The dynamic epsilon-equation (Source File: dissipationeq.F90)

INTERFACE:

    subroutine dissipationeq(nlev,dt,u_taus,u_taub,z0s,z0b,h,NN,SS)

DESCRIPTION:

The $k$-$\epsilon$ model in its form suggested by Rodi (1987) has been implemented in GOTM. In this model, the rate of dissipation is balanced according to

$$\dot{\epsilon} = {\cal D}_\epsilon + \frac{\epsilon}{k} ( c_{\epsilon 1} P + c_{\epsilon 3} G - c_{\epsilon 2} \epsilon ) \; , \quad$$ (163)

where $\dot{\epsilon}$ denotes the material derivative of $\epsilon$. The production terms $P$ and $G$ follow from (152) and ${\cal D}_\epsilon$ represents the sum of the viscous and turbulent transport terms.

For horizontally homogeneous flows, the transport term ${\cal D}_\epsilon$ appearing in (163) is presently expressed by a simple gradient formulation,

$${\cal D}_\epsilon = \dfrac{\partial}{\partial {z}} \left( \dfrac{... ...}{\sigma_\epsilon} \dfrac{\partial {\epsilon}}{\partial {z}} \right) \; , \quad$$ (164)

where $\sigma_\epsilon$ is the constant Schmidt-number for $\epsilon$.

It should be pointed out that not all authors retain the buoyancy term in (163), see e.g. Gibson and Launder (1976). Similar to the model of Mellor and Yamada (1982), Craft et al. (1996) set $c_{\epsilon 1}=c_{\epsilon 3}$. However, in both cases, the $k$-$\epsilon$ model cannot predict a proper state of full equilibrium in stratified flows at a predefined value of the Richardson number (see Umlauf et al. (2003) and discussion around (114)). Model constants are summarised in table 7.

Table 7: Constants appearing in (163) and (153).

	$c_\mu^0$	$\sigma_k$	$\sigma_\epsilon$	$c_{\epsilon 1}$	$c_{\epsilon 2}$
Rodi (1987)	$0.5577$	$1.0$	$1.3$	$1.44$	$1.92$

At the end of this routine the length-scale can be constrained according to a suggestion of Galperin et al. (1988). This feature is optional and can be activated by setting length_lim = .true. in gotmturb.nml.

USES:

    use turbulence, only: P,B,num
    use turbulence, only: tke,tkeo,k_min,eps,eps_min,L
    use turbulence, only: ce1,ce2,ce3plus,ce3minus
    use turbulence, only: cm0,cde,galp,length_lim
    use turbulence, only: epsilon_bc, psi_ubc, psi_lbc, ubc_type, lbc_type
    use turbulence, only: sig_e,sig_e0,sig_peps
    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: dissipationeq.F90,v $
    Revision 1.10  2007-01-06 11:49:15  kbk
    namelist file extension changed .inp --> .nml
    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 13:43:42  lars
    double definitions removed - to conform with DEC compiler
    Revision 1.3  2003/03/10 09:02:04  gotm
    Added new Generic Turbulence Model +
    improved documentation and cleaned up code