How does a neutron star "reheat" itself by ohmic dissipation of its magnetic field?

Question

@ProfRob's answer to Neutron star accurate visualization ends with:

...How their temperatures evolve after this is highly uncertain and none have been observed. The problem is that neutron stars have a very small heat capacity. They cool easily, but they can also be reheated easily by ohmic dissipation of their strong magnetic fields or accretion from the interstellar medium. This the surface temperature of most neutron stars is likely to be much lower than 10⁶ K.

I think of neutron stars as fairly rigid objects with magnetic fields "frozen in" and co-rotating with the object.

Ohmic heating would happen if a current were induced due to a changing magnetic flux. One simple source for this would be a decaying field due to electromagnetic dipole radiation¹, but I'm just grasping at straws here; there could be other more complicated mechanisms that are significant, see for example How do neutron stars maintain inhomogeneous surfaces and migrating "hot spots"? (e.g. SGR 1830-0645) So I'd like to ask:

Question: How does a neutron star "reheat" itself by ohmic dissipation of its magnetic field?

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¹See also Has anyone ever put a magnetic or electrostatic dipole on a rotating shaft, spun it and demonstrated reception of a propagating wave in the far-field?

Answer 1

Ohmic heating simply refers to the situation of ohmic losses $(I^2 R)$ caused by the finite conductivity of the neutron star interior combined with the currents responsible for generating their magnetic fields.

The deep interiors of neutron stars are probably superconducting fluids and expel the field to the outer crust (with a depth of $\sim 1$ km), which will have finite conductivity. In the absence of any regeneration mechanisms (i.e. a dynamo), then the currents will slowly dissipate and there will be a heating rate inversely proportional to the crust conductivity (e.g. Miralles et al. 1998).

Because the heat capacity of a neutron star is very low (a property of the degenerate fermion gases in most of its interior), even though the ohmic losses may be small, they can have profound consequences for the late thermal history of the neutron star, maintaining their surface temperatures at perhaps $\sim 10^5$K for $10^8$-$10^9$ years.