c c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c . . c . copyright (c) 1998 by UCAR . c . . c . University Corporation for Atmospheric Research . c . . c . all rights reserved . c . . c . . c . SPHEREPACK . c . . c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c c c c ... file sfvpgs.f c c this file includes documentation and code for c subroutine sfvpgs i c c ... files which must be loaded with sfvpgs.f c c sphcom.f, hrfft.f, vhags.f, shsgs.f, gaqd.f c c c subroutine sfvpgs(nlat,nlon,isym,nt,sf,vp,idv,jdv,br,bi,cr,ci, c + mdb,ndb,wshsgs,lshsgs,work,lwork,ierror) c c given the vector spherical harmonic coefficients br,bi,cr,ci, c computed by subroutine vhags for a vector field (v,w), sfvpgs c computes a scalar stream function sf and scalar velocity potential c vp for (v,w). (v,w) is expressed in terms of sf and vp by the c helmholtz relations (in mathematical spherical coordinates): c c v = -1/sint*d(vp)/dlambda + d(st)/dtheta c c w = 1/sint*d(st)/dlambda + d(vp)/dtheta c c where sint = sin(theta). w is the east longitudinal and v c is the colatitudinal component of the vector field from which c br,bi,cr,ci were precomputed. required associated legendre c polynomials are stored rather than recomputed as they are in c subroutine sfvpgc. sf(i,j) and vp(i,j) are given at the i(th) c gaussian colatitude point theta(i) (see nlat description below) c and east longitude lambda(j) = (j-1)*2*pi/nlon on the sphere. c c input parameters c c nlat the number of points in the gaussian colatitude grid on the c full sphere. these lie in the interval (0,pi) and are computed c in radians in theta(1) <...< theta(nlat) by subroutine gaqd. c if nlat is odd the equator will be included as the grid point c theta((nlat+1)/2). if nlat is even the equator will be c excluded as a grid point and will lie half way between c theta(nlat/2) and theta(nlat/2+1). nlat must be at least 3. c note: on the half sphere, the number of grid points in the c colatitudinal direction is nlat/2 if nlat is even or c (nlat+1)/2 if nlat is odd. c c nlon the number of distinct londitude points. nlon determines c the grid increment in longitude as 2*pi/nlon. for example c nlon = 72 for a five degree grid. nlon must be greater than c 3. the efficiency of the computation is improved when nlon c is a product of small prime numbers. c c c isym a parameter which determines whether the stream function and c velocity potential are computed on the full or half sphere c as follows: c c = 0 c c the symmetries/antsymmetries described in isym=1,2 below c do not exist in (v,w) about the equator. in this case st c and vp are not necessarily symmetric or antisymmetric about c the equator. sf and vp are computed on the entire sphere. c i.e., in arrays sf(i,j),vp(i,j) for i=1,...,nlat and c j=1,...,nlon. c c = 1 c c w is antisymmetric and v is symmetric about the equator. c in this case sf is symmetric and vp antisymmetric about c the equator and are computed for the northern hemisphere c only. i.e., if nlat is odd the sf(i,j),vp(i,j) are computed c for i=1,...,(nlat+1)/2 and for j=1,...,nlon. if nlat is c even then sf(i,j),vp(i,j) are computed for i=1,...,nlat/2 c and j=1,...,nlon. c c = 2 c c w is symmetric and v is antisymmetric about the equator. c in this case sf is antisymmetric and vp symmetric about c the equator and are computed for the northern hemisphere c only. i.e., if nlat is odd the sf(i,j),vp(i,j) are computed c for i=1,...,(nlat+1)/2 and for j=1,...,nlon. if nlat is c even then sf(i,j),vp(i,j) are computed for i=1,...,nlat/2 c and j=1,...,nlon. c c nt nt is the number of scalar and vector fields. some c computational efficiency is obtained for multiple fields. arrays c can be three dimensional corresponding to an indexed multiple c vector field. in this case multiple scalar synthesis will c be performed to compute sf,vp for each field. the c third index is the synthesis index which assumes the values c k=1,...,nt. for a single synthesis set nt = 1. the c description of the remaining parameters is simplified by c assuming that nt=1 or that all the arrays are two dimensional. c c idv the first dimension of the arrays sf,vp as it appears in c the program that calls sfvpgs. if isym = 0 then idv c must be at least nlat. if isym = 1 or 2 and nlat is c even then idv must be at least nlat/2. if isym = 1 or 2 c and nlat is odd then idv must be at least (nlat+1)/2. c c jdv the second dimension of the arrays sf,vp as it appears in c the program that calls sfvpgs. jdv must be at least nlon. c c br,bi, two or three dimensional arrays (see input parameter nt) c cr,ci that contain vector spherical harmonic coefficients c of the vector field (v,w) as computed by subroutine vhags. c c mdb the first dimension of the arrays br,bi,cr,ci as it c appears in the program that calls sfvpgs. mdb must be at c least min0(nlat,nlon/2) if nlon is even or at least c min0(nlat,(nlon+1)/2) if nlon is odd. c c ndb the second dimension of the arrays br,bi,cr,ci as it c appears in the program that calls sfvpgs. ndb must be at c least nlat. c c wshsgs an array which must be initialized by subroutine shsgsi. c once initialized, wshsgs can be used repeatedly by sfvpgs c as long as nlon and nlat remain unchanged. wshsgs must c not bel altered between calls of sfvpgs. c c c lshsgs the dimension of the array wshsgs as it appears in the c program that calls sfvpgs. define c c l1 = min0(nlat,(nlon+2)/2) if nlon is even or c l1 = min0(nlat,(nlon+1)/2) if nlon is odd c c and c c l2 = nlat/2 if nlat is even or c l2 = (nlat+1)/2 if nlat is odd c c then lshsgs must be at least c c nlat*(3*(l1+l2)-2)+(l1-1)*(l2*(2*nlat-l1)-3*l1)/2+nlon+15 c c c work a work array that does not have to be saved. c c lwork the dimension of the array work as it appears in the c program that calls sfvpgs. define c c l1 = min0(nlat,(nlon+2)/2) if nlon is even or c l1 = min0(nlat,(nlon+1)/2) if nlon is odd c c and c c l2 = nlat/2 if nlat is even or c l2 = (nlat+1)/2 if nlat is odd c c if isym is zero then lwork must be at least c c nlat*nlon*(nt+1) + nlat*(2*l1*nt + 1) c c if isym is nonzero then lwork must be at least c c l2*nlon*(nt+1) + nlat*(2*l1*nt + 1) c c ************************************************************** c c output parameters c c sf,vp two or three dimensional arrays (see input parameter nt) c that contains the stream function and velocity potential c of the vector field (v,w) whose coefficients br,bi,cr,ci c where precomputed by subroutine vhags. sf(i,j),vp(i,j) c are given at the i(th) gaussian colatitude point theta(i) c and longitude point lambda(j) = (j-1)*2*pi/nlon. the index c ranges are defined above at the input parameter isym. c c c ierror = 0 no errors c = 1 error in the specification of nlat c = 2 error in the specification of nlon c = 3 error in the specification of isym c = 4 error in the specification of nt c = 5 error in the specification of idv c = 6 error in the specification of jdv c = 7 error in the specification of mdb c = 8 error in the specification of ndb c = 9 error in the specification of lshsgs c = 10 error in the specification of lwork c ********************************************************************** c