Back to Summer 2004   Back: Introduction   Next: Bifurcation Structure d=0


Modelling the solar dynamo

The Equations

the equations   (1)
where a>0 and c<0.

Here the x-coordinate represents the toroidal field, and the y-coordinate the poloidal field. The hydrodynamic activity has been collapsed onto the z-axis, so that this coordinate represents the velocity.

The z-axis is invariant, as required, since a seed magnetic field is needed for dynamo action to set in. The term x2+y2 in equation 1(a) models the effect of the Lorenz force on the velocity and the equations have been normalised to set the coefficient of this term to 1.

In the absence of the magnetic field the hydrodynamics are described by the equation

dz/dt = μ -z2+cz3
for which μ acts as a control parameter. A change in the hydrodynamics of the star, such as in thermal forcing or differences due to rotation, should be reflected by changes in μ although it is not immediately clear what form these changes should take. The effect of μ and d on solutions is discussed later.

It is shown later that the constants a and c have the effect of distorting the bifurcation structure, but do not alter it qualitatively (as long as their sign remains the same.)

Setting z=0 in equations 1(b) and 1(c) gives the simple system

with solution
where C1 and C2 are constants which depend on the initial conditions. This suggests that ω may represent the cycle frequency, and λ the growth rate of the magnetic field. The effects of both ω and λ on solutions is discussed later.


Back to Summer 2004   Back: Introduction   Next: Bifurcation Structure d=0