  A framework for exponential time discretization of the multilayer rotating
shallow water equations is developed in combination with a mimetic
discretization in space. The method is based on a combination of existing
exponential time differencing (ETD) methods and a careful choice of approximate
Jacobians. The discrete Hamiltonian structure and conservation properties of
the model are taken into account, in order to ensure stability of the method
for large time steps and simulation horizons. In the case of many layers,
further efficiency can be gained by a layer reduction which is based on the
vertical structure of fast and slow modes. Numerical experiments on the example
of a mid-latitude regional ocean model confirm long term stability for time
steps increased by an order of magnitude over the explicit CFL, while
maintaining accuracy for key statistical quantities.
