Turbulence under breaking surface waves

In this scenario, it is demonstrated how the effect of breaking surface waves is parameterised in one- and two-equation models. This is usually done by injecting turbulent kinetic energy (TKE) at the surface, see Craig and Banner (1994) and Craig (1996). The rate of TKE injected is proportional to the surface friction velocity cubed, as defined in (209). Injection of TKE at the surface leads to a thin surface boundary layer, in which the vertical transport of TKE and its dissipation approximately balance. This layer is sometimes called the transport layer. Even though there can be considerable shear in this layer, shear-production of turbulence is negligible by definition (also see section 4.7.4).

Different types of models are available in GOTM for the wave-breaking
scenario. The key change in `gotmturb.nml` for runs with TKE
injection is to set `ubc_type = 2`, telling GOTM to set the type
of the upper boundary to TKE injection. The decay rates of the TKE and
the dissipation rate in the wave-affected layer are then an natural
outcome of the model. Note that with the KPP model, this scenario
cannot be run.

- For the one-equation models, as discussed in
Craig and Banner (1994), a linearly increasing macro length scale, ,
is postulated with a slope of
. This is analogous to the
law of the wall, even though there is no physical evidence for the
assumption that the length-scale under breaking waves behaves
identically as in wall-bounded shear-flows. As shown by
Craig and Banner (1994), an analytical solution for the one-equation
model can be derived, but only inside the transport layer, according
to which the TKE (and all other turbulence quantities) follows a
power-law (see discussion in section 4.7.3 and
section 4.7.4).
If you want to simulate wave breaking with a model of this type, simply copy all files from

`prescribed_nml/`to the current directoy, and run GOTM. A dynamic equation for is used, but the length scale is fixed, and prescribed by a triangular shape with slope (`length_scale_method = 2`in`gotmturb.nml`, see section 4.7.32). - For two-equation models, the slope of the length scale in the
transport layer is not simply prescribed and generally not equal to
. Umlauf et al. (2003) generalized the solution of
Craig and Banner (1994) and derived analytical solutions for the
non-linear system of equations describing the behaviour of
two-equation models for injection of TKE at the surface. They showed
that the TKE follows a power-law and that the length scale increases
linearly, however, with a slope
. They also compared
the spatial decay of turbulence in grid stirring experiments (thought
as an analogy to wave-breaking) to the results of several
two-equation models.
A numerical solution of the - model can be obtained by copying the files in

`kepspilon_nml`to the current directory, and insuring that`compute_kappa = .true.`and`sig_peps = .false.`in`gotmturb.nml`. Because the spatial decay rate of the TKE is very large for this model, the wave-affected layer is very small, and of the order of only a few tens of centimeters for this scenario. As discussed by Umlauf et al. (2003), this disadvantage can be overcome by using the - model with parameters given in`gotmturb.nml`in the directory`komega_nml/`. The decay rates of this model nicely correspond to those measured in the laboratory grid strirring experiments. The Mellor-Yamada model has also been investigated by Umlauf et al. (2003), but for this model, again, decay was shown to be too strong. In addition, the decay rate depends in an unphysical way on the wall-function required in this model. - As an alternative to the standard - model,
Burchard (2001a) suggested to make the turbulent Schmidt number
for the -equation, (163), a function of the
production-to-dissipation ratio,
. As shown in detail in
this paper, the variable Schmidt number can be used to ``force'' the
- model to compute for the slope of the length
scale, even under breaking waves. Then, obviously, the solution of
the - model corresponds to the solution of the simpler
one-equation model investigated by Craig and Banner (1994). Note again,
however, that there is no physical evidence for
in the wave-affected boundary layer.
If you want to simulate wave breaking with this model, simply copy the files from

`kepspilon_nml/`to the current directory, and make sure that you set`compute_kappa = .false.`and`sig_peps = .true.`in`gotmturb.nml`. Results are quite similar to those with the prescribed length scale. - Umlauf and Burchard (2003) analysed the properties of a whole
class of two-equation models for the case of TKE injection at the
surface. They suggested a `generic' model which could satisfy the
power-laws under breaking waves for any desired decay rate,
, and length scale slope, . This model is activated with
the input files from
`generic_nml/`. Users can select any reasonable values for and (and many others parameters like and ), and GOTM will automatically generate a two-equation model with exactly the desired properties. Parameters are computed according to the formulae described in section 4.7.3.

In all cases a surface-stress of
Nm was
applied. After a runtime of 2 days, a steady-state with a constant
stress over the whole water column of 20 m depth is reached. The wave
affected layer can be found in the uppermost meter or so, and because
of the strong gradients in this region we used a refined grid close to
the surface. The parameters for such a `zoomed grid' can be set in the
input file `gotmmean.nml` according to the decription in
section 3.2.4. If you want to compare the computed profiles
with the analytical solutions in (108), you'll need a
specification of the parameter . This parameter is computed in `k_bc()` to be found in `turbulence.F90`, where you can add a few
FORTRAN lines to write it out.

Karsten Bolding 2012-01-24