We consider direct and inverse problems of geophysical hydrodynamics, associated with prediction, posterior analysis, and variational assimilation of observational data. The focus is on numerical algorithms for solving problems with incomplete information (initial and boundary conditions). Along with the classical algorithms, an approach to setting and solving these problems developed in the works of G. I.Marchuk and his scientific school is presented. The approach is based on a combination of splitting methods and adjoint equations. This leads to the construction of flexible, hierarchically developed models of complex systems with a modular structure and efficient implementation. The main part of this approach is splitting a complex nonlinear system of equations for physical processes into a number of energetically balanced subsystems. Each individual subsystem can reuse splitting into subsystems of a simpler structure. The methodology is illustrated by solving the problems in hydrodynamics of the World Ocean and the Black Sea.

我们考虑与预测、后验分析和观测数据的变分同化相关的地球物理流体动力学的直接和逆问题。重点是解决信息不完整问题（初始和边界条件）的数值算法。除了经典算法外，还介绍了在 GIMarchuk 及其科学学校的作品中开发的设置和解决这些问题的方法。该方法基于分裂方法和伴随方程的组合。这导致构建具有模块化结构和高效实施的复杂系统的灵活、分层开发的模型。这种方法的主要部分是将物理过程的复杂非线性方程组分解为许多能量平衡的子系统。每个单独的子系统可以重用拆分为结构更简单的子系统。该方法通过解决世界海洋和黑海的流体动力学问题来说明。