Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy, but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2):495, 2013], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. Marsden and M. Desbrun. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena, 240(6):443-458, 2011] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.

翻译：天气和气候的数值模型关键取决于超双曲线保护法系统综合体的长期稳定性。 虽然这种稳定性通常是从(物理或数字)散射的(物理或数字)条件中获得的,但这种模拟的物理可靠性也取决于适当保存系统节能的数量,例如能源。为了解决这一明显的悖论,我们为浅水方程式开发了一个变式综合体,以节约能源,但会消散潜在的营养。我们的方法遵循了连续选择性的衰变框架[F. Gay-Balmaz和D. Holm. Casimir 隐视液的选择性衰变。非线性,26(2):495,2013年],这种模拟还取决于适当保存系统节能的节能数量。我们利用这个变式的离散法结合了[D. Pavlov, P. Mullen, Y. Tong, E. Kansoden和M. Desbrun. Desbrun. 将可抑制性液体的离散化结构化。Phyica-climal-dal-dal-dal-dal-dal-dal-dalbal-dal-dal-dal-dal-dal-dal-dal-dal-dal-dal-dal-dal-cal-dal-dal-dal-dal-dal-d-d-d-d-dalgalgal-d-d-d-dal-dal-dal-dal-d-d-d-d-dal-d-dal-d-d-d-d-d-d-d-dal-d-d-d-d-d-d-d-dal-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-dal-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-d-