Buckling of a floating fluid layer

Roll patterns on floating ice shelves have been suggested to arise from viscous buckling under compressive stresses. A model of this process is explored, allowing for a power-law fluid rheology for ice. Linear stability theory of uniformly compressing base flows confirms that buckling modes can be unstable over a range of intermediate wavelengths when gravity does not play a dominant role. The rate of compression of the base flow, however, ensures that linear perturbations have wavelengths that continually shorten with time. As a consequence, linear instability only ever arises over a certain window of time $t$ , and its strength can be characterised by finding the net amplification factor a buckling mode acquires for $t\to \infty$ , beginning from a given initial wavenumber. Bi-axial compression, in which sideways straining flow is introduced to prevent the thickening of the base flow, is found to be more unstable than purely two-dimensional (or uni-axial) compression. Shear-thinning enhances the degree of instability in both uni-axial and bi-axial flow. The implications of the theoretical results for the glaciological problem are discussed.