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MOM_EOS_Wright.F90
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MOM_EOS_Wright.F90
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!> The equation of state using the Wright 1997 expressions
module MOM_EOS_Wright
! This file is part of MOM6. See LICENSE.md for the license.
!***********************************************************************
!* The subroutines in this file implement the equation of state for *
!* sea water using the formulae given by Wright, 1997, J. Atmos. *
!* Ocean. Tech., 14, 735-740. Coded by R. Hallberg, 7/00. *
!***********************************************************************
!use MOM_hor_index, only : hor_index_type
implicit none ; private
#include <MOM_memory.h>
public calculate_compress_wright, calculate_density_wright, calculate_spec_vol_wright
public calculate_density_derivs_wright, calculate_specvol_derivs_wright
public calculate_density_second_derivs_wright
!public int_density_dz_wright, int_spec_vol_dp_wright
! A note on unit descriptions in comments: MOM6 uses units that can be rescaled for dimensional
! consistency testing. These are noted in comments with units like Z, H, L, and T, along with
! their mks counterparts with notation like "a velocity [Z T-1 ~> m s-1]". If the units
! vary with the Boussinesq approximation, the Boussinesq variant is given first.
!> Compute the in situ density of sea water (in [kg m-3]), or its anomaly with respect to
!! a reference density, from salinity (in psu), potential temperature (in deg C), and pressure [Pa],
!! using the expressions from Wright, 1997, J. Atmos. Ocean. Tech., 14, 735-740.
interface calculate_density_wright
module procedure calculate_density_scalar_wright, calculate_density_array_wright
end interface calculate_density_wright
!> Compute the in situ specific volume of sea water (in [m3 kg-1]), or an anomaly with respect
!! to a reference specific volume, from salinity (in psu), potential temperature (in deg C), and
!! pressure [Pa], using the expressions from Wright, 1997, J. Atmos. Ocean. Tech., 14, 735-740.
interface calculate_spec_vol_wright
module procedure calculate_spec_vol_scalar_wright, calculate_spec_vol_array_wright
end interface calculate_spec_vol_wright
!> For a given thermodynamic state, return the derivatives of density with temperature and salinity
interface calculate_density_derivs_wright
module procedure calculate_density_derivs_scalar_wright, calculate_density_derivs_array_wright
end interface
!> For a given thermodynamic state, return the second derivatives of density with various combinations
!! of temperature, salinity, and pressure
interface calculate_density_second_derivs_wright
module procedure calculate_density_second_derivs_scalar_wright, calculate_density_second_derivs_array_wright
end interface
!>@{ Parameters in the Wright equation of state
!real :: a0, a1, a2, b0, b1, b2, b3, b4, b5, c0, c1, c2, c3, c4, c5
! One of the two following blocks of values should be commented out.
! Following are the values for the full range formula.
!
!real, parameter :: a0 = 7.133718e-4, a1 = 2.724670e-7, a2 = -1.646582e-7
!real, parameter :: b0 = 5.613770e8, b1 = 3.600337e6, b2 = -3.727194e4
!real, parameter :: b3 = 1.660557e2, b4 = 6.844158e5, b5 = -8.389457e3
!real, parameter :: c0 = 1.609893e5, c1 = 8.427815e2, c2 = -6.931554
!real, parameter :: c3 = 3.869318e-2, c4 = -1.664201e2, c5 = -2.765195
! Following are the values for the reduced range formula.
real, parameter :: a0 = 7.057924e-4, a1 = 3.480336e-7, a2 = -1.112733e-7 ! a0/a1 ~= 2028 ; a0/a2 ~= -6343
real, parameter :: b0 = 5.790749e8, b1 = 3.516535e6, b2 = -4.002714e4 ! b0/b1 ~= 165 ; b0/b4 ~= 974
real, parameter :: b3 = 2.084372e2, b4 = 5.944068e5, b5 = -9.643486e3
real, parameter :: c0 = 1.704853e5, c1 = 7.904722e2, c2 = -7.984422 ! c0/c1 ~= 216 ; c0/c4 ~= -740
real, parameter :: c3 = 5.140652e-2, c4 = -2.302158e2, c5 = -3.079464
!>@}
contains
!> This subroutine computes the in situ density of sea water (rho in
!! [kg m-3]) from salinity (S [PSU]), potential temperature
!! (T [degC]), and pressure [Pa]. It uses the expression from
!! Wright, 1997, J. Atmos. Ocean. Tech., 14, 735-740.
subroutine calculate_density_scalar_wright(T, S, pressure, rho, rho_ref)
real, intent(in) :: T !< Potential temperature relative to the surface [degC].
real, intent(in) :: S !< Salinity [PSU].
real, intent(in) :: pressure !< pressure [Pa].
real, intent(out) :: rho !< In situ density [kg m-3].
real, optional, intent(in) :: rho_ref !< A reference density [kg m-3].
! *====================================================================*
! * This subroutine computes the in situ density of sea water (rho in *
! * [kg m-3]) from salinity (S [PSU]), potential temperature *
! * (T [degC]), and pressure [Pa]. It uses the expression from *
! * Wright, 1997, J. Atmos. Ocean. Tech., 14, 735-740. *
! * Coded by R. Hallberg, 7/00 *
! *====================================================================*
real, dimension(1) :: T0, S0, pressure0, rho0
T0(1) = T
S0(1) = S
pressure0(1) = pressure
call calculate_density_array_wright(T0, S0, pressure0, rho0, 1, 1, rho_ref)
rho = rho0(1)
end subroutine calculate_density_scalar_wright
!> This subroutine computes the in situ density of sea water (rho in
!! [kg m-3]) from salinity (S [PSU]), potential temperature
!! (T [degC]), and pressure [Pa]. It uses the expression from
!! Wright, 1997, J. Atmos. Ocean. Tech., 14, 735-740.
subroutine calculate_density_array_wright(T, S, pressure, rho, start, npts, rho_ref)
real, dimension(:), intent(in) :: T !< potential temperature relative to the surface [degC].
real, dimension(:), intent(in) :: S !< salinity [PSU].
real, dimension(:), intent(in) :: pressure !< pressure [Pa].
real, dimension(:), intent(inout) :: rho !< in situ density [kg m-3].
integer, intent(in) :: start !< the starting point in the arrays.
integer, intent(in) :: npts !< the number of values to calculate.
real, optional, intent(in) :: rho_ref !< A reference density [kg m-3].
! Original coded by R. Hallberg, 7/00, anomaly coded in 3/18.
! Local variables
real :: al0, p0, lambda
real :: al_TS, p_TSp, lam_TS, pa_000
integer :: j
if (present(rho_ref)) pa_000 = (b0*(1.0 - a0*rho_ref) - rho_ref*c0)
if (present(rho_ref)) then ; do j=start,start+npts-1
al_TS = a1*T(j) +a2*S(j)
al0 = a0 + al_TS
p_TSp = pressure(j) + (b4*S(j) + T(j) * (b1 + (T(j)*(b2 + b3*T(j)) + b5*S(j))))
lam_TS = c4*S(j) + T(j) * (c1 + (T(j)*(c2 + c3*T(j)) + c5*S(j)))
! The following two expressions are mathematically equivalent.
! rho(j) = (b0 + p0_TSp) / ((c0 + lam_TS) + al0*(b0 + p0_TSp)) - rho_ref
rho(j) = (pa_000 + (p_TSp - rho_ref*(p_TSp*al0 + (b0*al_TS + lam_TS)))) / &
( (c0 + lam_TS) + al0*(b0 + p_TSp) )
enddo ; else ; do j=start,start+npts-1
al0 = (a0 + a1*T(j)) +a2*S(j)
p0 = (b0 + b4*S(j)) + T(j) * (b1 + T(j)*(b2 + b3*T(j)) + b5*S(j))
lambda = (c0 +c4*S(j)) + T(j) * (c1 + T(j)*(c2 + c3*T(j)) + c5*S(j))
rho(j) = (pressure(j) + p0) / (lambda + al0*(pressure(j) + p0))
enddo ; endif
end subroutine calculate_density_array_wright
!> This subroutine computes the in situ specific volume of sea water (specvol in
!! [m3 kg-1]) from salinity (S [PSU]), potential temperature (T [degC])
!! and pressure [Pa]. It uses the expression from
!! Wright, 1997, J. Atmos. Ocean. Tech., 14, 735-740.
!! If spv_ref is present, specvol is an anomaly from spv_ref.
subroutine calculate_spec_vol_scalar_wright(T, S, pressure, specvol, spv_ref)
real, intent(in) :: T !< potential temperature relative to the surface [degC].
real, intent(in) :: S !< salinity [PSU].
real, intent(in) :: pressure !< pressure [Pa].
real, intent(out) :: specvol !< in situ specific volume [m3 kg-1].
real, optional, intent(in) :: spv_ref !< A reference specific volume [m3 kg-1].
! Local variables
real, dimension(1) :: T0, S0, pressure0, spv0
T0(1) = T ; S0(1) = S ; pressure0(1) = pressure
call calculate_spec_vol_array_wright(T0, S0, pressure0, spv0, 1, 1, spv_ref)
specvol = spv0(1)
end subroutine calculate_spec_vol_scalar_wright
!> This subroutine computes the in situ specific volume of sea water (specvol in
!! [m3 kg-1]) from salinity (S [PSU]), potential temperature (T [degC])
!! and pressure [Pa]. It uses the expression from
!! Wright, 1997, J. Atmos. Ocean. Tech., 14, 735-740.
!! If spv_ref is present, specvol is an anomaly from spv_ref.
subroutine calculate_spec_vol_array_wright(T, S, pressure, specvol, start, npts, spv_ref)
real, dimension(:), intent(in) :: T !< potential temperature relative to the
!! surface [degC].
real, dimension(:), intent(in) :: S !< salinity [PSU].
real, dimension(:), intent(in) :: pressure !< pressure [Pa].
real, dimension(:), intent(inout) :: specvol !< in situ specific volume [m3 kg-1].
integer, intent(in) :: start !< the starting point in the arrays.
integer, intent(in) :: npts !< the number of values to calculate.
real, optional, intent(in) :: spv_ref !< A reference specific volume [m3 kg-1].
! Local variables
real :: al0, p0, lambda
integer :: j
do j=start,start+npts-1
al0 = (a0 + a1*T(j)) +a2*S(j)
p0 = (b0 + b4*S(j)) + T(j) * (b1 + T(j)*((b2 + b3*T(j))) + b5*S(j))
lambda = (c0 +c4*S(j)) + T(j) * (c1 + T(j)*((c2 + c3*T(j))) + c5*S(j))
if (present(spv_ref)) then
specvol(j) = (lambda + (al0 - spv_ref)*(pressure(j) + p0)) / (pressure(j) + p0)
else
specvol(j) = (lambda + al0*(pressure(j) + p0)) / (pressure(j) + p0)
endif
enddo
end subroutine calculate_spec_vol_array_wright
!> For a given thermodynamic state, return the thermal/haline expansion coefficients
subroutine calculate_density_derivs_array_wright(T, S, pressure, drho_dT, drho_dS, start, npts)
real, intent(in), dimension(:) :: T !< Potential temperature relative to the
!! surface [degC].
real, intent(in), dimension(:) :: S !< Salinity [PSU].
real, intent(in), dimension(:) :: pressure !< pressure [Pa].
real, intent(inout), dimension(:) :: drho_dT !< The partial derivative of density with potential
!! temperature [kg m-3 degC-1].
real, intent(inout), dimension(:) :: drho_dS !< The partial derivative of density with salinity,
!! in [kg m-3 PSU-1].
integer, intent(in) :: start !< The starting point in the arrays.
integer, intent(in) :: npts !< The number of values to calculate.
! Local variables
real :: al0, p0, lambda, I_denom2
integer :: j
do j=start,start+npts-1
al0 = (a0 + a1*T(j)) + a2*S(j)
p0 = (b0 + b4*S(j)) + T(j) * (b1 + T(j)*((b2 + b3*T(j))) + b5*S(j))
lambda = (c0 +c4*S(j)) + T(j) * (c1 + T(j)*((c2 + c3*T(j))) + c5*S(j))
I_denom2 = 1.0 / (lambda + al0*(pressure(j) + p0))
I_denom2 = I_denom2 *I_denom2
drho_dT(j) = I_denom2 * &
(lambda* (b1 + T(j)*(2.0*b2 + 3.0*b3*T(j)) + b5*S(j)) - &
(pressure(j)+p0) * ( (pressure(j)+p0)*a1 + &
(c1 + T(j)*(c2*2.0 + c3*3.0*T(j)) + c5*S(j)) ))
drho_dS(j) = I_denom2 * (lambda* (b4 + b5*T(j)) - &
(pressure(j)+p0) * ( (pressure(j)+p0)*a2 + (c4 + c5*T(j)) ))
enddo
end subroutine calculate_density_derivs_array_wright
!> The scalar version of calculate_density_derivs which promotes scalar inputs to a 1-element array and then
!! demotes the output back to a scalar
subroutine calculate_density_derivs_scalar_wright(T, S, pressure, drho_dT, drho_dS)
real, intent(in) :: T !< Potential temperature relative to the surface [degC].
real, intent(in) :: S !< Salinity [PSU].
real, intent(in) :: pressure !< pressure [Pa].
real, intent(out) :: drho_dT !< The partial derivative of density with potential
!! temperature [kg m-3 degC-1].
real, intent(out) :: drho_dS !< The partial derivative of density with salinity,
!! in [kg m-3 PSU-1].
! Local variables needed to promote the input/output scalars to 1-element arrays
real, dimension(1) :: T0, S0, P0
real, dimension(1) :: drdt0, drds0
T0(1) = T
S0(1) = S
P0(1) = pressure
call calculate_density_derivs_array_wright(T0, S0, P0, drdt0, drds0, 1, 1)
drho_dT = drdt0(1)
drho_dS = drds0(1)
end subroutine calculate_density_derivs_scalar_wright
!> Second derivatives of density with respect to temperature, salinity, and pressure
subroutine calculate_density_second_derivs_array_wright(T, S, P, drho_ds_ds, drho_ds_dt, drho_dt_dt, &
drho_ds_dp, drho_dt_dp, start, npts)
real, dimension(:), intent(in ) :: T !< Potential temperature referenced to 0 dbar [degC]
real, dimension(:), intent(in ) :: S !< Salinity [PSU]
real, dimension(:), intent(in ) :: P !< Pressure [Pa]
real, dimension(:), intent(inout) :: drho_ds_ds !< Partial derivative of beta with respect
!! to S [kg m-3 PSU-2]
real, dimension(:), intent(inout) :: drho_ds_dt !< Partial derivative of beta with respcct
!! to T [kg m-3 PSU-1 degC-1]
real, dimension(:), intent(inout) :: drho_dt_dt !< Partial derivative of alpha with respect
!! to T [kg m-3 degC-2]
real, dimension(:), intent(inout) :: drho_ds_dp !< Partial derivative of beta with respect
!! to pressure [kg m-3 PSU-1 Pa-1]
real, dimension(:), intent(inout) :: drho_dt_dp !< Partial derivative of alpha with respect
!! to pressure [kg m-3 degC-1 Pa-1]
integer, intent(in ) :: start !< Starting index in T,S,P
integer, intent(in ) :: npts !< Number of points to loop over
! Local variables
real :: z0, z1, z2, z3, z4, z5, z6 ,z7, z8, z9, z10, z11, z2_2, z2_3
integer :: j
! Based on the above expression with common terms factored, there probably exists a more numerically stable
! and/or efficient expression
do j = start,start+npts-1
z0 = T(j)*(b1 + b5*S(j) + T(j)*(b2 + b3*T(j)))
z1 = (b0 + P(j) + b4*S(j) + z0)
z3 = (b1 + b5*S(j) + T(j)*(2.*b2 + 2.*b3*T(j)))
z4 = (c0 + c4*S(j) + T(j)*(c1 + c5*S(j) + T(j)*(c2 + c3*T(j))))
z5 = (b1 + b5*S(j) + T(j)*(b2 + b3*T(j)) + T(j)*(b2 + 2.*b3*T(j)))
z6 = c1 + c5*S(j) + T(j)*(c2 + c3*T(j)) + T(j)*(c2 + 2.*c3*T(j))
z7 = (c4 + c5*T(j) + a2*z1)
z8 = (c1 + c5*S(j) + T(j)*(2.*c2 + 3.*c3*T(j)) + a1*z1)
z9 = (a0 + a2*S(j) + a1*T(j))
z10 = (b4 + b5*T(j))
z11 = (z10*z4 - z1*z7)
z2 = (c0 + c4*S(j) + T(j)*(c1 + c5*S(j) + T(j)*(c2 + c3*T(j))) + z9*z1)
z2_2 = z2*z2
z2_3 = z2_2*z2
drho_ds_ds(j) = (z10*(c4 + c5*T(j)) - a2*z10*z1 - z10*z7)/z2_2 - (2.*(c4 + c5*T(j) + z9*z10 + a2*z1)*z11)/z2_3
drho_ds_dt(j) = (z10*z6 - z1*(c5 + a2*z5) + b5*z4 - z5*z7)/z2_2 - (2.*(z6 + z9*z5 + a1*z1)*z11)/z2_3
drho_dt_dt(j) = (z3*z6 - z1*(2.*c2 + 6.*c3*T(j) + a1*z5) + (2.*b2 + 4.*b3*T(j))*z4 - z5*z8)/z2_2 - &
(2.*(z6 + z9*z5 + a1*z1)*(z3*z4 - z1*z8))/z2_3
drho_ds_dp(j) = (-c4 - c5*T(j) - 2.*a2*z1)/z2_2 - (2.*z9*z11)/z2_3
drho_dt_dp(j) = (-c1 - c5*S(j) - T(j)*(2.*c2 + 3.*c3*T(j)) - 2.*a1*z1)/z2_2 - (2.*z9*(z3*z4 - z1*z8))/z2_3
enddo
end subroutine calculate_density_second_derivs_array_wright
!> Second derivatives of density with respect to temperature, salinity, and pressure for scalar inputs. Inputs
!! promoted to 1-element array and output demoted to scalar
subroutine calculate_density_second_derivs_scalar_wright(T, S, P, drho_ds_ds, drho_ds_dt, drho_dt_dt, &
drho_ds_dp, drho_dt_dp)
real, intent(in ) :: T !< Potential temperature referenced to 0 dbar
real, intent(in ) :: S !< Salinity [PSU]
real, intent(in ) :: P !< pressure [Pa]
real, intent( out) :: drho_ds_ds !< Partial derivative of beta with respect
!! to S [kg m-3 PSU-2]
real, intent( out) :: drho_ds_dt !< Partial derivative of beta with respcct
!! to T [kg m-3 PSU-1 degC-1]
real, intent( out) :: drho_dt_dt !< Partial derivative of alpha with respect
!! to T [kg m-3 degC-2]
real, intent( out) :: drho_ds_dp !< Partial derivative of beta with respect
!! to pressure [kg m-3 PSU-1 Pa-1]
real, intent( out) :: drho_dt_dp !< Partial derivative of alpha with respect
!! to pressure [kg m-3 degC-1 Pa-1]
! Local variables
real, dimension(1) :: T0, S0, P0
real, dimension(1) :: drdsds, drdsdt, drdtdt, drdsdp, drdtdp
T0(1) = T
S0(1) = S
P0(1) = P
call calculate_density_second_derivs_array_wright(T0, S0, P0, drdsds, drdsdt, drdtdt, drdsdp, drdtdp, 1, 1)
drho_ds_ds = drdsds(1)
drho_ds_dt = drdsdt(1)
drho_dt_dt = drdtdt(1)
drho_ds_dp = drdsdp(1)
drho_dt_dp = drdtdp(1)
end subroutine calculate_density_second_derivs_scalar_wright
!> For a given thermodynamic state, return the partial derivatives of specific volume
!! with temperature and salinity
subroutine calculate_specvol_derivs_wright(T, S, pressure, dSV_dT, dSV_dS, start, npts)
real, intent(in), dimension(:) :: T !< Potential temperature relative to the surface [degC].
real, intent(in), dimension(:) :: S !< Salinity [PSU].
real, intent(in), dimension(:) :: pressure !< pressure [Pa].
real, intent(inout), dimension(:) :: dSV_dT !< The partial derivative of specific volume with
!! potential temperature [m3 kg-1 degC-1].
real, intent(inout), dimension(:) :: dSV_dS !< The partial derivative of specific volume with
!! salinity [m3 kg-1 / Pa].
integer, intent(in) :: start !< The starting point in the arrays.
integer, intent(in) :: npts !< The number of values to calculate.
! Local variables
real :: al0, p0, lambda, I_denom
integer :: j
do j=start,start+npts-1
! al0 = (a0 + a1*T(j)) + a2*S(j)
p0 = (b0 + b4*S(j)) + T(j) * (b1 + T(j)*((b2 + b3*T(j))) + b5*S(j))
lambda = (c0 +c4*S(j)) + T(j) * (c1 + T(j)*((c2 + c3*T(j))) + c5*S(j))
! SV = al0 + lambda / (pressure(j) + p0)
I_denom = 1.0 / (pressure(j) + p0)
dSV_dT(j) = (a1 + I_denom * (c1 + T(j)*((2.0*c2 + 3.0*c3*T(j))) + c5*S(j))) - &
(I_denom**2 * lambda) * (b1 + T(j)*((2.0*b2 + 3.0*b3*T(j))) + b5*S(j))
dSV_dS(j) = (a2 + I_denom * (c4 + c5*T(j))) - &
(I_denom**2 * lambda) * (b4 + b5*T(j))
enddo
end subroutine calculate_specvol_derivs_wright
!> This subroutine computes the in situ density of sea water (rho in
!! [kg m-3]) and the compressibility (drho/dp = C_sound^-2)
!! (drho_dp [s2 m-2]) from salinity (sal in psu), potential
!! temperature (T [degC]), and pressure [Pa]. It uses the expressions
!! from Wright, 1997, J. Atmos. Ocean. Tech., 14, 735-740.
!! Coded by R. Hallberg, 1/01
subroutine calculate_compress_wright(T, S, pressure, rho, drho_dp, start, npts)
real, intent(in), dimension(:) :: T !< Potential temperature relative to the surface [degC].
real, intent(in), dimension(:) :: S !< Salinity [PSU].
real, intent(in), dimension(:) :: pressure !< pressure [Pa].
real, intent(inout), dimension(:) :: rho !< In situ density [kg m-3].
real, intent(inout), dimension(:) :: drho_dp !< The partial derivative of density with pressure
!! (also the inverse of the square of sound speed)
!! [s2 m-2].
integer, intent(in) :: start !< The starting point in the arrays.
integer, intent(in) :: npts !< The number of values to calculate.
! Coded by R. Hallberg, 1/01
! Local variables
real :: al0, p0, lambda, I_denom
integer :: j
do j=start,start+npts-1
al0 = (a0 + a1*T(j)) +a2*S(j)
p0 = (b0 + b4*S(j)) + T(j) * (b1 + T(j)*((b2 + b3*T(j))) + b5*S(j))
lambda = (c0 +c4*S(j)) + T(j) * (c1 + T(j)*((c2 + c3*T(j))) + c5*S(j))
I_denom = 1.0 / (lambda + al0*(pressure(j) + p0))
rho(j) = (pressure(j) + p0) * I_denom
drho_dp(j) = lambda * I_denom * I_denom
enddo
end subroutine calculate_compress_wright
!> This subroutine calculates analytical and nearly-analytical integrals of
!! pressure anomalies across layers, which are required for calculating the
!! finite-volume form pressure accelerations in a Boussinesq model.
! subroutine int_density_dz_wright(T, S, z_t, z_b, rho_ref, rho_0, G_e, HI, &
! dpa, intz_dpa, intx_dpa, inty_dpa, &
! bathyT, dz_neglect, useMassWghtInterp, rho_scale, pres_scale)
! type(hor_index_type), intent(in) :: HI !< The horizontal index type for the arrays.
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(in) :: T !< Potential temperature relative to the surface
! !! [degC].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(in) :: S !< Salinity [PSU].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(in) :: z_t !< Height at the top of the layer in depth units [Z ~> m].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(in) :: z_b !< Height at the top of the layer [Z ~> m].
! real, intent(in) :: rho_ref !< A mean density [R ~> kg m-3] or [kg m-3], that is subtracted
! !! out to reduce the magnitude of each of the integrals.
! !! (The pressure is calucated as p~=-z*rho_0*G_e.)
! real, intent(in) :: rho_0 !< Density [R ~> kg m-3] or [kg m-3], that is used
! !! to calculate the pressure (as p~=-z*rho_0*G_e)
! !! used in the equation of state.
! real, intent(in) :: G_e !< The Earth's gravitational acceleration
! !! [L2 Z-1 T-2 ~> m s-2] or [m2 Z-1 s-2 ~> m s-2].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(inout) :: dpa !< The change in the pressure anomaly across the
! !! layer [R L2 T-2 ~> Pa] or [Pa].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! optional, intent(inout) :: intz_dpa !< The integral through the thickness of the layer
! !! of the pressure anomaly relative to the anomaly
! !! at the top of the layer [R Z L2 T-2 ~> Pa m].
! real, dimension(HI%IsdB:HI%IedB,HI%jsd:HI%jed), &
! optional, intent(inout) :: intx_dpa !< The integral in x of the difference between the
! !! pressure anomaly at the top and bottom of the
! !! layer divided by the x grid spacing [R L2 T-2 ~> Pa].
! real, dimension(HI%isd:HI%ied,HI%JsdB:HI%JedB), &
! optional, intent(inout) :: inty_dpa !< The integral in y of the difference between the
! !! pressure anomaly at the top and bottom of the
! !! layer divided by the y grid spacing [R L2 T-2 ~> Pa].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! optional, intent(in) :: bathyT !< The depth of the bathymetry [Z ~> m].
! real, optional, intent(in) :: dz_neglect !< A miniscule thickness change [Z ~> m].
! logical, optional, intent(in) :: useMassWghtInterp !< If true, uses mass weighting to
! !! interpolate T/S for top and bottom integrals.
! real, optional, intent(in) :: rho_scale !< A multiplicative factor by which to scale density
! !! from kg m-3 to the desired units [R m3 kg-1 ~> 1]
! real, optional, intent(in) :: pres_scale !< A multiplicative factor to convert pressure
! !! into Pa [Pa T2 R-1 L-2 ~> 1].
! ! Local variables
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed) :: al0_2d, p0_2d, lambda_2d
! real :: al0, p0, lambda
! real :: rho_anom ! The density anomaly from rho_ref [kg m-3].
! real :: eps, eps2, rem
! real :: GxRho ! The gravitational acceleration times density and unit conversion factors [Pa Z-1 ~> kg m-2 s-2]
! real :: g_Earth ! The gravitational acceleration [m2 Z-1 s-2 ~> m s-2]
! real :: I_Rho ! The inverse of the Boussinesq density [m3 kg-1]
! real :: rho_ref_mks ! The reference density in MKS units, never rescaled from kg m-3 [kg m-3]
! real :: p_ave, I_al0, I_Lzz
! real :: dz ! The layer thickness [Z ~> m].
! real :: hWght ! A pressure-thickness below topography [Z ~> m].
! real :: hL, hR ! Pressure-thicknesses of the columns to the left and right [Z ~> m].
! real :: iDenom ! The inverse of the denominator in the weights [Z-2 ~> m-2].
! real :: hWt_LL, hWt_LR ! hWt_LA is the weighted influence of A on the left column [nondim].
! real :: hWt_RL, hWt_RR ! hWt_RA is the weighted influence of A on the right column [nondim].
! real :: wt_L, wt_R ! The linear weights of the left and right columns [nondim].
! real :: wtT_L, wtT_R ! The weights for tracers from the left and right columns [nondim].
! real :: intz(5) ! The integrals of density with height at the
! ! 5 sub-column locations [R L2 T-2 ~> Pa].
! real :: Pa_to_RL2_T2 ! A conversion factor of pressures from Pa to the output units indicated by
! ! pres_scale [R L2 T-2 Pa-1 ~> 1] or [1].
! logical :: do_massWeight ! Indicates whether to do mass weighting.
! real, parameter :: C1_3 = 1.0/3.0, C1_7 = 1.0/7.0 ! Rational constants.
! real, parameter :: C1_9 = 1.0/9.0, C1_90 = 1.0/90.0 ! Rational constants.
! integer :: is, ie, js, je, Isq, Ieq, Jsq, Jeq, i, j, m
! ! These array bounds work for the indexing convention of the input arrays, but
! ! on the computational domain defined for the output arrays.
! Isq = HI%IscB ; Ieq = HI%IecB
! Jsq = HI%JscB ; Jeq = HI%JecB
! is = HI%isc ; ie = HI%iec
! js = HI%jsc ; je = HI%jec
! if (present(pres_scale)) then
! GxRho = pres_scale * G_e * rho_0 ; g_Earth = pres_scale * G_e
! Pa_to_RL2_T2 = 1.0 / pres_scale
! else
! GxRho = G_e * rho_0 ; g_Earth = G_e
! Pa_to_RL2_T2 = 1.0
! endif
! if (present(rho_scale)) then
! g_Earth = g_Earth * rho_scale
! rho_ref_mks = rho_ref / rho_scale ; I_Rho = rho_scale / rho_0
! else
! rho_ref_mks = rho_ref ; I_Rho = 1.0 / rho_0
! endif
! do_massWeight = .false.
! if (present(useMassWghtInterp)) then ; if (useMassWghtInterp) then
! do_massWeight = .true.
! ! if (.not.present(bathyT)) call MOM_error(FATAL, "int_density_dz_generic: "//&
! ! "bathyT must be present if useMassWghtInterp is present and true.")
! ! if (.not.present(dz_neglect)) call MOM_error(FATAL, "int_density_dz_generic: "//&
! ! "dz_neglect must be present if useMassWghtInterp is present and true.")
! endif ; endif
! do j=Jsq,Jeq+1 ; do i=Isq,Ieq+1
! al0_2d(i,j) = (a0 + a1*T(i,j)) + a2*S(i,j)
! p0_2d(i,j) = (b0 + b4*S(i,j)) + T(i,j) * (b1 + T(i,j)*((b2 + b3*T(i,j))) + b5*S(i,j))
! lambda_2d(i,j) = (c0 +c4*S(i,j)) + T(i,j) * (c1 + T(i,j)*((c2 + c3*T(i,j))) + c5*S(i,j))
! al0 = al0_2d(i,j) ; p0 = p0_2d(i,j) ; lambda = lambda_2d(i,j)
! dz = z_t(i,j) - z_b(i,j)
! p_ave = -0.5*GxRho*(z_t(i,j)+z_b(i,j))
! I_al0 = 1.0 / al0
! I_Lzz = 1.0 / (p0 + (lambda * I_al0) + p_ave)
! eps = 0.5*GxRho*dz*I_Lzz ; eps2 = eps*eps
! ! rho(j) = (pressure(j) + p0) / (lambda + al0*(pressure(j) + p0))
! rho_anom = (p0 + p_ave)*(I_Lzz*I_al0) - rho_ref_mks
! rem = I_Rho * (lambda * I_al0**2) * eps2 * &
! (C1_3 + eps2*(0.2 + eps2*(C1_7 + C1_9*eps2)))
! dpa(i,j) = Pa_to_RL2_T2 * (g_Earth*rho_anom*dz - 2.0*eps*rem)
! if (present(intz_dpa)) &
! intz_dpa(i,j) = Pa_to_RL2_T2 * (0.5*g_Earth*rho_anom*dz**2 - dz*(1.0+eps)*rem)
! enddo ; enddo
! if (present(intx_dpa)) then ; do j=js,je ; do I=Isq,Ieq
! ! hWght is the distance measure by which the cell is violation of
! ! hydrostatic consistency. For large hWght we bias the interpolation of
! ! T & S along the top and bottom integrals, akin to thickness weighting.
! hWght = 0.0
! if (do_massWeight) &
! hWght = max(0., -bathyT(i,j)-z_t(i+1,j), -bathyT(i+1,j)-z_t(i,j))
! if (hWght > 0.) then
! hL = (z_t(i,j) - z_b(i,j)) + dz_neglect
! hR = (z_t(i+1,j) - z_b(i+1,j)) + dz_neglect
! hWght = hWght * ( (hL-hR)/(hL+hR) )**2
! iDenom = 1.0 / ( hWght*(hR + hL) + hL*hR )
! hWt_LL = (hWght*hL + hR*hL) * iDenom ; hWt_LR = (hWght*hR) * iDenom
! hWt_RR = (hWght*hR + hR*hL) * iDenom ; hWt_RL = (hWght*hL) * iDenom
! else
! hWt_LL = 1.0 ; hWt_LR = 0.0 ; hWt_RR = 1.0 ; hWt_RL = 0.0
! endif
! intz(1) = dpa(i,j) ; intz(5) = dpa(i+1,j)
! do m=2,4
! wt_L = 0.25*real(5-m) ; wt_R = 1.0-wt_L
! wtT_L = wt_L*hWt_LL + wt_R*hWt_RL ; wtT_R = wt_L*hWt_LR + wt_R*hWt_RR
! al0 = wtT_L*al0_2d(i,j) + wtT_R*al0_2d(i+1,j)
! p0 = wtT_L*p0_2d(i,j) + wtT_R*p0_2d(i+1,j)
! lambda = wtT_L*lambda_2d(i,j) + wtT_R*lambda_2d(i+1,j)
! dz = wt_L*(z_t(i,j) - z_b(i,j)) + wt_R*(z_t(i+1,j) - z_b(i+1,j))
! p_ave = -0.5*GxRho*(wt_L*(z_t(i,j)+z_b(i,j)) + &
! wt_R*(z_t(i+1,j)+z_b(i+1,j)))
! I_al0 = 1.0 / al0
! I_Lzz = 1.0 / (p0 + (lambda * I_al0) + p_ave)
! eps = 0.5*GxRho*dz*I_Lzz ; eps2 = eps*eps
! intz(m) = Pa_to_RL2_T2 * ( g_Earth*dz*((p0 + p_ave)*(I_Lzz*I_al0) - rho_ref_mks) - 2.0*eps * &
! I_Rho * (lambda * I_al0**2) * eps2 * (C1_3 + eps2*(0.2 + eps2*(C1_7 + C1_9*eps2))) )
! enddo
! ! Use Boole's rule to integrate the values.
! intx_dpa(i,j) = C1_90*(7.0*(intz(1)+intz(5)) + 32.0*(intz(2)+intz(4)) + 12.0*intz(3))
! enddo ; enddo ; endif
! if (present(inty_dpa)) then ; do J=Jsq,Jeq ; do i=is,ie
! ! hWght is the distance measure by which the cell is violation of
! ! hydrostatic consistency. For large hWght we bias the interpolation of
! ! T & S along the top and bottom integrals, akin to thickness weighting.
! hWght = 0.0
! if (do_massWeight) &
! hWght = max(0., -bathyT(i,j)-z_t(i,j+1), -bathyT(i,j+1)-z_t(i,j))
! if (hWght > 0.) then
! hL = (z_t(i,j) - z_b(i,j)) + dz_neglect
! hR = (z_t(i,j+1) - z_b(i,j+1)) + dz_neglect
! hWght = hWght * ( (hL-hR)/(hL+hR) )**2
! iDenom = 1.0 / ( hWght*(hR + hL) + hL*hR )
! hWt_LL = (hWght*hL + hR*hL) * iDenom ; hWt_LR = (hWght*hR) * iDenom
! hWt_RR = (hWght*hR + hR*hL) * iDenom ; hWt_RL = (hWght*hL) * iDenom
! else
! hWt_LL = 1.0 ; hWt_LR = 0.0 ; hWt_RR = 1.0 ; hWt_RL = 0.0
! endif
! intz(1) = dpa(i,j) ; intz(5) = dpa(i,j+1)
! do m=2,4
! wt_L = 0.25*real(5-m) ; wt_R = 1.0-wt_L
! wtT_L = wt_L*hWt_LL + wt_R*hWt_RL ; wtT_R = wt_L*hWt_LR + wt_R*hWt_RR
! al0 = wtT_L*al0_2d(i,j) + wtT_R*al0_2d(i,j+1)
! p0 = wtT_L*p0_2d(i,j) + wtT_R*p0_2d(i,j+1)
! lambda = wtT_L*lambda_2d(i,j) + wtT_R*lambda_2d(i,j+1)
! dz = wt_L*(z_t(i,j) - z_b(i,j)) + wt_R*(z_t(i,j+1) - z_b(i,j+1))
! p_ave = -0.5*GxRho*(wt_L*(z_t(i,j)+z_b(i,j)) + &
! wt_R*(z_t(i,j+1)+z_b(i,j+1)))
! I_al0 = 1.0 / al0
! I_Lzz = 1.0 / (p0 + (lambda * I_al0) + p_ave)
! eps = 0.5*GxRho*dz*I_Lzz ; eps2 = eps*eps
! intz(m) = Pa_to_RL2_T2 * ( g_Earth*dz*((p0 + p_ave)*(I_Lzz*I_al0) - rho_ref_mks) - 2.0*eps * &
! I_Rho * (lambda * I_al0**2) * eps2 * (C1_3 + eps2*(0.2 + eps2*(C1_7 + C1_9*eps2))) )
! enddo
! ! Use Boole's rule to integrate the values.
! inty_dpa(i,j) = C1_90*(7.0*(intz(1)+intz(5)) + 32.0*(intz(2)+intz(4)) + 12.0*intz(3))
! enddo ; enddo ; endif
! end subroutine int_density_dz_wright
! !> This subroutine calculates analytical and nearly-analytical integrals in
! !! pressure across layers of geopotential anomalies, which are required for
! !! calculating the finite-volume form pressure accelerations in a non-Boussinesq
! !! model. There are essentially no free assumptions, apart from the use of
! !! Boole's rule to do the horizontal integrals, and from a truncation in the
! !! series for log(1-eps/1+eps) that assumes that |eps| < 0.34.
! subroutine int_spec_vol_dp_wright(T, S, p_t, p_b, spv_ref, HI, dza, &
! intp_dza, intx_dza, inty_dza, halo_size, &
! bathyP, dP_neglect, useMassWghtInterp, SV_scale, pres_scale)
! type(hor_index_type), intent(in) :: HI !< The ocean's horizontal index type.
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(in) :: T !< Potential temperature relative to the surface
! !! [degC].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(in) :: S !< Salinity [PSU].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(in) :: p_t !< Pressure at the top of the layer [R L2 T-2 ~> Pa] or [Pa].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(in) :: p_b !< Pressure at the top of the layer [R L2 T-2 ~> Pa] or [Pa].
! real, intent(in) :: spv_ref !< A mean specific volume that is subtracted out
! !! to reduce the magnitude of each of the integrals [R-1 ~> m3 kg-1].
! !! The calculation is mathematically identical with different values of
! !! spv_ref, but this reduces the effects of roundoff.
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! intent(inout) :: dza !< The change in the geopotential anomaly across
! !! the layer [T-2 ~> m2 s-2] or [m2 s-2].
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! optional, intent(inout) :: intp_dza !< The integral in pressure through the layer of
! !! the geopotential anomaly relative to the anomaly
! !! at the bottom of the layer [R L4 T-4 ~> Pa m2 s-2]
! !! or [Pa m2 s-2].
! real, dimension(HI%IsdB:HI%IedB,HI%jsd:HI%jed), &
! optional, intent(inout) :: intx_dza !< The integral in x of the difference between the
! !! geopotential anomaly at the top and bottom of
! !! the layer divided by the x grid spacing
! !! [L2 T-2 ~> m2 s-2] or [m2 s-2].
! real, dimension(HI%isd:HI%ied,HI%JsdB:HI%JedB), &
! optional, intent(inout) :: inty_dza !< The integral in y of the difference between the
! !! geopotential anomaly at the top and bottom of
! !! the layer divided by the y grid spacing
! !! [L2 T-2 ~> m2 s-2] or [m2 s-2].
! integer, optional, intent(in) :: halo_size !< The width of halo points on which to calculate
! !! dza.
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed), &
! optional, intent(in) :: bathyP !< The pressure at the bathymetry [R L2 T-2 ~> Pa] or [Pa]
! real, optional, intent(in) :: dP_neglect !< A miniscule pressure change with
! !! the same units as p_t [R L2 T-2 ~> Pa] or [Pa]
! logical, optional, intent(in) :: useMassWghtInterp !< If true, uses mass weighting
! !! to interpolate T/S for top and bottom integrals.
! real, optional, intent(in) :: SV_scale !< A multiplicative factor by which to scale specific
! !! volume from m3 kg-1 to the desired units [kg m-3 R-1 ~> 1]
! real, optional, intent(in) :: pres_scale !< A multiplicative factor to convert pressure
! !! into Pa [Pa T2 R-1 L-2 ~> 1].
! ! Local variables
! real, dimension(HI%isd:HI%ied,HI%jsd:HI%jed) :: al0_2d, p0_2d, lambda_2d
! real :: al0 ! A term in the Wright EOS [R-1 ~> m3 kg-1]
! real :: p0 ! A term in the Wright EOS [R L2 T-2 ~> Pa]
! real :: lambda ! A term in the Wright EOS [L2 T-2 ~> m2 s-2]
! real :: al0_scale ! Scaling factor to convert al0 from MKS units [R-1 kg m-3 ~> 1]
! real :: p0_scale ! Scaling factor to convert p0 from MKS units [R L2 T-2 Pa-1 ~> 1]
! real :: lam_scale ! Scaling factor to convert lambda from MKS units [L2 s2 T-2 m-2 ~> 1]
! real :: p_ave ! The layer average pressure [R L2 T-2 ~> Pa]
! real :: rem ! [L2 T-2 ~> m2 s-2]
! real :: eps, eps2 ! A nondimensional ratio and its square [nondim]
! real :: alpha_anom ! The depth averaged specific volume anomaly [R-1 ~> m3 kg-1].
! real :: dp ! The pressure change through a layer [R L2 T-2 ~> Pa].
! real :: hWght ! A pressure-thickness below topography [R L2 T-2 ~> Pa].
! real :: hL, hR ! Pressure-thicknesses of the columns to the left and right [R L2 T-2 ~> Pa].
! real :: iDenom ! The inverse of the denominator in the weights [T4 R-2 L-4 ~> Pa-2].
! real :: hWt_LL, hWt_LR ! hWt_LA is the weighted influence of A on the left column [nondim].
! real :: hWt_RL, hWt_RR ! hWt_RA is the weighted influence of A on the right column [nondim].
! real :: wt_L, wt_R ! The linear weights of the left and right columns [nondim].
! real :: wtT_L, wtT_R ! The weights for tracers from the left and right columns [nondim].
! real :: intp(5) ! The integrals of specific volume with pressure at the
! ! 5 sub-column locations [L2 T-2 ~> m2 s-2].
! logical :: do_massWeight ! Indicates whether to do mass weighting.
! real, parameter :: C1_3 = 1.0/3.0, C1_7 = 1.0/7.0 ! Rational constants.
! real, parameter :: C1_9 = 1.0/9.0, C1_90 = 1.0/90.0 ! Rational constants.
! integer :: Isq, Ieq, Jsq, Jeq, ish, ieh, jsh, jeh, i, j, m, halo
! Isq = HI%IscB ; Ieq = HI%IecB ; Jsq = HI%JscB ; Jeq = HI%JecB
! halo = 0 ; if (present(halo_size)) halo = MAX(halo_size,0)
! ish = HI%isc-halo ; ieh = HI%iec+halo ; jsh = HI%jsc-halo ; jeh = HI%jec+halo
! if (present(intx_dza)) then ; ish = MIN(Isq,ish) ; ieh = MAX(Ieq+1,ieh); endif
! if (present(inty_dza)) then ; jsh = MIN(Jsq,jsh) ; jeh = MAX(Jeq+1,jeh); endif
! al0_scale = 1.0 ; if (present(SV_scale)) al0_scale = SV_scale
! p0_scale = 1.0
! if (present(pres_scale)) then ; if (pres_scale /= 1.0) then
! p0_scale = 1.0 / pres_scale
! endif ; endif
! lam_scale = al0_scale * p0_scale
! do_massWeight = .false.
! if (present(useMassWghtInterp)) then ; if (useMassWghtInterp) then
! do_massWeight = .true.
! ! if (.not.present(bathyP)) call MOM_error(FATAL, "int_spec_vol_dp_generic: "//&
! ! "bathyP must be present if useMassWghtInterp is present and true.")
! ! if (.not.present(dP_neglect)) call MOM_error(FATAL, "int_spec_vol_dp_generic: "//&
! ! "dP_neglect must be present if useMassWghtInterp is present and true.")
! endif ; endif
! ! alpha(j) = (lambda + al0*(pressure(j) + p0)) / (pressure(j) + p0)
! do j=jsh,jeh ; do i=ish,ieh
! al0_2d(i,j) = al0_scale * ( (a0 + a1*T(i,j)) + a2*S(i,j) )
! p0_2d(i,j) = p0_scale * ( (b0 + b4*S(i,j)) + T(i,j) * (b1 + T(i,j)*((b2 + b3*T(i,j))) + b5*S(i,j)) )
! lambda_2d(i,j) = lam_scale * ( (c0 + c4*S(i,j)) + T(i,j) * (c1 + T(i,j)*((c2 + c3*T(i,j))) + c5*S(i,j)) )
! al0 = al0_2d(i,j) ; p0 = p0_2d(i,j) ; lambda = lambda_2d(i,j)
! dp = p_b(i,j) - p_t(i,j)
! p_ave = 0.5*(p_t(i,j)+p_b(i,j))
! eps = 0.5 * dp / (p0 + p_ave) ; eps2 = eps*eps
! alpha_anom = al0 + lambda / (p0 + p_ave) - spv_ref
! rem = lambda * eps2 * (C1_3 + eps2*(0.2 + eps2*(C1_7 + C1_9*eps2)))
! dza(i,j) = alpha_anom*dp + 2.0*eps*rem
! if (present(intp_dza)) &
! intp_dza(i,j) = 0.5*alpha_anom*dp**2 - dp*(1.0-eps)*rem
! enddo ; enddo
! if (present(intx_dza)) then ; do j=HI%jsc,HI%jec ; do I=Isq,Ieq
! ! hWght is the distance measure by which the cell is violation of
! ! hydrostatic consistency. For large hWght we bias the interpolation of
! ! T & S along the top and bottom integrals, akin to thickness weighting.
! hWght = 0.0
! if (do_massWeight) &
! hWght = max(0., bathyP(i,j)-p_t(i+1,j), bathyP(i+1,j)-p_t(i,j))
! if (hWght > 0.) then
! hL = (p_b(i,j) - p_t(i,j)) + dP_neglect
! hR = (p_b(i+1,j) - p_t(i+1,j)) + dP_neglect
! hWght = hWght * ( (hL-hR)/(hL+hR) )**2
! iDenom = 1.0 / ( hWght*(hR + hL) + hL*hR )
! hWt_LL = (hWght*hL + hR*hL) * iDenom ; hWt_LR = (hWght*hR) * iDenom
! hWt_RR = (hWght*hR + hR*hL) * iDenom ; hWt_RL = (hWght*hL) * iDenom
! else
! hWt_LL = 1.0 ; hWt_LR = 0.0 ; hWt_RR = 1.0 ; hWt_RL = 0.0
! endif
! intp(1) = dza(i,j) ; intp(5) = dza(i+1,j)
! do m=2,4
! wt_L = 0.25*real(5-m) ; wt_R = 1.0-wt_L
! wtT_L = wt_L*hWt_LL + wt_R*hWt_RL ; wtT_R = wt_L*hWt_LR + wt_R*hWt_RR
! ! T, S, and p are interpolated in the horizontal. The p interpolation
! ! is linear, but for T and S it may be thickness wekghted.
! al0 = wtT_L*al0_2d(i,j) + wtT_R*al0_2d(i+1,j)
! p0 = wtT_L*p0_2d(i,j) + wtT_R*p0_2d(i+1,j)
! lambda = wtT_L*lambda_2d(i,j) + wtT_R*lambda_2d(i+1,j)
! dp = wt_L*(p_b(i,j) - p_t(i,j)) + wt_R*(p_b(i+1,j) - p_t(i+1,j))
! p_ave = 0.5*(wt_L*(p_t(i,j)+p_b(i,j)) + wt_R*(p_t(i+1,j)+p_b(i+1,j)))
! eps = 0.5 * dp / (p0 + p_ave) ; eps2 = eps*eps
! intp(m) = (al0 + lambda / (p0 + p_ave) - spv_ref)*dp + 2.0*eps* &
! lambda * eps2 * (C1_3 + eps2*(0.2 + eps2*(C1_7 + C1_9*eps2)))
! enddo
! ! Use Boole's rule to integrate the values.
! intx_dza(i,j) = C1_90*(7.0*(intp(1)+intp(5)) + 32.0*(intp(2)+intp(4)) + &
! 12.0*intp(3))
! enddo ; enddo ; endif
! if (present(inty_dza)) then ; do J=Jsq,Jeq ; do i=HI%isc,HI%iec
! ! hWght is the distance measure by which the cell is violation of
! ! hydrostatic consistency. For large hWght we bias the interpolation of
! ! T & S along the top and bottom integrals, akin to thickness weighting.
! hWght = 0.0
! if (do_massWeight) &
! hWght = max(0., bathyP(i,j)-p_t(i,j+1), bathyP(i,j+1)-p_t(i,j))
! if (hWght > 0.) then
! hL = (p_b(i,j) - p_t(i,j)) + dP_neglect
! hR = (p_b(i,j+1) - p_t(i,j+1)) + dP_neglect
! hWght = hWght * ( (hL-hR)/(hL+hR) )**2
! iDenom = 1.0 / ( hWght*(hR + hL) + hL*hR )
! hWt_LL = (hWght*hL + hR*hL) * iDenom ; hWt_LR = (hWght*hR) * iDenom
! hWt_RR = (hWght*hR + hR*hL) * iDenom ; hWt_RL = (hWght*hL) * iDenom
! else
! hWt_LL = 1.0 ; hWt_LR = 0.0 ; hWt_RR = 1.0 ; hWt_RL = 0.0
! endif
! intp(1) = dza(i,j) ; intp(5) = dza(i,j+1)
! do m=2,4
! wt_L = 0.25*real(5-m) ; wt_R = 1.0-wt_L
! wtT_L = wt_L*hWt_LL + wt_R*hWt_RL ; wtT_R = wt_L*hWt_LR + wt_R*hWt_RR
! ! T, S, and p are interpolated in the horizontal. The p interpolation
! ! is linear, but for T and S it may be thickness wekghted.
! al0 = wt_L*al0_2d(i,j) + wt_R*al0_2d(i,j+1)
! p0 = wt_L*p0_2d(i,j) + wt_R*p0_2d(i,j+1)
! lambda = wt_L*lambda_2d(i,j) + wt_R*lambda_2d(i,j+1)
! dp = wt_L*(p_b(i,j) - p_t(i,j)) + wt_R*(p_b(i,j+1) - p_t(i,j+1))
! p_ave = 0.5*(wt_L*(p_t(i,j)+p_b(i,j)) + wt_R*(p_t(i,j+1)+p_b(i,j+1)))
! eps = 0.5 * dp / (p0 + p_ave) ; eps2 = eps*eps
! intp(m) = (al0 + lambda / (p0 + p_ave) - spv_ref)*dp + 2.0*eps* &
! lambda * eps2 * (C1_3 + eps2*(0.2 + eps2*(C1_7 + C1_9*eps2)))
! enddo
! ! Use Boole's rule to integrate the values.
! inty_dza(i,j) = C1_90*(7.0*(intp(1)+intp(5)) + 32.0*(intp(2)+intp(4)) + &
! 12.0*intp(3))
! enddo ; enddo ; endif
! end subroutine int_spec_vol_dp_wright
end module MOM_EOS_Wright