Numerical advection of moments of the particle size distribution in Eulerian models

‘Invalid moment sets’—sets of quantities nominally moments but for which no underlying distribution could exist and therefore not moment sets—are generated in Eulerian models from valid sets by independent advection of each moment. Examination of invalid set generation by two representative advection schemes in one dimension for ensembles of 104 test cases spanning a range of initial moment sets and flow velocities reveals invalid set frequencies 0.7% for both schemes. Standard moment methods cannot accommodate invalid sets. Solutions to this problem are presented and evaluated for the ensembles and accurate moment advection free of invalid sets was obtained for both schemes. A new closure scheme insensitive to invalid sets using Lagrange interpolation of moment equation kernels is described and evaluated for condensation, dry deposition, and gravitational settling and found to match the high accuracy of quadrature.