  We present a numerical investigation of stochastic transport in ideal fluids.
According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principles
of transformation theory and multi-time homogenisation, respectively, imply a
physically meaningful, data-driven approach for decomposing the fluid transport
velocity into its drift and stochastic parts, for a certain class of fluid
flows. In the current paper, we develop new methodology to implement this
velocity decomposition and then numerically integrate the resulting stochastic
partial differential equation using a finite element discretisation for
incompressible 2D Euler fluid flows. The new methodology tested here is found
to be suitable for coarse graining in this case. Specifically, we perform
uncertainty quantification tests of the velocity decomposition of Cotter et al.
(2017), by comparing ensembles of coarse-grid realisations of solutions of the
resulting stochastic partial differential equation with the "true solutions" of
the deterministic fluid partial differential equation, computed on a refined
grid. The time discretization used for approximating the solution of the
stochastic partial differential equation is shown to be consistent. We include
comprehensive numerical tests that confirm the non-Gaussianity of the stream
function, velocity and vorticity fields in the case of incompressible 2D Euler
fluid flows.
