Solutions of the balance equation with minimum correction of the mass field

If the balance equation is solved for the stream function by conventional methods, the mass field has to satisfy a condition of ellipticity. In this paper we present a method to find a stream function when this condition is not satisfied. The method is an iterative method presented by Fjørtoft (1962). It is shown that if this method converges, it gives a solution where the residual of the balance equation is a minimum.

The method is tried on a height analysis of 300 mb which contains rather large hyperbolic areas. The convergence of the method is very slow, and a solution using a conventional method is used as a first guess. From the solution found we find a height field in balance with this stream function by solving the balance equation for the height field. It is shown that this field is a better approximation to the original height field than an elliptic field found after correction by a standard method. The new height field does not satisfy the condition of ellipticity with respect to the balance equation.