## 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, , follows immediately from the contraction of the Reynolds-stress tensor. In the case of a Boussinesq-fluid, this equation can be written as

 (152)

where denotes the material derivative of . and are the production of by mean shear and buoyancy, respectively, and the rate of dissipation. represents the sum of the viscous and turbulent transport terms. For horizontally homogeneous flows, the transport term appearing in (152) is presently expressed by a simple gradient formulation,

 (153)

where is the constant Schmidt-number for .

In horizontally homogeneous flows, the shear and the buoyancy production, and , can be written as

 (154)

see (50). Their computation is discussed in section 4.8.

The rate of dissipation, , 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, . Then, follows from the well-known cascading relation of turbulence,

 (155)

where 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)


Karsten Bolding 2012-12-28