The stochastic variational approach for geophysical fluid dynamics was introduced by Holm (Proc Roy Soc A, 2015) as a framework for deriving stochastic parameterisations for unresolved scales. This paper applies the variational stochastic parameterisation in a two-layer quasi-geostrophic model for a $ \beta $-plane channel flow configuration. We present a new method for estimating the stochastic forcing (used in the parameterisation) to approximate unresolved components using data from the high resolution deterministic simulation, and describe a procedure for computing physically-consistent initial conditions for the stochastic model. We also quantify uncertainty of coarse grid simulations relative to the fine grid ones in homogeneous (teamed with small-scale vortices) and heterogeneous (featuring horizontally elongated large-scale jets) flows, and analyse how the spread of stochastic solutions depends on different parameters of the model. The parameterisation is tested by comparing it with the true eddy-resolving solution that has reached some statistical equilibrium and the deterministic solution modelled on a low-resolution grid. The results show that the proposed parameterisation significantly depends on the resolution of the stochastic model and gives good ensemble performance for both homogeneous and heterogeneous flows, and the parameterisation lays solid foundations for data assimilation.
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Figure 2.
The series of snapshots shows the high-resolution solution
Figure 3.
The same as in Figure 2, but for
Figure 4.
The series of snapshots shows the dependence of the solution on the resolution for the low drag
Figure 5.
The series of snapshots in the figure shows the high-resolution solution
Figure 6.
Shown is a typical dependence of the area of the stochastic cloud
Figure 9.
Shown is the dependence of
Figure 10.
The same as in Figure 9, but for
Figure 11.
The same as in Figure 9, but for
Figure 12.
The same as in Figure 11, but for
Figure 13.
Evolution of
Figure 14.
Evolution of the root mean square error
Figure 15. The same as in Figure 14, but for an unbiased ensemble. The results for the homogeneous flow look very similar
Figure 16.
Evolution of the root mean square error,
Figure 17.
Shown is the dependence of
Figure 18.
The same as in Figure 17, but for
Figure 19.
Evolution of
Figure 20.
Typical rank histograms for heterogeneous flow velocity
Figure 21.
The series of snapshots shows the true deterministic solution
Figure 22.
The series of snapshots shows the true deterministic solution
Figure 23.
The same as in Figure 22, but for
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