{"Bibliographic":{"Title":"Decomposition of a wind field on the sphere","Authors":"","Publication date":"1976","Publisher":""},"Administrative":{"Date created":"08-20-2023","Language":"English","Rights":"CC 0","Size":"0000020826"},"Pages":["H\nQC\n851\n83\nOF\nCOMMENT\nU6N5\nV6\nno.\n59\n*\n*\nOAA Technical Memorandum NWS NMC-59\nANGELES\nSTATES\nOF\nDECOMPOSITION OF A WIND FIELD ON THE SPHERE\nNational Meteorological Center\nWashington, D. C.\nApril 1976\nnoaa\nNATIONAL OCEANIC AND\nNational Weather\nATMOSPHERIC ADMINISTRATION\nService","er Series\nThe National Met\n(NWS) produces weather anal-\nyses and forecast\nexpanded to include the entire\nglobe. The Center\ncy of forecasts, to provide\ninformation in the\nas practicable.\nNOAA Technical\nation of material of general\ninterest which\nformally elsewhere at a later\ndate. Publication\nTechnical Notes (TN), Na-\ntional Meterologi\nugh 48 are in the former series\nESSA Technical Mem\ninning with 49, publications\nare now part of th\nPublications lis\normation Service (NTIS), U.S.\nVa. 22151. Price: $3.00 paper\nDepartment of\nOFFICE\ncopy; $1.45 microfiche. Order\nnumber, when given, in parentheses.\nher Bureau Technical Notes\nTN 22 NMC 34 Tropospheric Heating and Cooling for Selected Days and Locations over the United States\nDuring Winter 1960 and Spring 1962. Philip F. Clapp and Francis J. Winninghoff, 1965.\n(PB-170-584)\nTN 30 NMC 35 Saturation Thickness Tables for the Dry Adiabatic, Pseudo-adiabatic, and Standard Atmo-\nspheres. Jerrold A. LaRue and Russell J. Younkin, January 1966. (PB-169-382)\nSummary of Verification of Numerical Operational Tropical Cyclone Forecast Tracks for\nTN 37 NMC 36\n1965. March 1966. (PB-170-410)\nTN 40 NMC 37 Catalog of 5-Day Mean 700-mb. Height Anomaly Centers 1947-1963 and Suggested Applica-\ntions. J. F. O'Connor, April 1966. (PB-170-376)\nESSA Technical Memoranda\nWBTM NMC 38 A Summary of the First-Guess Fields Used for Operational Analyses. J. E. McDonell, Feb-\nruary 1967. (AD-810-279)\nObjective Numerical Prediction Out to Six Days Using the Primitive Equation Model--A Test\nWBTM\nNMC\n39\nCase. A. J Wagner, May 1967. (PB-174-920)\nWBTM NMC 40 A Snow Index. R. J. Younkin, June 1967. (PB-175-641)\nNMC 41 Detailed Sounding Analysis and Computer Forecasts of the Lifted Index. John D. Stackpole,\nWBTM\nAugust 1967. (PB-175-928)\nNMC 42 On Analysis and Initialization for the Primitive Forecast Equations. Takashi Nitta and\nWBTM\nJohn B. Hovermale, October 1967. (PB-176-510)\nWBTM NMC 43 The Air Pollution Potential Forecast Program. John D. Stackpole, November 1967. (PB-176-\n949)\nNMC 44 Northern Hemisphere Cloud Cover for Selected Late Fall Seasons Using TIROS Nephanalyses.\nWBTM\nPhilip F. Clapp, December 1968. (PB-186-392)\nOn a Certain Type of Integration Error in Numerical Weather Prediction Models. Hans\nWBTM NMC\n45\nOkland, September 1969. (PB-187-795)\nNoise Analysis of a Limited-Area Fine-Mesh Prediction Model. Joseph P. Gerrity, Jr., and\nWBTM NMC\n46\nRonald D. McPherson, February 1970. (PB-191-188)\n47 The National Air Pollution Potential Forecast Program. Edward Gross, May 1970. (PB-192-\nWBTM\nNMC\n324)\nRecent Studies of Computational Stability. Joseph P. Gerrity, Jr., and Ronald D. McPher-\nWBTM\nNMC\n48\nson, May 1970. (PB-192-979)\n(Continued on inside back cover)","851\nU6N5\nno.59\nNOAA Technical Memorandum NWS NMC-59\nDECOMPOSITION OF A WIND FIELD ON THE SPHERE,\nClifford H. / Dey\nJohn A. Brown, Jr.\nNational Meteorological Center\nWashington, D. C.\nApril 1976\nCENTRAL\nLIBRARY\nMAR 7 1977\nN.O.A.A.\nU. S. Dept. of Commerce\nAND ATMOSPHERIC\nNOAA\nUNITED STATES\nNATIONAL OCEANIC AND\nNational Weather\nDEPARTMENT OF COMMERCE\nATMOSPHERIC ADMINISTRATION\nService\nElliot L. Richardson, Secretary\nRobert M. White. Administrator\nGeorge P. Cressman Director\nus OF country\n77\n0443","CONTENTS\nAbstract\nIntroduction\nStaggered and nonstaggered finite\ndifference formulations\nA staggered finite difference formulation\nof the spherical problem\nBoundary conditions at the poles\nSolution of the finite difference equations\nAccuracy of the solution\nSummary and conclusions\nAcknowledgment\n1\nReferences\n1\nAAOI\nenverame2","DECOMPOSITION OF A WIND FIELD ON THE SPHERE\nby\nClifford H. Dey and John A. Brown, Jr.\nNational Meteorological Center\nNational Weather Service, NOAA, Washington, D.C.\nABSTRACT\nWhen decomposing a horizontal wind field into its\nrotational and divergent components, care must be\ntaken to assure compensating truncation errors in\norder that the resulting wind components can be used\nto accurately reconstruct the original field. An\nexample is presented of a finite difference system\nwith second-order accuracy for a regular spherical\ngrid which yields results sufficiently accurate for\nuse in initialization procedures for a primitive\nequation forecast model used at the National\nMeteorological Center (NMC).\n1. Introduction\nThis paper is not presented to document a new idea in finite\ndifferencing, but rather to bring attention to a problem which is too\noften overlooked. When decomposing an atmospheric horizontal wind\nfield at a particular vertical level into its rotational and divergent\ncomponents, one should be careful to maintain a finite difference system\nwhich is internally consistent. By this we mean that the finite\ndifference forms of the wind components and the vorticity and divergence\nshould be established consistently from the stream function and velocity\npotential, both within the fluid and at the lateral boundaries.\nIn addressing the global forecast problem, personnel at the NMC have\ndeveloped a grid point 8-layer global primitive equation model (Stackpole,\nVanderman, and Shuman, 1973), hereafter referred to as the Global Model.\nRecent testing of this model has revealed the presence of large amplitude\nnonmeteorological oscillations in the forecast. In an attempt to reduce\nthe portion of the noise due to initial mass-wind imbalances, it was\nnecessary to develop a finite difference method for separating the analyzed\nwind (Flattery 1970) into its rotational and divergent components. The\nmethod presented here is an internally consistent finite difference system\nof second-order accuracy for a regular latitude-longitude grid. However,\nthe general procedures of this method should be applicable to other systems\nas well.","In section 2, the accuracies of two finite difference systems are\ncompared for a one-dimensional Cartesian grid. The details of the finite\ndifference system for the spherical grid are presented in section 3. The\nlateral boundary conditions at the poles are considered in section 4.\nThe numerical method for solving the problem is discussed in section 5.\nFinally, the results of testing the proposed technique on a real data case\nare presented in section 6.\n2. Staggered and Nonstaggered Finite Difference Formulations\nIn one dimension, the relative vorticity (E) and the wind (v) can be\nwritten as\n(1)\nv=\n(2)\nwhere 4 represents the stream function. Consider the problem where the\nanalyzed wind field (v) is given. Equation (1) will be used to obtain 4\nand Equation (2) to calculate the reconstructed wind field V. The finite\ndifference forms of (1) and (2) should be consistent for the reconstructed\nwind field v to be equal to the original wind field V.\nThe first finite difference system we consider is one in which the\nvorticity E applies at the grid points where the analyzed winds are located.\nThis will be termed the nonstaggered finite difference system. In this\nsystem, the finite difference forms of (1) and (2) are\n(3)\n(4)\nin which the notations\n(5)\nhave been used. Here, Ax is the spatial grid distance. Thus at point i,\nEquations (3) and (4) are, respectively,\n(4x) 2[¥i+1+ Yi-1\n(6)\n2","and\n(7)\nAssume solutions for the analyzed wind and the stream function are given\nby\n(8)\n(9)\n=\nwhere LAX is the wavelength. Substituting (8) and (9) into (6) yields\n(10)\nSubstituting this into (7) finally gives\n(11)\nEquation (11) shows that the reconstructed wind field Vi will underestimate\nthe analyzed wind field Vi due to noncompensating truncation errors.\nA finite difference system containing noncompensating truncation errors was\nused by Washington and Baumhefner (1974) in their search for a suitable\nglobal initialization scheme. After setting the vertically integrated\nmass-divergence to zero everywhere, their reconstructed wind fields had the\nsame patterns as the original fields but the magnitudes of the reconstructed\nfields were up to 13 m sec-1 smaller than those of the original. The\nlargest decreases were in the vicinity of the jet maxima.\nLet us now consider the second finite difference formulation in which the\nvorticity applies between the grid points where the winds are located.\nThis will be referred to as the staggered system. In this case, the\ndifferential equations (1) and (2) are approximated by\n(12)\nand\n(13)\n3","When v in (13) is substituted for V in (12), , a perfect equality is\nobtained. Thus, truncation error will be perfectly compensated in the\nstaggered system.\nA staggered finite difference system for decomposing global horizontal\nwind analyses is presented in the next section.\n3. A Staggered Finite Difference Formulation of the Spherical Problem\nThe decomposition of a horizontal wind field into its rotational and\ndivergent components is accomplished by solving the equations\n(14)\nand\n. = D,\n(15)\nin which E is the relative vorticity, D is the divergence, y is the stream\nfunction, X is the velocity potential, and is the horizontal\ngradient\noperator. The reconstruction of a wind field from such components is\naccomplished via Helmholtz' theorem\n(16)\n,\nwhere V is the reconstructed wind field.\nIn spherical horizontal coordinates ($ = latitude, l = longitude),\nequations (14) and (15) can be written as, respectively,\nas\nar\n(17)\ncoso]\nas ax\n(18)\nIn these equations, u and V are winds from the west and south, respectively,\nr is the radius of the earth,\nand 0 5 1 5 2TT .\n4","Both sides of (17) and (18) have been multiplied by [r2coso]. Likewise,\nthe wind components defined by (16) can be written in spherical coordinates\nas\n(19)\n(20)\ndo\nr\nThe staggered finite difference formulations of equations (17) through (20)\nare:\nFOR\n(21)\n(22)\n+\n(23)\nr\n(24)\nV\nThe grid arrangement and location of variables is shown in Figure 1. The\nfinite differences symbolized in equations (21)-(24) - have the following\nmeanings:\n11101\n(25)\n111(26\n+\n(27)\n+\n5","(\n[coso (\n(28)\nHere, AO = Dr, the grid point separation distance in radians. Equations\n(21) and (22) are solved by the accelerated Liebmann relaxation scheme,\nand a wind field can be reconstructed via equations (23) and (24). The\nrelations in this section are valid everywhere except at the poles. There\nsuitable boundary conditions must be used.\n4. Boundary Conditions at the Poles\nAt the North and South Poles, the grid arrangement is different from\nthat in the rest of the grid. This is shown in Figure 2 for the North\nPole. An assumption about the wind at the poles was made in order to\nobtain finite difference representations of equations (14) and (15) at\npoint 5 in Figure 2. The assumption is identified and the resulting\nfinite difference equations are presented in this section.\nLet circulation (C) be defined as\n(29)\nand vorticity by\nE = [C/AA] = [1/AA] $ . ds ,\n(30)\nwhere AA is the area enclosed by the circulation and 10 is a unit vector\ntangent to the line enclosing AA. In a similar manner, let divergence be\ndefined by\nD = [1/AA]\n(31)\nin which 14 is a unit outward vector normal to the line enclosing AA.\nThe convention used here is that positive & is counterclockwise circulation\nand positive D is net outflow. Equations (30) and (31) are approximated\nfor the area bounded by points a, b, and C in Figure 2 as follows:\n(32)\n6","(33)\nThe quantities h1, h2, h3, and AA are given by\nh2= =\n(34)\nAA =\nIt will be useful to express AA as\nAA = r20200 coso\n(35)\nFor this to be true,\nAO\nso that do is defined by\n(36)\n=\nSubstitution of (34) and (35) into (32) and (33) changes the latter into,\nrespectively,\n(37)\nD'}\n(38)\n,\nin which is and D' are given by\n(39)\n(40)\nThe winds at points b and C are defined in terms of 4 and X just as\nin the previous section. The boundary condition lies in the specification\n7","of the winds at the pole (point a). The NMC Global Model carries one\nvector wind at the pole and resolves that wind into u and V components\nat each longitude where gridpoints lie. In practice, therefore, ua and\nva do not differ greatly from ua and Va respectively. Because of this,\nwe assumed\n(41).\nand a\nThis reduces (37) and (38) to\nE = E*/r2cost, =\n(42)\nD = D*/r2cost,\n(43)\nSubstitution of the finite difference representations for u and V at\npoints b and C given in the previous section transforms (39) and (40)\ninto (after multiplying through by r2cosop\n(44)\nD' = V2x + R(y), =\n(45)\nwhere\n-2()5-()6+()7+2()+())\n(46)\n(47)\n5. Solution of the Finite Difference Equations\nThe complete system consists of equations (21) and (22) at all\ninterior gridpoints, equations (44) and (45) at the row of gridpoints\nnearest the North Pole (87.5°N for a 5° latitude-longitude grid), and\nequations similar to (44) and (45) at the row of gridpoints nearest the\nSouth Pole (87.5°S for a 5° latitude-longitude grid). The wind field\n8","was reconstructed from u and X fields via equations (23) and (24) for\nall gridpoints except at the poles. There, the averaging procedure\nused in the Global Model was invoked.\nThe consequences of equations (44) and (45) are that the solution\nfor 4 depends on the values of X at 87.5°N and S latitude, and the\nsolution for X depends on the values of y at 87.5°N and S latitude\n(again for a 5° grid). Thus, a special procedure was necessary to solve\nfor 4 and X.\nThe scheme used was to make an initial guess of R(X) = 0 at the\nnorthern and southernmost rows of gridpoints and solve for 4 by relaxation.\nHaving u, R(y) was computed at the row of gridpoints nearest each pole\nand a solution for X was obtained by relaxation. Although this procedure\ncould be iterated, it was found that additional cycles added little to\nthe accuracy of the reconstructed winds.\n6. Accuracy of the Solution\nIn order to test the accuracy of the method just described, a simple\nnumerical experiment was devised. First, an analyzed wind field was\nproduced by the global spectral analysis technique of Flattery (1970) on\na 5° latitude-longitude grid. Then this analyzed wind field was\ndecomposed into a stream function and a velocity potential, from which\na second wind field was constructed via Helmholtz' Theorem. The quality\nof the solution method was judged on the basis of root-mean-square (RMS)\ndifferences between the original and reconstructed wind fields.\nThe results of the experiment, presented in Table 1, indicate that\nthe reconstructed wind field is nearly identical to the original, the\nRMS differences being of the order of 0.01 m sec-1. The effect of the\nboundary condition can be seen in the tendency for the RMS differences\nto increase towards the poles. However, even there the largest individual\ndifferences were only 0.05 m sec-1. On the basis of these results, the\nsolution method is considered to be adequate for use in initialization\nprocedures for the Global Model.\n7. Summary and Conclusions\nThe necessity of using an internally consistent finite difference\nformulation to decompose a horizontal wind field into its rotational and\ndivergent components in order that these components can be used to\naccurately reconstruct the original field is demonstrated. Such an\ninternally consistent finite difference system is developed to decompose\na global wind field into a stream function and a velocity potential and\nreconstruct a global wind field from the components. The method requires\na boundary condition at the poles. The procedure is found to be\n9","Reconstructed (u,v, (in m sec-1\nTable 1.\nRMS [Original (u,v,\n-\nRMS u\nRMS V\nRMS\nLatitude\nDifference\nDifference\nDifference\n90°N\n0.0011\n0.0011\n0.0008\n85°N\n0.0245\n0.0233\n0.0245\n80°N\n0.0177\n0.0183\n0.0206\n75°N\n0.0162\n0.0158\n0.0172\nRMS u Difference\n70°N\n0.0141\n0.0140\n0.0127\n(whole grid)\n65°N\n0.0121\n0.0124\n0.0130\n0.0106\n60°N\n0.0107\n0.0112\n0.0105\n55°N\n0.0095\n0.0100\n0.0093\n50°N\n0.0084\n0.0092\n0.0092\n45°N\n0.0077\n0.0084\n0.0066\nRMS V Difference\n40°N\n0.0071\n0.0079\n0.0070\n(whole grid)\n35°N\n0.0065\n0.0074\n0.0070\n0.0109\n30°N\n0.0061\n0.0070\n0.0061\n25°N\n0.0057\n0.0067\n0.0054\n20°N\n0.0054\n0.0066\n0.0054\n15°N\n0.0052\n0.0064\n0.0055\nRMS\nDifference\n10°N\n0.0049\n0.0065\n0.0059\n(whole grid)\n5°N\n0.0048\n0.0064\n0.0047\n0.0105\n0°\n0.0047\n0.0066\n0.0056\n5°S\n0.0047\n0.0067\n0.0054\n10°S\n0.0047\n0.0069\n0.0061\n15°S\n0.0049\n0.0070\n0.0053\n20°S\n0.0049\n0.0073\n0.0057\n25°S\n0.0051\n0.0075\n0.0057\n30°S\n0.0051\n0.0079\n0.0057\n35°S\n0.0056\n0.0081\n0.0061\n40°S\n0.0057\n0.0085\n0.0062\n45°S\n0.0062\n0.0088\n0.0069\n50°S\n0.0067\n0.0094\n0.0072\n55°S\n0.0077\n0.0100\n0.0081\n60°S\n0.0088\n0.0109\n0.0086\n65°S\n0.0103\n0.0118\n0.0103\n70°S\n0.0123\n0.0133\n0.0121\n75°S\n0.0169\n0.0151\n0.0150\n80°S\n0.0222\n0.0183\n0.0198\n85°S\n0.0242\n0.0198\n0.0211\n90°S\n0.0003\n0.0003\n0.0002\n10","sufficiently accurate for use in initialization experiments with the\nNMC eight-layer global model. However, it should be possible to use\nthis solution method in conjunction with other numerical models as well\nby following the general procedure outlined here and tailoring the\ndetails to the model in question.\nAcknowledgment\nThe authors wish to thank Ronald D. McPherson for many useful\ndiscussions.\nReferences\nFlattery, T. W. , 1970: \"Spectral Models for Global Analysis and Fore-\ncasting,\" \" Proceedings of the Sixth AWS Technical Exchange Conference,\nU.S. Naval Academy, September 21-24, 1970, Air Weather Service\nTechnical Report 242.\nStackpole, J. D. , Vanderman, L. W., and Shuman, F. G. , 1974: \"The NMC\n8-layer Global Primitive Equation Model on a Latitude-Longitude\nGrid,\" Modelling for the First GARP Global Experiment, GARP\nPublication Series No. 14, June 1974, pp. 79-93.\nWashington, W. M., and Baumhefner, D. P. , 1974: \"A Method of Removing\nLamb Waves from Initial Data for Primitive Equation Models,\"\nJournal of Applied Meteorology, Vol. 14, No. 1, February 1975,\npp. 114-119.\n11","U.V\nU,V\nU,V\nU,V\nX\nX\n4X\nX\n4\nA\n(1+1,J+1)\n(i,J+1)\n(i-1,J+1)\nUV\nU,V\nU,V\nU,V\nCOS J+1/2\nX\nX\nX\n(i+1,J)\n(i-1,J)\n(i,J)\nU,V\nU,V\nU.V\nU,V\nCOS J-1/2\nX\nX\nX\n4\n(i-1,J-1)\n(i+1,J-1)\n(i,J-1)\nUV\nU2V\nuv\nU&V\nFigure 1. Grid arrangement and location of variables in the interior\nof the grid.\n12","Vd Va\nNORTH POLE\nUd\nUa\n4,X\nX\n,\nA\n6\n4\n4,\nx\n3\n5\nX\nX\nVc\nVb\n7\nUc\n9\nC\n2\n4,x\n8\nFigure 2. Grid arrangement and location of variables at the North Pole.\n13","(Continued from inside front cover)\nNOAA Technical Memoranda\nNWS NMC 49\nA Study of Non-Linear Computational Instability for a Two-Dimensional Model. Paul D.\nPolger, February 1971. (COM-71-00246)\nNWS NMC 50\nRecent Research in Numerical Methods at the National Meteorological Center. Ronald D.\nMcPherson, April 1971.\nNWS NMC 51\nUpdating Asynoptic Data for Use in Objective Analysis. Armand J. Desmarais, December\n1972. (COM-73-10078)\nNWS NMC 52\nToward Developing a Quality Control System for Rawinsonde Reports. Frederick G. Finger\nand Arthur R. Thomas, February 1973. (COM-73-10673)\nNWS NMC 53\nA Semi-Implicit Version of the Shuman-Hovermale Model. Joseph P. Gerrity, Jr., Ronald D.\nMcPherson, and Stephen Scolnik. July 1973. (COM-73-11323)\nNWS NMC 54\nStatus Report on a Semi-Implicit Version of the Shuman-Hovermale Model. Kenneth Campana,\nMarch 1974. (COM-74-11096/AS)\nNWS NMC 55\nAn Evaluation of the National Meteorological Center's Experimental Boundary Layer model.\nPaul D. Polger, December 1974. (COM-75-10267/AS)\nNWS NMC 56\nTheoretical and Experimental Comparison of Selected Time Integration Methods Applied to\nFour-Dimensional Data Assimilation. Ronald D. McPherson and Robert E. Kistler, April\n1975. (COM-75-10882/AS)\nNWS NMC 57\nA Test of the Impact of NOAA-2 VTPR Soundings on Operational Analyses and Forecasts.\nWilliam D. Bonner, Paul L. Lemar, Robert J. Van Haaren, Armand J. Desmarais, and Hugh M.\nO'Neil, February 1976. (PB-256075)\nNWS NMC 58\nOperational-Type Analyses Derived Without Radiosonde Data from NIMBUS 5 and NOAA 2 Temp-\nerature Soundings. William D. Bonner, Robert van Haaren, and Christopher M. Hayden, March\n1976. (PB-256099)"]}