DEPARTMENT OF COMMERCE SINCLAIR WEEKS, Secretary WEATHER BUREAU F. W. REICHELDEREER, Chief MONTHLY WEATHER REVIEW JAMES E. CASKEY, JR., Editor Volume 83 Number 3 MARCH 1955 Closed May 15, 1955 Issued June 15, 1955 ON THE NUMERICAL PREDICTION OF PRECIPITATION J. SMAGORINSKY Numerical Weather Prediction Unit, U. S. Weather Bureau, Washington. D. C. and G. 0. COLLINS I Short Range Forecast Development Section, U. S. Weather Bureau, Washington, D. C. [Manuscript received March 14. 19551 ABSTRACT With the three-dimensional field of velocity predicted by numerical methods it is possible to predict the moisture distribution and hence the occurrence of large-scale saturation. A three-parameter model was used to predict the 12-hour precipitation for the early stages of the storms of November 24, 1950 and November 5, 1953, neglecting cloud storage, supersaturation, a possible lack of condensation nuclei, evaporation from falling droplets, and moisture sources. Large-scale orographic influences were taken into account. A quantitative comparison of the predicted rainfall with the correspondingly large-scale smoothed observed precipitation indicates a skill comparable to that of the predicted flow. An examination of the small-scale observed rainfall indicates that in these cases convective instabilfty resulted in large standard deviations from the large-scale average. Numerical prediction of regions of convective instability, which is also shown, could for the time being be utilized for subjective interpretation. 1. 2. 3. 4. 5. 6. 7. 8. 9. Page 53 53 54 55 56 58 59 59 59 65 66 67 67 No. 4, Bultland, Md. 341693-6"1 1 Present affiliation: Joint Numerics1 Weather Prediction Unit, Federal Offloe Buildlng 1. INTRODUCTION Workers in the field of numerical prediction have con- cerned themselves almost exclusively with the prediction of changes in the three-dimensional mass field and hence, as a direct consequence, changes in the large-scale velocity and temperature fields. The prediction of these elements is necessary though not sufficient for the prediction of the large-scale precipitation fields. Thus onefinds in the literature (e. g. [7, lo]), for the most part only qualitative comparisons between numerically predicted vertical mo- tion fields and observed precipitation. The prediction of precipitation is a difEcult task mainly for two reasons: (i) a lack of detailed knowledge of the physics of formation of cloud particles and their precipita- tion; (ii) the fact that, unlike the normal situation with the other meteorological elements such as pressure, 63 54 M O N T H L Y W E A T H E MABCH 1955 temperature, and wind, small-scale precipitation often is of much greater magnitude than the large-scale precipita- tion. Our object will be to devise a dynamical large-scale precipitation model which by the application of numerical methods will enable us to predict. In what follows we will circumvent (i) by assuming [14] (a) there are always suffi- cient condensation nuclei, (b) no supersaturation, (c)no supercooling, (d) no non-adiabatic processesasidefrom those resulting from changes of state, and (e) and (f) cloud storage and evaporation from falling droplets are both negligible compared with significant amounts of precipitation. Furthermore, wewill assume no moisture source outside of the atmosphere, i. e., no evaporation from the surface of the earth. The justification, for shorter periods, lies in the fact that evaporation as a function of space is generally of much smaller amplitude and much more uniform than condensation, although in the large the two must balance? It is obvious that the present laminar lower boundary condition used in numer- ical prediction is incapable of permitting the transport of moisture across the lower boundary. In order to take evaporation sources into account we must ultimately assume a turbulent boundary layer to make possible eddy diffusion normal to the boundary. DifEculty(ii)will, for quantitative purposes, be ig- nored, but will be discussed again later. It must be pointed out that as a consequence of the large non-linear interaction, even the large-scale precipitation calculations must be in error. 2. CONDENSATION If T is the mixing ratio, then during the condensation process drldt>2, so that Both for small- and large-scale motions dpd/dt is given to good approximation by thus d In r8 dt bP " -A- b In W) wo, the side condition in (16) is required to insure that d In r8/dt<0 for the condensation process. Equations (l ), (2), (5)) and (16) taken together with a knowledge of V and w sate for the prediction of precipi- tation, given as initial conditions the spatial distribution of mass and mixing ratio. With the approximation that w.=wbp& where w is the vertical velocity when the height, z, is a vertical co- ordinate, that -= -g p d by the hydrostatic approxima- tion, and that the lapse rate is moist adiabatic, (16) reduces to a result deduced by Fulks [8]. Fulks did not intend to use this result prognostically but rather to cal- culate precipitation by means of an analogue of equation (1 ) from a known field of r, T, w, and p . However, we have shown equation (4) sufficient to calculate contem- porary precipitation. aP bz 3. VERTICAL MOTION The large-scalefield of vertical motion is not an ob- servable quantity. However, application of the hydro- static and geostrophic filtering approximations to the primitive hydrodynamic equations permits one to deduce the vertical motion field given only the three-dimensional mass (or pressure) field and appropriate boundary conditions. This can be seen from the following differ- ential equations [7] : thermodynamicenergyequation (IS) topredictthethl.ee-dimenslonal velocity. By * There is apparent inconsistency in the usf~ of the adiabatic approximation in the this, we assume that the heat of condensation does not materlally altar the fleld of flow overperiods of the order 24-36 hours. However,there Is evidence that for longer niflcant interactions [Ill. In the present situation we may regard the approxhation es periods, all non-adiabatic mums and sinks, orography and dch friction result in sig- a quasi-linearization with respect to the condensation in analogy with mall perturbation theory where one assumes that the basic flow determinea the propagation of a disturbance, but that this disturbance does not aifect the mean flow to the first order of amall quam tities. 56 MONTHLY WEATHER REVIEW in which we take 7" V"+f -f 1 In 8=const+" In p+ln C CP j is the Coriolis parameter and c, is the specific heat of air at constant volume. In deriving the vorticity equation (17) the vertical advection of Vorticity, oavlbp, and the turning of the -. . " vortex tubes in vertical planes, - --- - have been aw a v aw bu ax ap a y a p neglected. Since initially z is given as a function of x, y, and p , system (17), (18) and (19) may be regarded as two dserential equations in two unknowns: &/at and a. Customarily w is eliminated and the system solved for bz/bt. Then w can be determined explicitly by substi- tuting back either into (17) or (18). However, we may alternatively eliminate bz/bt and write the differential equation governing w : in which we abbreviated For simplicity, we have replaced q by.f when r ] occurs as a coefficient and also have assumed i3 $/a p , the static stability, to be constant in an isobaric surface, an approxi- mation effectively made in all two-parameter models. This three-dimensional Poisson equation may be solved by relaxation methods, once given the three-dimensional mass distribution which is sufficient to determine the inhomogeneous terms, and also given boundary conditions at p=po, p=O, and at the lateral boundaries. Consistent with a two-parameter model w may be expressed as a quadratic function of p . Taking w=wo, p=po w=o, p=o } Equation (20) may then be written as (24) We have thus reduced the problem to one of the solu- tions of a two-dimensional Helmholtz equation into which the boundary conditions at the top and bottom of the atmosphere already have been incorporated. The surface vertical velocity, wo, is induced by orogra- phy or by skin frictional action and is given by [5] wo =Vo-Vh+b [K sin (24 To (2 5 ) where h is the elevation of the large-scale orography, To is the geostrophic relative vorticity in the friction layer, K is the average eddy diffusivity and Y the angle between the wind and isobars. 4. SPECIALIZATION TO A THREE-PARAMETER MODEL For the purposes of actual calculation, the general pre- diction equations are reduced to a three-layered model aa described in general by Charney and Phillips [7] and in particular by Charney [4]. This model is equivalent to an atmosphere consisting of three divergent barotropic layers. The vertical velocity may be computed from the thermodynamic energy equation (18) which may be rewritten where D/Dt is the horizontal individual time derivative and a bar denotes the standard value. A quasi-lineari- zation has been performed so that when the stability appeared as a coefficient, the standard atmosphere sta- bility is used. This is consistent with the approximations made in deriving the prediction equations for this model. The numerical integration scheme used here carries the history of the motion in the potential vorticity. Hence b +/b t is never calculated explicitly. It is thus necessary to approximate b 4/a t in (26) by finite differences. In numerically integrating the potential vorticity equation, it is necessary to perform the initial time integration non- centrally over one finite difference time interval and thence to proceed by means of centered time differencesover double time intervals [7]. The result is that a small oscil- lation in 4 with a period of two time intervals is artificially induced.* This oscillation is barely detectable; however, then o may be expressed in terms of the vertical motion, a*, at the level p*=p0/2 : of the flrst barotropicpredictions [SI. The hourly forecast data wereavailable for in- 4 This was observed by one of the writers who participated in the ENIAC calculatlon spection since the internal memory limitations of the ENIAC required the output of all intermediate results. MUCH 1955 MONTHLY REVIEW 57 where time differences of the b, field are taken over an odd number of t,ime intervals, a significant error is intro- duced. In the case of differences over one time interval, the oscillation in some instances completely masks the physically real finite difference approximation to the con- tinuous time derivative. Therefore in the present calcu- lations, all time differences are ta.ken over a double interval. In finite difference form (i and j are horizontal coordi- nates, k pressure, and T time and the corresponding inter- vals between integral values of these coordinates are As, As, Ap, and At, respectively) equation (26) becomes where J A s=300 km., At=% hr., and mi? is taken to be unity for the latitude span used here on a Lambert conformal projection(see for example [12]), k=1, 13, 2, 2%, 3 refer to 200, 350, 500, 675, 850 mb., respectively, and the boundaries k=%, k=3% are placed at 25 mb. and 1000 ab., respectively. tu can first be computed cent,rally over a double time interval a t t=% hour using information at t=O, and t =l hour. The initial vertical velocities s t t =O are therefore taken to be the same as those at time t=% hour. Inspec- tionof the subsequent data showed that changes within a half hour interval are sufficiently small that no serious error is introduced by this assumption. Since the vertical velocities computed by the model, especially those a t 350 mb., are at rather high levels in the troposphere and above the maximum concentration of moisture, it is considered necessary, for the prediction of precipitation, to deduce vertical velocities a t lower levels. Predictions with models giving greater definition at lower levels, e. g., a three-layer model with p2 the vertical coordinate [4] which gives values at about 575 and 825 mh. or perhaps a five-level model, may eliminate this need to interpolate. Given w at four levels: 675, 350 mb. and the boundaries 1000 and 25 mb., one can fit a cubic with respect to pressure, giving a continuous function: 'w(P) =M(p)wm+N(p)~aro (30) in which M and N are the interpolation variables, and the anterior superscript denotes first approximation. As previously pointed out, the computations of vertical velocities in this particular model are based on the assump- tion t,hat the vertical velocity is zero at 1000 mb., the lower boundary. This assumption may not produce serious errors in the prediction of the geopotential field but does become more detrimental in the prediction of precipitation. Vertical velocities at the surface produced by forced ascent over orographic barriers can contribute considerably to the precipitation [lo]. The most logical and consistent way to include the effects of large-scale varying terrain would be to incorporate this lower boundary condition implicitly in the prediction equations. This can be done without great difficulty (see for example eq. (24)). Since the flow prediction equations used here do not take terrain into account, these effects are included a posteriori. Vertical velocit,ies at the lower boundary, 1000 mb., are computed from (see eq. (25)) wo=Vo*vh=Vgoo*vh (31) where VQO, is the geostrophic wind at 900 mb. Vgo, was extrapolated quadratically from information at 200, 500, and 850 mb. For a second approximation to the vertical velocity we define "(p)='w(p, P wo 1000 (32) While this procedure is somewhat arbitrary, it has the characteristic, somewhat similar to that of the atmosphere, that the effects of the lower boundary on the flow are damped out approximately linearly with decreasing pres- sure. Vertical velocities may be calculated in this manner for p=400(100)900mb. denoted by k'=l to 6, respectively. In the integration of the finite differenceform of the potential vorticity equation the choice of At was bounded by the requirement of computational stability with respect to As. Since there is no such restriction in using the results of the integration of the potential vorticity equation together with the finite difference form of the system (l), (2), (5), and (16), a time interval, A f t , in precipitation equations, may be chosen commensurate only with the time scale of the dependent variables. The b, field at the levels k', may be quadratically inter- polated and extrapolated from the values at k=1, 2, 3. The temperature @=-- - R dP ") is predicted only at k=1% and 2%. While a linear interpolation for inter- mediate levels may adequately determine the tempera- 58 MONTHLY REVIEW MAEOH 1955 ture, experiments in extrapolating to flanking levels prove inadequate for our purposes. This can be over- come to some extent by making selective use of the ob- served detail in the initial soundings. The observed initial temperatures at each level IC' are smoothed sub- jectively yielding T i J k ' . The predicted temperature change, AT, at k=1%, 2% over the time interval A't is then interpolated and extrapolated linearly to give the tem- perature change at the levels k'. This process may be iterated: T;,!$'=[T''+(AT)"+'] t j k ' (AT)~3:=[Uk~(Ar),,+bk~(AT),.]~'w } (33) where and b k t are determined from the interpolation formula. Since & is determined independently by a much simpler scheme, there will be some hydrostatic inconsistency with the corresponding Tkl field. This may be avoided by determining Tkt as described, but integrat- ing vertically to obtain &, using as a reference one of the &"S. The system of prediction equations (l ), (2 ), (3 ), (5 ) and (16) in finite difference form is thus: (34) (35) 1 and rfl-l=rO for /=O for r'>O For computationa,l convenience the density lapse rate is taken to correspond to that of the moist-adiabatic lapse rate, and hence is a function of height only. This approximation is not necessary with numerical prediction models of three or more parameters. I n t h e present three-parameter model the static stability, although constant in the vertical, may vary in time and in the hori- zontal. Hence the continuous form in finite differences becomes At the upper and lower boundaries &$lap must be de- termined non-centrally. In all calculations, for con- venience, L is taken as the latent heat of condensation, and e and r are taken with respect to water, irrespective of T. This restriction may easily be removed. Comparisons of predicted geopotential tendencies with observed 12-hour changes [2] indicate that a time interval, A't, of 12 hours represents an upper limit. However, ap- preciable error can be introduced by the assumption that the saturation state is constant over so long an interval. It is estimated that A't=3 hours would result in more tolerable truncation errors. It is possible, however, to use a larger Art if the truncation error resulting from the con- stant saturation state approximation is to some extent eliminated. This may be accomplished by estimating the time, t,, of occurrence of change of saturation state during A f t . One can determine from the initial mixing ratio, ro, the saturation state at that time. In the instance where To<(, then one may determine whether saturation will occur during A't from the non-central form of (37) between t =O and t,: t , is thus defined and determinable. If t,>Art then drldt=O over the entire interval. If not, dr,Jdt computed from (38) is to be weighted by the factor (1-tJA't). For initially saturated conditions oneuses a correspondingly derived criterion for the occurrence of unsaturated condi- tions during Art. Thompson and Collins [14] used a simi- lar technique for precipitation calculations in which A't was taken as 12 hours. They did not take horizontal moisture t.ransport into account. 5. THE EXPERIMENT The sit,uations chosenwere the early synoptically unspectacular stages of two cases of rapid cyclogenesis. The initial conditions were taken at 0300 OMT, November 24, 1950, and 1500 GMT, November 5 , 1 9 5 3 . The object was to predict the accumulated precipitation over a 12- hour period. For this purpose A't was first taken as 12 hours. With the scheme outlined in the previous section, the only predictive element is a determination of the time of occurrence of change of saturation state. Otherwise, only the initial w and r fields are needed. A secondcal- culation was prepared in which A't was taken as 3 hours and, therefore, the full set of prediction equations were used in 4 iterations for a 12-hour forecast. Numerical flow forecasts for the same two periods UARCH 1955 MONTHLY WEATHER REVIEW 59 treated here were performed by the Princeton group using a three-parameter 900-700400-mb. model [4], but 3-hourly vertical motion fieldswere not calculated. The fact that these situations were highly baroclinic in the lowest levels meant that the 850-500-200-mb. model was incapable of recognizing the large amount of potential energy available for conversion to kinetic energy. The result is that the predicted development by the 850-500- 200-mb. model was inferior to that by the 900-700-400- mb.model in both cases. This is particularly true for the November 5, 1953, case. 6. DISCUSSION OF RESULTS For the purposes of verifying the forecasts of large- scale precipitation accumulating over 12-hour periods, observations from all hourly precipitation reporting sta- tions were utilized. This of course gives a relatively fine- grained picture of the actual precipitation-with stations having an average separation of roughly 30miles. The observations were taken from Climatological Data for the United States [15] and from specially prepared listings supplied by the National Weather Records Center in Asheville, N. C. NOVEMBER 24, 1950 A comparison of the 850-mb. maps at initial time 0300 GMT (fig.2A) and 12 hours later (fig. 2E) shows that the depression centered south of Lake Superior filled but a secondary trough was forming to the south. This is predicted by the modelused(fig. 2E). At 500 mb. the Low centered in Wisconsin (fig. 2B) elongated and moved to southern Lake Michigan (fig. 2F). Again this evolu- tionwas adequately predicted (fig. 2F). At 200 mb. the flow forecast was not quite as good, but has less bearing on the present problem and is not shown. The calcu- lated vertical motion fields at t,he beginning and end of this 12-hour period at 675 mb. and at 350 mb. are shown in figure 2 (C, Dl G, and H). The predicted fields of 6 and w at the intermediate 3-hourly intervals ’are not shown. In figure 3A are plotted all non-zeroandnon-missing 12-hourly precipitation reports to the south and east of the heavy boundary. The results using A’t=12 hours (dashedlines) indicate a maximum of .23 inch at Lake Erie. However for A’t=3 hours (solid lines) there occurs a double maximum, one at Lake Erie of the same magni- tude and another in Tennessee with somewhat larger meximum value. A comparison of non-zero points and predicted precipitation greater than .01 inch indicates a good qualitative forecast for either model and thus a fairly reliable statement of the time of onset and cessation of precipitation. However examination of the individual reports indicates large deviations even from the results usingA’t=3 hours. Figure 3B summarizes the large- scale average observed precipitation and the standard deviation from this average. This was constructed by averaging all observations within the isohyetal channels of the prediction in whichA‘t=3 hours. These points were computed from over 1,300 observation stations in eastern United States. The model is seen to have over- predicted the large-scale precipitation by a factor of approximately 2 for the intermediate amounts and by a factor of 1.25 for the maximum. It is also important to bear in mind that the three- parameter modelused here (850-500-200-mb.)gives a flow forecast inferior to that of the 900-700400-mb. model even in the first 12 hours. I t is evident from a comparison of figure 2 (C, Dl G, und H) with figure 3A that upward vertical motion is only a necessary though not a sufficient condition for precipitation even for qualitative purposes. For in- stance one would not have judged from the vertical velocity field alone that the primary precipitation maxi- mum would be in Tennessee. NOVEMBER 5, 1953 This case deals with a nascent cyclone in the northeast Gulf of Mexico at 850 mb. (fig. 4A) which deepened and traveled up off the east coast to 31’ N. latitude in a 12-hour period (fig. 4E). Actually the 850-500-200-mb. model gave a forecast which was quite poor in describing the observed development. No deepening and only slight eastward motion were predicted at 850 mb. (fig. 4E); At 500 mb. the primary trough in northeastern United States (fig. 4B and F) was predicted to move too rapidly to the east and the development of a closed circulation wasmissed entirely. On the other hand the secondary in the Gulf, associated with the 850-mb. closed Low, was predicted to move to the east at the approximate speed observed. I t should be noted that the anticyclogenesis predicted in north-central United States at 850 mb. and 500 mb. but not observed does not directly affect the area of our precipitation forecasts. In figure 5Awe have, as before, the raw precipitation observations superimposed on forecasts computed using A‘t=12 and 3 hours. An essential difference in the observed precipitation between this case and that of November 1950is the oc- currence of large amounts along the Carolina and southern Virginia coasts. The observed 850-mb. flow at the begin- ning and end of this period showsonshore winds. This coastal effect is well known [I] but is not taken into account by the present theory. Fresumably this could be accom- plished by an extension of the theory resulting in equation (25) to apply to variable surface roughness. A significant difference is observed between the pre- dictions using A’t=12 hours and 3 hours. For A’t=12 hours the maximum is only half as large as that for A’t=3 60 MONTHLY M a B C H 1955 FIGURE 2.-November 24, 1950 case: initial, predicted, and verifying horizontal flow (A, B, E, F,); initial and predicted vertical flow (C, D, G, H) for the 12-hour period 0300-1500 GAIT, November 24, 1950. MONTHLY WEATHER REVIEW 61 Figure 2"Continued 341883-55-2 62 MONTHLY WEATHERREVIEW MAECH 1955 FIQURE 3.-(A) Observed precipitation during period 0300-1500 GMT November 24, 1950 with computed isohyets (A’t=3 hr., solid and A’t=12 hr., dashed) superimposed. R denotes precipitation 20.01 in. but amount is unknown and thus R is not included in the numerical verification. Missing reports and zero precipitation are not shown. (B) Summary of results with observations smoothed for large-scale verification. Numbers in parentheses at plotted points give number of stations averaged in the respective isohyetal channel. Dashed lines indicate standard deviation of observations in the respective channel. MARCH 1955 MONTHLY WEATIIER REVIEW 63 FIGURE 4.-November 5, 1953 case: initial, predicted, and verifying horizontal flow (A, B, E, F); initial and predicted vertical flow (c D, G, H) for the 12-hour period 1500 mrr November 5 to 0300 ~M T November 6, 1953. See p. 64 for parts E, F, G, H. 64 MONTHLY REVIEW MAEOH 1955 MARCH 1955 MONTHLY W E A T H E R R E V I E W 65 -- . hours and is also displaced to the north. Had the flow . ' prediction for this period been correct, as essentially resulted from the 900-700400-mb. model, the maximum . ' would have been elongated northeastward along the coast. , However one can speculate that, in spite of this elongation, the maximum would not have been reduced because of the larger vertical velocities associated with the increased icular case the vertical motion fields (fig. 4C, Dl G, and H) would have given a fairly realistic qualita- tive indication of the region of precipitation without referring to the moisture field and its changes. On the other hand, the magnitude of the vertical velocity would have been misleading. In the November 24, 1950 case a t 675 mb. an average maximum vertical velocity of"6.5 cm sec" yielded a predicted maximum precipitation of .23 in./12 hr., while in the November 5, 1953case an average maximum vertical velocity of"2.5cmsec" gave .43 in./l2 hr. ' In figure 5B we have a comparison of the large-scale smoothed observed precipitation vs. the computed. In this case the plotted points, representing a total of 184 observations within the Verification area chosen have more-or-less uniform scatter about the perfect forecast is interesting to observe the occurrence of much larger standard deviations than were found in the first case. This is a normal characteristic of precipitation at lower latitudes where the incidence of convective instability is greater. 184 STATIONS , o .m .x) .Y) .a .s a BO .m ea 90 OBYRVED PRECIPITATION 1IN.p 12HR) B (Mean within predicted Jsahyets with class interval .IO inches per 12 hr.) WGURE 5.-(A) Observed precipitation duringperiod1500 GMT November 5 to 0300 QMT November 6, 1953 with computed isohyets (A'6=3 hr., solid and A't=12 hr., dashed) superimposed. R denotes precipitation 20.01 in. but amount unknown and thus R is not included in the numerical verification. Missing reports 7. PREDICTION OF RELATED ELEMENTS Charts of the predicted and observed 12-hour changes in dew point at 900 mb. were prepared for the November 5, 1953 case (fig. 6). Cursory qualitative examination of the field distributions reveals an excellent correspondence. The 7.5' C. rise in western Kentucky is predicted quite well. Although the fall area in the Great Lakes region is correct, the minimum is predicted to be too far east. Howemr, although tho secondary minimum in western Virginia is correctly placed, the predicted minimum is too small in magnitude. A large discrepancy is noted over Arkansas where the observed rise is predicted as a sub- stantial fall. (.ln the other hand the negative band through the Gulf States, Oklahoma, Kansas is correct, although the observed minimum in the Gulf States is predicted farther to the northwest and twice too large. The rise off the southeast Atlantic Coast is correct but one-third the andzero precipitation are not shown. (B) Summary of results observed magnitude. Fortunately the discrepancies with observations smoothed for Iarge-scale verification. Num- pointed out above do not greatly affect the precipitation bers in parentheses at plotted points give number of stations averaged in the respective isohyetal channel. Dashed lines calculations* indicate standard deviation of observations in respective channel. The details of the moisture Prediction Caldations show 66 MONTHLY W E A T H E R R E V I E W MARCH 1955 FIGURE 6.-Predicted and observed 12-hourchange of dew point (“C.) at 900 mb. between 1500 GMT, November 5 and 0300 GMT, November 6, 1953. that on the average, the condensation, the local time derivative of mixing ratio, and the horizontal and vertical transports of mixing ratio are all of the same magnitude. Hence any approximation neglecting one of these would be invalid. We see that the large-scale average precipitation is on the whole predicted correctly. Thus onecould expect to be able quantitatively to predict total precipitation over large watersheds the scale of a unit mesharea- approximately 35,000 sq mi. The fine structure due to small-scale instability is not capable of being predicted by the models used. For a wholly dynamical prediction, small-scale non-linear theory of the type utilized by Tepper [13] wouldbe required. The obvious difficulty is that an extremely h e network of surface and aero- logical observations wouldbe required in order to ade- quately specify initial conditions. For the time being this is not economicallyfeasible operationally, even if the theory were adequately developed. For most pur- poses it might besufficient if statistical moments of precipitation higher than the mean (such as the standard deviation) weresomehow attainable by dynamical methods. Thus, what suggests itself, is a statistical-mechanical approach such as is used in other branches of physical science. The fine structure itself is not predicted but interpolated from Observations FIGURE 7.-Predicted and observed areas of convective instability (bOs/bp>O), 0300 GMT, November 6, 1953. rather the distribution of the statistical properties of the fine structure. Even for this modest requirement, it is necessary to understand the mechanism of convective processes in a moist atmosphere and the dependence of the small-scale dynamics on the ambient large-scale conditions. From parcel stability considerations we have the well known result that a necessary condition for convective instability is that the equivalent potential temperature decrease with Figure 7 shows a 12-hour forecast from the 1500 GMT, November 5 , 1953 situation of the occurrence of convective instability according to the parcel criterion. I t is seen that the verification with the observed occurrence is quite good,especiallysince the flow prediction at this time was already beginning to degenerate. The release of this instability could be accomplished, theoretically, by sufficient lifting. 8. FRICTIONAL VERTICAL VELOCITY Although the influence of frictional divergence is not included in the prediction model, it is of interest to study the magnitude of the vertical velocity so produced in one of the cases treated. Figure 8 shows the frictional vertical recent is an article by 0. H. B. Priestley, “Buoyant Motions and the Open Parcel”, 8 Parcel dynamics have often been viewed critically, and with good reason. The most 7he Metcorolopicul Magazine, vol. 83, No. 982, April 1954, pp. 107-114. MABCH 1955 MONTHLY WEATHER REVIEW 67 FIGURE 8.-Frictional vertical velocity (cm. sec. -*) at top of friction layer computed a t 1500 GMT, November 5, 1953. velocity at the top of the friction layer computed from the second term of equation (25) using the observed 850-mb. 00w at 1500 GMT November 5, 1953. The constants arechosen according to average values suggested by Brunt [3]: K=10 m.2 sec.” and ~~2 2 .5 ’ . We see that the magnitude of the forced frictional vertical velocity can attain values of over 1 cm.sec.“. Assuming that this boundary influence decreases linearly with decreasing pressure (as was done for the orographic influence), the computed free vertical velocities even at 675 mb. would change by not more than 25 percent. However there may result significant shifts in the location of centers of maximum or minimum. For example the frictional maximum of + 1.5 cm.sec.” south of Talla- hassee would result in shifting the free maximum at 675 mb. to the west over the northern Florida peninsula with little change in magnitude. On the other hand, fields of free vertical velocity not associated with large-scale storms, and which are thus moreof the magnitude of frictional vertical velocities, could undergo large percental changes, conceivably chang- ing sign. Under proper moisture conditions this could make the difference as to whether or not small amounts of precipitation would result over large areas. 9. CONCLUDING REMARKS The foregoing represents a rudimentary physical frame- work by which the occurrence of precipitation may be predicted. The reality of the results may be thought of as depending in part on the goodness of the numerical three-dimensionalflow prediction and in part on the physical model accounting for moisture changes and the prccipitation process. We have seen that the precipitation calculations were not too sensitive to the failings of the flow prediction in the two cases studied. To some extent this must be due to the time-wise integrating process (eq. (3:)). Since these were only 12-hour predictions, it is difficult to surmise the influence of a continually deteriorating flowprediction over a 24-hour period, On the other hand the predictions of dew point and convective instability, whichdo not’ involve such an integration, apparently have not suffered inordinately. The approximations regarding the condensation process discussed in the Introduction do not appear to be crucial in these cases. A more comprehensive study for a larger number of cases should disclose any systematic effect on the predictions. Relaxation of the constraints introduced by these approximations depends on progress in the field of cloudphysics. Without these constraints it should in principlebepossible to distinguish betweenlarge-scale cloudiness and clear skies. The approximations whichwere made only for com- putational convenience in that portion of the calculations done by hand, may of course be eliminated in program- ming the entire problem for a high-speed computing machine. Judging the results of the two caseschosen by the subjective standards normally used for precipitation verification, we conclude a skill comparable to that of the predicted flow. REFERENCES 1. T. Bergeron, “The Problem of Artificial Control of Rainfall on the Globe, 11. The Coastal Orographic Maxima of Precipitation in Autumn and Winter,” Tellus, vol. 1, No. 3, August 1949, pp. 15-32. 2. B. Bolin and J. G. Charney, “Numerical Tendency Computations from the Barotropic Vorticity Equa- tion,” Tellus, vol. 3, No. 4, November 1951, pp. 3. D. Brunt, Physical and Dynamical Meteorology, Cam- bridge University Press, Cambridge, 1939. 4. J. G. 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