On the Clague-Mathews relation for jökulhlaups

Abstract

In the empirical study of jökulhlaups, the peak discharge, Qmax, and water volume drained by the ice-dammed lake during the floods, Vt, appear to follow a power-law relation Qmax = KVtb, where K are b are constants determined from field data. First identified by Clague and Mathews (1973), this relation is a useful reference for predicting flood magnitude, but its physical origin remains unclear. Here, we develop the theory that connects it to contemporary models for simulating the flood hydrograph. We explain how the function Qmax = f(Vt) arises from Nye's (1976) theory of time-dependent water flow in a subglacial channel coupled to a lake, and we describe how discharge-volume data record the (monotonically increasing) form of this function so long as the lake is not emptied in the floods. The Grímsvötn jökulhlaups present an example where, because of partial draining of the lake, agreement between the model-derived f and data is excellent. It is documented that other lake systems drain completely, but we explain how the exponent b ≈ 2/3 observed for them collectively is due primarily to a scaling effect related to their size, modified by other factors such as the flood initiation process.