A Linear Equation for the Walker Cell Streamfunction and Its Analytic Solutions

We derive a linear streamfunction equation for the meridionally averaged, steady-state Walker circulation in a Boussinesq fluid. Assuming Rayleigh friction, Newtonian cooling, a no-motion basic state and constant static stability, the streamfunction is shown to obey Poisson's equation with zonal gradient of diabatic heating and vertical gradient of Coriolis force as the source terms. With the aid of this equation, we study the response of the Walker circulation to changes in the zonal and vertical extents of diabatic heating. In the non-rotating case, the problem has a close parallel in the field of electrostatics; the streamfunction plays a role analogous to an electric field due to a given charge distribution. Thus, taking advantage of previous results in electrostatics, a closed-form analytic formula for the circulation strength can be found for an elliptical heating shape. The solution indicates that the sensitivity of the circulation strength to the zonal and vertical scales can be very different; the Walker cell can rapidly transform between vertically-sensitive and zonally-sensitive regimes, once a critical aspect ratio (dependent on static stability, cooling and friction rates) is reached. We present some estimates of the sensitivity in the present-day climate, and discuss the limitations of our approach.