The dynamic psi-equation
This model has been formulated by Umlauf and Burchard (2003),
who introduced a `generic' variable,
where is the turbulent kinetic energy computed from (152) and
is the dissipative length-scale defined in (155).
For appropriate choices of the exponents , , and , the variable
can be directly identified with the classic length-scale determining
variables like the rate of dissipation, , or the product
used by Mellor and Yamada (1982) (see section 4.14
and section 4.15).
Some examples are compiled in table 8.
Exponents , , defined in (167), and
their relation to the variable of the second equation in some well-known
The transport equation for can written as
denotes the material derivative of ,
see Umlauf and Burchard (2003).
The production terms and follow from (154).
represents the sum of the viscous and turbulent
transport terms. The rate of dissipation can computed by solving
(167) for and inserting the result into (155).
For horizontally homogeneous flows, the transport terms
appearing in (168) are expressed by a simple
For appropriate choices of the parameters, most of the classic transport
equations can be directly recovered from the generic equation (168).
An example is the transport equation for the inverse turbulent time scale,
, which has been formulated by Wilcox (1988)
and extended to buoyancy affected flows by Umlauf et al. (2003). The precise
definition of follows from table 8, and its transport
equation can be written as
which is clearly a special case of (168). Model constants for this
and other traditional models are given in table 9.
Model constants of some standard models,
converted to the notation used here. The Schmidt-numbers for the model of
Mellor and Yamada (1982) are valid only in the logarithmic boundary-layer,
because the diffusion models (157) and (164)
are slightly different from (153) and (169).
There is no indication that one class of diffusion models is superior.
Apart from having to code only one equation to recover all of the
traditional models, the main advantage of the generic equation is its
flexibility. After choosing meaningful values for physically relevant
parameters like the von Kármán constant, , the temporal
decay rate for homogeneous turbulence, , some parameters related to
breaking surface waves, etc, a two-equation model can be generated,
which has exactly the required properties. This is discussed in
great detail in Umlauf and Burchard (2003). All algorithms have been
implemented in GOTM and are described in section 4.7.3.
use turbulence, only: P,B,num
use turbulence, only: tke,tkeo,k_min,eps,eps_min,L
use turbulence, only: cpsi1,cpsi2,cpsi3plus,cpsi3minus,sig_psi
use turbulence, only: gen_m,gen_n,gen_p
use turbulence, only: cm0,cde,galp,length_lim
use turbulence, only: psi_bc, psi_ubc, psi_lbc, ubc_type, lbc_type
use util, only: Dirichlet,Neumann
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
REALTYPE, intent(in) :: NN(0:nlev),SS(0:nlev)
Original author(s): Lars Umlauf and Hans Burchard