Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

Nonlinear dynamo t

The kinematic approximation becomes invalid when the magnetic field becomes strong enough to affect the fluid motions. In that case the velocity field becomes affected by the Lorentz force, and so the induction equation is no longer linear in the magnetic field. In most cases this leads to a quenching of the amplitude of the dynamo. Such dynamos are sometimes also referred to as hydromagnetic dynamos. Virtually all dynamos in astrophysics and geophysics are hydromagnetic dynamos.

Numerical models are used to simulate fully nonlinear dynamos. A minimum of 5 equations are needed. They are as follows. The induction equation, see above. Maxwell's equation:

 
abla cdot mathbf{B}=0

The (sometimes) Boussinesq conservation of mass:

 
abla cdot mathbf{u} = 0

The (sometimes) Boussinesq conservation of momentum, also known as the Navier-Stokes equation:

 frac{Dmathbf{u}}{Dt} = -
abla p + 
u 
abla^2 mathbf{u} + 
ho^'mathbf{g} + 2mathbf{Omega} 	imes mathbf{u} + mathbf{Omega} 	imes mathbf{Omega} 	imes mathbf{R} + mathbf{J} 	imes mathbf{B}

where 
u is the kinematic viscosity, 
ho^' is the density perturbation that provides buoyancy (for thermal convection 
ho^'=alphaDelta T, Omega is the rotation rate of the Earth, and mathbf{J} is the electrical current density.

Finally, a transport equation, usually of heat (sometimes of light element concentration):

 frac{partial T}{partial t} = kappa 
abla^2 T +epsilon

where T is temperature, kappa=k/
ho c_p is the thermal diffusivity with k thermal conductivity, c_p heat capacity, and 
ho density, and epsilon is an optional heat source. Often the pressure is the dynamic pressure, with the hydrostatic pressure and centripetal potential removed. These equations are then non-dimensionalized, introducing the non-dimensional parameters,

Ra=frac{galpha T D^3}{
u kappa} , E=frac{
u}{Omega D^2} , Pr=frac{
u}{kappa} , Pm=frac{
u}{eta}

where Ra is the Rayleigh number, E the Ekman number, Pr and Pm the Prandtl and magnetic Prandtl number. Magnetic field scaling is often in Elsasser number units B=
ho Omega/sigma.

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