#include"cppdefs.h" !----------------------------------------------------------------------- !BOP ! ! !ROUTINE: The dynamic k-equation \label{sec:tkeeq} ! ! !INTERFACE: subroutine tkeeq(nlev,dt,u_taus,u_taub,z0s,z0b,h,NN,SS) ! !DESCRIPTION: ! The transport equation for the turbulent kinetic energy, $k$, ! follows immediately from the contraction of the Reynolds-stress ! tensor. In the case of a Boussinesq-fluid, this equation can ! be written as ! \begin{equation} ! \label{tkeA} ! \dot{k} ! = ! {\cal D}_k + P + G - \epsilon ! \comma ! \end{equation} ! where $\dot{k}$ denotes the material derivative of $k$. $P$ and $G$ are ! the production of $k$ by mean shear and buoyancy, respectively, and ! $\epsilon$ the rate of dissipation. ${\cal D}_k$ represents the sum of ! the viscous and turbulent transport terms. ! For horizontally homogeneous flows, the transport term ${\cal D}_k$ ! appearing in \eq{tkeA} is presently expressed by a simple ! gradient formulation, ! \begin{equation} ! \label{diffusionTKE} ! {\cal D}_k = \frstder{z} \left( \dfrac{\nu_t}{\sigma_k} \partder{k}{z} \right) ! \comma ! \end{equation} ! where $\sigma_k$ is the constant Schmidt-number for $k$. ! ! In horizontally homogeneous flows, the shear and the buoyancy ! production, $P$ and $G$, can be written as ! \begin{equation} ! \label{PandG} ! \begin{array}{rcl} ! P &=& - \mean{u'w'} \partder{U}{z} - \mean{v'w'} \partder{V}{z} \comma \\[3mm] ! G &=& \mean{w'b'} \comma ! \end{array} ! \end{equation} ! see \eq{PG}. Their computation is discussed in \sect{sec:production}. ! ! The rate of dissipation, $\epsilon$, can be either obtained directly ! from its parameterised transport equation as discussed in ! \sect{sec:dissipationeq}, or from any other model yielding ! an appropriate description of the dissipative length-scale, $l$. ! Then, $\epsilon$ follows from the well-known cascading relation ! of turbulence, ! \begin{equation} ! \label{epsilon} ! \epsilon = (c_\mu^0)^3 \frac{k^{\frac{3}{2}}}{l} ! \comma ! \end{equation} ! where $c_\mu^0$ is a constant of the model. ! ! !USES: use turbulence, only: P,B,num use turbulence, only: tke,tkeo,k_min,eps use turbulence, only: k_bc, k_ubc, k_lbc, ubc_type, lbc_type use turbulence, only: sig_k use util, only: Dirichlet,Neumann IMPLICIT NONE ! ! !INPUT PARAMETERS: ! number of vertical layers integer, intent(in) :: nlev ! time step (s) REALTYPE, intent(in) :: dt ! surface and bottom ! friction velocity (m/s) REALTYPE, intent(in) :: u_taus,u_taub ! surface and bottom ! roughness length (m) REALTYPE, intent(in) :: z0s,z0b ! layer thickness (m) REALTYPE, intent(in) :: h(0:nlev) ! square of shear and buoyancy ! frequency (1/s^2) REALTYPE, intent(in) :: NN(0:nlev),SS(0:nlev) ! ! !REVISION HISTORY: ! Original author(s): Lars Umlauf ! (re-write after first version of ! H. Burchard and K. Bolding) ! ! $Log: tkeeq.F90,v $ ! Revision 20.0 2013/12/14 00:13:53 fms ! Merged revision 1.1.2.1 onto trunk ! ! Revision 1.1.2.1 2012/05/15 16:00:54 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 1.9 2005-11-15 11:35:02 lars ! documentation finish for print ! ! Revision 1.8 2005/11/03 20:53:37 hb ! Patankar trick reverted to older versions for ! stabilising 3D computations ! ! Revision 1.7 2005/08/11 13:11:50 lars ! Added explicit loops for diffusivities for 3-D z-level support. ! Thanks to Vicente Fernandez. ! ! Revision 1.6 2005/06/27 13:44:07 kbk ! modified + removed traling blanks ! ! Revision 1.5 2003/03/28 09:20:35 kbk ! added new copyright to files ! ! Revision 1.4 2003/03/10 09:02:06 gotm ! Added new Generic Turbulence Model + improved documentation and cleaned up code ! ! !EOP !------------------------------------------------------------------------ ! ! !LOCAL VARIABLES: REALTYPE :: DiffKup,DiffKdw,pos_bc REALTYPE :: prod,buoyan,diss REALTYPE :: prod_pos,prod_neg,buoyan_pos,buoyan_neg REALTYPE :: cnpar=_ONE_ REALTYPE :: avh(0:nlev) REALTYPE :: Lsour(0:nlev),Qsour(0:nlev) integer :: i ! !------------------------------------------------------------------------ !BOC ! tkeo=tke do i=1,nlev-1 ! compute diffusivity avh(i) = num(i)/sig_k ! compute production terms in k-equation prod = P(i) buoyan = B(i) diss = eps(i) if (prod+buoyan.gt.0) then Qsour(i) = prod+buoyan Lsour(i) = -diss/tke(i) else Qsour(i) = prod Lsour(i) = -(diss-buoyan)/tke(i) end if end do ! position for upper BC if (k_ubc.eq.Neumann) then ! flux at center "nlev" pos_bc = 0.5*h(nlev) else ! value at face "nlev-1" pos_bc = h(nlev) end if ! obtain BC for upper boundary of type "ubc_type" DiffKup = k_bc(k_ubc,ubc_type,pos_bc,z0s,u_taus) ! position for lower BC if (k_lbc.eq.Neumann) then ! flux at center "1" pos_bc = 0.5*h(1) else ! value at face "1" pos_bc = h(1) end if ! obtain BC for lower boundary of type "lbc_type" DiffKdw = k_bc(k_lbc,lbc_type,pos_bc,z0b,u_taub) ! do diffusion step call diff_face(nlev,dt,cnpar,h,k_ubc,k_lbc, & DiffKup,DiffKdw,avh,Lsour,Qsour,tke) ! fill top and bottom value with something nice ! (only for output) tke(nlev) = k_bc(Dirichlet,ubc_type,z0s,z0s,u_taus) tke(0 ) = k_bc(Dirichlet,lbc_type,z0b,z0b,u_taub) ! clip at k_min do i=0,nlev tke(i) = max(tke(i),k_min) enddo return end subroutine tkeeq !EOC !----------------------------------------------------------------------- ! Copyright by the GOTM-team under the GNU Public License - www.gnu.org !-----------------------------------------------------------------------