There are a number of re-occurring themes in numerical analysis
of differential equations; among these are conservation, stability
and accuracy. In my talk, I will start from the time-symmetric nature
of the fundamental (Euler) equations of geophysical fluid dynamics
to motivate conservation of volume under symmetric time-stepping
methods. Symmetric time-stepping methods are either expensive
and/or need to be implemented with small time-steps (CFL condition).
We demonstrate how a regularized form of the Euler equations
can circumvent this bottleneck. Finally, we come back to volume
conservation and the accuracy of (probabilistic) ensemble predictions.
Accurate prediction of ensemble statistics is essential for data
assimilation which leads us to an integrated view on dynamics and
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