Dynamics and Electrodynamics of the Equatorial Zone - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 96

and we find a new tensor given by

R

R − 1 σ = R · σ ·

For the simple case when geomagnetic and geographic north coincide and the
dip angle is I R corresponds to a rotation counterclockwise by the angle I about

the x axis and

,

⎛ ⎞

1 0 0

⎝ ⎠

R = 0 cos I − sin I

0 + sin I cos I

R to E determines J in the geographic coordinate system.
Following Forbes (1981), we denote the eastward direction as the

Application of σ

λ axis, the

northward as the θ axis, and the vertical as the z -axis. Then

⎛ ⎞

σ λλ σ λθ σ λ z

⎝ ⎠

R

σ = σ θλ σ θθ σ θ z

(3.15)

σ z λ σ z θ σ zz

with

σ λλ = σ P
σ λθ =− σ θλ =− σ H sin I
σ λ z =− σ z λ =+ σ H cos I
σ θθ = σ P sin 2 I

(3.16)

+ σ 0 cos 2 I
σ θ z = σ z θ = (σ 0 − σ P )

sin I cos I

σ zz = σ P cos 2 I + σ 0 sin 2 I

Note in this form that if we set I = 0, we recover a matrix identical to σ in (3.6).

Taking the divergence of J and substituting E =−∇ φ , we have the dynamo

equation

R

∇· σ · ( −∇ φ + × ) =

U B 0

where the quantities are all measured in the earth-fixed frame and expressed in
the rotated coordinate system. This may be written as,

R R

∇· σ ·∇ φ =∇· σ · ( U × B ) (3.17)

R as known functions, this equation is a complicated par-
tial differential equation with nonconstant coefficients. A common simplifying

Even taking U and σ