<p>In this work we consider the surface quasi-geostrophic (SQG) system under location uncertainty (LU) and propose a Milstein-type scheme for these equations. The LU framework, first introduced in [1], is based on the decomposition of the Lagrangian velocity into two components: a large-scale smooth component and a small-scale stochastic one. This decomposition leads to a stochastic transport operator, and one can, in turn, derive the stochastic LU version of every classical fluid-dynamics system.<span>&#160;</span></p><p>&#160; &#160; SQG is a simple 2D oceanic model with one partial differential equation, which models the stochastic transport of the buoyancy, and an operator which relies the velocity and the buoyancy.</p><p><span>&#160; &#160; </span>For this kinds of equations, the Euler-Maruyama scheme converges with weak order 1 and strong order 0.5. Our aim is to develop higher order schemes in time: the first step is to consider Milstein scheme, which improves the strong convergence to the order 1. To do this, it is necessary to simulate or estimate the L&#233;vy area [2].</p><p><span>&#160; &#160; </span>We show with some numerical results how the Milstein scheme is able to capture some of the smaller structures of the dynamic even at a poor resolution.<span>&#160;</span></p><p><strong>References</strong></p><p>[1] E. M&#233;min. Fluid flow dynamics under location uncertainty. <em>Geophysical & Astrophysical Fluid Dynamics</em>, 108.2 (2014): 119-146.<span>&#160;</span></p><p>[2] J. Foster, T. Lyons and H. Oberhauser. An optimal polynomial approximation of Brownian motion. <em>SIAM Journal on Numerical Analysis</em> 58.3 (2020): 1393-1421.</p>