In this paper, C1-conforming element methods are analyzed for the stream function formulation of a single layer non-stationary quasi-geostrophic equation in the ocean circulation model. In its first part, some new regularity results are derived, which show exponential decay property when the wind shear stress is zero or exponentially decaying. Moreover, when the wind shear stress is independent of time, the existence of an attractor is established. In its second part, finite element methods are applied in the spatial direction and for the resulting semi-discrete scheme, the exponential decay property, and the existence of a discrete attractor are proved. By introducing an intermediate solution of a discrete linearized problem, optimal error estimates are derived. Based on backward-Euler method, a completely discrete scheme is obtained and uniform in time a priori estimates are established. Moreover, the existence of a discrete solution is proved by appealing to a variant of the Brouwer fixed point theorem and then, optimal error estimate is derived. Finally, several computational experiments with benchmark problems are conducted to confirm our theoretical findings.

翻译：本文针对海洋环流模型中的单层非定常准地转方程的流函数形式，分析了C1-协调元方法。在第一部分，推导了一些新的正则性结果，这些结果表明当风切应力为零或呈指数衰减时，解具有指数衰减性质。此外，当风切应力与时间无关时，建立了吸引子的存在性。在第二部分，将有限元方法应用于空间方向，并针对由此得到的半离散格式，证明了指数衰减性质以及离散吸引子的存在性。通过引入离散线性化问题的中间解，推导出了最优误差估计。基于向后欧拉方法，得到了一个全离散格式，并建立了与时间无关的一致先验估计。此外，通过应用布劳威尔不动点定理的一个变体，证明了离散解的存在性，进而推导出最优误差估计。最后，通过多个基准问题的计算实验验证了我们的理论结果。