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Small modification of spherical harmonic SAL
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* Love number scaling coefficients multiplication is re-ordered to avoid
ambiguity and to follow MOM6 style.
* Coefficients for Pmm is simplified.

Both changes introduce bit-wise level answer change to previous SHT SAL
related unpublished commits.

* Patches for Doxygen
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@herrwang0
herrwang0 committed Sep 29, 2022
1 parent 7773971 commit 3f37f14
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Showing 3 changed files with 22 additions and 27 deletions.
2 changes: 1 addition & 1 deletion src/parameterizations/lateral/MOM_load_love_numbers.F90
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Expand Up @@ -5,7 +5,7 @@ module MOM_load_love_numbers

public Love_Data

integer, parameter :: lmax = 1440
integer, parameter :: lmax = 1440 !< Maximum degree of the stored Love numbers
real, dimension(4, lmax+1), parameter :: &
Love_Data = &
reshape((/ 0.0, 0.0000000000, 0.0000000000 , -1.0000000000 , &
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42 changes: 19 additions & 23 deletions src/parameterizations/lateral/MOM_spherical_harmonics.F90
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Expand Up @@ -23,12 +23,14 @@ module MOM_spherical_harmonics
!! [lmax=(ndegree+1)*(ndegree+2)/2] [nodim].
real, allocatable :: cos_clatT(:,:) !< Precomputed cosine of colatitude at the t-cells [nondim].
real, allocatable :: Pmm(:,:,:) !< Precomputed associated Legendre polynomials (m=n) at the t-cells [nondim].
real, allocatable :: cos_lonT(:,:,:), sin_lonT(:,:,:) !< Precomputed exponential factors at the t-cells [nondim].
real, allocatable :: cos_lonT_wtd(:,:,:), & !< Precomputed exponential factors at the t-cells weighted by a
sin_lonT_wtd(:,:,:) !! nondimensionalized cell area [nondim]
real, allocatable :: a_recur(:,:), b_recur(:,:) !< Precomputed recurrence coefficients [nondim].
real, allocatable :: Snm_Re_raw(:,:,:), & !< 3D array to store un-summed SHT coefficients
Snm_Im_raw(:,:,:) !! at the t-cells for reproducing sums [same as input variable]
real, allocatable :: cos_lonT(:,:,:), & !< Precomputed cosine factors at the t-cells [nondim].
sin_lonT(:,:,:) !< Precomputed sine factors at the t-cells [nondim].
real, allocatable :: cos_lonT_wtd(:,:,:), & !< Precomputed area-weighted cosine factors at the t-cells [nondim]
sin_lonT_wtd(:,:,:) !< Precomputed area-weighted sine factors at the t-cells [nondim]
real, allocatable :: a_recur(:,:), & !< Precomputed recurrence coefficients a [nondim].
b_recur(:,:) !< Precomputed recurrence coefficients b [nondim].
real, allocatable :: Snm_Re_raw(:,:,:), & !< Array to store un-summed SHT coefficients
Snm_Im_raw(:,:,:) !< at the t-cells for reproducing sums [same as input variable]
logical :: reprod_sum !< True if use reproducible global sums
end type sht_CS

Expand All @@ -45,8 +47,8 @@ subroutine spherical_harmonics_forward(G, CS, var, Snm_Re, Snm_Im, Nd)
type(sht_CS), intent(inout) :: CS !< Control structure for SHT
real, dimension(SZI_(G),SZJ_(G)), &
intent(in) :: var !< Input 2-D variable []
real, intent(out) :: Snm_Re(:), & !< Output real and imaginary SHT coefficients
Snm_Im(:) !! [same as input variable]
real, intent(out) :: Snm_Re(:) !< SHT coefficients for the real modes (cosine)
real, intent(out) :: Snm_Im(:) !< SHT coefficients for the imaginary modes (sine)
integer, optional, intent(in) :: Nd !< Maximum degree of the spherical harmonics
!! overriding ndegree in the CS [nondim]
! local variables
Expand Down Expand Up @@ -141,8 +143,8 @@ end subroutine spherical_harmonics_forward
subroutine spherical_harmonics_inverse(G, CS, Snm_Re, Snm_Im, var, Nd)
type(ocean_grid_type), intent(in) :: G !< The ocean's grid structure.
type(sht_CS), intent(in) :: CS !< Control structure for SHT
real, intent(in) :: Snm_Re(:), & !< Real and imaginary SHT coefficients with
Snm_Im(:) !! any scaling factors such as Love numbers []
real, intent(in) :: Snm_Re(:) !< SHT coefficients for the real modes (cosine)
real, intent(in) :: Snm_Im(:) !< SHT coefficients for the imaginary modes (sine)
real, dimension(SZI_(G),SZJ_(G)), &
intent(out) :: var !< Output 2-D variable []
integer, optional, intent(in) :: Nd !< Maximum degree of the spherical harmonics
Expand Down Expand Up @@ -269,17 +271,11 @@ subroutine spherical_harmonics_init(G, param_file, CS)
! Calculate the diagonal elements of the associated Legendre polynomials (n=m)
allocate(CS%Pmm(is:ie,js:je,m+1)); CS%Pmm(:,:,:) = 0.0
do m=0,CS%ndegree
! Pmm_coef = 1.0/(4.0*PI)
! do k=1,m ; Pmm_coef = Pmm_coef * real(2*k+1)/real(2*k); enddo
! Pmm_coef = sqrt(Pmm_coef)
! do j=js,je ; do i=is,ie
! CS%Pmm(i,j,m+1) = Pmm_coef * sin_clatT(i,j)**m
! enddo ; enddo
Pmm_coef = 1.0/(4.0*PI)
do k=1,m ; Pmm_coef = Pmm_coef * (real(2*k+1) / real(2*k)); enddo
Pmm_coef = sqrt(Pmm_coef)
do j=js,je ; do i=is,ie
CS%Pmm(i,j,m+1) = sqrt(1.0/(4.0*PI)) * sin_clatT(i,j)**m
do k = 1, m
CS%Pmm(i,j,m+1) = CS%Pmm(i,j,m+1) * sqrt(real(2*k+1)/real(2*k))
enddo
CS%Pmm(i,j,m+1) = Pmm_coef * (sin_clatT(i,j)**m)
enddo ; enddo
enddo

Expand Down Expand Up @@ -338,16 +334,16 @@ end function order2index
!! for forward and inverse transforms loosely follows Schaeffer (2013).
!!
!! In forward transform, a two-dimensional physical field can be projected into a series of spherical harmonics. The
!! spherical harmonic coefficient of degree n and order m for a field $f(\theta, \phi)$ is calculated as follows:
!! spherical harmonic coefficient of degree n and order m for a field \f$f(\theta, \phi)\f$ is calculated as follows:
!! \f[
!! f^m_n = \int^{2\pi}_{0}\int^{\pi}_{0}f(\theta,\phi)Y^m_n(\theta,\phi)\sin\theta d\theta d\phi
!! \f]
!! and
!! \f[
!! Y^m_n(\theta,\phi) = P^m_n(\cos\theta)\exp(im\phi)
!! \f]
!! where $P^m_n(\cos \theta)$ is the normalized associated Legendre polynomial of degree n and order m. $\phi$ is the
!! longitude and $\theta$ is the colatitude.
!! where \f$P^m_n(\cos \theta)\f$ is the normalized associated Legendre polynomial of degree n and order m. \f$\phi\f$
!! is the longitude and \f$\theta\f$ is the colatitude.
!! Or, written in the discretized form:
!! \f[
!! f^m_n = \sum^{Nj}_{0}\sum^{Ni}_{0}f(i,j)Y^m_n(i,j)A(i,j)/r_e^2
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5 changes: 2 additions & 3 deletions src/parameterizations/lateral/MOM_tidal_forcing.F90
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Expand Up @@ -79,7 +79,7 @@ module MOM_tidal_forcing
integer :: sal_sht_Nd !< Maximum degree for SHT [nodim]
real, allocatable :: Love_Scaling(:) !< Love number for each SHT mode [nodim]
real, allocatable :: Snm_Re(:), & !< Real and imaginary SHT coefficient for SHT SAL
Snm_Im(:) !! [Z ~> m]
Snm_Im(:) !< [Z ~> m]
end type tidal_forcing_CS

integer :: id_clock_tides !< CPU clock for tides
Expand Down Expand Up @@ -601,8 +601,7 @@ subroutine calc_love_scaling(nlm, rhoW, rhoE, Love_Scaling)

do m=0,nlm ; do n=m,nlm
l = order2index(m,nlm)
! Love_Scaling(l+n-m) = (3.0 / real(2*n+1)) * (rhoW / rhoE) * (1.0 + KDat(n+1) - HDat(n+1))
Love_Scaling(l+n-m) = (1.0 + KDat(n+1) - HDat(n+1)) / real(2*n+1) * 3.0 * rhoW / rhoE
Love_Scaling(l+n-m) = (3.0 / real(2*n+1)) * (rhoW / rhoE) * (1.0 + KDat(n+1) - HDat(n+1))
enddo ; enddo
end subroutine calc_love_scaling

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