The Heat Conduction Equation

The energy-transport counterpart of diffusion, governing how temperature relaxes through conductive heat flux.

Form

The heat equation is (3/2) n dT/dt = div(chi n grad T) + S in the simplest fluid closure, where T is temperature, chi is the thermal diffusivity, n is density and S is a heat source or sink. With constant coefficients and no source it is dT/dt = alpha d2T/dx2, alpha = chi being the thermal diffusivity. It is mathematically identical to the particle diffusion equation, but its coefficient chi carries distinct physics.

The (3/2) factor comes from the internal energy density (3/2)nT of an ideal gas. The conductive flux q = -chi n grad T is the Fourier-law closure. Sources include ohmic heating, auxiliary heating, and fusion alpha heating; sinks include radiation and transport losses across the boundary.

Parallel and Perpendicular Conduction

In magnetized plasma the electron parallel thermal conductivity is enormous and scales strongly with temperature (Spitzer conduction goes as T^{5/2}). Perpendicular conduction is far smaller. This anisotropy flattens temperature along field lines almost instantly while allowing steep gradients across them, which is precisely what sustains a confined temperature profile. Ion and electron channels have separate chi and exchange energy through collisions.

Confinement Consequences

Energy confinement time tau_E is essentially the stored thermal energy divided by the loss power, and it is governed by the effective cross-field chi. Because chi is usually turbulence-dominated rather than collisional, predicting it requires gyrokinetic simulation. A stiff temperature profile, where chi rises sharply above a critical gradient, is a widely observed nonlinear feature.

For a D-T device the alpha-particle heating term feeds the same equation that governs losses, so ignition analysis is fundamentally a balance in the heat-conduction equation between fusion self-heating and conductive plus radiative losses. Kronos design points such as the Hyperion Q of 3.424 are outputs of transport-balance modeling, not measured results.