The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 12

Fig. 1.4 Right-handed ( a )
and left-handed ( b )
spiral-helix motion

a b

z , ˉ z , ˉ

V V

∇× V ∇× V

ʱ < 0 ʱ > 0

When the radial component of the velocity is taken into account, it produces
the azimuthal component of the Coriolis force. However we now consider only the
components of the Coriolis force due to the perpendicular velocity constituent V
shown in Fig. 1.3 . As is seen from this figure, one of the components of the Coriolis
force is pointed “out of paper” (the circle with point) whereas another component
is oppositely directed (the circle with cross). These two forces create a moment,
which rotates the convective fluid element around the radius as shown in Fig. 1.3 .
The same is true for the convective element located at an arbitrary point of the
globe; that is, the Coriolis forces create a moment which rotates the element around
the axis/position vector drawn from the center of the globe. Similarly, the Coriolis
force can turn the oceanic flows, hurricanes, and tropical cyclones as it is seen from
satellite pictures.
It is obvious that applying a curl operator to this rotary motion gives the vector
pointed radially. We recall that the main movement of the element is radially
directed whence it follows that r V r O r D 0, where O r stands for unit vector. However,
the mean helicity is nonzero, i.e., h V . r V / i ¤ 0, due to the presence of the
perpendicular components of the velocity.
Once the convective element is dropped, the radial velocity becomes negative.
The perpendicular velocities V , V , and r V change their directions as well due
to compression of the element. In this case the Coriolis force moment rotates the
element in the opposite direction. This means that the vector r V changes sign
whereas the helicity keeps the sign unchanged.
For illustrative purposes, a right-handed spiral motion is shown in Fig. 1.4 a.
Consider, for example, a progressive motion with constant velocity V z >0positive
parallel to z axis that combines with rotation around this axis with constant angular
velocity ! D ! O z , where O z is a unit vector. Hence the azimuthal component of the

velocity is V D !r. Applying a curl operator to the velocity of this helical motion
yields r V D 2! O z . This vector is vertically upward as shown in Fig. 1.4 a, so that
h V . r V / i >0while the helicity Ǜ in Eq. ( 1.23 ) is negative. If there is a counter-
rotating helical motion as shown in Fig. 1.4 b, the vector r V is downward-directed
so that Ǜ>0.
The turbulent hydrodynamic and magnetic fields are random in character that
results in an extremely complicated pattern of MHD flow. In order to find the