Rikitake Model of Geomagnetic Reversal

time

v

a

x

0

y

0

z

0

The Rikitake model arose from the study of two connected identical frictionless disk dynamos, where reversals of electric current and magnetic field are observed.

An analogous model is proposed to explain the highly irregular reversals of the Earth’s magnetic ﬁeld over geological history.

Details

The original equations describing the system are given by:

L

1

d

I

1

dt

+

R

1

I

1

=

Ω

1

M

I

2

,

L

2

d

I

2

dt

+

R

2

I

2

=

Ω

2

M

I

1

,

C

1

d

Ω

1

dt

=

G

1

-

I

1,

I

2

,

C

1

d

Ω

1

dt

=

G

2

-

I

1,

I

2

,

where

L

1

and

L

2

are the self-inductances of the coils,

R

1

and

R

2

are the resistances,

I

1

and

I

2

are the electric currents,

Ω

1

and

Ω

2

are the angular velocities,

C

1

and

C

2

are the moments of inertia, and

G

1

and

G

2

are the driving forces.

Taking for simplicity

I

1

=

I

2

,

R

1

=

R

2

,

M=N

,

C

1

=

C

2

,

G

1

=

G

2

,

and making the substitutions

I

1

=

G

M

x

,

I

2

=

G

M

y

,

Ω

1

=

GL

CM

z

,

Ω

2

=

GL

CM

(z-a)

,

t=

CL

GM

t'

,

v=

C

LMG

with the parameter

v>0

, we obtain



x

=-vx+zy

,



y

=-vy+(z-a)x

,



z

=1-xy

.

The Rikitake system exhibits Lorenz-type chaos while orbiting around two unstable ﬁxed points.

References

[1] T. Rikitake, "Oscillations of a System of Disk Dynamos," Mathematical Proceedings of the Cambridge Philosophical Society, 54(1), 1958 pp. 89–105. doi: 10.1017/S0305004100033223.

[2] J. A. Jacobs, Reversals of Earth's Magnetic Field, New York: Cambridge University Press, 1994.

[3] I. Pehlivan, Y. Uyaroglu, and S. Coskun, "Simulation and Realization of the Rikitake Attractor's Chaotic Oscillator Circuit," Proceedings of the 5th International Advanced Technologies Symposium (IATS’09), May 13–15, 2009, Karabuk, Turkey. iats09.karabuk.edu.tr/press/bildiriler_pdf/IATS09_02-01_ 480.pdf.

External Links

Angular Velocity (ScienceWorld)

Moment of Inertia (ScienceWorld)

Current (ScienceWorld)

Self Inductance (ScienceWorld)

Dynamo (ScienceWorld)

Magnetic Field (ScienceWorld)

Geodynamo (ScienceWorld)

Lorenz Attractor (Wolfram MathWorld)

Permanent Citation

Enrique Zeleny

"Rikitake Model of Geomagnetic Reversal"
http://demonstrations.wolfram.com/RikitakeModelOfGeomagneticReversal/
Wolfram Demonstrations Project
Published: July 17, 2012