The long-term behaviour of the barotropic vorticity equation (BVE), which is beneficial for understanding long-term climate variations, has been investigated by applying an explicit quadratic conserved finite difference integration scheme. The results suggest that an initial pattern predominated at the beginning of time evolves to an equilibrium state after a long transition period in which the conserved prerequisites, such as the integral kinetic energy, square vorticity, and vorticity, are broken by accumulated errors. This could cause the BVE to be indeterministic. Similar phenomena could reappear in integration cases starting at the same initial pattern with different small perturbations, initial patterns, time and grid intervals, and difference schemes. The eventual equilibrium states for the above cases are different from each other but spatially similar. They all possess a single-wave structure in the horizontal plane but with random phases. The results further imply that despite the indeterministic characteristics due to the nonlinearity, the long-term behaviour of the BVE is deterministic in a spatial structure, shedding light on long-term climate change studies.

通过应用显式二次守恒有限差分积分方案，研究了正压涡度方程（BVE）的长期行为，这有助于理解长期气候变化。结果表明，在很长的过渡期后，以时间开始为主的初始模式演变为平衡状态，在该过渡期中，诸如累积动能，平方涡旋和涡旋之类的守恒前提因累积误差而破裂。这可能会导致BVE不确定。类似的现象可能会出现在以相同的初始模式开始，具有不同的小扰动，初始模式，时间和网格间隔以及差分方案的积分情况下。上述情况的最终平衡状态彼此不同，但在空间上相似。它们在水平面上均具有单波结构，但具有随机相位。结果进一步暗示，尽管由于非线性而具有不确定性特征，但BVE的长期行为在空间结构中是确定性的，这为长期气候变化研究提供了启示。