#include"cppdefs.h" !----------------------------------------------------------------------- !BOP ! ! !ROUTINE: The local, weak-equilibrium stability functions \label{sec:cmueC} ! ! !INTERFACE: subroutine cmue_c(nlev) ! ! !DESCRIPTION: ! ! This subroutine updates the explicit solution of ! \eq{bijVertical} and \eq{giVertical} with shape indicated ! by \eq{b13}. In addition to the simplifications discussed ! in \sect{sec:cmueB}, $P_b = \epsilon_b$ is assumed ! in \eq{giVertical} to eliminate the dependency on $\overline{T}$ ! according to \eq{Tequilibrium}. ! As discussed in \sect{sec:EASM}, this implies that the last of \eq{giVertical} ! is replaced by \eq{giVerticalEq}. Thus, the $\Gamma$-term in \eq{b13} drops ! out, and the solution is characterized by $c_\mu$ and $c'_\mu$ only. ! ! As a consequence, the numerators and the denominator appearing in \eq{cm} ! are of somewhat different form compared to the result in \sect{sec:cmueA}. ! They can be written as ! \begin{equation} ! \label{vdngEq} ! \begin{array}{rcl} ! D &=& d_0 ! + d_1 \overline{N}^2 + d_2 \overline{S}^2 ! + d_3 \overline{N}^2 \overline{S}^2 ! + d_4 \overline{N}^4 + d_5 \overline{S}^4 \comma \\[3mm] ! N_n &=& n_0 ! + n_1 \overline{N}^2 + n_2 \overline{S}^2 \comma \\[3mm] ! N_b &=& n_{b0} ! + n_{b1} \overline{N}^2 + n_{b2} \overline{S}^2 ! \point ! \end{array} ! \end{equation} ! ! The coefficients of $D$ are given by ! \begin{equation} ! \label{vdiEq} ! \begin{array}{rcl} ! d_0 &=& 36 {\cal N}^3 {\cal N}_b^2 \comma \\[3mm] ! d_1 &=& 84 a_5 a_{b3} {\cal N}^2 {\cal N}_b ! + 36 a_{b5} {\cal N}^3 {\cal N}_b \comma \\[3mm] ! d_2 &=& 9 (a_{b2}^2 - a_{b1}^2) {\cal N}^3 ! + 12 ( 3 a_3^2 - a_2^2) {\cal N} {\cal N}_b^2 \comma \\[3mm] ! d_3 &=& 12 (a_2 a_{b1} - 3 a_3 a_{b2} ) a_5 a_{b3} {\cal N} ! + 12 ( a_3^2 - a_2^2) a_5 a_{b3} {\cal N}_b \\[3mm] ! &+& 12 ( 3 a_3^2 - a_2^2) a_{b5} {\cal N} {\cal N}_b \comma \\[3mm] ! d_4 &=& 48 a_5^2 a_{b3}^2 {\cal N} + 36 a_5 a_{b3} a_{b5} {\cal N}^2 \comma \\[3mm] ! d_5 &=& 3 ( 3 a_3^2 - a_2^2)(a_{b2}^2 - a_{b1}^2) {\cal N} ! \comma ! \end{array} ! \end{equation} ! and the coefficients of the numerators are ! \begin{equation} ! \label{vniEq} ! \begin{array}{rcl} ! n_0 &=& 36 a_1 {\cal N}^2 {\cal N}_b^2 \comma \\[3mm] ! n_1 &=&-12 a_5 a_{b3} (a_{b1}+a_{b2}) {\cal N}^2 ! - 8 a_5 a_{b3} (-6 a_1+a_2+3 a_3) {\cal N} {\cal N}_b \\[3mm] ! &+& 36 a_1 a_{b5} {\cal N}^2 {\cal N}_b \comma \\[3mm] ! n_2 &=& 9 a_1 (a_{b2}^2 - a_{b1}^2){\cal N}^2 ! \end{array} ! \end{equation} ! and ! \begin{equation} ! \label{vnbiEq} ! \begin{array}{rcl} ! n_{b0} &=& 12 a_{b3} {\cal N}^3 {\cal N}_b \comma \\[3mm] ! n_{b1} &=& 12 a_5 a_{b3}^2 {\cal N}^2 \comma \\[3mm] ! n_{b2} &=& 9 a_1 a_{b3} (a_{b1}-a_{b2}) {\cal N}^2 ! + a_{b3} (6 a_1 (a_2-3 a_3) ! - 4 (a_2^2-3 a_3^2) ) {\cal N} {\cal N}_b \comma ! \end{array} ! \end{equation} ! ! These polynomials correspond to a slightly generalized form of the solution ! suggested by \cite{Canutoetal2001a} and \cite{Chengetal2002}. For cases with ! unstable stratification, the same clipping conditions on $\alpha_N$ is applied ! as described in \sect{sec:cmueD}. For the cases of extreme shear, the ! limiter described in the context of \eq{simpleShearCondition} is active. ! ! !USES: use turbulence, only: an,as,at use turbulence, only: cmue1,cmue2 use turbulence, only: cm0 use turbulence, only: cc1 use turbulence, only: ct1,ctt use turbulence, only: a1,a2,a3,a4,a5 use turbulence, only: at1,at2,at3,at4,at5 IMPLICIT NONE ! ! !INPUT PARAMETERS: ! number of vertical layers integer, intent(in) :: nlev ! !DEFINED PARAMETERS: REALTYPE, parameter :: asLimitFact=1.0d0 REALTYPE, parameter :: anLimitFact=0.5d0 ! ! !REVISION HISTORY: ! Original author(s): Lars Umlauf ! ! $Log: cmue_c.F90,v $ ! Revision 20.0 2013/12/14 00:13:30 fms ! Merged revision 1.1.2.1 onto trunk ! ! Revision 1.1.2.1 2012/05/15 16:00:53 smg ! initial cvs ci for these modules to mom5. ! AUTHOR:Griffies ! REVIEWERS: ! TEST STATUS: ! CHANGES PUBLIC INTERFACES? ! CHANGES ANSWERS? ! ! Revision 1.1.2.1.390.1 2012/04/23 20:30:29 smg ! updated to the gotm-2012.03.09 CVS tag. ! AUTHOR:Martin Schmidt ! REVIEWERS:Griffies ! TEST STATUS: ! CHANGES PUBLIC INTERFACES? ! CHANGES ANSWERS? ! ! Revision 1.1 2005-06-27 10:54:33 kbk ! new files needed ! ! !EOP !----------------------------------------------------------------------- ! !LOCAL VARIABLES: ! integer :: i REALTYPE :: N,Nt REALTYPE :: d0,d1,d2,d3,d4,d5 REALTYPE :: n0,n1,n2,nt0,nt1,nt2 REALTYPE :: dCm,nCm,nCmp,cm3_inv REALTYPE :: tmp0,tmp1,tmp2 REALTYPE :: asMax,asMaxNum,asMaxDen REALTYPE :: anMin,anMinNum,anMinDen !----------------------------------------------------------------------- !BOC N = 0.5*cc1 Nt = ct1 d0 = 36.* N**3. * Nt**2. d1 = 84.*a5*at3 * N**2. * Nt + 36.*at5 * N**3. * Nt d2 = 9.*(at2**2.-at1**2.) * N**3. - 12.*(a2**2.-3.*a3**2.) * N * Nt**2. d3 = 12.*a5*at3*(a2*at1-3.*a3*at2) * N + 12.*a5*at3*(a3**2.-a2**2.) * Nt & + 12.*at5*(3.*a3**2.-a2**2.) * N * Nt d4 = 48.*a5**2.*at3**2. * N + 36.*a5*at3*at5 * N**2. d5 = 3.*(a2**2.-3.*a3**2.)*(at1**2.-at2**2.) * N n0 = 36.*a1 * N**2. * Nt**2. n1 = - 12.*a5*at3*(at1+at2) * N**2. + 8.*a5*at3*(6.*a1-a2-3.*a3) * N * Nt & + 36.*a1*at5 * N**2. * Nt n2 = 9.*a1*(at2**2.-at1**2.) * N**2. nt0 = 12.*at3 * N**3. * Nt nt1 = 12.*a5*at3**2. * N**2. nt2 = 9.*a1*at3*(at1-at2) * N**2. + ( 6.*a1*(a2-3.*a3) & - 4.*(a2**2.-3.*a3**2.) )*at3 * N * Nt cm3_inv = 1./cm0**3. ! mininum value of "an" to insure that "as" > 0 in equilibrium anMinNum = -(d1 + nt0) + sqrt((d1+nt0)**2. - 4.*d0*(d4+nt1)) anMinDen = 2.*(d4+nt1) anMin = anMinNum / anMinDen do i=0,nlev ! clip an at minimum value an(i) = max(an(i),anLimitFact*anMin) ! compute maximum as for shear instability asMaxNum = d0*n0 + (d0*n1+d1*n0)*an(i) + (d1*n1+d4*n0)*an(i)*an(i) + d4*n1*an(i)*an(i)*an(i) asMaxDen = d2*n0 + (d2*n1+d3*n0)*an(i) + d3*n1*an(i)*an(i) asMax = asMaxNum / asMaxDen ! clip as at miximum value as(i) = min(as(i),asLimitFact*asMax) dCm = d0 + d1*an(i) + d2*as(i) + d3*an(i)*as(i) + d4*an(i)*an(i) + d5*as(i)*as(i) nCm = n0 + n1*an(i) + n2*as(i) nCmp = nt0 + nt1*an(i) + nt2*as(i) cmue1(i) = cm3_inv*nCm /dCm cmue2(i) = cm3_inv*nCmp/dCm end do return end subroutine cmue_c !----------------------------------------------------------------------- !EOC !----------------------------------------------------------------------- ! Copyright by the GOTM-team under the GNU Public License - www.gnu.org !-----------------------------------------------------------------------