The Stefan Problem: Polar Exploration and the Mathematics of Moving Boundaries∗

A geophysicist and a mathematician are having coffee. The mathematician queries, “what are you working on these days?” and the geophysicist responds, “I am studying the growth of ice.” The mathematician says, “I see, the Stefan Problem!”, to which his colleague says, “Oh, so you know about Stefan and Weyprecht?”...“Certainly, Stefan must be a mathematician, but Weyprecht–who is Weyprecht?” We are all keenly aware of the difficulties associated with understanding the subtle aspects of the problems that capture the attention of our colleagues in other disciplines even if those areas historically emerged from the same guiding principles. The disparity between the contemporary mathematical studies of “ice growth”, and its origins in polar exploration and Austrian science are the focus of this contribution. In physics, we know of Josef Stefan as an academic advisor to Ludwig Boltzmann in Vienna. The former is noted for having experimentally discovered, in 1879, the blackbody radiation law which relates the power/area of radiation emitted by an opaque body, PR, to the absolute temperature T through PR = σT , where σ = 5.6703×10−8 W m−2K−4 is now called Stefan’s constant. Boltzmann derived his advisor’s relation five years later from statistical mechanics, and it is now called the Stefan-Boltzmann law. This relation is, of course, also of great importance in polar science, allowing us to determine the surface temperature of sea ice by measurement of the emitted radiation, or vice versa. The seemingly banal phenomenon of the growth of a solid from a cooled boundary comes under the rubric of socalled Stefan problems. However, their rich nonlinear behavior, has attracted substantial mathematical interest (e.g. [1]), and their ubiquity in fields ranging from geology to metallurgy stimulates continual rediscovery of Stefan’s work, but rarely a scrutiny of its curious history. It not widely appreciated that Stefan’s interest in the thermodynamics of moving boundaries originated with field observations made during various nineteenth century polar expeditions. An interesting question of history concerns precisely which set of observations originally motivated Stefan’s consideration of the problem. The curiosity derives from Stefan’s pioneering paper of 1889, “Über einige Probleme der Theorie der Wärmeleitung” [2], which begins with a laconic description of the case to be treated: