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T^{3} ( i . e. , the periodic motion). In the fluid dynamics there appear many systems of non-linear differential equations involving parameters such as the Mach number and the Alfv\’en number etc. One problem on the singular limit is to determine the limiting system which has a completely different property comparing with the original system, as such a parameter tends to some value. When the system is hyperbolic, this problem has been studied in C. Browning–H.-O. Kreiss [2], S. Klainerman–A. Majda [4], A. Majda [5] and S. Schochet [6]. In particular, Browning and Kreiss studied the Alfv\’en limit for the compressible magnet0-fluid motion as an example of their theorems. However, to show this, they needed more assumptions on the initial data than those in other papers above. The purpose of this paper is to determine the limiting system for the incompressible magnet0-fluid motion under the natural assumptions on the initial data. The limiting system becomes essentially the system of the two dimensional motion (see (1. 6)). We state main results in Section 1. In Section 2, we show the uniform estimates with respect to the Alfv\’en number, which are obtained by the energy method. The convergence of the solutions is generally proved in Section 3. Especially, Lemma 3. 3 is employed to determine the limiting system. The proof of our theorem is finally completed in Section 4.

Singular limit of the incompressible ideal magneto-fluid motion with respect to the Alfvén number