  An optimisation scheme is developed to accurately represent the sub-grid
scale forcing of a high dimensional chaotic ocean system. Using a simple
parameterisation scheme, the velocity components of a 30km resolution shallow
water ocean model are optimised to have the same climatological mean and
variance as that of a less viscous 7.5km resolution model. The 5 day
lag-covariance is also optimised, leading to a more accurate estimate of the
high resolution response to forcing using the low resolution model.
  The system considered is an idealised barotropic double gyre that is chaotic
at both resolutions. Using the optimisation scheme, we find and apply the
constant in time, but spatially varying, forcing term that is equal to the time
integrated forcing of the sub-mesoscale eddies. A linear stochastic term,
independent of the large-scale flow, with no spatial correlation but a
spatially varying amplitude and time scale is used to represent the transient
eddies. The climatological mean, variance and 5 day lag-covariance of the
velocity from a single high resolution integration is used to provide an
optimisation target. No other high resolution statistics are required.
Additional programming effort, for example to build a tangent linear or adjoint
model, is not required either.
  The focus of this paper is on the optimisation scheme and the accuracy of the
optimised flow. The method can be applied in future investigations into the
physical processes that govern barotropic turbulence and it can perhaps be
applied to help understand and correct biases in the mean and variance of a
more realistic coarse or eddy-permitting ocean model. The method is
complementary to current parameterisations and can be applied at the same time
without modification.
