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subs.F90
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subs.F90
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MODULE SUBS
!-----------------------------------------------------------------------------
! This module contains functions used to perform spherical harmonics
! transformation and writing output files
!-----------------------------------------------------------------------------
#include <fftw3.f>
! OPTIONS:
! FFTW_ESTIMATE - a fast but probably inefficient FFT plan
! FFTW_MEASURE - does a few FFTs to gauge the machine speed
! FFTW_PATIENT - searches a range of algorithms to find the best one
! FFTW_EXHAUSTIVE - does a complete algorithm search, including some unlikely ones, to find the fastest. NB - may be slow to compute the plan.
INTEGER, PARAMETER :: TRANSFORM_OPTIMISATION = FFTW_PATIENT
INTEGER(KIND=8) :: PLAN_FFT_R2HC
INTEGER, PARAMETER :: NTHETA_GRID_CONTOUR_GMT = 100, NPHI_GRID_CONTOUR_GMT = 200 !number of points for contour plots in GMT
INTEGER, PARAMETER :: GMT_THETA = 30, GMT_PHI = 30 !grid spacing for arrows in GMT plots
INTEGER, PARAMETER :: NUMBER_RANDOM_START_PTS = 1000
INTEGER, PARAMETER :: TOROIDAL_FLOW = 1, POLOIDAL_FLOW = 2
TYPE HARMONIC_STRUCTURE
INTEGER L, M
INTEGER SINCOS
END TYPE HARMONIC_STRUCTURE
INTEGER, PARAMETER :: EXTRA_LONG_REAL = 8, LONG_REAL = 8
REAL(KIND=LONG_REAL), ALLOCATABLE, DIMENSION(:, :) :: FFT_TRANSFORM_ARRAY
REAL(KIND=8), PARAMETER :: CMB_RADIUS = 3485.0E3_8, ES_RADIUS = 6371.0E3_8, Pi = 3.14159265358979_8
INTEGER, PARAMETER :: COSINE_HARMONIC = 1, SINE_HARMONIC = 2
CONTAINS
SUBROUTINE GAUWTS(NTHPTS, GAUX, GAUW)
IMPLICIT NONE
!-----------------------------------------------------------------------------
! Computes the integration points and weights needed for Legendre transforms:
!
! INPUTS:
! NTHPTS (INTEGER)
! number of points in theta direction
! GAUX ( REAL( KIND= LONG_REAL) ), dimension(1:NTHPTS)
! The abscissae of the Gauss-Legendre integration, in cos(theta)
! GAUW ( REAL( KIND= LONG_REAL) ), dimension(1:NTHPTS)
! The weights of the Gauss-Legendre integration
!
!-----------------------------------------------------------------------------
INTEGER NTHPTS
REAL(KIND=EXTRA_LONG_REAL) GAUX(1:NTHPTS), GAUW(1:NTHPTS)
INTENT(IN) NTHPTS
INTENT(OUT) GAUX, GAUW
!
INTEGER M, I, J
REAL(KIND=EXTRA_LONG_REAL) XM, XL, P1, P2, P3, EPS, PP, Z, Z1
PARAMETER(EPS=1.0E-13_LONG_REAL)
REAL(KIND=LONG_REAL), PARAMETER :: X1 = -1.0_LONG_REAL, X2 = 1.0_LONG_REAL
! ....... the roots are symmetric in the interval so need only find
! half of them.
M = (NTHPTS + 1)/2
XM = 0.5_LONG_REAL*(X2 + X1)
XL = 0.5_LONG_REAL*(X2 - X1)
! ........start looping over the desired roots .......................
DO I = 1, M
Z = COS(PI*(REAL(I, KIND=EXTRA_LONG_REAL) - 0.25_EXTRA_LONG_REAL)/ &
(REAL(NTHPTS, KIND=EXTRA_LONG_REAL) + 0.5_EXTRA_LONG_REAL))
! ..... starting with this approximation to the Ith root, we
! enter the main loop of refinement by Newton's method.
100 CONTINUE
P1 = 1.0_LONG_REAL
P2 = 0.0_LONG_REAL
! ........... Loop up the recurrence relation to get the
! legendre Polynomial evaluated at Z.
DO J = 1, NTHPTS
P3 = P2
P2 = P1
P1 = ((2.0_EXTRA_LONG_REAL*J - 1.0_EXTRA_LONG_REAL)*Z*P2 - (J - 1.0_EXTRA_LONG_REAL)*P3)/REAL(J, KIND=EXTRA_LONG_REAL)
END DO
! ..................... finish recurrence relation loop ...
! ... P1 is now the desired Legendre Polynomial. We now compute PP,
! its derivative by a standard relation involving also P2, the
! polynomial of one order lower.
PP = NTHPTS*(Z*P1 - P2)/(Z*Z - 1.0_EXTRA_LONG_REAL)
Z1 = Z
Z = Z1 - P1/PP
IF (ABS(Z - Z1) .GT. EPS) GOTO 100
! ...........scale the root to the desired interval .................
GAUX(I) = XM - XL*Z
! ...........and add its symmetric counterpart ......................
GAUX(NTHPTS + 1 - I) = XM + XL*Z
! ...........calculate the weight ...................................
GAUW(I) = 2.0_EXTRA_LONG_REAL*XL/((1.0_EXTRA_LONG_REAL - Z*Z)*PP*PP)
! ...........and add its symmetric counterpart ......................
GAUW(NTHPTS + 1 - I) = GAUW(I)
END DO
! ......... end looping over the desired roots .......................
RETURN
END SUBROUTINE GAUWTS
!*********************************************************************
SUBROUTINE GET_LEGENDRE_FUNCTIONS(COSTHETA_ARR, LMAX, MMAX, Y, dY)
IMPLICIT NONE
!-----------------------------------------------------------------------------
! Computes the required derivatives of the associated Legendre functions in EXTRA_LONG_REAL precision
! The arrays Y and dY are returned, which contain the elements
! P_l^m and d P_l^m/d theta in the harmonic ordering where
! the l's are consecutive
! Dimensions of Y etc. are: (grid, HARMONICS)
!
! Uses the recursion relations in A&S modified for fully normalised associated Legendre functions to compute
! P_m^m, P_m+1^m and then P_l^m.....
!
! Uses the recursion relation involving the derivative and multiple orders to compute d P_l^m / d(theta)
!-----------------------------------------------------------------------------
REAL(KIND=EXTRA_LONG_REAL) :: COSTHETA_ARR(1:)
INTEGER :: LMAX, MMAX
REAL(KIND=EXTRA_LONG_REAL) :: Y(1:, 1:), dY(1:, 1:)
REAL(KIND=EXTRA_LONG_REAL) :: SINTHETA, PMM, PMM1, COSTHETA, FAC1, FAC2, SINSQINV, FACTOR
INTEGER :: NUM, INDEX, I, M, J, L
DO I = 1, SIZE(COSTHETA_ARR)
COSTHETA = COSTHETA_ARR(I)
SINTHETA = SQRT((1.0_EXTRA_LONG_REAL - COSTHETA)*(1.0_EXTRA_LONG_REAL + COSTHETA))
SINSQINV = 1.0_EXTRA_LONG_REAL/((1.0_EXTRA_LONG_REAL - COSTHETA)*(1.0_EXTRA_LONG_REAL + COSTHETA))
DO M = 0, MMAX
! Compute P_m^m
PMM = 1.0_EXTRA_LONG_REAL
NUM = 1
DO J = 1, M
PMM = PMM*SINTHETA*REAL(NUM, KIND=EXTRA_LONG_REAL)/SQRT(REAL(NUM*(NUM + 1), KIND=EXTRA_LONG_REAL))
NUM = NUM + 2
END DO
! renormalise to "fully normalised" functions
PMM = PMM*SQRT(0.5_EXTRA_LONG_REAL*REAL(2*M + 1, KIND=EXTRA_LONG_REAL))
IF (M .eq. LMAX) THEN
INDEX = SIZE(Y, 2)
Y(I, INDEX) = PMM !P_LMAX^LMAX
EXIT !end of loop
END IF
L = M
INDEX = M*LMAX + (M*(3 - M))/2 + 1 + (L - M)
Y(I, INDEX) = PMM
! Compute P_{m+1}^m
PMM1 = COSTHETA*REAL(2*M + 1, KIND=EXTRA_LONG_REAL)*PMM
! renormalise to "fully normalised" functions
PMM1 = PMM1*SQRT(REAL(2*M + 3, KIND=EXTRA_LONG_REAL))/REAL(2*M + 1, KIND=EXTRA_LONG_REAL)
IF (M .eq. LMAX - 1) THEN
INDEX = SIZE(Y, 2) - 1
Y(I, INDEX) = PMM1 ! P_LMAX^LMAX-1
CYCLE !next M
END IF
L = M + 1
INDEX = M*LMAX + (M*(3 - M))/2 + 1 + (L - M)
Y(I, INDEX) = PMM1 ! P_M+1^M
DO L = M + 2, LMAX
INDEX = M*LMAX + (M*(3 - M))/2 + 1 + (L - M)
FAC1 = REAL((2*L + 1)*(L - M), KIND=EXTRA_LONG_REAL)/REAL((2*L - 1)*(L + M), KIND=EXTRA_LONG_REAL)
FAC1 = SQRT(FAC1)
FAC2 = REAL((2*L + 1)*(L - M)*(L - M - 1), KIND=EXTRA_LONG_REAL)/ &
REAL((2*L - 3)*(L + M)*(L + M - 1), KIND=EXTRA_LONG_REAL)
FAC2 = SQRT(FAC2)
Y(I, INDEX) = (FAC1*COSTHETA*REAL(2*L - 1, KIND=EXTRA_LONG_REAL)*Y(I, INDEX - 1) - &
FAC2*REAL(L + M - 1, KIND=EXTRA_LONG_REAL)*Y(I, INDEX - 2))/REAL(L - M, KIND=EXTRA_LONG_REAL)
END DO
END DO !M
! Compute derivatives:
DO M = 0, MMAX
DO L = M, LMAX
INDEX = M*LMAX + (M*(3 - M))/2 + 1 + (L - M)
IF (M .eq. L) THEN
dY(I, INDEX) = REAL(L, KIND=EXTRA_LONG_REAL)*COSTHETA/SINTHETA*Y(I, INDEX)
ELSE
dY(I, INDEX) = REAL(L, KIND=EXTRA_LONG_REAL)*COSTHETA/SINTHETA*Y(I, INDEX) - &
REAL(L + M, KIND=EXTRA_LONG_REAL)* &
SQRT(REAL((2*L + 1)*(L - M), KIND=EXTRA_LONG_REAL)/REAL((2*L - 1)*(L + M), KIND=EXTRA_LONG_REAL))* &
Y(I, INDEX - 1)/SINTHETA
END IF
END DO
END DO
END DO !I
! Renormalise so that the harmonics are Schmidt quasi-normalised.
DO M = 0, MMAX
DO L = M, LMAX
INDEX = M*LMAX + (M*(3 - M))/2 + 1 + (L - M)
IF (M .eq. 0) THEN
FACTOR = SQRT(2.0_EXTRA_LONG_REAL/REAL(2*L + 1, KIND=EXTRA_LONG_REAL))
ELSE
FACTOR = SQRT(4.0_EXTRA_LONG_REAL/REAL(2*L + 1, KIND=EXTRA_LONG_REAL))
END IF
Y(:, INDEX) = Y(:, INDEX)*FACTOR
DY(:, INDEX) = DY(:, INDEX)*FACTOR
END DO
END DO
! PRINT*, 'here', Y(1,9), COSTHETA_ARR(1), ACOS( COSTHETA_ARR(1))
RETURN
END SUBROUTINE GET_LEGENDRE_FUNCTIONS
SUBROUTINE EVALUATE_B_R_GRID(NTHETA_GRID, &
NPHI_GRID, &
GAUSS, &
HARMONICS, &
LMAX_B_OBS, &
LMAX, &
B_R, &
GRAD_H_B_R, &
ALF, &
DALF, &
ONE_DIV_SINTHETA)
!-----------------------------------------------------------------------------
! Evaluates B_r on the CMB on the defined (theta,phi) grid, along with its horizontal derivatives.
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: NTHETA_GRID, NPHI_GRID, LMAX, LMAX_B_OBS
REAL(KIND=EXTRA_LONG_REAL) :: GAUSS(1:), B_R(1:, 0:), GRAD_H_B_R(1:, 0:, 1:)
REAL(KIND=EXTRA_LONG_REAL) :: ALF(1:, 1:), DALF(1:, 1:), ONE_DIV_SINTHETA(1:)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
INTEGER :: I, INDEX_PLM, I_THETA, I_PHI
REAL(KIND=EXTRA_LONG_REAL) :: PHI_DEP, DERV_PHI_DEP, PHI
GRAD_H_B_R(:, :, :) = 0.0_EXTRA_LONG_REAL
B_R(:, :) = 0.0_EXTRA_LONG_REAL
DO I_THETA = 1, NTHETA_GRID
DO I_PHI = 0, NPHI_GRID - 1
PHI = I_PHI*2.0_EXTRA_LONG_REAL*Pi/REAL(NPHI_GRID, KIND=EXTRA_LONG_REAL)
DO I = 1, SIZE(HARMONICS)
IF (HARMONICS(I)%L .GT. LMAX_B_OBS) CYCLE
IF (HARMONICS(I)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(I)%M*PHI)
DERV_PHI_DEP = HARMONICS(I)%M*COS(HARMONICS(I)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(I)%M*PHI)
DERV_PHI_DEP = -HARMONICS(I)%M*SIN(HARMONICS(I)%M*PHI)
END IF
INDEX_PLM = HARMONICS(I)%M*LMAX + (HARMONICS(I)%M*(3 - HARMONICS(I)%M))/2 + 1 + (HARMONICS(I)%L - HARMONICS(I)%M)
B_R(I_THETA, I_PHI) = B_R(I_THETA, I_PHI) + &
GAUSS(I)*(ES_RADIUS/CMB_RADIUS)**(HARMONICS(I)%L + 2)* &
(HARMONICS(I)%L + 1)*ALF(I_THETA, INDEX_PLM)*PHI_DEP
GRAD_H_B_R(I_THETA, I_PHI, 1) = GRAD_H_B_R(I_THETA, I_PHI, 1) + &
GAUSS(I)*(ES_RADIUS/CMB_RADIUS)**(HARMONICS(I)%L + 2)* &
(HARMONICS(I)%L + 1)*DALF(I_THETA, INDEX_PLM)*PHI_DEP/CMB_RADIUS
GRAD_H_B_R(I_THETA, I_PHI, 2) = GRAD_H_B_R(I_THETA, I_PHI, 2) + &
GAUSS(I)*(ES_RADIUS/CMB_RADIUS)**(HARMONICS(I)%L + 2)* &
(HARMONICS(I)%L + 1)*ALF(I_THETA, INDEX_PLM)*DERV_PHI_DEP* &
ONE_DIV_SINTHETA(I_THETA)/CMB_RADIUS
END DO
END DO
END DO
RETURN
END SUBROUTINE EVALUATE_B_R_GRID
SUBROUTINE EVALUATE_B_SURF_GRID(NTHETA_GRID, &
NPHI_GRID, &
GAUSS, &
HARMONICS, &
LMAX_B_OBS, &
LMAX, &
B_R_SURF, &
B_T_SURF, &
B_P_SURF, &
ALF, &
DALF, &
ONE_DIV_SINTHETA)
!-----------------------------------------------------------------------------
! Evaluates B at the Earth's surface on the defined (theta,phi) grid
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: NTHETA_GRID, NPHI_GRID, LMAX, LMAX_B_OBS
REAL(KIND=EXTRA_LONG_REAL) :: GAUSS(1:), B_R_SURF(1:, 0:)
REAL(KIND=EXTRA_LONG_REAL) :: B_T_SURF(1:, 0:), B_P_SURF(1:, 0:), ALF(1:, 1:)
REAL(KIND=EXTRA_LONG_REAL) :: DALF(1:, 1:), ONE_DIV_SINTHETA(1:)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
INTEGER :: I, INDEX_PLM, I_THETA, I_PHI
REAL(KIND=EXTRA_LONG_REAL) :: PHI_DEP, DERV_PHI_DEP, PHI
B_R_SURF(:, :) = 0.0_EXTRA_LONG_REAL
B_T_SURF(:, :) = 0.0_EXTRA_LONG_REAL
B_P_SURF(:, :) = 0.0_EXTRA_LONG_REAL
DO I_THETA = 1, NTHETA_GRID
DO I_PHI = 0, NPHI_GRID - 1
PHI = I_PHI*2.0_EXTRA_LONG_REAL*Pi/REAL(NPHI_GRID, KIND=EXTRA_LONG_REAL)
DO I = 1, SIZE(HARMONICS)
IF (HARMONICS(I)%L .GT. LMAX_B_OBS) CYCLE
IF (HARMONICS(I)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(I)%M*PHI)
DERV_PHI_DEP = HARMONICS(I)%M*COS(HARMONICS(I)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(I)%M*PHI)
DERV_PHI_DEP = -HARMONICS(I)%M*SIN(HARMONICS(I)%M*PHI)
END IF
INDEX_PLM = HARMONICS(I)%M*LMAX + (HARMONICS(I)%M*(3 - HARMONICS(I)%M))/2 + 1 + (HARMONICS(I)%L - HARMONICS(I)%M)
B_R_SURF(I_THETA, I_PHI) = B_R_SURF(I_THETA, I_PHI) + &
GAUSS(I)*(HARMONICS(I)%L + 1)*ALF(I_THETA, INDEX_PLM)*PHI_DEP
B_T_SURF(I_THETA, I_PHI) = B_T_SURF(I_THETA, I_PHI) - &
GAUSS(I)*DALF(I_THETA, INDEX_PLM)*PHI_DEP
B_P_SURF(I_THETA, I_PHI) = B_P_SURF(I_THETA, I_PHI) - &
GAUSS(I)*ALF(I_THETA, INDEX_PLM)*DERV_PHI_DEP*ONE_DIV_SINTHETA(I_THETA)
END DO
END DO
END DO
RETURN
END SUBROUTINE EVALUATE_B_SURF_GRID
SUBROUTINE EVALUATE_NABLA2_B_R_GRID(NTHETA_GRID, NPHI_GRID, GAUSS, HARMONICS, LMAX_B_OBS, LMAX, NABLA2_B_R, ALF)
!-----------------------------------------------------------------------------
! caclulate horizontal laplacian of function B_r at the CMB from the gauss coefficients
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: NTHETA_GRID, NPHI_GRID, LMAX, LMAX_B_OBS
REAL(KIND=EXTRA_LONG_REAL) :: GAUSS(1:), NABLA2_B_R(1:, 0:), ALF(1:, 1:)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
INTEGER :: I, INDEX_PLM, I_THETA, I_PHI
REAL(KIND=EXTRA_LONG_REAL) :: PHI_DEP, DERV_PHI_DEP, PHI
NABLA2_B_R(:, :) = 0.0_LONG_REAL
DO I_THETA = 1, NTHETA_GRID
DO I_PHI = 0, NPHI_GRID - 1
PHI = I_PHI*2.0_LONG_REAL*Pi/REAL(NPHI_GRID, KIND=EXTRA_LONG_REAL)
DO I = 1, SIZE(HARMONICS)
IF (HARMONICS(I)%L .GT. LMAX_B_OBS) CYCLE
IF (HARMONICS(I)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(I)%M*PHI)
DERV_PHI_DEP = HARMONICS(I)%M*COS(HARMONICS(I)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(I)%M*PHI)
DERV_PHI_DEP = -HARMONICS(I)%M*SIN(HARMONICS(I)%M*PHI)
END IF
INDEX_PLM = HARMONICS(I)%M*LMAX + &
(HARMONICS(I)%M*(3 - HARMONICS(I)%M))/2 + &
1 + (HARMONICS(I)%L - HARMONICS(I)%M)
NABLA2_B_R(I_THETA, I_PHI) = NABLA2_B_R(I_THETA, I_PHI) - &
GAUSS(I)*(1/CMB_RADIUS)**2*(ES_RADIUS/CMB_RADIUS)**(HARMONICS(I)%L + 2)* &
HARMONICS(I)%L*(HARMONICS(I)%L + 1)**2*ALF(I_THETA, INDEX_PLM)*PHI_DEP
END DO
END DO
END DO
END SUBROUTINE EVALUATE_NABLA2_B_R_GRID
SUBROUTINE EVALUATE_U_H_SINGLE_HARMONIC(NTHETA_GRID, &
NPHI_GRID, &
I_HARMONIC, &
I_TYPE, &
U_H, &
DIV_H_U_H, &
ALF, &
DALF, &
HARMONICS, &
LMAX, &
ONE_DIV_SIN_THETA, &
COSTHETA_GRID)
!-----------------------------------------------------------------------------
! Evaluates u_H on the defined (theta,phi) grid, along with its horizontal divergence.
! if harmonic is -1, then evaluate all harmonics.
! div_h . u_h = d(u_theta)/d theta + cot(theta) * u_theta + 1/sin(theta) * d(u_phi)/d phi
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: NTHETA_GRID, NPHI_GRID, I_HARMONIC, I_TYPE, LMAX
REAL(KIND=EXTRA_LONG_REAL) :: U_H(1:, 0:, 1:), DIV_H_U_H(1:, 0:), ALF(1:, 1:), DALF(1:, 1:)
REAL(KIND=EXTRA_LONG_REAL) :: ONE_DIV_SIN_THETA(1:), COSTHETA_GRID(1:)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
INTEGER :: INDEX, I_THETA, I_PHI, J, LOWER_HARMONIC, UPPER_HARMONIC
REAL(KIND=EXTRA_LONG_REAL) :: PHI, DERV_PHI_DEP, PHI_DEP
U_H(:, :, :) = 0.0_LONG_REAL
IF (I_TYPE .eq. TOROIDAL_FLOW) THEN
DO I_THETA = 1, NTHETA_GRID
DO I_PHI = 0, NPHI_GRID - 1
PHI = I_PHI*2.0_LONG_REAL*Pi/REAL(NPHI_GRID, KIND=EXTRA_LONG_REAL)
J = I_HARMONIC
INDEX = HARMONICS(J)%M*LMAX + (HARMONICS(J)%M*(3 - HARMONICS(J)%M))/2 + 1 + (HARMONICS(J)%L - HARMONICS(J)%M)
IF (HARMONICS(J)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(J)%M*PHI)
DERV_PHI_DEP = HARMONICS(J)%M*COS(HARMONICS(J)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(J)%M*PHI)
DERV_PHI_DEP = -HARMONICS(J)%M*SIN(HARMONICS(J)%M*PHI)
END IF
U_H(I_THETA, I_PHI, 1) = DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
U_H(I_THETA, I_PHI, 2) = -PHI_DEP*DALF(I_THETA, INDEX)
END DO
END DO
DIV_H_U_H(:, :) = 0.0_LONG_REAL
! surface divergence of toroidal flow is zero.
ELSE ! poloidal flow
DO I_THETA = 1, NTHETA_GRID
DO I_PHI = 0, NPHI_GRID - 1
PHI = I_PHI*2.0_LONG_REAL*Pi/REAL(NPHI_GRID, KIND=EXTRA_LONG_REAL)
J = I_HARMONIC
INDEX = HARMONICS(J)%M*LMAX + (HARMONICS(J)%M*(3 - HARMONICS(J)%M))/2 + 1 + (HARMONICS(J)%L - HARMONICS(J)%M)
IF (HARMONICS(J)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(J)%M*PHI)
DERV_PHI_DEP = HARMONICS(J)%M*COS(HARMONICS(J)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(J)%M*PHI)
DERV_PHI_DEP = -HARMONICS(J)%M*SIN(HARMONICS(J)%M*PHI)
END IF
U_H(I_THETA, I_PHI, 1) = PHI_DEP*DALF(I_THETA, INDEX)
U_H(I_THETA, I_PHI, 2) = DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
DIV_H_U_H(I_THETA, I_PHI) = -ALF(I_THETA, INDEX)*PHI_DEP* &
REAL(HARMONICS(J)%L*(HARMONICS(J)%L + 1), KIND=LONG_REAL)/CMB_RADIUS
! surface divergence of poloidal flow is -l(l+1)/r for each harmonic
END DO
END DO
! IF( I_THETA .eq. 1 .and. I_PHI .eq. 1) THEN
! PRINT*, U_H(1,1,1:2), DIV_H_U_H(1,1)
! STOP
! ENDIF
END IF
!PRINT*,'HERE', DERV_PHI_DEP , -M * SIN( HARMONICS(I_HARMONIC)%M * PHI)
RETURN
END SUBROUTINE EVALUATE_U_H_SINGLE_HARMONIC
SUBROUTINE EVALUATE_U_H(NTHETA_GRID, &
NPHI_GRID, &
SPEC_TOR, SPEC_POL, &
U_H, &
DIV_H_U_H, &
ALF, &
DALF, &
HARMONICS, &
LMAX_U, &
LMAX_SV, &
ONE_DIV_SIN_THETA)
!-----------------------------------------------------------------------------
! Evaluates u_H on the defined (theta,phi) grid
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: NTHETA_GRID, NPHI_GRID, I_HARMONIC, I_TYPE, LMAX_U, LMAX_SV
REAL(KIND=EXTRA_LONG_REAL) :: U_H(1:, 0:, 1:), ALF(1:, 1:), DALF(1:, 1:), ONE_DIV_SIN_THETA(1:), SPEC_TOR(1:), SPEC_POL(1:)
REAL(KIND=EXTRA_LONG_REAL) :: DIV_H_U_H(1:, 0:)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
INTEGER :: INDEX, I_THETA, I_PHI, J
REAL(KIND=EXTRA_LONG_REAL) :: PHI, DERV_PHI_DEP, PHI_DEP
U_H(:, :, :) = 0.0_LONG_REAL
DIV_H_U_H(:, :) = 0.0_LONG_REAL
DO I_THETA = 1, NTHETA_GRID
DO I_PHI = 0, NPHI_GRID - 1
PHI = I_PHI*2.0_LONG_REAL*Pi/REAL(NPHI_GRID, KIND=EXTRA_LONG_REAL)
DO I_HARMONIC = 1, LMAX_U*(LMAX_U + 2)
INDEX = HARMONICS(I_HARMONIC)%M*LMAX_SV + &
(HARMONICS(I_HARMONIC)%M*(3 - HARMONICS(I_HARMONIC)%M))/2 + &
1 + (HARMONICS(I_HARMONIC)%L - HARMONICS(I_HARMONIC)%M)
IF (HARMONICS(I_HARMONIC)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = HARMONICS(I_HARMONIC)%M*COS(HARMONICS(I_HARMONIC)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = -HARMONICS(I_HARMONIC)%M*SIN(HARMONICS(I_HARMONIC)%M*PHI)
END IF
! toroidal
U_H(I_THETA, I_PHI, 1) = U_H(I_THETA, I_PHI, 1) + &
SPEC_TOR(I_HARMONIC)*DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
U_H(I_THETA, I_PHI, 2) = U_H(I_THETA, I_PHI, 2) - &
SPEC_TOR(I_HARMONIC)*PHI_DEP*DALF(I_THETA, INDEX)
! poloidal
U_H(I_THETA, I_PHI, 1) = U_H(I_THETA, I_PHI, 1) + &
SPEC_POL(I_HARMONIC)*PHI_DEP*DALF(I_THETA, INDEX)
U_H(I_THETA, I_PHI, 2) = U_H(I_THETA, I_PHI, 2) + &
SPEC_POL(I_HARMONIC)*DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
DIV_H_U_H(I_THETA, I_PHI) = DIV_H_U_H(I_THETA, I_PHI) - &
SPEC_POL(I_HARMONIC)*ALF(I_THETA, INDEX)*PHI_DEP* &
REAL(HARMONICS(I_HARMONIC)%L*(HARMONICS(I_HARMONIC)%L + 1), KIND=EXTRA_LONG_REAL)/CMB_RADIUS
END DO
END DO
END DO
RETURN
END SUBROUTINE EVALUATE_U_H
SUBROUTINE REAL_2_SPEC(REAL_SPACE, SPEC, LMAX, LMAX_TRANSFORM, HARMONICS, NPHI_GRID, NTHETA_GRID, LEGENDRE_INV)
!-----------------------------------------------------------------------------
! Projects a scalar quantity defined on a grid onto Schmidt quasi-normalised spherical harmonics
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: NTHETA_GRID, NPHI_GRID
REAL(KIND=EXTRA_LONG_REAL) :: REAL_SPACE(1:, 0:), SPEC(1:), LEGENDRE_INV(1:, 1:)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
! Computes the spherical harmonic coefficients from Y_0^0 up to L,M = LMAX_TRANSFORM.
! LMAX is the maximum degree with which the array SPEC is defined. If LMAX_TRANSFORM < LMAX then the SPEC array is padded with zeros.
REAL(KIND=EXTRA_LONG_REAL) :: PHI_FAC
INTEGER :: INDEX_PLM, I
INTEGER :: LMAX, LMAX_TRANSFORM, MMAX, I_HARMONIC
! REAL_SPACE(:,:) = 1.0
! PRINT*, 'SLOW1', SUM( REAL_SPACE(1,:) ) / NPHI_GRID
! FFT
FFT_TRANSFORM_ARRAY(1:NTHETA_GRID, 0:NPHI_GRID - 1) = REAL_SPACE(:, :)
CALL DFFTW_EXECUTE(PLAN_FFT_R2HC)
REAL_SPACE(:, :) = FFT_TRANSFORM_ARRAY(1:NTHETA_GRID, 0:NPHI_GRID - 1)
!PRINT*, 'FAST1', REAL_SPACE(1,0)
! PRINT*, REAL_SPACE(:,1); PRINT*, REAL_SPACE(:,1) * 1.0_LONG_REAL * Pi / REAL( NPHI_GRID, KIND = LONG_REAL); STOP !; PRINT*, LEGENDRE_INV(1:NTHETA_GRID,19); PRINT*, SUM( REAL_SPACE(:,2) * LEGENDRE_INV(1:NTHETA_GRID,19)); STOP
! Legendre transform
SPEC(:) = 0.0_LONG_REAL
DO I = 1, LMAX_TRANSFORM*(LMAX_TRANSFORM + 2)
! Recall that
! cos(m phi), m=1,2,3,4 is stored in the m^{th} element as 0.5
! sin(m phi), m=1,2,3,4 is stored in the (NPHI_GRID-m)^{th} element as -0.5
! cos(0 phi) is stored in the 0^{th} element as 1
!PHI_FAC = 2.0_LONG_REAL * Pi / REAL( NPHI_GRID, KIND = EXTRA_LONG_REAL)
!PHI_FAC = PHI_FAC / ( 4.0_LONG_REAL * Pi / REAL( 2 * HARMONICS(I)%L + 1, KIND = EXTRA_LONG_REAL ) )
PHI_FAC = 0.5_LONG_REAL*REAL(2*HARMONICS(I)%L + 1, KIND=EXTRA_LONG_REAL)/REAL(NPHI_GRID, KIND=LONG_REAL)
INDEX_PLM = HARMONICS(I)%M*LMAX + (HARMONICS(I)%M*(3 - HARMONICS(I)%M))/2 + 1 + (HARMONICS(I)%L - HARMONICS(I)%M)
IF (HARMONICS(I)%SINCOS .EQ. COSINE_HARMONIC) THEN
SPEC(I) = PHI_FAC*SUM(REAL_SPACE(1:NTHETA_GRID, HARMONICS(I)%M)*LEGENDRE_INV(1:NTHETA_GRID, INDEX_PLM))
ELSE
SPEC(I) = -PHI_FAC*SUM(REAL_SPACE(1:NTHETA_GRID, NPHI_GRID - HARMONICS(I)%M)*LEGENDRE_INV(1:NTHETA_GRID, INDEX_PLM))
END IF
END DO
RETURN
END SUBROUTINE REAL_2_SPEC
SUBROUTINE WRITE_U_GMT(GMT_THETA, GMT_PHI, EVEC_TOR, EVEC_POL, HARMONICS, LMAX, SCALE_FACTOR)
!-----------------------------------------------------------------------------
! Writes the flow to a file that GMT can read. Expects u to describe a flow in units of km/yr
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: GMT_THETA, GMT_PHI, LMAX
REAL(KIND=EXTRA_LONG_REAL) :: EVEC_TOR(1:), EVEC_POL(1:)
REAL(KIND=LONG_REAL) :: SCALE_FACTOR
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: COSTHETA_GRID(:), ALF(:, :), DALF(:, :), ONE_DIV_SIN_THETA(:), U_H(:, :, :)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: TEMP_THETA_TRANSFORM_ALF(:, :), TEMP_THETA_TRANSFORM_DALF(:, :)
REAL(KIND=EXTRA_LONG_REAL) :: PHI, ANGLE_VEC, MOD_VEC, PHI_DEP, DERV_PHI_DEP
INTEGER :: I_THETA, I_PHI, I_HARMONIC, INDEX
! setup grids
ALLOCATE (COSTHETA_GRID(1:GMT_THETA))
DO I_THETA = 1, GMT_THETA
COSTHETA_GRID(I_THETA) = COS(I_THETA/REAL(GMT_THETA + 1, KIND=EXTRA_LONG_REAL)*Pi)
END DO
ALLOCATE (TEMP_THETA_TRANSFORM_ALF(1:GMT_THETA, (LMAX + 1)*(LMAX + 2)/2), &
TEMP_THETA_TRANSFORM_DALF(1:GMT_THETA, (LMAX + 1)*(LMAX + 2)/2), &
ALF(1:GMT_THETA, (LMAX + 1)*(LMAX + 2)/2), &
DALF(1:GMT_THETA, (LMAX + 1)*(LMAX + 2)/2), &
ONE_DIV_SIN_THETA(1:GMT_THETA), U_H(1:GMT_THETA, 1:GMT_PHI, 2))
CALL GET_LEGENDRE_FUNCTIONS(REAL(COSTHETA_GRID, KIND=EXTRA_LONG_REAL), &
LMAX, &
LMAX, &
TEMP_THETA_TRANSFORM_ALF, &
TEMP_THETA_TRANSFORM_DALF)
ALF = REAL(TEMP_THETA_TRANSFORM_ALF, KIND=EXTRA_LONG_REAL)
DALF = REAL(TEMP_THETA_TRANSFORM_DALF, KIND=EXTRA_LONG_REAL)
ONE_DIV_SIN_THETA(:) = 1.0_LONG_REAL/SQRT(1.0_LONG_REAL - COSTHETA_GRID(:)**2)
U_H(:, :, :) = 0.0_LONG_REAL
DO I_HARMONIC = 1, SIZE(EVEC_TOR)
INDEX = HARMONICS(I_HARMONIC)%M*LMAX + &
(HARMONICS(I_HARMONIC)%M*(3 - HARMONICS(I_HARMONIC)%M))/2 + &
1 + (HARMONICS(I_HARMONIC)%L - HARMONICS(I_HARMONIC)%M)
DO I_THETA = 1, GMT_THETA
DO I_PHI = 1, GMT_PHI
PHI = (I_PHI - 1)*2.0_LONG_REAL*Pi/REAL(GMT_PHI, KIND=LONG_REAL)
IF (HARMONICS(I_HARMONIC)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = HARMONICS(I_HARMONIC)%M*COS(HARMONICS(I_HARMONIC)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = -HARMONICS(I_HARMONIC)%M*SIN(HARMONICS(I_HARMONIC)%M*PHI)
END IF
! Toroidal flow
U_H(I_THETA, I_PHI, 1) = U_H(I_THETA, I_PHI, 1) + &
EVEC_TOR(I_HARMONIC)*DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
U_H(I_THETA, I_PHI, 2) = U_H(I_THETA, I_PHI, 2) - &
EVEC_TOR(I_HARMONIC)*PHI_DEP*DALF(I_THETA, INDEX)
! Poloidal flow
U_H(I_THETA, I_PHI, 1) = U_H(I_THETA, I_PHI, 1) + &
EVEC_POL(I_HARMONIC)*PHI_DEP*DALF(I_THETA, INDEX)
U_H(I_THETA, I_PHI, 2) = U_H(I_THETA, I_PHI, 2) + &
EVEC_POL(I_HARMONIC)*DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
!IF( I_HARMONIC .eq. 1) PRINT*, EVEC( I_HARMONIC) , PHI_DEP , DALF(I_THETA, INDEX),DERV_PHI_DEP , ALF(I_THETA, INDEX) ,ONE_DIV_SIN_THETA(I_THETA)
END DO
END DO
END DO
!PRINT*, U_H
!STOP
OPEN (15, FILE='FLOW.DAT', STATUS='REPLACE', FORM='FORMATTED')
DO I_THETA = 1, GMT_THETA
DO I_PHI = 1, GMT_PHI
PHI = (I_PHI - 1)*2.0_LONG_REAL*Pi/REAL(GMT_PHI, KIND=LONG_REAL)
ANGLE_VEC = ATAN(-U_H(I_THETA, I_PHI, 2)/U_H(I_THETA, I_PHI, 1))*180.0d0/PI
! correction as atan has principal angle -90 to +90 degrees.
if (U_H(I_THETA, I_PHI, 1) .gt. 0.0d0) then
if (U_H(I_THETA, I_PHI, 2) .lt. 0.0d0) then
ANGLE_VEC = ANGLE_VEC - 180.0d0
else
ANGLE_VEC = ANGLE_VEC + 180.0d0
end if
end if
MOD_VEC = SQRT(U_H(I_THETA, I_PHI, 1)**2 + U_H(I_THETA, I_PHI, 2)**2)
WRITE (15, '(2F8.2,2F15.5)'), PHI*180.0/Pi, 90.0d0 - ACOS(COSTHETA_GRID(I_THETA))*180.0/Pi, ANGLE_VEC, MOD_VEC*SCALE_FACTOR
!WRITE(15,*), PHI*180.0/Pi ,90.0d0-ACOS(COSTHETA_GRID(I_THETA))*180.0/Pi, ANGLE_VEC
!PRINT*, U_H(I_THETA, I_PHI,1:2)
END DO
END DO
CLOSE (15)
RETURN
END SUBROUTINE WRITE_U_GMT
SUBROUTINE WRITE_U_CENTRED(GMT_THETA, GMT_PHI, EVEC_TOR, EVEC_POL, HARMONICS, LMAX)
!-----------------------------------------------------------------------------
! Writes the flow to a grid centred on 0 longitude
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: GMT_THETA, GMT_PHI, LMAX
REAL(KIND=EXTRA_LONG_REAL) :: EVEC_TOR(1:), EVEC_POL(1:)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: COSTHETA_GRID(:), ALF(:, :), DALF(:, :), ONE_DIV_SIN_THETA(:), U_H(:, :, :)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: TEMP_THETA_TRANSFORM_ALF(:, :), TEMP_THETA_TRANSFORM_DALF(:, :)
REAL(KIND=EXTRA_LONG_REAL) :: PHI, ANGLE_VEC, MOD_VEC, PHI_DEP, DERV_PHI_DEP
INTEGER :: I_THETA, I_PHI, I_HARMONIC, INDEX
! setup grids
ALLOCATE (COSTHETA_GRID(1:GMT_THETA))
DO I_THETA = 1, GMT_THETA
COSTHETA_GRID(I_THETA) = COS(I_THETA/REAL(GMT_THETA + 1, KIND=EXTRA_LONG_REAL)*Pi)
END DO
ALLOCATE (TEMP_THETA_TRANSFORM_ALF(1:GMT_THETA, (LMAX + 1)*(LMAX + 2)/2), &
TEMP_THETA_TRANSFORM_DALF(1:GMT_THETA, (LMAX + 1)*(LMAX + 2)/2), &
ALF(1:GMT_THETA, (LMAX + 1)*(LMAX + 2)/2), &
DALF(1:GMT_THETA, (LMAX + 1)*(LMAX + 2)/2), &
ONE_DIV_SIN_THETA(1:GMT_THETA), U_H(1:GMT_THETA, 1:GMT_PHI, 2))
CALL GET_LEGENDRE_FUNCTIONS(REAL(COSTHETA_GRID, KIND=EXTRA_LONG_REAL), &
LMAX, &
LMAX, &
TEMP_THETA_TRANSFORM_ALF, &
TEMP_THETA_TRANSFORM_DALF)
ALF = REAL(TEMP_THETA_TRANSFORM_ALF, KIND=EXTRA_LONG_REAL)
DALF = REAL(TEMP_THETA_TRANSFORM_DALF, KIND=EXTRA_LONG_REAL)
ONE_DIV_SIN_THETA(:) = 1.0_LONG_REAL/SQRT(1.0_LONG_REAL - COSTHETA_GRID(:)**2)
U_H(:, :, :) = 0.0_LONG_REAL
DO I_HARMONIC = 1, SIZE(EVEC_TOR)
INDEX = HARMONICS(I_HARMONIC)%M*LMAX + &
(HARMONICS(I_HARMONIC)%M*(3 - HARMONICS(I_HARMONIC)%M))/2 + &
1 + (HARMONICS(I_HARMONIC)%L - HARMONICS(I_HARMONIC)%M)
DO I_THETA = 1, GMT_THETA
DO I_PHI = 1, GMT_PHI
! splits up into equal segments away from the end points.
PHI = -Pi + Pi/REAL(GMT_PHI, KIND=EXTRA_LONG_REAL) + 2.0*Pi/REAL(GMT_PHI, KIND=EXTRA_LONG_REAL)*(I_PHI - 1)
IF (HARMONICS(I_HARMONIC)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = HARMONICS(I_HARMONIC)%M*COS(HARMONICS(I_HARMONIC)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = -HARMONICS(I_HARMONIC)%M*SIN(HARMONICS(I_HARMONIC)%M*PHI)
END IF
! Toroidal flow
U_H(I_THETA, I_PHI, 1) = U_H(I_THETA, I_PHI, 1) + &
EVEC_TOR(I_HARMONIC)*DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
U_H(I_THETA, I_PHI, 2) = U_H(I_THETA, I_PHI, 2) - &
EVEC_TOR(I_HARMONIC)*PHI_DEP*DALF(I_THETA, INDEX)
! Poloidal flow
U_H(I_THETA, I_PHI, 1) = U_H(I_THETA, I_PHI, 1) + &
EVEC_POL(I_HARMONIC)*PHI_DEP*DALF(I_THETA, INDEX)
U_H(I_THETA, I_PHI, 2) = U_H(I_THETA, I_PHI, 2) + &
EVEC_POL(I_HARMONIC)*DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
!IF( I_HARMONIC .eq. 1) PRINT*, EVEC( I_HARMONIC) , PHI_DEP , DALF(I_THETA, INDEX),DERV_PHI_DEP , ALF(I_THETA, INDEX) ,ONE_DIV_SIN_THETA(I_THETA)
END DO
END DO
END DO
!PRINT*, U_H
!STOP
OPEN (15, FILE='FLOW_VECTORS_CENTRED.DAT', STATUS='REPLACE', FORM='FORMATTED')
DO I_THETA = GMT_THETA, 1, -1
DO I_PHI = 1, GMT_PHI
PHI = -Pi + Pi/REAL(GMT_PHI, KIND=EXTRA_LONG_REAL) + 2.0*Pi/REAL(GMT_PHI, KIND=EXTRA_LONG_REAL)*(I_PHI - 1)
WRITE (15, '(2F8.2,2F15.5)'), PHI*180.0/Pi, 90.0d0 - ACOS(COSTHETA_GRID(I_THETA))*180.0/Pi, U_H(I_THETA, I_PHI, 1:2)
END DO
END DO
CLOSE (15)
RETURN
END SUBROUTINE WRITE_U_CENTRED
SUBROUTINE WRITE_U_RANDOM(EVEC_TOR, EVEC_POL, HARMONICS, LMAX)
!-----------------------------------------------------------------------------
! Writes the flow to a quasi-uniformly sampled spherical grid
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: LMAX
REAL(KIND=EXTRA_LONG_REAL) :: EVEC_TOR(1:), EVEC_POL(1:)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: COSTHETA_GRID(:), ALF(:, :), DALF(:, :), U_H(:)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: TEMP_THETA_TRANSFORM_ALF(:, :), TEMP_THETA_TRANSFORM_DALF(:, :)
REAL(KIND=EXTRA_LONG_REAL) :: PHI, PHI_DEP, DERV_PHI_DEP, ONE_DIV_SIN_THETA, RANDOM_NUMBERS(1:2), THETA
INTEGER :: I_THETA, I_PHI, I_HARMONIC, INDEX, i, I_LOC
INTEGER, ALLOCATABLE :: SEED(:)
! Generate LAT/LONG grid
CALL RANDOM_SEED(SIZE=i)
ALLOCATE (SEED(1:i))
SEED(:) = 1
CALL RANDOM_SEED(PUT=SEED)
ALLOCATE (COSTHETA_GRID(1))
OPEN (15, FILE='FLOW_VECTORS_RANDOM.DAT', STATUS='REPLACE', FORM='FORMATTED')
ALLOCATE (TEMP_THETA_TRANSFORM_ALF(1, (LMAX + 1)*(LMAX + 2)/2), &
TEMP_THETA_TRANSFORM_DALF(1, (LMAX + 1)*(LMAX + 2)/2), &
ALF(1, (LMAX + 1)*(LMAX + 2)/2), &
DALF(1, (LMAX + 1)*(LMAX + 2)/2), &
U_H(2))
DO I_LOC = 1, NUMBER_RANDOM_START_PTS
CALL RANDOM_NUMBER(RANDOM_NUMBERS(1:2))
!PRINT*, RANDOM_NUMBERS
PHI = RANDOM_NUMBERS(1)*2.0_8*Pi
THETA = ASIN(SQRT(1.0_8 - (2.0_8*RANDOM_NUMBERS(2) - 1.0_8)**2))
IF ((2.0_8*RANDOM_NUMBERS(2) - 1.0_8) < 0.0_8) THETA = Pi - THETA
COSTHETA_GRID(1) = COS(THETA)
!PRINT*, THETA, PHI
CALL GET_LEGENDRE_FUNCTIONS(REAL(COSTHETA_GRID, KIND=EXTRA_LONG_REAL), &
LMAX, &
LMAX, &
TEMP_THETA_TRANSFORM_ALF, &
TEMP_THETA_TRANSFORM_DALF)
ALF = REAL(TEMP_THETA_TRANSFORM_ALF, KIND=EXTRA_LONG_REAL)
DALF = REAL(TEMP_THETA_TRANSFORM_DALF, KIND=EXTRA_LONG_REAL)
ONE_DIV_SIN_THETA = 1.0_LONG_REAL/SQRT(1.0_LONG_REAL - COSTHETA_GRID(1)**2)
U_H(:) = 0.0_LONG_REAL
DO I_HARMONIC = 1, SIZE(EVEC_TOR)
INDEX = HARMONICS(I_HARMONIC)%M*LMAX + &
(HARMONICS(I_HARMONIC)%M*(3 - HARMONICS(I_HARMONIC)%M))/2 + &
1 + (HARMONICS(I_HARMONIC)%L - HARMONICS(I_HARMONIC)%M)
IF (HARMONICS(I_HARMONIC)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = HARMONICS(I_HARMONIC)%M*COS(HARMONICS(I_HARMONIC)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = -HARMONICS(I_HARMONIC)%M*SIN(HARMONICS(I_HARMONIC)%M*PHI)
END IF
! Toroidal flow
U_H(1) = U_H(1) + EVEC_TOR(I_HARMONIC)*DERV_PHI_DEP*ALF(1, INDEX)*ONE_DIV_SIN_THETA
U_H(2) = U_H(2) - EVEC_TOR(I_HARMONIC)*PHI_DEP*DALF(1, INDEX)
! Poloidal flow
U_H(1) = U_H(1) + EVEC_POL(I_HARMONIC)*PHI_DEP*DALF(1, INDEX)
U_H(2) = U_H(2) + EVEC_POL(I_HARMONIC)*DERV_PHI_DEP*ALF(1, INDEX)*ONE_DIV_SIN_THETA
!IF( I_HARMONIC .eq. 1) PRINT*, EVEC( I_HARMONIC) , PHI_DEP , DALF(I_THETA, INDEX),DERV_PHI_DEP , ALF(I_THETA, INDEX) ,ONE_DIV_SIN_THETA(I_THETA)
END DO
WRITE (15, '(2F8.2,2F15.5)'), PHI*180.0/Pi, 90.0 - THETA*180.0/Pi, U_H(1:2)
END DO
CLOSE (15)
RETURN
END SUBROUTINE WRITE_U_RANDOM
SUBROUTINE CALC_U_GRID(THETA_GRID_SIZE, PHI_GRID_SIZE, COSTHETA_GRID, PHI_GRID, EVEC_TOR, EVEC_POL, HARMONICS, LMAX, USQ)
!-----------------------------------------------------------------------------
! Computes u^2 on a grid
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: THETA_GRID_SIZE, PHI_GRID_SIZE, LMAX
REAL(KIND=EXTRA_LONG_REAL) :: EVEC_TOR(1:), EVEC_POL(1:), USQ(:, :), PHI_GRID(PHI_GRID_SIZE), COSTHETA_GRID(1:THETA_GRID_SIZE)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: ALF(:, :), DALF(:, :), ONE_DIV_SIN_THETA(:), U_H(:, :, :)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: TEMP_THETA_TRANSFORM_ALF(:, :), TEMP_THETA_TRANSFORM_DALF(:, :)
REAL(KIND=EXTRA_LONG_REAL) :: PHI, ANGLE_VEC, MOD_VEC, PHI_DEP, DERV_PHI_DEP
INTEGER :: I_THETA, I_PHI, I_HARMONIC, INDEX
!PRINT*, PHI_GRID_SIZE, THETA_GRID_SIZE
! setup grids
ALLOCATE (TEMP_THETA_TRANSFORM_ALF(1:THETA_GRID_SIZE, (LMAX + 1)*(LMAX + 2)/2), &
TEMP_THETA_TRANSFORM_DALF(1:THETA_GRID_SIZE, (LMAX + 1)*(LMAX + 2)/2), &
ALF(1:THETA_GRID_SIZE, (LMAX + 1)*(LMAX + 2)/2), &
DALF(1:THETA_GRID_SIZE, (LMAX + 1)*(LMAX + 2)/2), &
ONE_DIV_SIN_THETA(1:THETA_GRID_SIZE), U_H(1:THETA_GRID_SIZE, 1:PHI_GRID_SIZE, 2))
CALL GET_LEGENDRE_FUNCTIONS(REAL(COSTHETA_GRID, KIND=EXTRA_LONG_REAL), &
LMAX, &
LMAX, &
TEMP_THETA_TRANSFORM_ALF, &
TEMP_THETA_TRANSFORM_DALF)
ALF = REAL(TEMP_THETA_TRANSFORM_ALF, KIND=EXTRA_LONG_REAL)
DALF = REAL(TEMP_THETA_TRANSFORM_DALF, KIND=EXTRA_LONG_REAL)
ONE_DIV_SIN_THETA(:) = 1.0_LONG_REAL/SQRT(1.0_LONG_REAL - COSTHETA_GRID(:)**2)
U_H(:, :, :) = 0.0_LONG_REAL
DO I_HARMONIC = 1, SIZE(EVEC_TOR)
INDEX = HARMONICS(I_HARMONIC)%M*LMAX + &
(HARMONICS(I_HARMONIC)%M*(3 - HARMONICS(I_HARMONIC)%M))/2 + &
1 + (HARMONICS(I_HARMONIC)%L - HARMONICS(I_HARMONIC)%M)
DO I_THETA = 1, THETA_GRID_SIZE
DO I_PHI = 1, PHI_GRID_SIZE
PHI = (I_PHI - 1)*2.0_LONG_REAL*Pi/REAL(PHI_GRID_SIZE, KIND=EXTRA_LONG_REAL)
IF (HARMONICS(I_HARMONIC)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = HARMONICS(I_HARMONIC)%M*COS(HARMONICS(I_HARMONIC)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = -HARMONICS(I_HARMONIC)%M*SIN(HARMONICS(I_HARMONIC)%M*PHI)
END IF
! Toroidal flow
U_H(I_THETA, I_PHI, 1) = U_H(I_THETA, I_PHI, 1) + &
EVEC_TOR(I_HARMONIC)*DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
U_H(I_THETA, I_PHI, 2) = U_H(I_THETA, I_PHI, 2) - &
EVEC_TOR(I_HARMONIC)*PHI_DEP*DALF(I_THETA, INDEX)
! Poloidal flow
U_H(I_THETA, I_PHI, 1) = U_H(I_THETA, I_PHI, 1) + &
EVEC_POL(I_HARMONIC)*PHI_DEP*DALF(I_THETA, INDEX)
U_H(I_THETA, I_PHI, 2) = U_H(I_THETA, I_PHI, 2) + &
EVEC_POL(I_HARMONIC)*DERV_PHI_DEP*ALF(I_THETA, INDEX)*ONE_DIV_SIN_THETA(I_THETA)
!IF( I_HARMONIC .eq. 1) PRINT*, EVEC( I_HARMONIC) , PHI_DEP , DALF(I_THETA, INDEX),DERV_PHI_DEP , ALF(I_THETA, INDEX) ,ONE_DIV_SIN_THETA(I_THETA)
END DO
END DO
END DO
DO I_THETA = 1, THETA_GRID_SIZE
DO I_PHI = 1, PHI_GRID_SIZE
USQ(I_THETA, I_PHI) = U_H(I_THETA, I_PHI, 1)**2 + U_H(I_THETA, I_PHI, 2)**2
END DO
END DO
RETURN
END SUBROUTINE CALC_U_GRID
SUBROUTINE WRITE_TOR_STREAM_FN(N_THETA, N_PHI, EVEC_TOR, HARMONICS, LMAX)
!-----------------------------------------------------------------------------
! Writes the stream function of the toroidal flow to a grid centred on 0 longitude
!-----------------------------------------------------------------------------
IMPLICIT NONE
INTEGER :: N_THETA, N_PHI, LMAX
REAL(KIND=EXTRA_LONG_REAL) :: EVEC_TOR(1:)
TYPE(HARMONIC_STRUCTURE) :: HARMONICS(1:)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: COSTHETA_GRID(:), ALF(:, :), DALF(:, :), ONE_DIV_SIN_THETA(:), STREAMFN(:, :)
REAL(KIND=EXTRA_LONG_REAL), ALLOCATABLE :: TEMP_THETA_TRANSFORM_ALF(:, :), TEMP_THETA_TRANSFORM_DALF(:, :)
REAL(KIND=EXTRA_LONG_REAL) :: PHI, ANGLE_VEC, MOD_VEC, PHI_DEP, DERV_PHI_DEP
INTEGER :: I_THETA, I_PHI, I_HARMONIC, INDEX
! setup grids
ALLOCATE (COSTHETA_GRID(1:N_THETA))
DO I_THETA = 1, N_THETA
COSTHETA_GRID(I_THETA) = COS(I_THETA/REAL(N_THETA + 1, KIND=EXTRA_LONG_REAL)*Pi)
END DO
ALLOCATE (TEMP_THETA_TRANSFORM_ALF(1:N_THETA, (LMAX + 1)*(LMAX + 2)/2), &
TEMP_THETA_TRANSFORM_DALF(1:N_THETA, (LMAX + 1)*(LMAX + 2)/2), &
ALF(1:N_THETA, (LMAX + 1)*(LMAX + 2)/2), &
DALF(1:N_THETA, (LMAX + 1)*(LMAX + 2)/2), &
ONE_DIV_SIN_THETA(1:N_THETA), STREAMFN(1:N_THETA, 1:N_PHI))
CALL GET_LEGENDRE_FUNCTIONS(REAL(COSTHETA_GRID, KIND=EXTRA_LONG_REAL), &
LMAX, &
LMAX, &
TEMP_THETA_TRANSFORM_ALF, &
TEMP_THETA_TRANSFORM_DALF)
ALF = REAL(TEMP_THETA_TRANSFORM_ALF, KIND=EXTRA_LONG_REAL)
DALF = REAL(TEMP_THETA_TRANSFORM_DALF, KIND=EXTRA_LONG_REAL)
ONE_DIV_SIN_THETA(:) = 1.0_LONG_REAL/SQRT(1.0_LONG_REAL - COSTHETA_GRID(:)**2)
STREAMFN(:, :) = 0.0_LONG_REAL
DO I_HARMONIC = 1, SIZE(EVEC_TOR)
INDEX = HARMONICS(I_HARMONIC)%M*LMAX + &
(HARMONICS(I_HARMONIC)%M*(3 - HARMONICS(I_HARMONIC)%M))/2 + &
1 + (HARMONICS(I_HARMONIC)%L - HARMONICS(I_HARMONIC)%M)
DO I_THETA = 1, N_THETA
DO I_PHI = 1, N_PHI
! splits up into equal segments away from the end points.
PHI = -Pi + Pi/REAL(N_PHI, KIND=EXTRA_LONG_REAL) + 2.0*Pi/REAL(N_PHI, KIND=EXTRA_LONG_REAL)*(I_PHI - 1)
IF (HARMONICS(I_HARMONIC)%SINCOS .EQ. SINE_HARMONIC) THEN
PHI_DEP = SIN(HARMONICS(I_HARMONIC)%M*PHI)
DERV_PHI_DEP = HARMONICS(I_HARMONIC)%M*COS(HARMONICS(I_HARMONIC)%M*PHI)
ELSE
PHI_DEP = COS(HARMONICS(I_HARMONIC)%M*PHI)