The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 6

This equation can be reduced to the form

d
dt C . r V / D 0;

(1.11)

where the total time derivative

d=dt D @ t C V r : (1.12)

The principle of conservation of momentum relates the fluid velocity to the
forces acting on the fluid. This equation must include the familiar terms such as
the pressure gradient, the gravitational and “viscous” forces. The presence of an
electrical conductivity of the fluid requires the inclusion of volume force on the
fluid given by j B ; that is, by Ampére force. In the reference frame fixed to the
Earth one should take into account the internal terms such as Coriolis force and
centrifugal force resulted from the Earth spin. The equation of momentum in its
general form can be written as

2 V C j B C F ;

d V =dt Dr P C g C r (1.13)

where g describes the gravitational field, is called the dynamic viscosity
coefficient, and F stands for all the inertial forces acting on the fluid.
As the conducting fluid is immersed in an external magnetic field, the electric
currents and fields can be developed from the hydrodynamic motion of the medium.
The magnetic field forces the electric currents and thus may greatly affect the
medium motion. The electric currents in turn change the magnetic field. In this
notation, the interaction between the magnetic and hydrodynamic fields results in
a complicated picture of Magnetohydrodynamic (MHD) flow. The set of MHD
equations ( 1.2 )-( 1.4 ), ( 1.9 ), ( 1.10 ), and ( 1.13 ) can be supplemented by the equation
of state and by the equation which is derived from the principle of conservation
of energy. The interested reader is referred to the text by Kelley ( 1989 ) for details
about these equations and for a more complete treatise on dissipative and viscous
processes in conducting fluids.
The MHD approximation can be applied to a variety of physical objects such as
a flow of conducting molten matter inside the Earth core, cosmic plasma in the solar
wind, solar flares, the Earth's magnetosphere electrodynamics, and etc.
Applying a curl operator to Eq. ( 1.9 ) and substituting Eq. ( 1.2 )for r E into
Eq. ( 1.9 ) yields

2 B Cr . V B /;

@ t B D m r (1.14)

where m D . 0 / 1 is the magnetic diffusion coefficient measured in m 2 /s.
Consider first the case of a still fluid so that Eq. ( 1.14 ) simplifies to

2 B :

@ t B D m r (1.15)