 1 L 2, the wedge is asymmetric.

MARCH 2006 R O E A N D B A K E R 863 FIG. 1. Schematic illstration of model framework. Black lines are hydrometeor trajectories. The dashed lines are sorce lines that delineate all points within the sorce region where condensation ends p as precipitation at the same srface point. See text for more details. b. Airflow Or model is time independent. The incoming air is assmed to be moving at constant speed, netrally stratified, and satrated. We frther assme that flow ndergoes mechanically forced ascent on the windward flank and forced descent to the lee. Ths the rate of vertical ascent (or descent) is given by the strength of the prevailing wind mltiplied by the slope (i.e., w dz s /dx s ), independent of height. Strictly, sch a flow response cold only be possible for a netrally stable atmosphere in which the horizontal scales are mch larger than the vertical scales. In reality, changes in vertical motion (e.g., crossing from the windward to the leeward flank) case vertical accelerations that, in a stably stratified flid, set p gravity waves that propagate away from the sorce (e.g., Smith 1979, 2003; Drran 1986, 2003, Colle 2004). Gravity waves will case the biggest departre from the mechanically forced response for small steep montain ranges, weak flow regimes, or strongly stable conditions when the modification of the flow by the gravity wave response wold be largest. In the context of orographic precipitation, a sense of the departre from the mechanically forced pattern is presented in the work of Hobbs et al. (1973), Smith and Barstad (2004), and Colle (2004). Omitting gravity waves precldes, for example, any lifting pwind of the montain. They can also trigger nstable convection, which can modify precipitation patterns (e.g., Browning et al. 1974; Medina and Hoze 2003). Moist netral, or near-netral, flows have been shown to be relevant to orographic precipitation in several recent stdies (e.g., Rotnno and Ferretti 2001; Medina and Hoze 2003). We will discss the conseqences of a more realistic flow response in section 5. A third effect of stable stratification is that for smaller horizontal scales the inflence of the montain on the circlation decays with height (e.g., Drran 1986). This last aspect can be incorporated into the model by the choice of moistre scale height described below. c. Condensation rate On the windward flank, the ascent of satrated air leads to a steady condensation rate per nit volme, S, which can be written as Skg m 3 s 1 dq s dz dz dt. 2 Here, q s is the satration specific hmidity, dz/dt is the vertical velocity, and is the density of air. This expression can be developed sing the relations dzdt w 1 dz s HL dx 1, s and also by making an exponential approximation for the specific hmidity so that (z) q s (z) 0 q 0 exp(z/ H m ), independent of x; 0 and q 0 are the density and the satration specific hmidity at z 0, and q 0 is a fnction of srface temperatre and pressre. Here H m is the moistre scale height and typically varies between 2 and 4 km, being greatest in the Tropics (e.g., Peixoto and Oort 1992), and cold also be regarded as

864 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63 incorporating the decrease with height of the inflence of the montain on the atmospheric flow. Hence (2) can be rewritten as Sz 0q 0 H H m L 1 e zh m. 3 Condensation occrring at altitde reaches the grond only after clod droplets are converted to hydrometeors and only after those hydrometeors have reached the grond. (In this discssion we se terms appropriate to warm clods. Or model, however, is applicable to mixed phase and/or entirely glaciated clods, which we represent by different choices of model parameters. We do not consider clods with a phase change with altitde althogh or framework allows for this extension.) We assme that a finite period of time, g, mst elapse for the growth of clod droplets to form hydrometeors and that, thereafter, they fall with a characteristic terminal velocity, f. In essence this assmes a single representative size scale for hydrometeors and, so, neglects any size distribtion. Collection or coalescence of clod water droplets on the way down is also not treated explicitly, so the changes in the trajectories that this mass accmlation reslts in are not inclded either. These assmptions give the following characteristic pathways, or trajectories, for the falling hydrometeors depending on whether the precipitation ends p falling on the windward or leeward flanks of the range, illstrated in Figs. 1a,b. For the windward flank A I to A II : growth of droplets dring which they are carried with the ascending flow. A II to A III : hydrometeors fall throgh ascending air, no evaporation. For the leeward flank B I to B II : as for A I to A II. B II to B III : as for A II to A III. B II to B IV : hydrometeors fall throgh descending air, some evaporation occrs. Finally, for x s L 2, the last part of the trajectory is throgh air with no vertical velocity. These assmptions reslt in a sorce region on the windward flank. Becase of the finite formation time for hydrometeors, it ends at x g (as indicated by Fig. 1). The finite growth time also means that no precipitation reaches the grond for x s L 1 g. For every point on the srface downwind of this, a sorce line can be defined as the locs of points in the sorce region from which condensation ends p as precipitation at that srface point. Examples are shown as dashed lines in Figs. 1a,b. The total condensation rate integrated along any sorce line can be written as z max Sz dl Sz1 sorce z min dx 2 12 dz dz. 4 Here dl is an infinitesimal increment along the sorce line, and z min and z max are the intersections of the sorce line with the bondaries of the sorce region. To convert this integral into the precipitation rate, R(x s ), we note that (4) gives the precipitation flx per nit area perpendiclar to the direction of the sorce line, and a geometric factor needs to be inclded to convert this into a flx per nit area in the horizontal. Moreover, for points to the lee of the range crest, evaporation of falling hydrometeors can be crdely and simply incorporated according to the time spent otside of the orographic clod: precipitation exponentially diminishes with a characteristic time scale, e. Incorporating these two aspects, the precipitation rate per nit horizontal area can be written as Rx s tan z max 1 Sz dz tan 1 1z min L 1 g x s 0 Rx s tan 2 tan 2 tan 1 z min for exp z 0 z s e f 5a z max Sz dz for 0 x s L 2 5b Rx s tan 3 tan 1e z 0 z s z max e f Sz dz for L 2 x s, z min 5c where z 0 is the altitde of the point where the trajectory crosses the crest (i.e., at x 0). The are the windward and leeward montain slopes, and the are the angles of the hydrometeor trajectories make to the horizontal (Fig. 1). Note the evaporation factor can be omitted entirely by setting e. In the absence of evaporation water is conserved: all condensation occrring within the sorce region ends p as precipitation at the srface. Eqation (5) ses the sorce lines in Fig. 1, which are dictated by hydrometeor growth times and trajectories, to map condensation occrring in the sorce region onto precipitation falling at the srface. Ths the determination of the precipitation rate at a srface point reqires calclation of the sorce line for that point in order to find the appropriate z max and z min. The eqations for the sorce lines and expressions for z max and z min are developed in appendix A.

MARCH 2006 R O E A N D B A K E R 865 For the windward flank there is no evaporation ( e ), and combining Eqs. (3), (5a), (A8), (A10) gives Rx s 0q 0 H L 1 L1f 1 H L 1 f H exp z min H m exp L 1 f z min H H m 6a z min z s gh. 6b L 1 For the leeward flank, (3), (5b) or (5c), (A11), and (A13) give Rx s q oh L 1 L1f H L 1 f exp z min 1exp H H m x s e 1 exp f H m L 1 g 1 H f L 1 7a z min H x s f gh. 7b L 1 d. Nondimensionalization Eqations (6) and (7) nondimensionalize natrally by introdcing five physically meaningfl parameters. Sbscripts (1, 2) denote windward and leeward sides, respectively: R 0 0q 0 H L 1 1,2 L 1,2 f H H H m 1,2 L 1,2 g H f e Vertically integrated condensation rate in a windward air colmn. The ratio of the raindrop trajectories slope to the orographic slopes. The ratio of montain height to moistre scale height. The ratio of montain length to the formation length scale. The ratio of montain height to the evaporation height scale. 8 A sixth parameter, L 1 /L 2, cold be introdced that wold govern the asymmetry of the montain range. The system wold then be described completely in nondimensional terms, bt for the sake of clarity we leave this last parameter ot of the eqations shown. The precipitation eqations can be expressed in terms of these parameters [after a lot of rearrangement and sbstitting from (1) for the shape of the montain]. For 1 1 1 x s L 1 0: For x s L 2 0: Rx s R 0 1 1 1exp z min exp 1 z min H z min H 1 1 1 x s L 1. H Rx s R 0 1 1 1exp 2 x s L 2exp z min H 1 exp1 1 1 1 1 9a 9b 10a z min H 1 1 x s 2. 10b 1 L 2 These parameters have the following physical constraints. For precipitation to fall at all, the terminal velocity mst exceed the vertical velocity of the air, which reqires that 1 1. As the sorce lines get closer to horizontal 1 tends to 1, bt z max tends to z min,sor(x s ) remains finite (as it mst physically). A second constraint, necessary for any precipitation to occr, is that the windward width mst be greater than the growth length scale: 1 1. This is related to the atoconversion limit arged for by Robichad and Astin (1988) in which clod water within the orographic clod prodces precipitation withot an external seeder clod. It is also the threshold behavior, noted in Jiang and Smith (2003), in which clod droplet conversion mst occr dring the residence time within the sorce region. The seeder feeder mechanism can be imitated in this model by taking a short growth time, which leads to efficient rain ot on the windward flank (e.g., Sinclair 1994). 3. Reslts To present the basic precipitation pattern we take the following set of parameters: H 2.5 km, L 1,2 30 km,

866 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63 FIG. 2. Precipitation pattern for standard choice of parameters (eqal to midvales in Table 1). Topography is shown as gray line. 10 m s 1, f 4.0 m s 1, g 1000 s, e 2000 s, H m 3.0 km, and q 0 4.0gkg 1. Appendix B provides jstification for these choices from microphysical calclations. They are also broadly consistent with observations in case stdies of orographic precipitation (e.g., Sinclair 1994; Smith and Barstad 2004). Colle (2004) integrates a mesoscale forecast model in a twodimensional (height-width) setp. The pattern is shown in Fig. 2. Becase of the finite growth time, precipitation is offset from the toe of the windward flank. It increases rapidly to a rate of abot 8mmh 1, abot halfway along the windward flank. There is a gradal drop-off toward the crest, which is followed by a mch stronger decrease after crossing to the lee, becoming negligible for x s 20 km. This shape is consistent with the qintessential pattern for orographic precipitation on large montain ranges, fond in nmeros observational and modeling stdies. The sharp kink in the pattern at the divide reflects the trianglar shape of the topography. For a topographic slope eqivalent to ors, Colle (2004) find maximm precipitation rates between 3 and 13 mm h 1 for winds varying between 10 and 30 m s 1 for their weak atmospheric stability cases. Comparing Fig. 2 with Colle (2004), significant spill-over precipitation reaches over abot the same distance beyond the divide. The biggest difference with Colle (2004) is the absence of significant precipitation pwind of the topography. In Colle (2004) (and also Smith and Barstad 2004) this occrs becase of ascent of air windward of the montain de to the dynamical response of the atmosphere. A freqently employed description of orographic precipitation is precipitation efficiency, PE (e.g., Smith 1979; Smith et al. 2003). It can be defined at each point as the actal precipitation rate divided by the vertically integrated condensation rate above that point if the vertical velocity everywhere eqaled its vale at the srface. This measre emerges natrally from or analyses as PE Rx s R 0 e z s H m. 11 Becase of the finite growth time, advection, and slanting hydrometeor trajectories in the model, it is possible for localized precipitation efficiency to exceed 100% (see also Smith and Barstad 2004). For the standard case, precipitation efficiency actally reaches 145%. (That is, the precipitation at the grond at a given location can be greater than the total amont of water condensed in the vertical colmn above. This highlights the nonphysical natre of the localized precipitation efficiency.) Averaged over the windward domain, however, it is 65%, which is not atypical in observations (e.g., Sawyer 1956; Myers 1962; Colton 1976; Smith et al. 2003). Sensitivity stdies in section 3a show that a wide range of precipitation efficiencies is possible (Fig. 3). Bt, all else being eqal, smaller montains (height and/or width) have lower precipitation efficiencies, which is consistent with the observations cited above and, for example, Robichad and Astin (1988) and Jiang and Smith (2003). The location of the maximm precipitation rate can be nderstood directly from Fig. 1; it will be at the

MARCH 2006 R O E A N D B A K E R 867 FIG. 3. Sensitivity of precipitation pattern (i.e., R/R 0 ) to variations in nondimensional parameters. Different panels show sensitivity to the different parameters. Each panel shows the effect of varying the indicated parameter across its range of plasible vales (given in Table 2), while all other parameters are held at their midrange vales. See the text for more details. Note that precipitation efficiency can be inferred from this figre: it is eqal to R/R 0 exp(z s /H ), and for (a) max it is close to 100%. (b), (c) When the precipitation maximm is offset from the crest, its magnitde does not depend on or. srface point with the longest, moistest sorce line, which for the assmptions and geometry of the model means it will always occr on the windward flank. We compte d dx s Rx s R 0 from (9) and set it to zero to find x s L 1max min0, 1 1 1 ln 1 1 1. 12 If the growth time is sfficiently large that the growth phase occrs over a large fraction of the windward flank (i.e., 1 tends to 1), the width of the sorce region decreases and the maximm precipitation rate tends to occr close to the divide. For small vales of the parameters (i.e.,, 1 3), the location of the maximm is most sensitive to. That is, when the slopes of hydrometeor trajectories are almost as steep as the montain slopes, the most effective way to shift the location of the precipitation maximm is by changing the ratio of the moistre height scale to the montain height. Conversely, for large vales of and, the sensitivity to dominates. The vale of the maximm precipitation rate can be fond by sbstitting (12) into (9): R max R 0 1 1 1exp ln 1 1 1 exp 1ln 1 1 1 for x s L 1max 0,

868 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63 R max R 0 1 1 1exp1 1 1 exp 1 1 1 1 for x s L 1max 0. 13 Interestingly, if it is not located at the crest, the maximm precipitation rate does not depend on either or 1 even thogh its location does (shown in Fig. 3). This is becase the vale of 1 does not affect the location of the sorce line giving maximm precipitation, and the maximm difference between two exponentials of the form of (9a) depends only on the factor by which the exponents differ (and ths not on ). In contrast, when the precipitation maximm is located at the crest, its magnitde decreases when either the growth time increases or the moistre scale height decreases. a. Sensitivity of precipitation pattern to parameter variations Table 1 gives a range of plasible physical parameters with low, mid, and high vales. By selecting from these vales, we can create low, mid, and high vales for each of the nondimensional parameters, shown in Table 2, and adjst where extreme vales wold be otside the bonds of physically meaningfl soltions. The reslts are shown in Fig. 3 and discssed below. The precipitation pattern is most sensitive to plasible variations in 1,2 (the ratio of montain slopes to trajectory slopes). If the srface slopes are shallow compared to the hydrometeor trajectories ( 1,2 large), most of the precipitation occrs on the windward flank. In this case, precipitation efficiency is close to 100%. Conversely for low vales of (e.g., from strong horizontal winds or low fall speeds), trajectories are shallow and precipitation is carried far over the divide. This is essentially the behavior noted and explained by Hobbs et al. (1973) for the Cascades in Washington State. Note TABLE 2. Table of nondimensional parameters calclated from vales in Table 1 and Eq. (8). An asterisk denotes a vale that was adjsted to the minimm vale needed to get a physically meaningfl precipitation pattern. Parameter Low Mid High 1, 2 1.01* 4.8 800 1, 2 1.01* 3.0 500 0 0.31 40 0.25 0.83 2.0 also that the precipitation efficiency plmmets in this case. The vale of also affects the pattern considerably. Where the moistre height scale is deep compared to the montain height scale, precipitation maximizes at or near the crest, and the rain shadow is not strong. If moistre is confined to low elevations, the sorce line for the crest accesses low moistre air, and the precipitation rate maximizes far down on the windward flank. The hydrometeor growth time has an impact on the pattern when g is comparable to the montain halfwidth. In this case, precipitation is significantly displaced from the foot of the windward flank, and the maximm occrs at the crest. Moreover, becase of the restricted width of the sorce region, points to the lee of the montain sample air at relatively high altitde, and conseqently there is a rapid drop-off of precipitation to the lee. Evaporation impacts only the leeward pattern by constrction, bt it can have an important impact on the strength of the rain shadow if its characteristic time scale is comparable to the fall time of the hydrometeors. We next present reslts from a somewhat broader range of parameter choices, focsing on three aspects of the precipitation pattern: the maximm precipitation rate, the location of the maximm, and the strength of the rain shadow. We refer to section 5 and to Colle (2004) and Smith and Barstad (2004) for a description of how parameter variations can lead to dynamically indced changes in the precipitation pattern. TABLE 1. Table of dimensional parameters sed in model sensitivity analysis. Appendix B presents microphysical calclations for warm rain, grapel, and lightly rimed ice sed to gide these ranges. For simplicity a symmetric montain was assmed. Parameter (nits) Symbol Low Mid High Montain height (km) H 1.0 2.5 4.0 Montain width (km) L 1, L 2 10 30 100 Wind speed (s 1 ) 1.0 10.0 30.0 Terminal fall seed (m s 1 ) f 1.0 4.0 8.0 Growth time (s) g 200 1000 2000 Evaporation time scale (s) e 100 2000 Inf Moistre height scale (km) H m 2.0 3.0 4.0 b. Maximm precipitation rate Figre 4 shows contors of the maximm precipitation as a fnction of and f, and of H and L 1,2. Interestingly, for a given fall speed f, the maximm precipitation rate reaches its largest vale at an intermediate vale of. This is fndamentally de to the model framework representing the vertical dimension. As increases from small vales, the condensation rate increases becase of greater ascent on the windward flank. However advection increases too. This redces the width of the sorce region and the slopes of the sorce lines and hydrometeor trajectories become shal-

MARCH 2006 R O E A N D B A K E R 869 FIG. 4. Sensitivity of the maximm precipitation rate to plasible variations in parameters. Shown are contors of the maximm precipitation rate as a fnction of (a) and f, (b) L 1,2 and H. In each panel all other vales were held at their standard vales. In (a) the area nderneath the dashed gray line has 1, and so vales are not meaningfl. Contor interval is 2 mm h 1. lower (Fig. 1). This has two effects: the lengths of sorce lines decrease, which redces the line integral [i.e., Eq. (5)], and more hydrometeors formed at altitde are advected over the crest. As wind speeds increase, these effects come to dominate over the increased condensation rate. This is particlarly prononced at low vales of f (Fig. 4a), sggesting the effect may be stronger for snowfall than for rain. Demonstrating this behavior, the precipitation patterns for three vales of are shown in Fig. 5. A similar althogh sbtler effect can be seen in Fig. 4b: for a given montain height there is an optimm half-width (L 1 ) for the precipitation rate, cased again by the decreasing length of the sorce region for a finite growth time. Figre 4b also shows that away from this small L 1 regime, the maximm precipitation is the same for montains of the same srface slope. Bt importantly we note that, in contrast to a simple pslope model (i.e., precipitation proportional to slope), changes in srface slope do not prodce proportional changes in maximm precipitation rate. c. Location of maximm precipitation The location of the maximm precipitation rate is, not srprisingly, sensitive to the vale of, bt only p to the vale at which the precipitation maximm reaches the crest (Fig. 6a). Depending on the vale of f (and of corse the dimensions of the montain), the precipitation maximm may be at the crest for a wide range of. This reslt has some potential vale: in sch FIG. 5. Precipitation pattern as a fnction of. All other parameters at standard vales. The maximm precipitation rate attains its largest vale at an intermediate vale of.

870 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63 FIG. 6. Sensitivity of the location of the maximm precipitation rate to plasible variations in parameters. As in Fig. 4 bt contors are of the location of the maximm precipitation rate [i.e., (x s /L 1 ) max ]. A vale of 1 means the maximm precipitation rate is at the toe of the windward flank, a vale of 0 means it is at the crest. a regime, the pattern of the precipitation wold be robst nder a wide variety of storm strengths. The location of the maximm does not depend strongly on the montain height bt is sensitive to the half-width (Fig. 6b). d. Strength of rain shadow The strength of the rain shadow is determined by integrating (9) and (10) to give the total precipitation falling on the windward and leeward flanks, P W and P L, respectively. The reslting expressions are given in section c of appendix A, and Figs. 7a,b shows their ratio, P W /P L, as a fnction of and f, g and e. On its own, P W looks a lot like the maximm precipitation rate: as increases, P W increases p to a maximm and then decreases as precipitation is increasingly advected over the divide. Conseqently, P L contines to increase ntil, for very strong winds, significant precipitation gets advected beyond even the leeward flank. The strength of the rain shadow ths decreases as increases. Interestingly, for high wind speeds or low fall speeds, there is a sbstantial region of parameter space where the rain shadow is reversed (P W /P L 1). As is to be expected, the rain shadow is stronger for higher fall speeds. For or standard montain, the strength of the rain shadow is relatively insensitive to variations of the growth and evaporation time scales (Fig. 7b) over the range considered. Of corse, P W is independent of e, and only weakly dependent on g. For smaller montain ranges than or standard case, P W, and, hence the rain shadow, is mch more sensitive to g (not shown). Evaporation does have the potential to significantly affect the rain shadow when the evaporation height scale FIG. 7. Sensitivity of rain shadow to plasible variations in selected parameters. Graphs show contors of the total precipitation on the windward flank divided by the total precipitation on the leeward flank, P W /P L, for (a) varying and f and (b) varying g and e. Otherwise all vales held at standard vales. Note in (a) that the rain shadow is reversed (i.e., more precipitation on leeward flank) for P W /P L 1.

MARCH 2006 R O E A N D B A K E R 871 ( f e ) is comparable to the montain height scale, which means e 600 s in Fig. 7b. 4. Other aspects of the precipitation pattern a. Vertical wind shear So far we have not considered the effects of vertical wind shear, which cases condensate formed at higher elevations to be advected even frther than that formed at low levels (e.g., Colle 2004). This stretches ot the precipitation pattern, shown schematically in Fig. 8. In appendix C the eqations for sorce lines are derived in the case of niform vertical shear and g 0. The geometrical factors mltiplying the integrals in (5) are no longer constant with height. They are replaced by their mean vales, which is a reasonable approximation provided 3. Figre 9 shows the calclated precipitation pattern for the standard parameter set and a vertical shear of 0.003 s 1. The peak precipitation rate is redced by arond 25%, and the downwind precipitation rate increases appreciably. We note that jst calclating trajectories does not take into accont how vertical shear wold change the condensation rate in the sorce region (or, in a fller treatment, the impacts of changes on the phase and amplitde of gravity waves). Since typical ses of a simplified model like this might mean that wind speed at only one level is specified, it is sefl to ask if the effect of vertical shear can be imitated by changing the vale of one of the other parameters. Changing the vale of does not work becase it has an effect on both the distribtion and amplitde of the precipitation rate (Fig. 9b). By analogy with the natral nondimensional parameter is L f 0 z H m H. 14 For the shear case with z 3 10 3 s 1 and standard parameters, 0.25. Withot shear f mst be eqal to 2.1 m s 1 to prodce the same. Figre 9c shows that this vale of f qite closely reprodce the effects of wind shear, althogh the location of the precipitation maximm is offset. In the model development we have neglected coalescence of falling hydrometeors, a major effect of which is to increase fall speeds dring descent. The reslting parabolic trajectories wold be similar to those shown in Figs. 8 and B1. The reslts in Fig. 9 sggests sch effects may be emlated by tning the vale of f. b. Asymmetry of montain range Last, we present the precipitation patterns for asymmetric montain ranges. Figre 10 shows the pattern FIG. 8. Schematic illstration of hydrometeor trajectories inclding vertical wind shear. Zero growth time is assmed, so sorce lines and trajectories are coincident. reslting from the standard parameter set bt with L 1 / L 2 1/3, 1, and 3. If the height and total width of the montain range are fixed, the asymmetry of the montain shape controls the steepness of the windward slopes, the condensation rate via (3), and in trn the location and magnitde of the maximm precipitation rate. As the asymmetry varies, changes in windward slope are offset by changes in the sorce region width. In the case of no growth time and no evaporation these exactly balance, and total precipitation reaching the srface remains constant. This mst be so since the total lifting of the colmn is a fnction of crest height only. However a finite growth time means that the narrower the windward flank (sch as the Sothern Alps in New Zealand), the narrower the sorce region width, and so less the total precipitation that falls as is seen in Fig. 10. Note thogh, that the maximm precipitation rate is largest for the narrow windward flank. 5. Smmary and discssion We have soght to characterize the microphysical and geometrical controls on patterns of orographic precipitation. The model presented is an extension of the classic pslope model to inclde a representation of the vertical dimension, and to accont for both the growth time for hydrometeors and their advection by the prevailing wind. In essence the model is best thoght of as providing scaling relationships for the pattern and allowing for a straightforward evalation of the relative

872 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63 FIG. 9. Effect on the precipitation pattern of adding vertical wind shear. Standard parameters are sed except for g e 0 and, as indicated in the legend, (a) impact of increasing the vertical wind shear, (b) selecting different does not reprodce the effect of vertical shear, and (c) the vale of the fall speed can be tned to closely match the effect of shear. importance of the varios parameters within a self consistent framework. For the range of vales considered (Tables 1 and 2) all the parameters can have a significant control on the precipitation pattern (Figs. 3 to 7). If the moistre scale height is low compared to the montain height ( H/H m 1), precipitation will maximize low on the windward flank (and precipitation efficiency will be high); if the moistre layer is deep, precipitation will occr nearer the crest. Evaporation is effective when its characteristic depth scale is small compared to montain height ( H/ e f 1), and reslts in a strong rain shadow. If the montain width is large compared to growth length scale for hydrometeors (i.e., 1 L 1 / g 1), the maximm precipitation rate will occr along the windward flank. If it is small, precipitation efficiency is redced. Indeed, there may be no precipitation at all if clod particles are advected beyond the sorce region before reaching threshold size to fall. If the trajectory slopes are steep compared to the montain slopes ( 1 L 1 f /H 1), precipitation efficiency can be at, or exceed, 100%, and the maximm will be located down on the windward divide.

MARCH 2006 R O E A N D B A K E R 873 FIG. 10. Asymmetry of the montain range. The ratio of L 1 to L 2 is varied (bt with L 1 L 2 60 km). Otherwise the standard parameter set is sed. The maximm precipitation rate and the strength of the rain shadow depend on the asymmetry and becase of the finite formation time and evaporation time scale, so does the total precipitation rate. Shallower trajectories redce precipitation efficiency, psh the maximm toward the divide, and prodce significant leeward precipitation. The sensitivity of the pattern to variations in these parameters means that even dring a single storm the pattern of the orographic component of the precipitation mst be expected to change. This can also be inferred from observations (e.g., Hobbs et al. 1980; Herzegh and Hobbs 1980; Yter and Hoze 2003; Yter et al. 2004, manscript sbmitted to J. Atmos. Sci.; A04), and mesoscale models (e.g., Lang and Barros 2002; Smith et al. 2003; A04; Colle 2004). Therefore, even for a single storm and certainly on a climatological basis, model parameters mst be regarded as characteristic or effective vales incorporating the integrated effects of some complicated interactions, rather than reflecting actal physical vales. There are two main conceptal categories of simplified models of orographic precipitation. One focses on the seeder feeder mechanism in which washot of droplets (or ice) from the orographic clod occrs throgh accretion onto preexisting hydrometeors (e.g., Bader and Roach 1977; Carrthers and Cholarton 1983; Cholarton and Perry 1986; Robichad and Astin 1988). The others, like ors, (pslope models), concentrate on the rate of condensation dring the ascent

874 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63 of air and then distribte that as precipitation over the srface according to parameterization of microphysics schemes, or with some more arbitrary downwind smoothing of the condensation patterns (e.g., Sawyer 1956; Alpert 1986; Roe 2002; Smith and Barstad 2004). To a certain extent the distinction between these approaches is blrred and somewhat artificial. The common theme of these models is that ascent on a windward flank prodces a condensation region (i.e., or sorce region). How (and if) that condensation reaches the grond as precipitation depends on the dominant conversion mechanisms operating at the time. Selection of the right parameters can emlate these different mechanisms. For example, in the seeder feeder process, condensate is immediately incorporated into hydrometeors throgh accretion. Or model and those of, for example, Sinclair (1994) and Smith and Barstad (2004) are able to represent this by a short growth time, g. There are, thogh, some potentially important differences in formlation of these models. For example, most simple pslope models integrate the condensation rate in a vertical colmn. In actality, the slanting of hydrometeor trajectories means that precipitation arriving at a srface point did not all originate from the same horizontal distance pwind. Some aspects of the precipitation pattern might allow for some discrimination between models. The finite formation time in or model creates a maximm precipitation rate at intermediate horizontal wind speed: althogh the condensation rate increases with wind strength, the width of the sorce region decreases. At some point, as the winds increase, this latter effect comes to dominate, and precipitation decreases. This effect is present in the nonlinear treatment of Jiang and Smith (2003), bt does not occr in a seeder feeder model or in typical pslope formlations (e.g., Alpert 1986; Sinclair 1994) in which precipitation increases linearly with wind strength. Secondly, with strong enogh winds (or low enogh fall speeds), or model can give rise to a reverse rain shadow with more precipitation falling leeward than windward (Fig. 7). Last, or model also leads to apparent localized precipitation efficiencies well in excess of 100%, again de to the finite formation time and slanting hydrometeor trajectories. Ascertaining whether any of these effects are seen in a dense network of raingage data, or nder controlled conditions in mesoscale models wold be an important test. A confonding factor in this type of analysis is that, in reality, model parameters are not necessarily independent. If, for example, as increases, the precipitation changes from stratiform to convective, the appropriate vale of g likely changes, in trn affecting the pattern. Smith and Barstad (2004) present a highly adaptable orographic precipitation model incorporating a linear atmospheric response and capable of efficient calclation over complex terrain. In the configration analogos to ors (trianglar ridge, dynamics switched off) their precipitation always maximizes on, or leeward of, the crest (nlike or model), and precipitation rates diminish mch more slowly as increases. In their approach, condensation in the sorce region is integrated in a vertical colmn and redistribted as srface precipitation exponentially by a characteristic conversion time scale and a characteristic fallot time scale. By contrast, this redistribtion occrs in or model becase of a finite growth time and slanting hydrometeor trajectories. In these two approaches then, precipitation has arrived at the srface from different regions within the sorce region, and this gives rise to the differences noted above. It remains to be seen, however, whether this effect can practically be discerned in observations. A clear way forward wold be to compare these simpler models with a mesoscale model akin to that of Colle (2004) and to identify where the precipitation arriving at the grond originated, and to evalate the relative importance of the different cases of the pattern. The analytical soltion presented was obtainable only for a simplified montain geometry. In reality, parcels ndergo mltiple ascents and descents over individal ridges and valleys and can become well mixed dring their jorney. Among other things this makes the physical meaning of a local measre like precipitation efficiency nclear (Smith et al. 2003). The assmed flow response in or model is also nrealistic bt it enabled s to specify a niform windward sorce region. The details of the atmospheric response can in some cases importantly modify the shape and strength of the sorce region and hence also the pattern of precipitation (e.g., Smith 1979; Robichad and Astin 1988; Smith and Barstad 2004; Colle 2004). Frthermore, the model has also omitted many other processes that have been shown to be important in orographic precipitation: blocking (e.g., Katzfey 1995a,b; Rotnno and Feretti 2001), modification of the atmospheric flow by latent heating (e.g., Jiang 2003), orographic triggering of convective instability (e.g., Medina and Hoze 2003), evaporative cooling (Barros and Lettenmeier 1994b), valley circlations (Steiner et al. 2003), and mltiphase clods (e.g., Yter and Hoze 2003). This latter sitation can be incorporated in a version of or model with two layers, each with different growth times, wind speeds, and fall velocities; the freezing level then becomes another parameter in the model [Colle (2004) also considers the effect of the freezing level]. Even in sitations where these other mechanisms ap-

MARCH 2006 R O E A N D B A K E R 875 ply, the model reslts for the relative importance of formation time, advection time, and evaporation as a fnction of montain scale, for example, are likely to still have meaning. It also remains to be asked whether these other mechanisms are important in setting the climatological patterns of precipitation or if the sensitivity to changes in forcing can be qalitatively nderstood in terms of a modified pslope model sch as that of Smith and Barstad (2004) or the one presented here. The reslts of this and other stdies show that even greatly simplified representations of orographic precipitation contain a rich and interesting set of possibilities, the investigation of which is a step toward a greater nderstanding of the controls on what is one of the most important interactions between the land srface and the atmosphere. Acknowledgments. The athors are gratefl to Ron Smith, Socorro Medina, Dale Drran, and Sandra Yter for insightfl and instrctive conversations. And to Matthias Steiner and two anonymos reviewers for thoghtfl and constrctive reviews which improved the manscript, and to George Craig, the editor. MBB acknowledges spport from NSF Grant ATM 02-11247. B II B III : z w 1 g z 0 0 x g B III B IV : z 0 z s x s 0 f w 2 f w 1 f 1 H f 1 H f L 2, f L 1 A3 where z 0 is the altitde of the trajectory at x 0. Eliminating z 0 and rearranging gives z f 1 H f L 1x z s f 1 H f L 2x s g f. A4 Last, for x s L 2, the last portion of the raindrop trajectory is throgh air with no vertical velocity C II C III : z w 1 g z 0 0 x g C III C IV : z 0 z L2 L 2 0 f w 2 C IV C V : z L 2 0 x s L 2 f, f w 1 f 1 H f 1 H f L 2 f L 1 A5 APPENDIX A Analytical Expressions for Windward and Leeward Precipitation Totals a. Sorce line eqations We here develop the eqations for the sorce lines sing Fig. 1. There are three sorce line fnctions to consider: one for x s 0, one for 0 x s L 2, and one for x s L 2. First, for condensation occrring at (x, z) the point A II can be expressed as (x g, z w 1 g ). Then, from its geometry, the line A II A III mst satisfy the relation z w 1 g z s x s x g f w 1, A1 which on sbstitting w 1 H/L 1 and rearranging gives z f 1 H f L 1x z s f 1 H f L 1x s f g. A2 Eqation (A2) gives the sorce line for any windward (x s, z s ). Next, for the sorce line for the region to the lee of the crest 0 x s L 2. Again from the geometry of the trajectories where z L2 is the altitde of the raindrop trajectory at z L 2. Eliminating z 0 and z L2 gives z f 1 H f L 1 x H x s f f g. A6 Eqations (A6) and (A4) are in, fact, the same, which can be seen by sbstittion of Eq. (1). Eqations (A2) and (A6) therefore give sorce lines, locations of sorces of precipitation which will fall at (x s, z s ). b. Calclations of z max and z min Using these sorce lines we can calclate the z min and z max for (5a) (5c). From Fig. 1, the windward z min comes from the intersection of the sorce line [i.e., (A2)] and the topography. That is, z min z, where z (H/L 1 )(L 1 x), which gives z min L 1 f H z s f x s1 H f L 1 f g L 1 f 1 H f L 1, which on sbstittion from Eq. (1) simplifies to A7 z min z s gh L 1 ; A8

876 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63 z max is given by (A2), where z max z and x L 1 z max z s f x s1 H f L 1 f g L 1 f 1 H f L 1: A9 hence, z max L 1 f H z min. A10 For the leeward flank it has already been noted that there is a single sorce line eqation. Starting with Eq. (A6), z min z, where x g, and z max z, where x L 1 z min H x s f gh L 1. A11 From the geometry of the sorce line, z max can be obtained, z max z min g L 1 f w 1 which on rearranging yields f 1 H f L 1, z max z min L 1 g f 1 H f L 1. A12 A13 These expressions can be sbstitted into Eq. (5) to give the precipitation rate. c. Windward and leeward precipitation totals Using Eqs. (9) and (10), expressions can also be obtained for the integrated windward and leeward precipitation totals 0 P W L 1 g Rx s dx s, which on sbstittion from (9) and tidying p gives P W A14 1 1 1 R 0L 1 1 exp1 1 1 1 1 1 1 exp 11 1 1. For the leeward flank A15 P L 0L 2 Rx s dx s, A16 which sing (10) prodces P L 1 1 1 R 0L 2 2 1 exp1 1 1 1 1 exp1 1 11 exp 2. A-17 The strength of the rain shadow can be defined as P W /P L. APPENDIX B Microphysical Parameters Here we provide approximations to the microphysical eqations that allow s to estimate vales of the following parameters: g, f, evap, and moist the time scale for moistening the leeward air de to hydrometeor evaporation into it. Note that, if the radii of the hydrometeors in the clod approaching the montain are already above their threshold vales at x L 1, they will prodce precipitation with no preliminary growth period. Since we are concerned here with orographic enhancement of precipitation, we ignore this contribtion to the precipitation pattern. We assme two growth mechanisms: vapor deposition and collection of clod particles. Ths we neglect interactions among falling hydrometeors. The mass of a clod drop of radis r at altitde z changes in time according to the eqation dm dm r, z r,z dm r, z. dt dt dtcoll a. Vapor deposition For a single hydrometeor B1 dm r, z rgq s zz 1 B2 dt where G 10 5 m 2 s 1 is a slowly varying fnction of temperatre T(z); (z) is the spersatration (relative to ice or water) in the clod. In the case of liqid drops growing in a warm clod and ice crystals growing in a flly glaciated one, we assme that the depositional growth rate jst balances the rate of liberation of vapor in the rising air so that Eq. (B2) can be replaced by