Supplementary material to "How well do different tracers constrain the firn diffusivity profile?"

Abstract

Firn model equationsHere we describe the equations for firn diffusion, convective mixing, advection and bubble trapping in a reference frame that moves downwards with the ice as used in the CSIRO firn model. The equations are related to the equations in a 5 fixed reference frame (relative to the surface) from Rommelaere et al. (1997) and are more rigorously derived than in Trudinger et al. (1997), giving more accurate representation of the physics and allowing us to avoid some of the assumptions that were used in the previous version of the CSIRO firn 10 model (most notably the upward flux of air due to compression). We use the following notation:t. Time (years) t . Time in moving coordinates (years) Although t = t , we distinguish them to clarify partial derivatives Ice conservation and coordinate systemsUnit volume has proportion s of air and 1 − s of ice, since mass (per unit volume) is (1 − s) × ρ ice = ρ, giving total porosityThe mass of ice (per unit area) above z isand vertical velocity of ice (relative to the surface) is 60We wish to use a moving coordinate system, so care is needed in addressing transformations from a fixed coordinate system. This is done in terms of a generic function F (.,.), with the need to consider that as coordinates change, the functional relation will also change, and that the notation 65 should reflect this.Here we define a number of coordinate systems, along with the transformations and notation for functions. We are interested in the final coordinate system (y,t ), and the others are given to show how (y,t ) relates to (z,t).