Numerical analysis of lava lake cooling models; Part I, Description of the method

Abstract

A computer method for numerical solution of problems in transient heat flow has been developed and applied to cooling models of Hawaiian lava lakes. This report and part II by Peck, Hamilton, and Shaw apply the method to Alae lava lake as the first detailed test against a systematic program of temperature measurements in lava lakes carried out by the Hawaii Volcano Observatory. The method is neither a finite element nor finite difference method, as the terminology of numerical analysis is currently employed, though there are similarities to both. It is an adaptation of a computer method that we call the method of explicit cell balances. Input-output balances are computed between cells in a grid system simulating the lava lake geometry. Computation explicitly performs the cell balances sequentially in the same way the "cooling wave" moves into the lava body without recourse to numerical solutions of sets of simultaneous equations. The procedure clearly identifies the physical processes governing thermal effects: the effects of the latent heat of crystallization, effects of rainfall permeating the lava surface, and effects of large variations in density and thermal conductivity. The accuracy of numerical approximations was tested against exact solutions for an extrusive sheet without sources. Deviations were less than 1 degrees C in all cases. Agreement with data for Alae lava lake given in part II indicates that the method can be used with good confidence for refinement of thermal properties and identification of heat transfer mechanisms in other lava lakes as well as for predictive tests of cooling histories for hypothetical systems involving complicated heat transfer mechanisms.