对气象常用坐标系中位涡形式的探讨

An investigation into the expression of potential vorticity in the common meteorological coordinate systems

气象常用垂直坐标系中的位涡方程及位涡形式是位涡理论及位涡诊断技术的基础.本文依据坐标转换的观点,分别用两种不同的方法推导出等压坐标和等熵坐标中的位涡方程及相应的位涡表达式.一是从三维矢量运动方程出发,由三维涡度方程、连续方程和热力学方程推导位涡方程;二是直接从等压坐标和等熵坐标中的标量运动方程组出发推导位涡方程.结果表明,用两种方法所得到的等压坐标中的位涡方程和位涡表达式形式有所不同,而等熵坐标中用两种方法所得到的位涡方程和位涡形式相同.对垂直坐标系的物理本质分析表明,采用第一种方法时尽管矢量运动方程中的气压梯度力项作了坐标转换,但本质上仍是将等压坐标和等熵坐标作为数学坐标来处理.而第二种方法是将等压坐标和等熵坐标作为静力平衡成立条件下的物理坐标系,不是将其作为数学坐标系来使用.等压坐标系中从标量运动方程出发得到的位涡方程才是严格的位涡方程.由于在等熵坐标系中,位温梯度与等熵面垂直,因此无论将等熵坐标作为物理坐标还是数学坐标来考虑,其位涡方程和位涡表达式都是一致的.

The potential vorticity theory and diagnostic techniques are based on the potential vorticity equation and expression in the common meteorological coordinate systems.In this paper,the potential vorticity equation and expression in the isobaric and isoentropic coordinates are gotten via coordinate transformation with the two methods.First,starting from the threedimensional vector motion equation,the potential vorticity equations and expressions are gotten by the combination of the three-dimensional vorticity equation,continuity equation,and thermodynamic equation.Second,the potential vorticity equations and expressions are directly gotten from the corresponding scalar motion equations in the isobaric and isoentropic coordinates.The results show that potential vorticity expression is different with one method from that with the other in the isobaric coordinate system,and it is the same as each other in the isoentropic coordinate system.It was found,based on further analysis of the physical nature of the coordinates,that the isobaric and isoentropic coordinates are essentially treated as a mathematical coordinate system with the first method despite the coordinate transformation made for the term of pressure gradient force in the vector motion equation.From the procedure for the second method it is clearly seen that the isobaric and isoentropic coordinate systems are the physical coordinate system under the assumption of static equilibrium,which are not simply used as a mathematical coordinate system.As far as the isobaric coordinate is concerned,only the potential vorticity equation obtained from the scalar motion equations is the strict potential vorticity equation.As for the isoentropic coordinate,owing to the potential temperature gradient perpendicular to the isoentropic plane,the potential vorticity equation and expression are the same regardless of the coordinate being viewed as the physical or the mathematical.