 Dispersion relation of an equal-diffusivity interface dynamo

ID: dispersion-relation-of-an-equal-diffusivity-interface-dynamo

Let

p = s - iω

and

q^{2} = k^{2} + p / η

, with

Re q > 0

. The upper poloidal amplitude is

(A_{0} + C z) e^{- q z}

and the lower toroidal amplitude is

(B_{0} + cz) e^{q z}

2 η qC = α B_{0}

and

- 2 η q c = ikV A_{0}

; interface matching gives

C = 2 q A_{0}

and

c = - 2 q B_{0}

. Eliminating amplitudes yields the dispersion relation. Its potentially growing branch has

s = η k^{2} (D /32 - 1)

and

ω = - η k^{2} D /32

with the convention

e^{ik x - iω t}

. The opposite branch decays.

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