1. Quantification of spurious dissipation and mixing in ocean models:
discrete variance decay
in a finite-volume framework
Hans Burchard1, Carsten Eden2, Ulf Gräwe1,
Knut Klingbeil1, Mahdi Mohammadi-Aragh1
and Nils Brüggemann2
1. Leibniz Institute for Baltic Sea Research Warnemünde, Germany
2. ClimaCampus, University of Hamburg, Germany
Numerical variance decay of tracers: numerical mixing.
Numerical variance decay of velocity: numerical dissipation.

2. Method I (Burchard & Rennau, 2008) in a nutshell
Generalisation by Burchard & Rennau (2008)
for any advection scheme:
Numerical mixing is the advected tracer square
minus the square of the advected tracer, divided by Dt.
First-order upstream (FOU) for s:
1D advection equation for S:
1D advection equation for s2:
FOU for s is equivalent to FOU for s² with variance decay :
numerical diffusivity
Salinity gradient squared
Morales Maqueda & Holloway (2006)

3. Method II: Klingbeil et al. (under review), also in a nutshell
Reconstruct
decomposition
Evolve
Starting from the discrete advection eq. and the initial state, advection in a FV framework can be described by several steps.
(X) In a first step the advection schemes approximate an interfacial value.
This interfacial value is associated with a subvolume that will be advected out to the neighbouring FV cell.
(X) The subvolume V* that remains in the FV cell, is obtained by a conservative decomposition of the original FV cell.
(X) In the next step the subvolumes are advected.
(X) In the last step the subvolume that remained in FV-cell and the subvolume advected into the FV-cell are conservatively recombined into a new FV cell.
The existence of subvolumes now offers the math. sound definition of variance inside a single FV-cell.
The initial and final FV-cells do not contain any subvolumes and thus no variance.
However, after the decomposition and before the recombination the variance can be calculated based on the corresponding 2nd moments. (X)
(X) The net variance decay can now be quantified by considering the variance gain during the decomposition and the variance loss during the recombination.
(X) And the so obtained expression for the DVD rate is equivalent to the diagnostic equation, if the discrete advective fluxes of 2nd moment therein are given by the K14 method.
In addition the this sound definition of K14, it will now be demonstrated that K14 also gives more reliable results. 1
Average
recombination
adapted from Morales Maqueda and Holloway (2006)

4. Comparing methods I & II in 1D
Variance loss due
to recombination
Variance gain due
to decomposition
Klingbeil et al. (under review)

5. Salinity mixing analysis in Western Baltic Sea (adaptive coordinates)
Klingbeil et al. (under review)

6. Meso-scale dynamics and stratification in Eady channel
SST zonally averaged q
Restratification occurs due to extraction of kinetic energy from
geostrophically balanced flow to eddy kinetic energy (due to momentum advection),
a process which critically depends on numerical mixing and dissipation.
Mohammadi-Aragh et al. (in preparation)

7. Meso-scale dynamics and stratification in Eady channel
stratification numerical dissipation background potential energy
Mohammadi-Aragh et al. (in preparation)

8. Take home message
Numerical mixing & dissipation are specifically critical in regimes
with low physical mixing & dissipation and strong eddy dynamics.
Accurate methods for the local quantification of numerical mixing
and dissipation have been introduced.
For realistic ocean modelling, good parameterisations of physical
mixing and dissipation need to be combined with accurate advection
schemes and an optimal choice for the numerical grid (isopcynal or
adaptive coordinates help).
To derive new meso-scale or sub-meso-scale mixing parameterisations
using numerical experiments, reference experiments with high numerical
accuracy need to be employed.
All simulations carried out with the
General Estuarine Transport Model (GETM,