Modern weather and climate prediction models are based on a system of nonlinear partial differential equations called the primitive equations. Lie symmetries of the primitive equations with zero external heating rate are computed and the structure of its maximal Lie invariance algebra, which is infinite-dimensional, is studied. The maximal Lie invariance algebra for the case of a nonzero constant Coriolis parameter is mapped to the case of vanishing Coriolis force. The same mapping allows one to transform the constantly rotating primitive equations to the equations in a resting reference frame. This mapping is used to obtain exact solutions for the rotating case from exact solutions for the nonrotating equations. Another important result of the paper is the computation of the complete point symmetry group of the primitive equations using the algebraic method.

现代的天气和气候预测模型基于称为原始方程的非线性偏微分方程组。计算了外部加热速率为零的原始方程的Lie对称性，并研究了其最大Lie不变性代数的无穷维结构。非零常数科里奥利参数情况下的最大李不变性代数映射到科里奥利力消失的情况。相同的映射允许将不断旋转的原始方程式转换为静止参考系中的方程式。该映射用于从非旋转方程的精确解中获得旋转情况的精确解。