Finger Rafting

Finger Rafting: A Generic Instability of Floating Elastic Sheets

Dominic Vell and J.S. Wettlaufer

PRL 98, 088303 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e088303

Another good recommendation from the editors of PRL.

The authors show that the phenomenon of finger rafting observed in ice floes is not related to an intrinsic property of water, but is a general occurrence for floating elastic sheets. Finger rafting describes the interlocking protrusions that form when two ice sheets collide. As shown in Figure 1, it looks very similar to interlocking fingers.

The authors analyze the plate equation for a two dimensional semi-infinite sheet floating on water with a delta function source at the edge. The plate equation describes vertical displacements of the sheet. It is similar to Laplace's equation for an electrostatic potential. However, Laplace's equation is quadratic in derivatives while the plate equation is quartic. This leads to solutions that both decay exponentially and oscillate.

I was able to solve the one-dimensional version of the equation using a Fourier transform, and it illustrates the origin of this phenomenon. Laplace's equation is quadratic in k. Inverting the transform, one has two simple poles which lie either on the real axis or the imaginary axis. If the poles lie on the real axis, the function oscillates; if they lie on the imaginary axis, the function decays exponentially. When inverting the transform for the quartic equation, there are four poles, two of which must be included in the contour integral. For the plate equation, the poles lie on the lines Re(z) = +/- Im(z) --- i.e., they have both real and imaginary parts. This leads to functions that decay exponentially and oscillate.

The oscillations are what give rise to finger rafting. The authors model the collision process as two semi-infinite plates colliding, with one plate having a slight protrusion that rises over the other. This leads to a localized deformation in each plate, but the forcing terms in the respective plate equations have opposite signs. As a result, the oscillations are exactly out of phase: the maxima of one plate face the minima of the other plate. The plates only touch where the displacement is zero, so the zeros of the displacement function determine the size of the fingers. Since the function decays exponentially, the authors only consider the first zero.

They find reasonable agreement with experiment, although none their experimental data for wax do not fall on the theoretical line. They attribute discrepancies to reasonable causes like nonuniform edges and buckling. The authors also point out that the formation of fingers would be a dynamics process --- the deformations travel at the wave velocity of the floating sheet. For sea ice, they estimate this to be about 5 m/s. That would be a neat process to watch!

An interesting feature of the results is their universality. The characteristic length scale and pressure are set by material properties, and the results are given in terms of these scaled variables. As a result, the theory can be applied to elastic sheets in microscopic systems as well as to the dynamics of tectonic plates.

A final note: The two-dimensional version of the plate equation involves an integral very similar to the one I carried out in the one-dimensional case. The exponential is replaced by a Bessel function, the measure becomes k dk, and the limit of integration is 0 to infinity rather than all space. I have not been able to work this one out yet.