Nonlinear Balance and Potential‐Vorticity Thinking At Large Rossby Number

Two rational approximations are made to the divergence, potential‐vorticity, and potential‐temperature equations resulting in two different nonlinear balance models. Semi‐balance is similar (but not identical to) the nonlinear balance model of Lorenz. Quasi‐balance is a simpler model which is equivalent to quasi‐geostrophy at low Rossby number and to the barotropic model at high Rossby number. Practical solutions involving methods that work for all Rossby numbers are outlined for both models. A variety of simple initial‐value problems are then solved with the aim of fortifying our insight into the behaviour of flows at large Rossby numbers. the flows associated with a potential‐vorticity anomaly at very large Rossby numbers differ in significant ways from the corresponding low Rossby number results. In particular, an isolated anomaly has zero vertical radius of influence at infinite Rossby number, while the induced tangential velocity in a horizontal plane containing the anomaly decreases inversely as, rather than inversely as the square of, the radius. the effects of heating and frictional forces are approached from a point of view somewhat different from that recently expressed by Haynes and McIntyre, though the physical content is the same.