Geomagnetically Induced Currents

[Johnston]

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Hello, I have a question about geomagnetically induced currents.

I understand that these currents involve ten to hundreds of amps flowing through the earth's crust.

For the current to be that high, considering the crust would have a fairly high resistance compared to a standard conductor, the voltages involved must also be quite large.

(Ohm's Law, V = IR)

Shouldn't there also be heat generated in the upper mantle and lithosphere where these currents flow?

(Ohmic Heating, Q is proportional to I^2 R)
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[Peter]

As far as I am aware, such currents are only believed to flow in the iron core.

Since no currents are believed to run through the mantle, there is no heating due to such currents.