The dynamic k-equation

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

$$\dot{k} = {\cal D}_k + P + G - \epsilon \; , \quad$$ (150)

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 (150) is presently expressed by a simple gradient formulation,

$${\cal D}_k = \dfrac{\partial}{\partial {z}} \left( \dfrac{\nu_t}{\sigma_k} \dfrac{\partial {k}}{\partial {z}} \right) \; , \quad$$ (151)

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

$$\begin{array}{rcl} P &=& - {\langle u'w' \rangle} \dfrac{\par... ...ad \\ [3mm] G &=& {\langle w'b' \rangle} \; , \quad \end{array}$$ (152)

see (48). Their computation is discussed in section 4.8.

The rate of dissipation, $\epsilon$, can be either obtained directly from its parameterised transport equation as discussed in section 4.15, 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,

$$\epsilon = (c_\mu^0)^3 \frac{k^{\frac{3}{2}}}{l} \; , \quad$$ (153)

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 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