Kinetic undercooling makes the speed of a phase boundary proportional to the difference between its equilibrium and actual interface temperatures. A curvature correction raises the required undercooling for protrusions and therefore stabilizes short wavelengths.
A planar front growing into a supercooled melt can amplify corrugations because protrusions enter colder liquid. Interface curvature opposes this effect. Linearization couples the perturbation growth rate to the decaying thermal exponent ahead of the front.
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