Radiation conditions for the lateral boundaries of limited‐area numerical models

Attention is focused on the use of numerical formulations of the Sommerfeld radiation condition δΦ/δt + cδΦ/δx = 0 as a lateral boundary condition. This numerical boundary condition which originally used the leap frog scheme (Orlanski 1976) is shown to be made more accurate by the use of the upstream method. In this form it is tested in a two‐dimensional model of a density current and is stable. the boundary becomes effectively transparent to incident disturbances, even under the severe test of allowing the density current head to propagate upstream and through the boundary, thus changing initial inflow to strong outflow. An analysis is presented of various extrapolation and radiation boundary conditions, involving calculation of their accuracy for waves and more general solutions. the leading terms in the truncation error are compared, and for general disturbances some have second order accuracy, while for solutions of the wave equation, it is shown that the radiation condition is third order. It can be made fourth order, for both leapfrog and upstream differencing, by the use of an extrapolation formula to give a better numerical estimate for c. This condition derives its greater accuracy from the use of values at seven (space and time) gridpoints, in contrast to the original scheme which used four, and standard extrapolation formulae which use, at most, three points.