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 sfvpec.f c c this file includes documentation and code for c subroutine sfvpec i c c ... files which must be loaded with sfvpec.f c c sphcom.f, hrfft.f, vhaec.f, shsec.f c c c subroutine sfvpec(nlat,nlon,isym,nt,sf,vp,idv,jdv,br,bi,cr,ci, c + mdb,ndb,wshsec,lshsec,work,lwork,ierror) c c given the vector spherical harmonic coefficients br,bi,cr,ci, c computed by subroutine vhaec for a vector field (v,w), sfvpec 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 recomputed rather than stored as they are in c subroutine sfvpes. sf(i,j) and vp(i,j) are given at colatitude c c theta(i) = (i-1)*pi/(nlat-1) c c and east longitude c c lambda(j) = (j-1)*2*pi/nlon c c on the sphere. c c c input parameters c c nlat the number of colatitudes on the full sphere including the c poles. for example, nlat = 37 for a five degree grid. c nlat determines the grid increment in colatitude as c pi/(nlat-1). if nlat is odd the equator is located at c grid point i=(nlat+1)/2. if nlat is even the equator is c located half way between points i=nlat/2 and i=nlat/2+1. c nlat must be at least 3. note: on the half sphere, the c number of grid points in the colatitudinal direction is c nlat/2 if nlat is even or (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 sf 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 sfvpec. 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 sfvpec. 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 vhaec. c c mdb the first dimension of the arrays br,bi,cr,ci as it c appears in the program that calls sfvpec. 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 sfvpec. ndb must be at c least nlat. c c wshsec an array which must be initialized by subroutine shseci. c once initialized, wshsec can be used repeatedly by sfvpec c as long as nlon and nlat remain unchanged. wshsec must c not bel altered between calls of sfvpec. c c c lshsec the dimension of the array wshsec as it appears in the c program that calls sfvpec. 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 lshsec must be at least c c 2*nlat*l2+3*((l1-2)*(nlat+nlat-l1-1))/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 sfvpec. 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*((nt*nlon+max0(3*l2,nlon)) + 2*l1*nt+1) c c if isym is not zero then lwork must be at least c c l2*(nt*nlon+max0(3*nlat,nlon)) + 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 vhaec. sf(i,j),vp(i,j) c are given at the colatitude point c c theta(i) = (i-1)*pi/(nlat-1) c c and longitude point c c lambda(j) = (j-1)*2*pi/nlon c c the index 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 lshsec c = 10 error in the specification of lwork c ********************************************************************** c