Editing Dynamo theory (section)

===Energy conversion between magnetic and kinematic energy===
The scalar product of the above form of Navier-Stokes equation with <math>\;\rho_0 \mathbf{u}\;</math> gives the rate of increase of kinetic energy density, <math>\; \tfrac{1}{2} \rho_0 u^2 c \;</math>, on the left-hand side. The last term on the right-hand side is then <math>\; \mathbf{u} \cdot ( \mathbf{J} \times \mathbf{B} ) \;</math>, the local contribution to the kinetic energy due to [[Lorentz force#Continuous charge distribution|Lorentz force]].

The scalar product of the induction equation with <math display="inline">\tfrac{1}{\mu_0}\mathbf{B}</math> gives the rate of increase of the magnetic energy density, <math>\;\tfrac{1}{2} \mu_0 B^2\;</math>, on the left-hand side. The last term on the right-hand side is then <math display="inline">\tfrac{1}{\mu_0} \mathbf{B} \cdot \left( \nabla \times \left( \mathbf{u} \times \mathbf{B} \right) \right) \;.</math> Since the equation is volume-integrated, this term is [[integration by parts|equivalent up to a boundary term]] (and with the double use of the [[Triple product#Scalar triple product|scalar triple product]] identity) to <math display="inline">\; -\mathbf{u} \cdot \left( \tfrac{1}{\mu_0} \left( \nabla \times \mathbf{B} \right) \times \mathbf{B} \right) = -\mathbf{u} \cdot \left( \mathbf{J} \times \mathbf{B} \right) ~</math> (where one of Maxwell's equations was used). This is the local contribution to the magnetic energy due to fluid motion.

Thus the term <math>\;-\mathbf{u} \cdot ( \mathbf{J} \times \mathbf{B} ) \;</math> is the rate of transformation of kinetic energy to magnetic energy. This has to be non-negative at least in part of the volume, for the dynamo to produce magnetic field.<ref name=Kono2002/>

From the diagram above, it is not clear why this term should be positive. A simple argument can be based on consideration of net effects. To create the magnetic field, the net electric current must wrap around the axis of rotation of the planet. In that case, for the term to be positive, the net flow of conducting matter must be towards the axis of rotation. The diagram only shows a net flow from the poles to the equator. However mass conservation requires an additional flow from the equator toward the poles. If that flow was along the axis of rotation, that implies the circulation would be completed by a flow from the ones shown towards the axis of rotation, producing the desired effect.