界面PDE约束的控制问题是由数值天气预报驱动的重要的数学科学问题，是计算地球流体力学的核心问题之一。海气界面的蒸发率和蒸发方式会影响海陆面的能量平衡，从而会驱动大量的气候变化过程。由于界面处环境条件复杂，界面蒸发率的准确预测一直是具有挑战性的难题。本项目在数学上可以归结为界面PDE约束的最优控制问题，重点以准地转方程组为研究对象，研究如何精准预测界面处的蒸发率，揭示造成数值预报结果不确定性的机制，找到提高预报准确率的数学方法。针对界面PDE约束的最优控制问题，重点构造保物理特性的新型能量守恒有限体积元方法，以便同时高效稳定地求解非线性耦合的状态、伴随和控制方程组，发展界面PDE约束的最优控制问题新的逼近理论和数值分析方法，在实际天气监测和预报业务中发挥计算数学的积极作用。

The optimal control problem of PDE constraints with interface is an important mathematical science problem driven by numerical weather forecast and one of core problems in computational geophysical fluid dynamics. Evaporation rates and patterns affect the energy balance of terrestrial surfaces and drive a multitude of climatic processes. The prediction of evaporative drying rates from porous media remains a challenge due to the ambient conditions at the interface. The modeling of the proposal in mathematics can be described as an optimal control problem of PDE constraints with interface. This proposal will focus on qusigeostrophic equations to study how we accurately predict the evaporation rates on the interface, reveal mechanisms that yield the uncertainty of numerical prediction and obtain mathematical methods to improve predictability. For the optimal control problem of PDE constraints with interface, we will focus on constructing the new finite volume element method of physical properties preserving, which efficiently solve nonlinear coupled systems by the state, the adjoint state and the control, and develop new approximate theory and numerical analysis such that computational mathematics plays an active role in the actual weather monitoring and forecasting business.