Proc . R . Soc . Lond . A 414 , 83-102 (1987) Printed in Great Britain Convection in thawing subsea permafrost B y G. P. G aldi 1, L. E. P ayne 2, M. R. E. P roctor 3 and B. S traughan 4 1 D i p a r t i m e n t o d i Mcitematica, Universitd di Ferrara, 44100 Ferrara, Italy 2 Department of Mathematics, Cornell University, Ithaca, New York 14853, U.S.A. 3 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K. 4 Department of Mathematics, University of Glasgow, Glasgow GUI 3QW, U.K. (Communicated by I. N. Sneddon, F.R.S. - Received 13 August 1986 - Revised 6 May 1987) Detailed quantitative values are obtained for the critical values of the salt Rayleigh number for both linear and nonlinear stability, for a simplified model appropriate to the onset of buoyant, relatively fresh water motion in a layer of salty subsea sediments. The geophysical problem that motivates this work arises because of the formation of substantial permafrost around the Earth’s shores some 18000 years ago. W ith the rise of sea levels the perm afrost has responded to the relatively warm and salty sea, which has created a thawing front and a layer of salty sediments beneath the sea bed. This phenomenon has been studied extensively off the coast of Alaska by W. Harrison and co- workers and our analysis is based on a model developed by W. Harrison and D. Swift. From the mathematical viewpoint the analysis reduces to studying convection in a porous medium with a nonlinear boundary condition. We find the critical Rayleigh number for convection according to linear theory, but our main thrust is directed toward the nonlinear problem. Here we use an energy method to determine a critical Rayleigh number below which convection cannot develop. We first show there is a critical Rayleigh number close to that of linear theory, which guarantees unconditional nonlinear stability. Then we demonstrate conditional nonlinear stability (i.e. conditional upon the existence of some finite threshold amplitude, which we calculate) provided the critical Rayleigh number of linear theory is not exceeded. The latter analysis requires two approaches according to whether the two-dimensional or three- dimensional problem is considered. In particular, a novel energy has to be introduced to make the three-dimensional problem tractable. i. I ntroduction During the Earth’s history, the sea level has depended directly on the size of glaciers and the amount of water bound as ice. In particular, about 18000 years ago (see Miiller-Beck 1966, p. 1193) the level was some 100-110 m lower than it presently is; the ambient air temperatures were much colder too, reaching a [ 83 j Downloaded from https://royalsocietypublishing.org/ on 16 January 2022