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            <funder>The authors also acknowledge the support of the ERC EU project 856408-STUOD and the Inria challenge SURF (Sea Uncertainty Representation  and Forecast).</funder>
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              <p>To address the complexity of numerically simulating turbulent flows at large scale, stochastic modelling approaches, particularly those incorporating transport noise, have gained a lot of attraction. Building on a Newtonian framework derived from stochastic transport, the Location Uncertainty (LU) model introduces stochastic contributions into fluid dynamics while keeping naturally energy conservation through a fluctuationdissipation equilibrium. From a numerical point of view, recent studies have highlighted the benefits of simplifying high-order temporal schemes for stochastic simulations by omitting certain terms, resulting in discrete schemes that closely mirror deterministic models. This work focuses on developing efficient discrete schemes for a stochastic shallow water model, particularly emphasizing "second-order" methods. These methods involve iterated double advection of noise, ensuring an implicit implementation of noise-associated diffusion, thus achieving a balanced energy dissipation at the discrete level. The proposed schemes extend entropy-conserving methods, providing a robust framework for modeling under location uncertainty. These schemes have been numerically assessed for both idealized and realistic test cases over ensembles of realizations. In the realistic case, the stochastic simulations have been compared against observational data, while in the idealized case they have been evaluated against ground-truth fields obtained from a higher-resolution deterministic simulation.</p>
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