Boundary conditons for the k-equation (Mellor-Yamada style)


   REALTYPE function  q2over2_bc(bc,type,zi,z0,u_tau)

Computes prescribed and flux boundary conditions for the transport equation (156). The formal parameter bc determines whether Dirchlet or Neumann-type boundary conditions are computed. Depending on the physical properties of the boundary-layer, the parameter type relates either to a visous, a logarithmic, or an injection-type boundary-layer. In the latter case, the flux of TKE caused by breaking surface waves has to be specified. Presently, there is only one possibility to do so implemented in GOTM. It is described in section 4.34. All parameters that determine the boundary layer have to be set in gotmturb.nml.

Note that in this section, for brevity, $ z$ denotes the distance from the wall (or the surface), and not the standard coordinate of the same name used in GOTM.

Viscous boundary-layers

This type is not implemented yet in GOTM.

Logarithmic boundary-layers

The Dirichlet (prescribed) boundary condition follows from (103) and (159) as

$\displaystyle q^2/2= \dfrac{u_*^2 B_1^\frac{2}{3}}{2} \quad .$ (131)

The Neumann (flux) boundary condition can be derived from the constancy of $ q^2/2$ in the logarithmic region. This fact can be written as

$\displaystyle F_q = - S_q q l \dfrac{\partial {k}}{\partial {z}} = 0 \quad .$ (132)

Shear-free boundary-layers with injection of TKE

The Dirichlet (prescribed) boundary condition follows simply from the power-law in (110),

$\displaystyle \frac{q^2}{2} = k = K (z+z_0)^\alpha \quad .$ (133)

The Neumann (flux) boundary condition can be written as

$\displaystyle F_q = - S_q q l \dfrac{\partial {k}}{\partial {z}} = - \sqrt{2} S_q K^\frac{3}{2} \alpha L (z+ z_0)^{\frac{3}{2} \alpha} \; , \quad$ (134)

which follows immediately from (110). The parameter $ K$ can be determined from an evaluation of (134) at $ z=0$. The result is

$\displaystyle K = \left( - \dfrac{F_q}{\sqrt{2} S_q \alpha L} \right)^\frac{2}{3} \dfrac{1}{z_0^\alpha} \; , \quad$ (135)

where the specification of the flux $ F_q$ and the value of $ z_0$ have to be determined from a suitable model of the wave breaking process.


  integer, intent(in)                  :: bc,type
  REALTYPE, intent(in)                 :: zi,z0,u_tau
   Original author(s): Lars Umlauf

Karsten Bolding 2012-12-28