Abstract
The semi‐geostrophic solution to the Eady frontogenesis problem is believed to represent an asymptotic limit of the full Euler equations as the Rossby number tends to zero. This is investigated using numerical solutions. The exercise is non‐trivial because the semi‐geostrophic solution is discontinuous, and so may not be found as the limit of the Eulerian form of the equations. It is shown that solutions of the two‐dimensional compressible Euler equations converge to a singular limit with geostrophic balance in the cross‐front direction. The convergence is at the expected second‐order rate in Rossby number up to the formation of the discontinuity and at a first‐order rate for some time afterwards. Numerical robustness can be assured and the degree of convergence improved if a quasi‐monotone form of interpolation is used in the semi‐Lagrangian advection. Direct integrations of a semi‐geostrophic model using standard numerical methods require significant computational compromises, and give a somewhat smoothed version of this limit. © Crown Copyright 2008. Reproduced with the permission of HMSO. Published by John Wiley & Sons, Ltd.