Solution of a Model for the Oceanic Pycnocline Depth: Scaling of   Overturning Strength and Meridional Pressure Difference
  We present an analysis of the model by Gnanadesikan [1999] for the pycnocline depth in the ocean. An analytic solution for the overturning strength as a function of the meridional pressure difference is derived and used to discuss their mutual scaling. We show that scaling occurs only in two unphysical regimes of the model. In the absence of the Southern Ocean (SO) processes, i.e. for a northern overturning cell, the volume transport is proportional to the square root of the pressure difference. Linear scaling is seen when the overturning is restricted entirely to the SO, i.e. when no northern downwelling exists. For comparison, we present simulations with the coupled climate model CLIMBER-3$\alpha$ which show linear scaling over a large regime of pressure differences in the North Atlantic (NA). We conclude that the pycnocline model is not able to reproduce the linear scaling between its two central variables, pressure and volume transport. 
Anders Levermann and Alexa Griesel