TC en I LIBRARY lairVTEKSITY OF CALIFORNIA PAVIS ^ ^ OF MLIf ORKIA •RARY Wvis " >PY 2 STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION BULLETIN NO. 88 PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION EDMUND G. BROWN Governor MAY, 1960 HARVEY O. BANTCS Director of Water Resources SEP I'^mo LIBRARY STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION BULLETIN NO. 88 PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION EDMUND G. BROWN Yi^^^^f^ HAR\'EY O. BANKS Governor pf^^.-^.jSiSI Director of Water Resources MAY, 1960 LlBKAKi UNIVERSITY OF CALIFOKMIA DAVIS STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION BULLETIN NO. 88 PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION EDMUND G. BROWN ^2^^ShP\ HAR\'EY O. BANKS Governor IIv.^'^TSbS Director of Water Resources MAY, 1960 STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION BULLETIN NO. 88 PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION EDMUND G. BROWN H^PS^P\ HARVEY O. BANKS Governor M\^^^ t^m^l Director of Water Resources MAY, 1960 PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION by Dr. Joseph B. Knox Meteorological Consultant Affiliated with the Lawrence Radiation Laboratory. University of California, Livertnore, California. JOSEPH B. KNOX 37 La Gonda Court Danville, California Majr 10, I960 I'r^ Ilar-vTc" C. lianks, Director California State DepartiTicnt of Water Resources P. 0. Box 386 Sacramento 2, California Dear Mr. Banks: Submitted herewith is a report on procedures for estimating maximum possible precipitation for large watersheds in California. The study and report were developed at the request of the De- partment of Water Resources to assist the Depai-tment in its determina- tion of necessary reservoir and spillway capacities in mountainous watersheds. Specifically, the investigation applies the extreme values of known meteorological parameters to an atmospheric model in order to estimate the maximum accumulated precipitation and its distribution for the Feather River Basin. To obtain fairly detailed geographic distributions of the maxi- mum possible (accumulated) precipitation, an electronic computing pro- gram was developed. The practical advantage of such a program is that the model and the methods are readily applicable to other mountainous watersheds in the State. Sincerely, Joseph B. Knox Meteorological Consultant -I- TABLE OF CONTENTS Page Letter of Transmittal i Organization v Acknowledgements vi List of Symbols vii l.l Introduction 1 2.1 Model for Orographic Rainfall 2 2.2 The u, w Computation 3 2.3 The Trajectories of Precipitation Products 6 2.4 The Rate of Orographic Precipitation . . . 9 2.5 The Vorticity Budget Check on the u, w Solution 9 3.1 The Model for Rainfall from Large-scale Synoptic Disturbances 11 4. 1 A Model of the Time Distribution of the Precipitation Rate 13 5.1 The Case Study of December 1955 — the Feather River Basin 15 6. 1 The "Near-Maximum" Storm 17 7.1 The Estimated Maximum Possible Storm 18 8.1 Concluding Remarks 20 References 23 TABLES 3.1 The Hourly Rate of Synoptic-Scale Precipitation 12 4.1 Values of 17 Corresponding to Critical Periods for Maximum Precipitation in the Interval t^ 14 5.1 Maximum Values of Meteorological Parameters — December 1955 Storm 16 7.1 Average 72-Hour Precipitation, Feather River Basin 20 APPENDIXES .Appendix A. The Computation of the Topography of the Sacramento Valley Inversion 25 Appendix B. The Computational Stability of the Finite Difference Approximation in Sec 2.2 27 FIGURES Figure 1 Schematic Cross-section of Terrain, Showing Valley Inversion 2a Location Map - Feather River Basin 2b Location Map - Profile Strips A, B, and C 2c Detailed Location of Profiles Used in Computer Calculations 3a Average Elevation of Profile A 3b Average Elevation of Profile B 3c Average Elevation of Profile C 4 Schematic Diagram of Trajectories of Precipitation Products 5 Schematic Diagram of a Raindrop Trajectory - Illustrating the Computing Scheme 6 Plot of Raindrop Trajectories No. 36 and No. 55 on Profile A 7a Distribution of Hourly Rate of Orographic Precipitation on Profile A, Dec. 1955 Storm (4 m/s) 7b Distribution of Hourly Rate of Orographic Precipitation on Profile A, Dec. 1955 Storm (6 m/s) 7c Distribution of Hourly Rate of Orographic Precipitation on Profile A, Dec. 1955 Storm (8 m/s) 8 A Vorticity Budget of the Layer - 1000 to 1500 Meters (Dec. 1955 Storm) 9 The Adiabatic Double Couette Flow Model 10 Graphical Solution for Adverse Periodicity 11a The Calculated 72-hour Isohyetal Map, December 1955 Storm (8 m/s), Feather River Basin lib The Calculated 72-hour Isohyetal Map, December 1955 Storm (6 m/s). Feather River Basin lie The Calculated 72-hour Isohyetal Map, December 1955 Storm (4 m/s), Feather River Basin 12 The 72-hour Isohyetal Map of Observed Precipitation, December 1955 Storm, Feather River Basin 13 The 72-hour Isohyetal Map, "Near Maximum" Storm, Feather River Rasin 14 The 72-hour Isohyetal Map, Estimated Maximum Possible Storm, Feather River Basin -IV- ORGANIZATION HARVEY O. BANKS DIRECTOR. DEPARTMENT OF WATER RESOURCES RALPH M. BRODY DEPUTY DIRECTOR. DEPARTMENT OF WATER RESOURCES JAi'.lES F. WRIGHT DEPUTY DIRECTOR. DEPARTMENT OF WATER RESOURCES WALTER G. SCHULZ CHIEF. DIVISION OF DESIGN & CONSTRUCTION CARL A. WERNER CHIEF, OPERATIONS BRANCH This Report Was Prepared By: DR. JOSEPH B. KNOX METEOROLOGICAL CONSULTANT With Principal Assistance From: JOSEPH L BURNS SUPERVISING HYDRAULIC ENGINEER WILLIAM A. ARVOLA METEOROLOGIST n WAYNE J. KAMMERER ASSISTANT HYDRAULIC ENGINEER WILLIAM B. MIERKE ASSISTANT HYDRAULIC ENGINEER PAUL L. BARNES CHIEF. DIVISION OF ADMINISTRATION PORTER TOWNER CHIEF COUNSEL ISABEL C. NESSLER COORDINATOR OF REPORTS -V- ACKNOWLEDGEMENTS Several members of the Department of Water Resources have been of great assistance to the research project: Mr. William Arvola, Mr. Wayne Kammerer, and Mr. William Mierke for their respective services in meteorology, computation, and programming. The author is also grateful for the interest of Mr. Carl A. Werner in this study, and for the contributions, interest, and frequent discussions on the part of Mr. Joseph I. Burns. -VI- LIST OF SYMBOLS The symbols frequently used in the study are listed below with their definitions: D — the diameter of a raindrop. E ■ the export of the vorticity q. i — the indexing jf the vertical in the computational plane. j — the indexing of the discretized vertical coordinate. m - the indexing of the straight line segment fitted to the average terrain. n — the vertical number (0, 1, 2, ) within a given m— region. p — the pressure. P, — the rate of precipitation Pj - the hourly rate of precipitation from the synoptic-scale disturbances. P^g^ - the maximum hourly rate of precipitation. P f„ - the minimum hourly rate of precipitation. Pg - the hourly orographic rate of precipitation. P(Xj) -- the mean rate of precipitation at the location x^ during a prescribed period. q — the vorticity of the two-dimensional motion in the vertical plane. t^ — the saturation mixing ratio. S(x) — the simplified terrain slope t - time. t, - an interval of time. T — the period of the time function used to describe the rate of precipitation. u — the component of the horizontal wind normal to the mountain barrier. u, — the value of u at the simplified terrain level. w — the vertical velocity. Wp - the terminal fall velocity (in still air) of raindrops of size D. Wo.W;,.. — the terminal fall velocity of raindrops of a particular size, and passing through spe- cial grid points in the computational net. Wo'* I- • ~ ^^^ f^ll velocity of raindrops relative to the earth. X - the space coordinate normal to the barrier. Xj — the X— coordinate of the computational verticals. x„ — the X— coordinate of discontinuities in the terrain slope. z — the vertical coordinate. -vi;- LIST OF SYMBOLS (Continued) Zj — the z— coordinate of the simplified terrain. z (i— l,i) — the elevation at the i— 1 vertical of the raindrop trajectory terminating at Xj. fl — the saturation potential temperature. ^ — the mesh constant in the x— direction, for m equal a constant. p — the density of dry air. 0) — the individual rate of change of pressure on a moving air parcel. -viii- PROCEDURES FOR ESTIMATENG MAXUMUM P0SS5BLE PRECilPITATION by Josepli B. Knox LI SNTRODUCTiON By way of introduction, the objectives of this investigation on the estimation of maxi- mum possible precipitation are outlined as follows: 1. To devise a physical model for estimating maximum possible precipitation over large watersheds. (In the model devised, the orographic precipitation, the precipitation from large-scale fields of vertical motion associated with synoptic-scale disturbances, and the spillover can be quantitatively evaluated.) 2. To describe, in some detail, the geographical distribution of maximum possible pre- cipitation over a watershed. 3. To develop proper computing methods so that electronic computers may be used to at- tain these objectives. 4. To apply the model to the meteorological case study of the December 1955 storm in the Feather River Basin as a verification of the model's capability to specify accu- mulated precipitation and its geographic distribution. 5. To maximize the parameters of the model, thereby producing an estimate of the maxi- mum possible precipitation. Precipitation-producing mechanisms that lead to excessive winter rainfall in California are (a) orographic lifting, (b) lifting due to large-scale fields of vertical motion (including frontal lifting), and (c) vertical instability. Great deluges, such as the December 1955 storm in California, occur through the simultaneous operation of the first two factors for a watershed whose horizontal area is of the order of thousands of square miles. In the proposed model, tlie hourly rate of precipitation is a function of a number of me- teorological variables — (a) the intensity of the wind normal to the mountain range, (b) the avail- able moisture, (c) the intensity of traveling disturbances, (d) the raindrop size assumed in the model, and (e) the height of the low-level inversion overlying the Central Valley of California. Once models of orographic and frontal precipitation are constru' ! d, the parameters representing the five above physical features are physically or statistically adjusted to extreme values; in this way, a quanti- tative estimate of the maximum hourly precipitation is calculated. From the maximum hourly rate of precipitation for both orographic and frontal sources, we will construct a simple model of the time distribution of the precipitation rate in which the ac- cumulated precipitation depends not only on the five prior mentioned parameters but also on the storm periodicity (or the periodicity with which disturbances approach the watershed). To maximize the accumulated precipitation, the most adverse storm periodicity is selected. With aa electronic computer GBM 650), we can readily perform the following computations: (a) the detailed geographi- cal distribution of maximum possible precipitation, and (b) the variation of the maximum possible precipitation with the assumed raindrop size. The methods to be described later permit the compu- l- tation of the distribution of spillover into the leeward basin. In this study, spillover is defined as that portion of the rain (orographically produced) that falls over onto the lee side of the basin ot onto a high, flat plateau from the windward side. To estimate the hourly orographic rate of precipitation, we shall use a two-dimensional orographic precipitation model, suggested by Professor J. Bjerknes in the 1940's, and adapted for use in the Sacramento Valley and the Feather River Basin. The estimated maximum rate of precipitation due to large-scale synoptic processes shall be calculated using a vertical velocity distribution from a dynamical model of the atmosphere. ' During the last decade dynamical models of the atmosphere have been devised and their uses ex- plored. This brief experience, supported by diagnostic vertical motion studies, indicates that cal- culations of extreme rates of rainfall by these methods will be fruitful. 2.1 MODEL FOR OROGRAPHIC RAINFALL The first step in devising a simple orographic rainfall mudel is the selection of a pro- file describing the terrain and the topography of the inversion surface over which the moist mari- time air ascends. A schematic profile nomial to the Sierra Nevada is shown in Figure 1. A sloping temperature inversion is depicted over the Sacramento Valley. Beneath this inversion, cold air moves from south to north (or south-southeast to north-northwest) during the approach of a cyclone from the southwest. Since this shallow layer of air does not impinge on western Sierra slopes, it does not contribute to the precipitation in basins like the Feather River. Rather, this shallow layer of cold air is forced to ascend orographically at the north end of the Sacramento Valley, between Mt. Shasta and Crater Peak, as discussed by J. Bjerknes (1946). However, the air mass that is forced to ascend the western Sierra slopes is the warm moist air of maritime origin, moving from a west- southwesterly (or southwesterly) direction. The flow in this warm air mass may be modelled by a uni-directional flow normal to the Sierra. The sloping inversion over the Sacramento Valley in the model represents a narrow zone separating the low-level flow parallel to the Sierra from the warm upper flow normal to the mountains. The inversion slope may be computed from meteorological data, using the procedure outlined in Appendix A. Since the December 1955 stonn represents an impor- tant test case for this study, it should be mentioned that the three-dimensional analysis of the wind field during this storm, described in a recent paper by Myers (1959), confirms the structure of the orographic precipitation model proposed by Bjerknes (1946). We shall now consider the computa- tion of the maximum orographic rate of precipitation from a two-dimensional model. A two-dimensional model of orographic precipitation is constructed following the method of Bjerknes (1940); however, in addition, the proposed model contains the following features: (a) a computationally stable computing scheme, (b) a computation of spillover, and (c) a vorticity bud- get "check" on the proposed steady-state solution. To devise the model we proceed as follows. Far upwind of the mountain barrier in es- sentially undisturbed flow, the wind normal to the barrier (designated by u) is known as a function of height. This wind data serves as information on the inflow boundary for computation of an ap- proximate steady-state solution of the flow of saturated adiabatic air over the valley inversion and the mountain barrier. In a manner similar to Bjerknes (1940), we select the following distribution of vertical velocity, w (x, z)- w (x, z J (V2) (1) in which the vertical velocity, w, decreases by a factor of two cvciy 1000 meters. Coriii>iiM. /. Cl> with the equation of continuity for steady-state two-dimensional motion (in which tin.- horizonlnl variation of the density is neglected), (9x o 'iz ^ it is possible to compute numerically the w (x, z) and u (x, z) distribution for the region of interest. 2.2 THE o, w COMPUTATION To compute the distribution of the horizontal wind, u, and the vertical velocity, w, in a vertical plane normal to the mountain barrier, we proceed as follows: 1. Consider the Feather River Basin, shown in Figure 2a. A major part of this basirt can be covered by three profile strips, each 16 miles in width. The locations of these pro- file strips are shown in Figures 2b and 2c. Lines 16 miles in length are drawn normal to the axes of the strips at one-mile intervals along the strips (see Figure 2c). The average terrain elevations along these lines are obtained, and then are plotted to form the average terrain p.ofiles (see Figures 3a, 3b, and 3c). 2. The average terrain profiles produced in this manner are still quite complex for com- putational purposes; for simplification, straight-line segments are fitted to the average profiles. In this fitting process, the main windward slope is preserved, while minor features are smoothed. The interior mountain valleys are assumed to be filled with cold air capped by a horizontal inversion. Figures 3a, b, c show the average terrain heights, the simplified profile, and the Sacramento Valley inversion for Profiles A, B, and C. 3. To compute the distribution of u and w from Equations (1) and (2) in the vertical plane along a simplified profile, this two-dimensional space is discretized so that "„,.„ " , 1 2Az (12) 6. The calculations outlined in Paragraphs 4 and S (above) are continued until a vertical of discontinuous terrain slope is reached. With the horizontal wind component u^ known on this vertical and ignoring the slope discontinuity in the calculation of the vertical velocity, we obtain w, ^ u, "S , where S is the average slope, '/2(S, + Sj). To compute the remaining vertical veloci- ties on this vertical, w, is substituted in Equation '1), and the resulting values of ver- tical velocity are subsequently substituted into (12) in order to calculate the horizon- tal wind as a function of height on the (i+1) vertical. With the horizontal wind known -5- on the (i-tl) vertical, the computing routine begins again with calculations discussed in Paragraph 4. In this manner, the horizontal wind and the vertical velocity are cal- culated on successive verticals until the region of interest is covered. At the conclusion of the u, w computation, one is confronted with the question as to whether the computing program is error free. To determine this, a series of three checks is set up to examine the solution: 1. Since the orographic model is assumed to be a steady state, the u, w numerical solu- tion must, if correct, have no net inflow of dry air into the region between the first ver- tical and the last (downwind) vertical. In the test problem (composed of over fifty ver- ticals) the mass outflow of dry air exceeded the mass inflow by three per cent. Both the sign and magnitude of this error compare well with the estimated truncation error (of five per cent) discussed in Appendix B. 2. The vertical velocity decreases by a factor of two every thousand meters. 3. The behavior of the first and second differences (in the x-direction) of the horizontal wind, u, must be orderly, and changes in these differences from one vertical to another must have a physical basis. 2.3 THE TRAJECTORIES OF PRECIPITATION PRODUCTS The next problem in calculating the hourly rate of orographic precipitation is the de- termination of the trajectories of raindrops of a given size where the raindrops terminate their earth- ward fall at the bottom of verticals in the computational grid. Given the horizontal and vertical velocities of the air in the model, and the terminal fall velocity of the selected raindrop, an approx- imate trajectory determination is possible with the following assumptions: 1. The terminal fall velocity of a raindrop of arbitrary shape can be approximated by the terminal fall velocity of a spherical drop of the same mass. 2. The dependence of the terminal fall velocity on atmospheric density can be neglected. 3. The effect of change of phase of the precipitation products on the trajectory can be neglected in calculations of maximum possible precipitation. 4. The condensation products are assumed to be large drops of 2100 micron diameter, re- ported by Byers (1944) to correspond to excessive rain. Physically, assumptions 3 and 4 maximize the windward precipitation by bringing con- densation products to the ground quickly. The oblique raindrop trajectories, shown schematically in Figure 4, define skew-shaped volumes of air from which the rain is falling onto both the windward and leeward slopes. For exam- ple, precipitation arriving on the windward slope between X^ and X^ originates from the skew- slmped volume contained between trajectories T^ and T, shown in Figure 4. The rate at which pre- cipitation falls on a unit horizontal area from regions of positive vertical motion in the atmosphere is given by the following expression, derived by Smagorinsky and Collins (1955), ^ (Pi) Pi - -L /-f>w.5r,, (13) where r^ is the saturation inixiiit; ratio, 5r^ is Vi\r change in the obturation nixir.j; ratio tiurir.f; I'r.-.: saturated adiabatic ascent of tlie air through v. pressure interval dp, w is th'j vxTticnl velocity, p is the density of air, and p^ the density of water. The rate of precipitation, computed from Equation (13), is subject to the following assumptions: (a) sufficient condensation nuclei are present, (b) no super-saturation, (c) no super-cooling, (d) no non-adiabatic p'ocesses other than those occurring from changes in state, and (e) both cloud storage and evaporation, from falling droplets are negligible compared to P. . The assumptions, with the exception of (d), maximize the rate of precipitation Pj . It is proposed that the orographic rate of precipitation be calculated by applying Equation (13) to the skew-shaped volumes terminating on the windward slope, and that the spillover (occurring, for ex- ample, at vertical X_ in Figure 4), be computed in an analogous way. To devise a simple numerical scheme for calculating the coordinates of n raindrop tra- jectory, we proceed as follows. Consider the raindrop of diameter D (where D-2100/i and tlie ter- minal fall velocity W = 6m/s) that strikes the terrain at the i-th vertical (see Figure 5). Suppose we designate the height of the intersection of this raindrop trajectory on the (i-1) vertical by In addition, two hypothetical drops, whose trajectories are shown in Figure 5, are of interest; name- ly, (1) the raindrop passing through the (i-1) vertical at height z^ and striking the terrain at the i-th vertical, and (2) the raindrop passing through the (i-1) vertical at the height z, and striking the ter- rain at the i-th vertical. These two raindrops are of quite different size; from their fall velocities rela- tive to the earth, designated by W^ and Wj , the corresponding terminal tail velocities VV^ and W, are computed from the u, w solution as follows: ■"( W„ =0=% w[i, z„ (i)| t w[.--l, z, (i)l +W„ (14) and V,' = 'A /w[i, z, (i)l t w[i-l, jUl I -t W, (15) where W. [zG^l)-2. (i)l/Atj, (16) and At, =^ '/v. |u[i. Z3 (i)i t u[i-i, i+n (17) Once Wj and Wj, are calculated, we determine if W^ (selected as 6 m 's) lies between '.V arid W, If |Wq I<|VVJ-.|W,|, then z (i-1, i) may be calculated from the divided difference formula. .o(i-l,0 = z„._^HWJ_ |Wi| - I Wo I z + 1) - ^ Zo (18) -7- In the event W exceeds the magnitude of both W and W^ , then the calculation on the (i-1) vertical is repeated using hypothetical raindrops passing through the levels (j+1) and 0+2); then zi,(i-i, i) = za+i)+ ^~'^^'' |vv,| -|w,| L (j+i)| z(J+2)-z(j+l) (19) Once the height of this trajectory on the (i-1) vertical is calculated, we repeat the computation in order to determine z (i-2, i-1). Equations similar to (18) and (19) are used for this purpose; for ex- ample, z (i-2, i-1) = z (i-1, i) + 6-|Wzp(.-,.i)| _ r (i_i_ i)l ° (20) = zp (i-1, i) + |8 (i-2) [z,+ J - z^ (i-1, i)l , where |Wz^ (i-l.l)l<|WD|<|Wj,5|. This portion of the machine program (designated as the trajectory subroutine) calculates and out- puts (a) the raindrop trajectory by tracing the locus of the intersection of a given trajectory with successive verticals, and (b) the interpolated value of vertical motion at these intersection points. The calculation of a particular trajectory ceases when it intersects the isobaric tropopause in the model. The computational stability of the trajectory routine may be examined in the following way: Equation (20) is of the form Zj, (i-k-1) = A (i-k) z„ (i-k) + B (i-k). If the index k=i, 2 ... k, we obtain the following set of equations; z^ (i-2) = A (i-1) z^ (i-1) + B (i-1) Zj, (i-3) = A (i-2) zj, (i-2) + B (i-2) = A (i-2) [ A (i-1) z^ (i-1) + B (i-1) 1 + B (i- 2) Zo (i-4) = A (i-3) z^ (i-3) + B (i-3) = A (i-3) A (i-2) / A (i-l) Zp (i-1) + B (i-1)] + B (i-3) Zq (i-k) = A (i-k+1) Zp (i-k+1) + B (i-k+1) = Ia (i-k+1) A (i-3) A (i-2)| /a (i-1) Zp (i-1) + B (i-1) | + B (i-k+1) The functions A and B, coniir tide from u, w, nre i.,;,.". .-kd us known in tlir nhovt .sc.. i- i last equation in the set, the influence of an ertor S; trajectory height at the (i-k) vertical is computed as last equation in the set, the influence of an ertor Sz (i-1), introduced on the (i-l) vertical, on lU.. Szp (i-k) = JA (i-1) A (i-2) A (i-3) A (i-k+l)| ^z^ (i-1) . (21) The error propagated to the (i-k) vertical is less than (or equal to) the error introduced on the (i-1) vertical if all the values of A are less than or equal to one in magnitude. From Equation (20), it can be seen that if |/3|<1, the magnitude of A (i-ktl) is less than or equal to one. In this case the pro- pagated error becomes smaller (or remains the same) as the calculation jsroceeds from one vertical to the next. Hence, the trajectory computing routine is stable. An illustration is shown in Figure 6 of raindrop trajectories computed for verticals No. 36 and No. 55 on Profile A in the Feather River Basin. Consider the trajectory intersecting the terrain at vertical No. 36. On the portion of the trajectory between the points A and B the orogra- phic component of vertical motion produces (in the model) precipitation products falling earthward ut a known terminal fall velocity. Similarly, on the trajectory segment CD forced orographic ascent provides additional precipitation, which (in the model) also falls along this trajectory.* However, on the trajectory segment EF the orographic component of vertical motion is zero; no new orograph- ic precipitation products are formed in the air parcels momentarily located on this segment. But those precipitation products formed on the trajectory between A and E free fall along EF, striking the giound at the 36th vertical. The precipitation deposited on the ground at vertical No. 36 is (as we have previously defined) spillover. Precipitation deposited on the ground upwind of vertical No. 36 (for example, verticals 21 through 34) is the familiar orographic precipitation on the wind- ward slope. 2.4 THE RATE OF OROGRAPHiC PRECIPITATION The rate of orographic precipitation on the windward slopes and the spillover are readi- ly computed from Equation (13) with the data from the trajectory subroutine. Figures 7a, b, and c show the hourly rate of orographic precipitation calculated for the storm of December 1955. as h function of x for tenninal fall velocities of 4, 6, and 8ni/s on Profile A. The precipitation rate on the windward slope is, of course, the greatest for the largest terminal fall velocity, 3m/s, while the precipitation rate on the lee slope, or the spillover, is the largest for the 4m/s tenninal fall velocity. Since the field of vertical vi^locity is the same in all three cases, and the condensation rate in the free atmosphere is thereby the same, the change in rainfall intensity is only a matter of redistribution of precipitation products. 2.5 THE VORTICITY BUDGET CHECK ON THE u, w SOLUTION In calculating the u, w solution for the orographic rate of precipitation, we have used only the y-component of the equation of motion. One possible check on the proposed u, w solution is to ascertain how well the proposed solution satisfies the x- and z-components of the equation of * It should bi! noled Ihiit in the computation of the w-fifl>l ulong a vertiuul at a terroiii-slope dlsconUnuhy un avumi^e of the slope on both sides of tliis veiti'-al is used; thus, between verticals 32 and M and between verticals 31 and 35, the spillover idmputation uses positive w-values on verticals .^2, 3.1, and 34. Orov;raphic preciplt.itiim is, by virtue of this appruximiition, produced along trujeclory segments UC and DE, even thcui.-h these set;mt?!its lie above Mat terrain. -9- motion. These two component equations car. be replaced by a vorticity theorem for the motion in the X, 2 plane for steady-state conditions, V . (qv) = A (uq) + A (wq) = . where q is the vorticity of the two-dimensional motion in the x,z plane, q = £u_ _ 5w By means of Gauss' Thaorem, given in Holrrboe (1944), the above steady-state requirement on the vorticity q is Eq = Jv • :(q v) 6A = /q v • id n oi the net export of vorticity, E , through a closed curve bounding the area A must be zero. We shall now compute the export of vorticity for a series of small areas (Az^ on Profile B. By examining the E field afterwards, we will be able to discover the manner in which the steady-state solution aif- q fers from the proposed u,w fields. For the storm of December 1955, the net export of vorticity, E^, is computed for the small areas shown in Figure 8. The products uq and wq are plotted in the figure. It should be noted that (a) E is negative in all the areas computed, (b) both contributions to the convergence of qv are negative, (c) the computed E is small compared to the large contributions to the export, and (d) the computed E^^is of the same order as the small contributions of wq to the export. It is clear that in the 1000-meter to 1500-meter layer that we are somewhat removed from a steady-state solu- tion. From the computed E„ field we are able to find the proper behavior of u,w in this layer. To do this, we shall first simplify the u-profile on verticals 18 and 25 as shown in Figure 8. Suppose g^ (uq) remains negative in the region under the u-profile kink (at 1000 meters); we could thar ask what distribution of w is required in this layer to balance the vorticity budget. It is readily seen that to achieve vorticity balance (under the profile kink) the vertical velocity must increase with height up to the u-profile kink. This latter result obtained by physical inference is supported by a theoretical study. Using the analytical results of Doos (1958) and methods similar to Holmboe (1953), the author has studied steady-state, two-dimensional flow in an adiabatic atmosphere,* containing a double Cou- ette flow profile, and bounded (above and below) by rigid planes (see Figure 9). The result of the analysis shows that the vertical velocity increases upwards to the sinusoidal interface between the two Couette flows, and that the vertical velocity decreases with height above the interface. Estimates made from this theoretical model indicate that vertical velocities at the sinusoidal in- terface are of the order of 1.1 to 2.0 times the vertical velocities on a sinusoidal upwind slope, for conditions appropriate to the December 1955 storm. Since we have demonstrated a probable departure of the proposed vertical velocity from reality, we should pause and consider the principal sources of error for the orographic rate of • An adiabatic atmosphere Is a "n'.odel atmosphere" in which the potential temperature, 6, is independent of the three space coordinutus. -10- precipitation. The principal sources of error are: (a) the z-dependency of the vertical velocity, (b) the assumed homogeneously large raindrop size, and (c) the neglect of the effect of change of state on the trajectory of the precipitation products. The effects of these errors on the orographic rate of precipitation are as follows: 1. The assumption that w halves every one thousand meters underestimates the rate of precipitation in the layer under the u-profile kink. 2. The assumption of homogeneously large raindrop size overestimates the steepness of the trajectory, and hence overestimates the rate of precipitation. 3. With the neglect of the effect of change of state on the trajectory of precipitation pro- ducts, the assumed terminal fall velocities of 4, 6, and 8 m/s exceed the snowflake terminal fall velocities of 0.5 to 3 m/s, reported by Douglas, Gunn, and Marshall (1957). Since snow traverses a trajectory more nearly horizontal than assumed, by overesti- mating the upper trajectory slope we in turn overestimate the orographic rate of pre- cipitation. It is quite possible that these contributing errors nearly compensate, or cancel, one another. The capability of the proposed model and methods to specify accumulated precipitation distributions can be determined empirically in meteorological case studies. The necessary com- putations can be readily performed with the electronic computing program developed. When the capability of the model to specify accumulated precipitation (or rate of precipitation) is confirmed, the parameters of the model can be maximized to obtain the estimated maximum possible precipita- tion. 3.1 THE MODEL FOR RAINFALL FROM LARGE-SCALE SYNOPTIC DISTURBANCES Since the development of weather forecasting by dynamical methods, models depicting the evolution of large-scale disturbances have been proposed in which the rate of precipitation (due to large-scale vertical motion) can be computed. This evolution is physically governed by the prin- ciple of conservation of mass, the first law of thermodynamics, and the three equations of motion. Once the differential equations expressing these principles are "tailored" to describe synoptic (or large) scale disturbances, a partial differential equation in vertical velocity is obtained [see Eliassen (1955, 1957) and Smebye (1958)]. The adequate solution of the complete partial differential equation for the vertical velocity in three dimensions has not yet, to the author's knowledge, been achieved. However, Sm&gorinsky and Collins (1955) and Smebye (1958) have reported on the use of a two-level model for the calculation of vertical velocity and the prediction of precipitation. By maximizing the physical parameters in the two-level model used by these investigators, we may ob- tain an estimate of the extreme precipitation rate from large-scale processes. In one case study of precipitation prediction involving both showery and large-scale precipitation, it was shown that the predicted precipitation was in good agreement with the observed average in zones defined by the predicted isohyets. The vertical velocity distribution, as mentioned, may be obtained from the two-level model, Charney and Phillips (1953), and Sawyer and Bushby (1953). In the two-level model it is as- sumed that the individual rate of cbaaee of pressure on an ascending air parcel, (y(p) = dp/dt, is Tj ). The total precipitation, Pj , at X, during the time interval t, is P^(x,.t,) = V(x,)t,+^P^(x^)JsLsm(Mi]^^P^(x,) Jl sin f^!!:^^ . 2b \T^J 2ff \ "^2 / Subtracting R, from Equation (25) and assuming that the maximum hourly rate of precipitation is the same in the two idealized "storms" [that is,AP(x,) = AP^,(x,) + APj (x,)] , we find P-P, = APj Since, by assumption, T^ > Tj and sin (2 n tj/Tj.) = 1, it is seen that Pj equals P only if APj is zero. Under these conditibns, the practical result is that the total precipitation at x, during t, is a maximum for a simple sine distribution corresponding to the most adverse period. 5.1 THE CASE STUDY OF DECEMBER 1955 - THE FEATHER RIVER BASIN In order to test the capability of the model to specify accumulated precipitation and its geographic distribution, we shall apply the model to the December 1955 storm. This storm is marked by three periods of intense rainfall occurring at times nearly coi^ responding with upper air observations on (a) 1500 GCT, 19 December, (b) 0300 GCT, 22 Decem- ber, and (c) 0300 GCT, 23 December. From these observations, it is possible to measure (or else compute) the parameters entering the models for orographic precipitation and frontal precipitation. The observed parameters are (a) the saturation potential temperature at 850 mb, (b) the observed wind for Oakland (and in particular the wind at 850 mb), (c) the height of the Central Valley inver- sion, (d) the slope of the valley inversion, and (e) the vertical velocity at the 500 mb level as de- termined by the two-level prediction model. The extreme value of each of these parameters found in the December 1955 storm is tabulated below: .15- Table 5.1 Maximum Values of Meteorological Parameters — December 1955 Storm Ot at 850 mb ; 70 °F u at 850 mb . ;• 65 knots Inversion height at -beginning of profile stripy,,., ,,^ 364 m Inversion slope ,/'V 0.001 w (at 500 mb) 11.2 cm/sec No correction for adverse storm periodicity is used in the December 1955 stoim since power spectrum analysis of the precipitation records at Brush Creek (a representative recording precipitation station on the windward slope) shows no evidence for significant spectral maxima at the critical periods. To eliminate the adverse periodicity correction from our study of this storm, we set T^=48 hours in Equation (25); thus, the contribution of the periodic term in Equation (25) is zero. With the substitution of the above values of the parameters (shown in Table 5.1) into the models discussed in Sections 2, 3, and 4, we obtain, by machine calculation, the 72-hout ac- cumulated precipitation as a function of x on the three profiles of the Feather River Basin. Figures 11a, lib, and lie represent the calculated 72-hour isohyetal maps, corresponding to terminal fall velocities of 8 m/s, 6 m/s, and 4 m/s, respectively (where P„j„ in the time distribution is zero, and P, is 0.12 inch per hour). The calculated 72-hour precipitation amounts are plotted at each vertical for the three profiles. Consider the 72-hour precipitation pattern computed for a terminal fall velocity of 8 m/s. In this precipitation pattern, the orographic maxima stand out most dramatically; namely, along the western slope of the Sierra Nevada and to the rear of the basin on Profiles A and B. The lar- gest 72-hour totals occur, of course, on the windward Sierra slope, since the assumed "large-sized" precipitation falls quickly from the moving air parcels. Spillover from the windward Sierra slope ceases on Profiles A and C near Mountain Meadows Reservoir and Last Chance Creek (located re- spectively at miles 92 and 95)— here the precipitation is frontal. However, on Profile B spillover and/or direct orographic precipitation exist the whole length of the profile, making the precipita- tion shadow less marked on Profile B than on Profiles A and C. The precipitation maxima at the northeastern ends of Profiles A and B are produced by ascent over a small, sharp rise in terrain on the basin rim. As the terminal fall velocity of the raindrops is decreased, the spillover into the up- per Feather River Basin increases at thi^ WlUpense of the orographic maxima on the windward slope. By extrapolating the isohyetal pattern to the portions of the leeward basin excluded from the cal- culation, the total basin precipitation increases slightly as the terminal fall velocity decreases. This is understandable in terms of the increased spillover and the geometry of the basin, in which the area of the "leeward" basin exceeds that of the "windward" basin significantly. The observed 72-hour precipitation pattern for the December 1955 storm (for the period ending 0700 PST on December 22, 1955) is shown in Figure 12. A comparison of the computed pre- cipitation patterns with the observed pattern gives the following salient points: 1. The observed precipitation pattern is not as detailed as the computed pattern; this a- rises from the sparce density of precipitation stations. It should be noted that only one precipitation station is located in a computed precipitation maximum. Because of -16- the large interval between the precipitation stations, many of the features in the com- puted pattern can not be directly confirmed. (The features might be confirmed, however, with the addition of a few new strategically-located precipitation stations.) The pre- cipitation maximum to the rear of the leeward basin is unobserved by the current pre- cipitation network. 2. The calculated pattern for the December 1955 storm with a raindrop terminal fall velo- city W = 6 m/s compares quite well with the observed pattern for the portion of the basin covered by the profile strips. It should be noted that there exists reasonable a- greement between the observed basin average precipitation (10.1 inches) and the cal- culated basin average precipitation (11.8 inches). 3. On windward slopes, the precipitation maxima in the observed pattern are smaller than in the predicted pattern. The observed precipitation may well be underestimated due to: (a) the influence of terrain slope on the precipitation catch as reported by Lands- berg (1957) and Hamilton (1954), (b) the non-vertical fall of raindrops through the air above the rain gage, (c) the exposure of the gage, and (d) the intensity of the air flow past the rain gage. Using Hamilton's equation to estimate the precipitation measure- ment error, it is possible that the observed precipitation on upper windward slopes is underestimated by 20-30 per cent. In view of this possibility, the computed windward maxima appear to be quite acceptable. 4. Certain observed precipitation amounts may be seen to differ considerably from the computed amounts for the same location. In this regard, the following stations— Storrie, Strawberry Valley, BuUards Bar, and Greenville— should be checked for exposure of the rain gage and representativeness of the local terrain slope, in order to determine if there is any obvious reason for the discrepancy. From the test case of December 1955, we conclude that the simple physical models proposed and the numerical methods employed are able to compute the rainfall distribution for a given intense storm. In the next two sections we shall consider the maximization of the parameters of the model in order to estimate the distribution of the 72-hour maximum possible precipitation for the Feather River Basin. 6.1 THE "NEAR-MAXIMUM" STORM The purpose of this section is to obtain the distribution of the 72-hour precipitation in the Feather River Basin by estimating the extreme values of all parameters in the model with the exception of the horizontal wind field; this precipitation distribution is designated as the "near- maximum" storm. In the "near-maximum" storm, we will assume the horizontal wind field is that of the December 1955 storm. The parameters to be simultaneously maximized are given below with their estimated extreme values, and comments concerning their selection. 1. The critical periodicity of the synoptic-scale disturbance for the "near-maximum" storm is selected as 59 hours for the time distribution No. I. A critical period of 59 hours corresponds to the most adverse frequency short of a period of 162 hours or longer. 2. The saturation potential temperature of the moist tropical air mass (i.e., the air mass above the low-level inversion) is selected as 74°F. This extreme value of the satura- tion potential temperature exceeds the value of d for the December 1955 storm by four degrees. To estimate the maximum value of we will consider data concerning the -17- extreme ran^e in sea surface temperature in winter for particular months and for one- degree intervals of latitude, as compired by Bemett (I^zpQ. Although the sea surface temperature data cited is for the southern hemispheric winter, by selecting a latitude corresponding to the latitude of the northern hemispheric tropical source region (like 20° North), we can obtain an estimate of the range of for maritime tropical air in its source region. Palmer (1958). The interval between extreme sea surface tempera- tures in degrees Fahrenheit is shown below for the months of June, July, and August at 20°S latitude, from Bernett (1944), June July August e,op 6°F T^F From this data, the saturation potential temperature of maritime tropical air could be as much as three or four degrees higher than that in the December 1955 storm, or0, = 74°F. This latter value exceeds (by one degree) the highest value of 0^ at 700 and 850 mb during December (irrespective of wind direction) as reported from ten years of upper air data for Oakland, California (1946-1955), by the U. S. Weather Bureau (1958). 3. Assume P (the minimum hourly rate of precipitation in the time distribution model) is 0.05 inch per hour. 4. The terminal fall velocity of the precipitation products is maximized at 8 m/s, since this selection places the largest precipitation amounts on the windward slope where it is available for rapid surface runoff. 5. The frontal hourly rate of precipitation is assumed to be twice that of the December 1955 storm. [The validity of this assumption could be tested by the application of Equations (22) and (23) to the extreme cases of precipitation associated with SW storm types during the decade (1946-1955).! The 72-hour isohyetal map for the "near-maximum" storm is shown in Figure 13. The geometry of the pattern is similar to that in Figures 11a, b, c, which have been discussed in de- tail; however, the depths are significantly larger than in the December 1955 storm. Once the 850 mb horizontal wind speed is maximized, the next step is to scale the "near-maximum" isohyetal map to this extreme wind condition. The question of how to scale the "near-maximum" storm to the estimated maximum possible storm is considered in the next section. 7.1 THE ESTIMATED MAXIMUM POSSIBLE STORM Suppose we define the maximum possible storm as the largest accumulated precipita- tion that can reasonably be expected to occur for a given watershed during a 72-hour period. The specific problem considered in this section is that of reasonably scaling the "near-maximum" storm to the maximum possible storm. We shall proceed in three steps: (a) the estimated maximum possi- ble storm will be calculated by simultaneously maximizing all the parameters in the model, (b) a critique of this procedure will then be offered, and (c) a more reasonable method of scaling to the maximum possible storm will be proposed. The simultaneous maximization of ail parameters in the model may be achieved in the following way; if the 850 mb extreme wind exceeds the 850 mb wind used in the December 1955 storm by a factor C, and if it is assumed that the wind at all levels is similarly scaled, then the orographic precipitation scales by a factor of C. In addition, if it is assumed that the perturbation -18- vorticity is the same as in the "near-maximum" storm, the frontal contribution to the precipitation also scales by a factor of C. Under these conditions, the isohyetal map of the estimated maximum possible storm (as well as the mean depth of water applied to the basin) can be produced by multi- plying the isohyets of the "near-maximum" storm (or its mean depth of water) by the factor C. An adaptation of Gumbel's extreme value theory, reported by Gringorten (1959), enables us to compute the extreme wind at 850 mb for periods of 50, 100, and 500 years. The results of the calculation, with a 95 per cent confidence limit, are tabulated below with the corresponding scaling factor C. The simultaneous maximization of all parameters leads to the 72-hour accumulated precipitation (Column 4 below) for the portion of the Feather River Basin covered by the profile strips. Scaling Period u max Factor C P(72)„„ SOyrs 89 knots 1.34 27.9 inches 100 yrs 93 knots 1.41 29.4 inches 500 yrs 103 knots 1.56 32. 5 inches If the u^^j, for the 500-year period is selected for the development of the maximum possible storm, the 72-hour mean accumulated precipitation with simultaneous maximization of parameters is 32.5 inches in the portion of the Feather River Basin covered by the profile strips. This accumulated precipitation exceeds the greatest observed mean basin depth by a factor of 2.75. Since the simul- taneous maximization of all parameters leads to a very large mean basin depth, it might be well to consider conditions under which simultaneous maximization is physically inconsistent. During southwest storm types, it may well be argued that when the basic flow is very strong, any frontal wave disturbance forming in this current will move very rapidly towards the northeast. The rapid northeast movement is important for two reasons. First, the travel time of the disturbance from its place of birth to the Sierra Nevada will be so short that the wave will have little chance for development. Secondly, the wave disturbance, by virtue of its rapid movement, will spend little time over the Sierra watersheds. It therefore seems physically inconsistent to maxi- mize simultaneously the wind normal to the barrier and the precipitation from synoptic-scale disturb- ances. Suppose we now consider a way to estimate maximum possible precipitation devoid of simultaneous parameter maximization. All parameters with the exception of the precipitation of synoptic-scale disturbances will be maximized, and then the precipitation from the synoptic scale is approximated by a reasonable upper limit consistent with other extreme parameter values. We shall estimate the upper limit of the precipitation from the synoptic scale by the observed synoptic- scale precipitation, 0.12 inch per hour, in the December 1955 storm. To scale the "near-maximum" storm to maximum possible precipitation for non-simultaneous parameter maximization, we proceed as follows. From Equation (25) we obtain the change in P (x, , 72), designated by 5 P (x,, 72), for a change in frontal precipitation, 5 P,, X I I dP (x,, 72) = 36 F>P, + Jl SP, Art -19- The change in P^ is equal to -0.255 inch per hour,* so that the change in P(x,, 72) is -10.36 in- ches. So, to obtain the 72-hour isohyetal map of the maximum possible precipitation from the "near- maximum" storm, we use P (x., 72) = 1.56 P (x,, 72) - 10.36. max ^i' ' n.max^l (30) The resulting estimated 72-hour maximum possible precipitation for the Feather River Basin (e.g. those portions covered by the profile strips) is reduced from 32.5 inches to a more reasonable esti- mate of 22.1 inches. The corresponding 72-hour isohyetal map of the maximum possible storm for the Feather River Basin is shown in Figure 14. The numerical results obtained in this study for the Feather River Basin are shown in Table 7.1; the upper part summarizes the 72-hour accumulated precipitation for the portion of the basin covered by the profile strips. The lower part of Table 7.1 gives the results extended to cover the entire basin. The estimated maximum possible 72-hour storm (extended to the entire Feather River Basin) has a mean basin depth of 19.3 inches. This result is ten percent less than the pre- liminary (December) estimate by the Hydrometeorologica! Section of the U. S. Weather Bureau (un- published paper, March 1959). Table 7.1 Average 72-Hoor Precipitation Feather River Basin Storm Terminal Depth Depth Depth Fall Velocity (Windward) (Leeward) (Windward & Leeward) FOR PORTION OF BASIN INCLUDED WITHIN PROFILE STRIPS. Dec. 19-22, 1955 observed* 16.0 in 7.0 in 10.1 in Dec. 1955 calculated 8 m/s 18.5 in 8.3 in 11.9 in Dec. 1955 calculated 6 m/s 16.6 in 9.1 in 11.8 in Dec. 1955 calculated 4 m/s 14.1 in 9.9 in 11.4 in "Near-Maximum" 8 m/s 28.5 in 16.7 in 20.8 in Estimated Maximum Possible 8 m/s 34.1 in 15.8 in 22. 1 in FOR ENTIRE BASIN. Dec. 19-22, 1955 observed* 16.0 in 6.5 in 8.9 in "Near-Maximum" 8 m/s 28.5 in 15.8 in 19.0 in Estimated Maximum Possible 8 m/s 34.1 in 14.3 in 19.3 in *For the 72-hour period ending 0700 PST, December 22, 1955. 8.1 CONCLUDING REMARKS The objectives accomplished in this hydrometeorological study may be summarized as follows: * When the scaling factor of 1.56 is applied to P, = 0.24, we get 0.24 « 1.56 = 0.375. Thus, if we adopt Pj 0. 12, the change in Pj would be 0. 12 - 0.375 = - 6.255 inch per hour. -20- 1. A physical model and numerical methods have been devised for the computation of maxi- mum possible precipitation produced by orographic ascent and large-scale synoptic pro- cesses over large watersheds; 2. The spillover and its geographical distribution can be calculated with the assumption of uniform but extreme drop size; 3. The application of the model and the computing methods to the December 1955 storm shows the capability of the model in describing the distribution of precipitation over a large watershed like the Feather River Basin; 4. The 72-hour isohyetal map for the "near-maximum" storm has been calculated; this isohyetal map maybe simply scaled to the estimated maximum possible storm by means of the "wind scaling" factor C and Equation (30); 5. A machine program now exists so that the distribution of maximum possible precipita- tion over a given large watershed may be computed, given the geometry of the terrain in simplified profiles, the extreme values of certain meteorological parameters (enum- erated in Table 5.1), and the critical period T^- (t,). It should be stressed, however, that the model and the machine program should not be applied to watersheds in regions where the major precipitation results from intense convective activity. In such regions, the extreme value and duration of high intensity precipitation of small spatial scale must be estimated in other ways. Certain limitations in the methodology of this study exist and represent areas for future endeavor. First, no completely adequate method currently exists for computing the three-dimensional spatial distribution of vertical velocity of large-scale atmospheric disturbances; however, future basic research may furnish the necessary computing methods and meteorological case studies. Sec- ondly, a statistical study of the annual extreme vorticityin southwesterly type storms (in California) is needed in order to insure that this parameter is maximized consistently in the proposed model. Thirdly, the assumption that condensation products form in a uniform, large drop size is physically unrealistic; however, this assumption maximizes the precipitation on the windward slope. It may well be appropriate to outline, briefly, areas in which future hydrometeorological research would be advantageous: 1. The capability of the proposed model to delineate the distribution of 24- to 72-hour pre- cipitation should be tested in other hydrometeorological case studies. 2. In computing the wind scaling factor (using Gumbel's theory of extreme values) the sample of annual extremes numbers only thirteen; because of the small sample size, the scaling factor may be overestimated. We may be able to increase the size of the sample by applying the model diagnostically to the extreme annual storms for the past 20 to 50 years. 3. A statistical study of the annual extreme value of vorticityin southwesterly type storms is needed to insure that the vorticity is maximized consistently. 4. A power spectrum analysis of the hourly precipitation in several major storms should be performed to determine the relative importance of different frequencies to total pre- cipitation, and to aid in the extrapolation of the results from large watersheds to water- sheds of small area. -21- 5. Sea surface temperature anomalies should be studied on a monthly basis in order to determine their magnitude, spatial extent, and time duration; work of this type is being performed by Mr. J. F. T. Saur and Mr. L. E. Eber of the Bureau of Commercial Fish- eries, Biological Laboratory, Stanford University. Liaison with this group may well be advantaeeous. 6. Since the proposed model for predicting the "maximum possible" precipitation results in an isohyetal map of greater detail than can be dbserved in the Feather River Basin, it would be interesting to increase the density of precipitation stations in this basin. The purpose of such a program would be to verify the features in the predicted isohye- tal maps (for southwest storm types) that are currently unobservable. 7. To facilitate future applications of the code to other basins and to meteorological case studies, a manual for computers and programmers might be profitably prepared, describing the input, the operations required between computing phases, and the pro- gram structure. -22- REFERENCES 1. Bernett, M. A. F. (1944), "Note on Sea Surface T emperature between New Zealand and Samoa". Misc. Met. Notes, Meteorological Office, Air Department, Wellington, New Zealand. 2. Bjerknes, J. (1940), "A Method of Approximate Computation of Orographic Precipitation". Un- published report for the U. S. Engineers. 3. Bjerknes, J. (1946), "Report to the Chief of the U. S. Engineers about Methods in use at tlie Hydrometeorological Section of the U. S. Weather Bureau for Computation of Maximum Possible Precipitation". Unpublished. 4. Burns, Joseph I. (1959). Private communication. 5. Byers, H. (1944), "General Meteorology". McGraw-Hill Co., p. 386. 6. Charney, J. and Phillips, N. A. (1953), "Numerical Integration of tlie Quasi-Geostrophic Equa- tion for Barotropic and Simple Baroclinic Flow". Journal of Meteor., Vol. 10, No. 2, p. 71. 7. Doos, Bo R. (1958), "The Effect of the Vertical Variation of Wind and Stability on Mountain Waves". Technical Report No. 13, Project NR-082-071, Contract NONR-1600(00), Office of Naval Research. 8. Douglas, R. H., et al (1957), "Pattern in the Vertical of Snow Generation". Journal of Meteor., Vol. 14, No. 2, pp. 95-114. 9. Eliassen, A. (1955), "Lectures in Physical Weather Prediction". University of California at Los Angeles. 10. Eliassen, A. (1957), "Handbuch derPhysik", Vol. 48. 11. Gates, W. L. (1958). Paper presented to the American Meteorological Society Seminar, Los Angeles, California. 12. Gringorten, I. I. (1959), "Extreme Value Statistics— A Revised Application". Paper presented to the San Diego Meeting of the AMS, June 1959. 13. Hamilton, E. L. (1954), "Rainfall Sampling on Rugged Terrain". U. S. Department of Agricul- ture, Tech. Bulletin No. 1096, 41 pp. 14. Holmboe, J. (1944), "Dynamic Meteorlogy". John Wiley and Sons, Inc., New York, New York. 15. Holmboe, J. (1953), "The Upper Level Winds Project". Final Report, Contract W 28-099 ac- 403, Geophysical Research Directorate, AFCRC, Cambridge, Mass. 16. Landsberg, H. E. (1957), "Review of Climatology, 1951-1957". Meteorological Monographs, Vol. 13, No. 12, pp. 1-43. 17. Levee, R. (1959). Private communication. 18. Meinzer, O. E. (1942), "Hydrology". Dover Publications, Inc., New York, New York. 19. Myers, V. (1959), "Factors in California Orographic Precipitation". Paper presented at the 177th Meeting of the AMS, San Diego, California, June 1959. -23- 20. Palmer, C. E. (1958). Private communication. 21. Richtmyer, R. D. (1957), "Difference Methods for Initial Value Problems". Interscience Pub- lishers, Inc., New York, New York. 22. Sawyer, J. S. and Bushby, F. H. (1953), "A Baroclinic Model Atmosphere Suitable for Numeri- cal Integration". Journal of Meteor., Vol. 10, No. 1, p. 54. 23. Smagorinsky, J. and Collins, C. 0. (1955), "On the Numerical Prediction of Precipitation". Monthly Weather Review, Vol. 83, No. 3, pp. 53-68. 24. Smebye, S. J. (1958), "Computation of Precipitation from Large-Scale Vertical Motion". Jour- nal of Meteor., Vol. 15, No. 5, pp. 547-560. 25. Weaver, R. L. (1959), "Types and Synoptic Characteristics of Major California Storms". Paper presented to the 177th Meeting of the AMS, San Diego, California, June 1959. 26. U. S. Weather Bureau, Technical Paper No. 32 (195B), "Upper Air Climatology of the United States, Part 2 — Extremes and Standard Deviations of Average Heights and Temperatures" . -24- APPENDIX A. THE COMPUTATION OF THE TOPOGRAPHY OF THE SACRAMENTO VALLEY INVERSION The topography of the inversion, overlying the Sacramento Valley, may be calculated as follows. Consider the (x, z) profile, through verticals A, B, and C, shown in the following figure: z B Schematic Sketch of Height Relationships used for Computing Inversion Slope At the vertical B, a radiosonde station reports wind, temperature, pressure and mixing ratio as a function of height. From this data the inversion height on vertical B is known. Above the inversion the wind is normal to the mountains in the model; thus, the intersection of the isobaric surface with the (x, z) plane, above the inversion, is horizontal. Let p(z) denote the density below the inver- sion, and p'(z), the density above; then the station pressure at C is P = P c c (Zj) + /p'gSz+ fp gSz , (A.l) and the station pressure at A is Pa=Pa(=^ ' / r ,) + /pg^z = P^ (Zj) + /pgSz + I pg^z (A. 2) -25- Subtracting (A. I) from (A. 2) we obtain ''a "" Pc ^ JCp-^) g5z - ypgSz . From the above equation the vertical distance (z — h„) is ! C^ 2, -h 1 c (p-p')g Pa - Pc +ypBSz -26- APPENDIX B. THE COMPUTATIONAL STABILITY OF THE FINITE DIFFERENCE APPROXI- MATION IN SECTION 2.2 In this appendix we shall explore (a) the stability of the finite difference approxima- tion. Eauation (11), and (b) the error introduced into the solution by the replacement of the partial differential equation, (8), by the finite difference equation, (11). The finite difference equation, ap- proximating the first order differential equation, is said to be stable if the error introduced on i-th vertical is not percentually increased when propagated along the constant level z to the (i-t-1) ver^ tical. To test the stability we proceed as follows: consider Eauation (8) 4iL = -u,(x)F(x, z). (8) dx For a segment of the simplified profile the slope of the terrain is known, between the i and the (i-t-1) vertical, and the above equation is of the form ^ = - Au , (9x whose solution is u = u<,e-^', (B.l) wnere u^ is a constant determined by boundary conditions. The solution to the finite difference equation, (11). "i+i - Ui = - ^Au,+ j , (B.2) is found by substituting Ui = "o^' (B.3) in Equation (B.2), solving for r in terms of ^A, and then substituting r in the general solution (B.3) In tnt-t way UfA and the sohition to the finite difference equation is "■•■="» (uT*)'*"- *■''' -27- Consider the error Su ; the finite difference equation governing the way this error is propagated to the (i+1) vertical is Su„,-Su, = -fA«u,,, The solution to this difference equation, with the boundary condition i = 0, 5u = Su 5u,^, = « (B.S) With Equations (B.4) and (B.S), the percentual error in the solution on the i=0, i and i+1 verticals is calculated as It is seen that an error introduced into the solution on the first vertical propagates through the solution without becoming percentually larger; thus, the proposed finite difference ap- proximation satisfies a definition of stability, given by Levee (1959). When a partial differential equation is replaced by a finite difference equation, an error, known as truncation error, is incurred. Thetruncation error, as defined by Richtmeyer (1957), is "u i-"i /f'" (B.6) and this error ( may be calculated from the above equations in the following way. The substitution of (B.l) and (B.2) in Equation (B.6) gives f =-Au,^,- -A(i+'/2)^ — Au e (B.7) From Equation (B.4), we can calculate a series expansion for ln(^= - (i+1) InM I- - (i+l) (aa - (m\ AU8E Sierra Nevnda CALIFORNIA WATER RESOURCES AND CONSTRUCTION IS BRANCH OR ESTIMATING _E PRECIPITATION x> — SECTION OF TERRAIN, .EY INVERSION Fig. I ISOBARIC TROPOPAUSE E Z 3 O STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION SCHEMATIC CROSS -SECTION OF TERRAIN, SHOWING VALLEY INVERSION Fig I STATE or CALirOHNI* OePAHTMCNT OF WATER RESOUWCCS DIVISION or OCSION AND CONS TNUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION LOCATION MAP FEATHER RIVER BASIN Fig . 2a STATE or C*LIPO)INI« OCPAKTMCNT or WATEf) RESOUKCCS DIVISION or OCSISN AND CONSTRUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION LOCATION MAP FEATHER RIVER BASIN \ Fig . 2q STATE or CAtlFORNIA OCPARTMCNT OF WATER RESOUftCCS DIVISION or DCSION AND CONSTflUCTION LOCATI RIVER STATE OF CALIFORNIA Fig. 2a STATE OF CALIFOKNI* OCPARTMCNT OF WATEN RESOURCES DIVISION OF DESION «N0 CONSTRUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION -+■ LOCATION MAP PROFILE STRIPS A, B. and C \ M B X 1 Fig. 2b STATE or CALirOKNIA OCPARTMENT OF WATEN RESOURCES DIVISION OF OESIOM AND CONSTRUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION ■+■ LOCATION MAP PROFILE STRIPS A, B. and C \ M B ^ 1 ^ Fig. 2b STATt OF CALIFORNIA OEPARTMCNT OF WATER RESOURCES DrvrSION OF DESIGN AND CONSTRUCTION PROCEDURES FOR ESTIMATING PROFILE STATE OF CALIFORNIA M ^ X / Fig, 2b STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION PROCEDURES FOR ESTIMATING I MAXIMUM POSSIBLE PRECIP ITATION DETAILED LOCATION OF PROFILES USED IN COMPUTER CALCULATIONS f •CALI or HILCt Fig 2c STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES OVISOW 0F0ESK3N AND CONSTRUCTION PROCEDURES FOR ESTIMATING t MAXIMUM POSSIBLE PRECIPITA TION DETAILED LOCATION OF PROFILES USED J IN COMPUTER CALCULATIONS ICALI or MILCt I t > 4 • Fig 2c STATE OF CALIFORNIA DEPARTMENT OF WATER RESOUNCCS DIVISION OF DESIGN AND CONSTRUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITA T ION DETAILED LOCATION OF PROFILES USED IN COMPUTER CALCULATIONS Fig 2c 8000 7000 6000 PROFILE " ft" 1 BASIN 1 BOUNDARY /^ \ JT . /^ V. A V A A .. A./'^^'^x:^ / \ J^ / A -^ \ /^ • • c o o > • Ul 5000 4000 /\ K > ^' V - ^ f V (■ X^^ v^^* J ^ \ A A/ ^ / V \" / 3000 2000 1000 ° / L EGEND / r ^_y^ Averaga Tvr^ Elevation / ^^ Simplified ■^^ Profile / 1T»T( OF C*LI'0««.l« OC»*"TMENT Qf WITER HESOuACES DIVISION Of OE*iG" iNO CONSIBJCTlON PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION ' 1 o«i 1 FY INVEftSlOM ^^^^^^ -/ / / AVERAGE ELEVATION OF PROFILE "A", Showing Simplified Profile Used In Computer Calculations \'>'>.N\\\\\\\ ;v\\\v\v\v\ s.-<:vcvt,%\-\-<.^g^ cvrvw^' 3 10 20 30 40 50 60 70 80 90 Milts From X. 100 no 120 130 140 150 160 170 Fig J STATE OF CALIFORNIA OEPARTMENI OF WATER RESOURCES DIVISION OF OESION AND CONSTRUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION + AVERAGE ELEVATION OF PROFILE Showing Simplified Profile Used In Computer Calculations. "B", 170 Fig 3b \ STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESION AND CONSTRUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION ■+■ AVERAGE ELEVATION OF PROFILE B Showing Simplified Profile Used In Computer Calculations. 170 Fig 3b 9000 8000 7000 6000 5000 PROFILE "B" 1 SASI L_ N BOUNDARY / ^/^ F'^^^ M t^^^^ ^ l\ P A / V / f"^^ ^ "H ll 4 ^ ^\^ ^ / "^ i \ i • • c o o • 3000 / V i J LEGEND A ^>F^ Av«roge -^t;^^?^^^ Elevotion 2000 // ^^a Simplified ^^^^ Profile r ST*T( OF c»iire*Hi« OEMHTMCHI or WATER NISOUncES OIVISION or OeSIOH *NC CONSTRUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION 1 ,.LLI. '"Vt 1 f 1000 1 ,<oo ^ yi OF TRAJECTORIES nON PRODUCTS Fig. 4 )PAUSE \ CALIFORNIA JWATER RESOURCES 4 AND CONSTRUCTION NS BRANCH OR ESTIMATING iLE PRECIPITATION oo i/l OF TRAJECTORIES nON PRODUCTS Fig. 4 Tl T2 T3 T4 TS Te ISOBARIC TROPOPAUSE \ \ \ \ \ \ \ \\ \ \ \ \ \ \\ \ Z 3 o < > UJ , I ■■ w \\.. \ V (Coost \ TEMPERATURE INVERSION'S. .'Ronqe"'-->-!. ^coLO, MOIST l^'^^'^. _ \ I • I XI X2 X3 X4XS X6 XT STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH PROCEDURES FOR MAXIMUM POSSIBLE ESTIMATING PRECIPITATION SCHEMATIC DIAGRAM OF TRAJECTORIES OF PRECIPITATION PRODUCTS Fig. 4 " v-------^ Terrain 'Surface OF CALIFORNIA DF WATER RESOURCES IGN AND CONSTRUCTION IONS BRANCH FOR ESTIMATING IBLE PRECIPITATION — oOo- A RAINDROP TRAJECTORY : COMPUTING SCHEME Fig 5 Ttrrain 'Surfact OF CALIFORNIA 3F WATER RESOURCES IGN AND CONSTRUCTION IONS BRANCH FOR ESTIMATING IBLE PRECIPITATION —000- A RAINDROP TRAJECTORY COMPUTING SCHEME Fig 5 .Trajectory J-t-2 \ \ \ \ \ \ ^'' .-^^ Terrain /'Surface J + 1 ''P^ ,ff^"^ s. i liTA .--" i- -2 i -1 STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AWD CONSTRUCTION OPERATIONS BRANCH PI MA) SCHEMATIC D ILLUSTR ROCEDURES FOR ESTIMA 1 UJ 1 \. ■\ A; UJ ''000 ■E I \ \ ^^ '! * 500 — ~ — y <^ 10 20 3 40 5lO 60 7 C « 9 ,0 " 12 13 14 15 16 170 MILES FROM Xo PROFILE "A" NUMBER OF MILES FROM ELEVATION ON VERTICAL Xo PROFILE (Meters) 364 00 21 4 1 72 431 14 32 64,0 1692 25 33 66 1692 25 34 67 1750-77 48 95 28 1750 77 49 98,0 1866 29 55 MOO 1866 29 STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION -COO- PLOT OF RAINDROP TRAJECTORIES No 36 AND No 55 ON PROFILE "A" -I ; - - ; 1 • i i • iili iMi i i t ■ i ! : ■ Mir I ; : ■ i 1 I- i Mil ! ; t ! ; ; ! : i ■ . 1 1 ,s i ! • : ^ i ; : ; 1 i : \ r ^. H 1 \ \ 1 \ |;.'-7 : : -^ T- ■ 1 \ • ■ i i : \ \ '■ j M ■ j I'M \\\' \ ' i r^-t.i \\ \ \ -J I! \ f \ w \ •rH i \ 1 ' '. \ \ ■ \l "r ' ■; 1 T ■ t V ■ r f - ■ 1. 1-1 ' i. ■ t - - m !-)-r ' -i:ijl.. ■/' ■ ' 1 . -. ; ' . ; : : - ; k^j -!■:;.[- i- 1-1 I i I LI: -I-h! - • ;— 1^-4— r 1 : ■ ' ■ : f 1 . . :■■ ' ' ' i ! : • ■ .'■ • : ■ '--1 /ELOCITY - 4"^/s : •_ .---- ' i-t - rH:t- xzHiF -Hit -It ■ ! » 1 ■ ; 1 5 30 35 40 45 ERTICAL ON PR0FILE"A" 50 55 TC OF CtLIFO*NI« NT OF WATER RCSOUKCCS OESION *N0 CONSTRUCTION ^ES FOR ESTIMATING PSSIBLE PRECIPITAT ION !- oc»- ^^ ^ URLY RATE OF OROGRAPH ROFILE"A", DEC. 1955 STOR IC M ; : I . 1 1 . . : ' i : |i if i 1 i-nl ^ : i i i : : ' ( 1 L i !-lr! i 1 ! ! ' ■ : I ■ ; 1 , ■ i ' j:, UJ 001 2 000 1000 1tr 1-tl i}] TERMINAL FALL VELOCITY - 4 /$ thi. ^n- -!:.l±r 10 20 25 30 35 40 45 II . II 50 55 NUMBER OF VERTICAL ON PROFILE A STiTt OF CALIFO«NU DEPAttTyENTOF WATER HCSOURCES DIVISION OF DESIGN AMD CONSTRUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION Fig. 7o -oOo- DISTR PRECI IB PI UTION OF HOURLY RATE OF OROGRAPHIC TATION ON PROFILE"A", DEC. 1955 STORM »rE Of CaLIFORNI* ENTOF WATER HESOuRCES F DESIGN AND CONSTRUCTION RES FOR ESTIMATING 'OSSIBLE PRECIPITATION -oOo- lURLY RATE OF OROGRAPH ROFILE"A", DEC. 1955 STOR IC M 5 30 35 40 45 pRTICAL ON PROFILE "A" 5 c 5 •t:; r. ': -F'\: < 1-f- n^-i- "r: i.t -*'-.■ k' p ; "T"i .-T .:-r l^'. '■' •1 ■; ' t ' * 1 1 1 1 1 1 1 1 1 1 ' 1 J— \i^ y ~" j^ i^ !-:• PROFILE "a » u; '■':k -'TW- — 1-' - {!- "Vi .•£H ■+ ■ V.r:. -W:. "1* ■ ■?■ ■t-r- vJ.'z :4:i '. ' '. " ' *"■ * :':i- ■-r '. I '.^ i:;' ;^: :iU ;:; J- n f^^i nii- ^Ti .:^^ . -"^ ]^i I. t- ftTE OF C«LIF0»N1« ENT OF WATER RESOuSCES F DESIGN iNO COnSTRucriON IRES FOR ESTIMATING 'OSSIBLE PRECIPITATION Fig. 7b - LiJ 10 15 20 25 30 35 40 45 NUMBER OF VERTICAL ON PROFILE "A" STATE OF CflLIFOf?NIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION Fig, 7b -- DISTRIBUTION OF HOURLY RATE OF OROGRAPHIC PRECIPITATION ON PR0FILE"A", DEC. 1955 STORM ; , ' / / . ■ i r ': \ ' "'. '. \ ; : 1 ; .\ I ^ 1 -; - 1 - - 1 i ^ j_uL. \ 1 .!.;:■: i ; i q \ \ ^ : -\ • . 1 ; . 1 : A i : - y ' j> A \ r... ■ !>..•.■ \ ^... . f. L. 4. -J- '-i.. f i ^ '- -T : ■ i - I ' : ] ■ ' 1 :- • k: i ■ .'. ELOCITY - e'^/s 1 1 i- ' - tf]. 1- -'4- 4.; ,' 1 ■;:fft x4-_.. 1 I 30 35 40 45 RTICLE ON PROFILE "a" 50 55 ,4^ £44 M. IX XJ. XI PROFILE"A" r .:•„'. i]:4tijl't^-^4:^4^;iiii:j::i 7c re OF CALIFORNIA NT OF WATER RESOURCES OESION AND CONSTRUCTION !ES FOR ESTIMATING Fig )SSIBLE PRECIPITATION f; -oOo- RLY RATE OF OROGRAPH ^OFILE"A", DEC. 1955 STOR IC M TE OF CALIFORNIA WT OF WATER RESOURCES OESISN AND CONSTRUCTION >ES FOR ESTIMATING )SSIBLE PRECIPITATION c -oOo- RLY RATE OF OROGRAPH ^OFILE"A", DEC. 1955 STOR IC M 3 O ■c \ w d) SI o c z O < 1— Q. O tr Q- o I CL < a: O tr o Ll. o LlJ t- < >- _i CE O I « o E Z O 1- < > _l UJ 10 — 9 - 8 - .7 - 6 - • 5 - - = "■: 1 1 ' • J ' ' ; : : ; ■ i I" ; ■ !-i ' ■; . 1 i i 1 • I ! i ■'. \ : ' ; \ '■ ;^ ' ^ I -T : 1 / 'N ' i t j : ■ ■ / ^ 1 : . ■ :■ ' ; ! / i f ■ ■ 1 ■ i ! i M 3 - r " ■ 1 ' : 1- 1 ■ \ ': '. \ -: -\-'.--- J ;. ■" ^.■ - I r i : : ; t ! : i j i i :/ - 14-11 ! : : - -i . 1 ; ';_ . : ; 7 ■ '. \ ' ' ; : i : -'-• i ■ * - : ' ; • ' m -1 f-I""*. • 1 1 1 ■ J 1 i 1 . ; ■„ . 1 i. , -1 1. : ■ . . : ! [S: ' i"; , - 1 ■ . ■ '■\ *■ -in' i 1 • ll A ■ ' % ''.'■■' : ; : ^ ' 08 i : ■ ^ i ; ■ \ \ 07 ■ \ ^ \ ■ • - ■ '"■ ^ \ \ ; A i : - I • 03 : . 1 -'. ■ 1 ; ■ ■ -■;!-:: A 02 ; ■ 1 I \ ; - ; I i : ■ ] ' '. , \ 1 >;-;_; ■ n 1 . . : : ' I L r : ':'■ I ; ;:" - - ( 1 ^ -— 1 '■ ' 'ill 4 ' - ] r_ -! -' 4^ 11 1- "'": i ' ' i "i 1 ; ' ■ ' : \ ■ . r I ' '. \ \ , ' 1 1 : ■ ; ■ j A i' i- 1 i wWJ-<><»- 1 yo-tn:^ 1 I - ! j , ' : : t : . ; ■ ■ 1 ' i 1 . 1 ' ' 1 1 t ; -■" ! - 002 : ' t : \ i ; ■; - ■ 'i ' ; : . :l . ; ; ' ' ■- i- i ; j_ '■ :~ ':_; ■ 1 ; !- i r ; i TERN 1INAL FALL VELOCITY - 8"^/s 1 ; ■ r 1 i-T: :^-Jai '!:,i" n. _4 .. j_ i--!-- : mJ-i ■ ■■: • ■ 1 ;:fF iz -t- ■ 2000 - D ":t: 5 1 ;;!t 7 X ^1 1 Nl 5 JME 2 3ER tTt — £U 2 - V ^l: 1 iiiii: 5 ER TIC LE ■3 0I\ ,'X S5 P i T RO 1^; i , FIL 4 E ' 5 A" ■ i 1: 5 44, 55 '■P Tq^ 1000 — #1 13 f n ^4- m tttJ: :.ui t3= I }■ 5 it Trn -'Tf Sit 1 ■4fT ^ ^ Vt PF (OF ILE :"A # i ^0 Eti , ..4: iff -Cff - -\h\ ■;Tt^ i::- ;.. -,n£- _ 01 P WAX DEP< VISI ?0C IML rli; ST RTM ON EOl _M ATE ENT F DE JRE = 0S OF C OF » SION s i_ 3IBI ALIF «TE AND OR ^ F 0«N H «E CON EST RE( lA SOU STRL IMA ;ipi RCES CTIC TIN TAT N G ION t!F: - t;'; r -iL:j Fig iiiJ 7c Dl P SI RE rR CI IB PI L Jl ri( Tl Oh OF HOURLY RATE OF ( J ON profile"a", dec. )R I9J oc 55 ;r s AF T( >H DR IC M 22 23 VERTICAL ;alifornia 'ater resources and construction I S BRANCH or estimating Ie precipitation iT OF THE LAYER - S (DEC. 1955 STORM ) Fig 8 22 23 VERTICAL ;alifornia 'ater resources and construction I S BRANCH OR ESTIMATING LE PRECIPITATION iT OF THE LAYER - S (DEC. 1955 STORM ) Fig 8 2000 20 21 22 23 NUMBER OF VERTICAL STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION A VORTICITY BUDGET OF THE LAYER - 1000 TO 1500 METERS (DEC. 1955 STORM) Fig 8 -Interfoce Solidified stream line representing ttie windward slope • CALIFORNIA WATER RESOURCES N AND CONSTRUCTION )NS BRANCH FOR )LE ESTIMATING PRECIPITATION JOo- IIABATIC IE FLOW MODEL Fig. 9 r^/f^-i ' J • -Interfoce -Solidified stream line representing the windward slope ■ CALIFORNIA WATER RESOURCES N AND CONSTRUCTION )NS BRANCH FOR JLE ESTIMATING PRECIPITATION 300- IIABATIC TE FLOW MODEL Fig. 9 = Constant i Wind Profile STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH -Interface Solidified stream line representing ttte windward slope PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION ooo THE ADIABATIC DOUBLE COUETTE FLOW MODEL Fig. 9 ASYMPTOTE OF COT(^%) • n ROOTS FOR MAXIMA IN DISTRIBUTION I X UROOTS FOR MAXIMA IN DISTRIBUTION H ^[i^y];;iii:i[;Uaibj;iii:j:Hi.feii3i:.:H;^!i-;!j^L;^^ STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION 000 GRAPHICAL SOLUTION FOR ADVERSE PERIODICITY I .2 1.3 I 4 Fig 10 ASYMPTOTE 0FC0T(^7n) • n ROOTS FOR MAXIMA IN DISTRIBUTION I X UROOTS FOR MAXIMA IN DISTRIBUTION H ^ill!j.ti;!ilHniiEiLyj;iiD:HiHliiri:.:n;iii!:-;:JrL;il^^ STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH PROCEDURES FOR ESTIMATING MAXIMUM POSSIBLE PRECIPITATION 000 GRAPHICAL SOLUTION FOR ADVERSE PERIODICITY I .2 1.3 I ^ 4 Fig 10 .20 [ -■>; -:;1^^ -4-^- -r'" 7^- ^■^|ii^t: _---.;- ;:ri .1 -:; ■/it -r; : •l^f ft].' ■ [-. 1 *1:r ,^^ ^lai-yH i... -- ifj- Tr^ 'X' tifrjr ^Hi ^ il-' i T. i."" 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' ■ 1 ', ■ I : -nA- /T ■ 1 ■ t; ;' ■; ;:1 ■ Mi '"■'■ .,li Ifj^ ;i-i| ti , li . ■ i1 rti; t' i: : ' '■X\- 'i; '. i I 5.'" r z -- - — ~r Trf p-: r r^ ;" ■''.\ ^^^ :i.|t .1-1. . tC; Syt ij !%'-^ ^^ ' ' K 1 - • 1 ::!! r i"-'I ivr: t - - ■ 1.' ih; - ; .■-, L;:; it.itirril 1 :;rr -; f . : ' ■;i: ••;; . , ■ ^ ■ ; ;■ '.;■.. : *•! \w- %' '\s\ Ji : ' [ 1 ■ , 1 ■ ' :':'i! illjl^ : :;;.■:::■ 3 ■:"■ r : i- ' ■ ' lid i j ! j i'l- i ^ 1 ;- ' ii ;. - ''4; ;; : 1 ■ ' 1 ; i , 1 - ■ ";; iM li : ', *' i ^it^-l> • / ii^ T-->^ •:'- t-: -+- .•■ Tit i: ' liil -■!-.' r!' r- ' '- r;U i'." ;.l. . ^fr-^;i '■ '■'' -!i \\\ Ijil j";.i' i-'l' . iti! ■i^' ii-i. ■ i;, ■■:l — f '■rr. ^ jt r Tz. - : ■■ Ita .-.h ::, ■ i. J rn ■ ; *' '^"'1 ■ ■ ['■ - !-■ - '■1 1' ^t!^t> ^'"^I'.i 1 i 1 ' :. ' 1 -r 'U- 1' 'it; t, "ii it ijl Mr, TM; .%, 'i- ril: '-iir iit -;.;:,-: ilii lI, i 1 iii' i'-':t|f Z p .-n j:: i;.-. , _- '. " '^4 :i";i -;r, i. :'r 1.;- -"r ■ 1 ■ ' nii' .1: ,1! , ..J.. 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'^ ^: ^ ■ - -ir s tii- '■r-' 1 ■ :i ^l-i :■■[' :ii' ' r 1 W% 1 i!'i il •i4' !IH %^ 1 r i IF 1 '' ' ■n: iiti STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH tj - ;■ ■^:T .04 4'i yf^ rrii: T^-. :4t-; ^-KT -- IS" .■^'^f — *■.>■■ '■ ' T ■ [•■■ 1 41 i — H-;- .11;: .1!:. 11- : :!■ :i|-'- . 1; , t.4i^- 1: "'' •X ! ifi 1:1: 1; 1^4 itftttt r.J' iiii _l^:l i ■C^ • ■H .- ^-r- 7^' 7 ^- --- -- + M - m ■ll- -rr- ■r- :■::;,,. : --+--— - — f-T- 1 '"I ;1|j ^•l i. ■•t ' i j-i; Ijl :';i i;li — t- 1 ii- i -^^:. ii', : L' ■ — A: - - -:\- iVf \'x pff ■H'-t '4^ ■4-i :j t; '•'!' i 1- " ' ^ '' aii ji ij ':;. :,: ■ :Mv :*', r^i •! i]il i^ii i!>^ -ii- ;!;■ ' PROCEDURES FOR ESTIMATING - ^-, :.. r..: .^/^ .02 1- —-4 !■:: -i— ^ i,i' -;- ■ ' -\ '-: ::-"^^ -• : i' !' 2-: ■:' .■;7 ^!f k i l^:^ -!■* 'lit Vm\ ' ■'' 'ill !;i! , .r, ; i-i' "'' ;'!| 'III ijU -' ■ ■ 1 1 MAXIML IM POSSIBLE PRECIPITi fVTION '^; 7 : .: ^ ;~ ;■; . ■ : r- :: ; 4'i .^hI lift ■i t =-ii ■ 1 ■ ■ 1 ' ;:| 1 1 ■ 1' 1!^ : : : 1 ;''! ' '1 ' fii'l \ CO) fei.^ j: ■ y it T- rn :" : . ■ i i' i_ V ■ ':':[] ' ■ ! " !!;■■ i;!i ll'' ^: :i; li4 li:^ i'^itj- i-i!'i;i;!:i- ■i h' t '1 Ii it-" ;r.i *H| iii ' 1 t 1 ■[-1 i;l Mil it: GRAPHICAL SOLUTION FOR ADVERSE PERIODICITY "+' U'^ A- '.' ■'vL •::;t V( t Str ; J ^ :, 1 . ., . >'■ -T- t;: 1 1,-7' # iri' i4 ■fl- tf;. r- : itr! .; ■ ■ i il:,! R:. i|U Ifr -> it 1 '-i a t-i' •I- t[ ■' H 1 :^-- •'-- irn .-u :, . " . :: ;: t':t 1 1 ■—1 M - -Si' -t -L1-. :hi ff] 1^ iir: ' !.■ Hi Hi- ^ F'! ■M%\ M't ]■ Il '■' ''r' 1 H t| lir-; ';ii lit ^;ii II11I 1 2 n — 3 0,4 ^■ 5 ,6 0.7 8 0.9 I 1 . 1 1 2 1 3 1 A Fig 10 STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH PROCEDURES FOR ESTIMATING AXIMUM POSSIBLE PRECIPITATION -000- .CULATED 72-HOUR ISOHYETAL MAP, CEMBER 1955 STORM (8 "^/s ), FEATHER RIVER BASIN Fig lio STATE OF CALIFORNIA DEPARTMENT OF WATER RESOURCES DIVISION OF DESIGN AND CONSTRUCTION OPERATIONS BRANCH PROCEDURES FOR ESTIMATING AXIMUM POSSIBLE PRECIPITATION -000- .CULATED 72-HOUR ISOHYETAL MAP, CEMBER 1955 STORM (8 ^/s ), FEATHER RIVER BASIN Fig. Ila (t Profile "a' na DAVIS 240512 o ^ \