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 vhaec.f c c this file contains code and documentation for subroutines c vhaec and vhaeci c c ... files which must be loaded with vhaec.f c c sphcom.f, hrfft.f c c c subroutine vhaec(nlat,nlon,ityp,nt,v,w,idvw,jdvw,br,bi,cr,ci, c + mdab,ndab,wvhaec,lvhaec,work,lwork,ierror) c c subroutine vhaec performs the vector spherical harmonic analysis c on the vector field (v,w) and stores the result in the arrays c br, bi, cr, and ci. v(i,j) and w(i,j) are the colatitudinal c (measured from the north pole) and east longitudinal components c respectively, located at colatitude theta(i) = (i-1)*pi/(nlat-1) c and longitude phi(j) = (j-1)*2*pi/nlon. the spectral c representation of (v,w) is given at output parameters v,w in c subroutine vhsec. 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 c than zero. the axisymmetric case corresponds to nlon=1. c the efficiency of the computation is improved when nlon c is a product of small prime numbers. c c ityp = 0 no symmetries exist about the equator. the analysis c is performed on the entire sphere. i.e. on the c arrays v(i,j),w(i,j) for i=1,...,nlat and c j=1,...,nlon. c c = 1 no symmetries exist about the equator. the analysis c is performed on the entire sphere. i.e. on the c arrays v(i,j),w(i,j) for i=1,...,nlat and c j=1,...,nlon. the curl of (v,w) is zero. that is, c (d/dtheta (sin(theta) w) - dv/dphi)/sin(theta) = 0. c the coefficients cr and ci are zero. c c = 2 no symmetries exist about the equator. the analysis c is performed on the entire sphere. i.e. on the c arrays v(i,j),w(i,j) for i=1,...,nlat and c j=1,...,nlon. the divergence of (v,w) is zero. i.e., c (d/dtheta (sin(theta) v) + dw/dphi)/sin(theta) = 0. c the coefficients br and bi are zero. c c = 3 v is symmetric and w is antisymmetric about the c equator. the analysis is performed on the northern c hemisphere only. i.e., if nlat is odd the analysis c is performed on the arrays v(i,j),w(i,j) for c i=1,...,(nlat+1)/2 and j=1,...,nlon. if nlat is c even the analysis is performed on the the arrays c v(i,j),w(i,j) for i=1,...,nlat/2 and j=1,...,nlon. c c = 4 v is symmetric and w is antisymmetric about the c equator. the analysis is performed on the northern c hemisphere only. i.e., if nlat is odd the analysis c is performed on the arrays v(i,j),w(i,j) for c i=1,...,(nlat+1)/2 and j=1,...,nlon. if nlat is c even the analysis is performed on the the arrays c v(i,j),w(i,j) for i=1,...,nlat/2 and j=1,...,nlon. c the curl of (v,w) is zero. that is, c (d/dtheta (sin(theta) w) - dv/dphi)/sin(theta) = 0. c the coefficients cr and ci are zero. c c = 5 v is symmetric and w is antisymmetric about the c equator. the analysis is performed on the northern c hemisphere only. i.e., if nlat is odd the analysis c is performed on the arrays v(i,j),w(i,j) for c i=1,...,(nlat+1)/2 and j=1,...,nlon. if nlat is c even the analysis is performed on the the arrays c v(i,j),w(i,j) for i=1,...,nlat/2 and j=1,...,nlon. c the divergence of (v,w) is zero. i.e., c (d/dtheta (sin(theta) v) + dw/dphi)/sin(theta) = 0. c the coefficients br and bi are zero. c c = 6 v is antisymmetric and w is symmetric about the c equator. the analysis is performed on the northern c hemisphere only. i.e., if nlat is odd the analysis c is performed on the arrays v(i,j),w(i,j) for c i=1,...,(nlat+1)/2 and j=1,...,nlon. if nlat is c even the analysis is performed on the the arrays c v(i,j),w(i,j) for i=1,...,nlat/2 and j=1,...,nlon. c c = 7 v is antisymmetric and w is symmetric about the c equator. the analysis is performed on the northern c hemisphere only. i.e., if nlat is odd the analysis c is performed on the arrays v(i,j),w(i,j) for c i=1,...,(nlat+1)/2 and j=1,...,nlon. if nlat is c even the analysis is performed on the the arrays c v(i,j),w(i,j) for i=1,...,nlat/2 and j=1,...,nlon. c the curl of (v,w) is zero. that is, c (d/dtheta (sin(theta) w) - dv/dphi)/sin(theta) = 0. c the coefficients cr and ci are zero. c c = 8 v is antisymmetric and w is symmetric about the c equator. the analysis is performed on the northern c hemisphere only. i.e., if nlat is odd the analysis c is performed on the arrays v(i,j),w(i,j) for c i=1,...,(nlat+1)/2 and j=1,...,nlon. if nlat is c even the analysis is performed on the the arrays c v(i,j),w(i,j) for i=1,...,nlat/2 and j=1,...,nlon. c the divergence of (v,w) is zero. i.e., c (d/dtheta (sin(theta) v) + dw/dphi)/sin(theta) = 0. c the coefficients br and bi are zero. c c c nt the number of analyses. in the program that calls vhaec, c the arrays v,w,br,bi,cr, and ci can be three dimensional c in which case multiple analyses will be performed. c the third index is the analysis index which assumes the c values k=1,...,nt. for a single analysis set nt=1. the c discription of the remaining parameters is simplified c by assuming that nt=1 or that all the arrays are two c dimensional. c c v,w two or three dimensional arrays (see input parameter nt) c that contain the vector function to be analyzed. c v is the colatitudnal component and w is the east c longitudinal component. v(i,j),w(i,j) contain the c components at colatitude theta(i) = (i-1)*pi/(nlat-1) c and longitude phi(j) = (j-1)*2*pi/nlon. the index ranges c are defined above at the input parameter ityp. c c idvw the first dimension of the arrays v,w as it appears in c the program that calls vhaec. if ityp .le. 2 then idvw c must be at least nlat. if ityp .gt. 2 and nlat is c even then idvw must be at least nlat/2. if ityp .gt. 2 c and nlat is odd then idvw must be at least (nlat+1)/2. c c jdvw the second dimension of the arrays v,w as it appears in c the program that calls vhaec. jdvw must be at least nlon. c c mdab the first dimension of the arrays br,bi,cr, and ci as it c appears in the program that calls vhaec. mdab 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 ndab the second dimension of the arrays br,bi,cr, and ci as it c appears in the program that calls vhaec. ndab must be at c least nlat. c c wvhaec an array which must be initialized by subroutine vhaeci. c once initialized, wvhaec can be used repeatedly by vhaec c as long as nlon and nlat remain unchanged. wvhaec must c not be altered between calls of vhaec. c c lvhaec the dimension of the array wvhaec as it appears in the c program that calls vhaec. define c c l1 = min0(nlat,nlon/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 lvhaec must be at least c c 4*nlat*l2+3*max0(l1-2,0)*(nlat+nlat-l1-1)+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 vhaec. define c c l2 = nlat/2 if nlat is even or c l2 = (nlat+1)/2 if nlat is odd c c if ityp .le. 2 then lwork must be at least c c nlat*(2*nt*nlon+max0(6*l2,nlon)) c c if ityp .gt. 2 then lwork must be at least c c l2*(2*nt*nlon+max0(6*nlat,nlon)) c c ************************************************************** c c output parameters c c br,bi two or three dimensional arrays (see input parameter nt) c cr,ci that contain the vector spherical harmonic coefficients c in the spectral representation of v(i,j) and w(i,j) given c in the discription of subroutine vhsec. br(mp1,np1), c bi(mp1,np1),cr(mp1,np1), and ci(mp1,np1) are computed c for mp1=1,...,mmax and np1=mp1,...,nlat except for np1=nlat c and odd mp1. mmax=min0(nlat,nlon/2) if nlon is even or c mmax=min0(nlat,(nlon+1)/2) if nlon is odd. 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 ityp c = 4 error in the specification of nt c = 5 error in the specification of idvw c = 6 error in the specification of jdvw c = 7 error in the specification of mdab c = 8 error in the specification of ndab c = 9 error in the specification of lvhaec c = 10 error in the specification of lwork c c c ******************************************************************* c c subroutine vhaeci(nlat,nlon,wvhaec,lvhaec,dwork,ldwork,ierror) c c subroutine vhaeci initializes the array wvhaec which can then be c used repeatedly by subroutine vhaec until nlat or nlon is changed. 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 c than zero. the axisymmetric case corresponds to nlon=1. c the efficiency of the computation is improved when nlon c is a product of small prime numbers. c c lvhaec the dimension of the array wvhaec as it appears in the c program that calls vhaec. define c c l1 = min0(nlat,nlon/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 lvhaec must be at least c c 4*nlat*l2+3*max0(l1-2,0)*(nlat+nlat-l1-1)+nlon+15 c c c dwork a double precision work array that does not have to be saved. c c ldwork the dimension of the array dwork as it appears in the c program that calls vhaec. ldwork must be at least c 2*(nlat+2) c c c ************************************************************** c c output parameters c c wvhaec an array which is initialized for use by subroutine vhaec. c once initialized, wvhaec can be used repeatedly by vhaec c as long as nlat or nlon remain unchanged. wvhaec must not c be altered between calls of vhaec. 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 lvhaec c = 4 error in the specification of ldwork c c c **********************************************************************