Nearly Symmetric Kinematic and Hydromagnetic Dynamos

Certain aspects of the hydromagnetic and kinematic dynamo theory related to the Earth's magnetic field are considered. The work of Tough and Roberts [J. G. Tough and P. H. Roberts, Physics of the Earth and Planetary Interiors (North-Holland, Amsterdam, 1968), Vol. 1, p. 288] on the hydromagnetic dynamo is extended in two respects. First, the most general form of the prescribed body force which is consistent with the model is considered. Secondly, the mean quantity 〈Sφ〉 [see Eqs. (18)] is determined up to order R−1/2, where R is the magnetic Reynolds number. It is shown that the remarkable effective variables first introduced by Braginskii {S. I. Braginskii, Zh. Eksp. Teor. Fiz. 47, 1084 (1964) [Sov. Phys. JETP 20, 726 (1965)]} are relevant in the hydromagnetic dynamo in the second approximation. It is also shown by considering a related problem concerning the heat conduction equation that effective variables are unlikely to have any relevance in the kinematic dynamo theory in the third approximation.