An analytical approach to predictability based on potential vorticity gradients

The initial perturbations which are responsible for the error growth in a given flow can be viewed as perturbations of its initial potential vorticity field. As such, their dynamics are closely linked in physical space to the regions which present the largest growth in potential vorticity gradients. For all tracer fields, exact equations can be derived for their gradients growth, and this can be applied to the potential vorticity field as well. In this context, one can identify key regions as well as directions of most rapid growth of the initial flow, which are found to depend on its stirring properties. We have compared favorably our analytical prediction of the forecast error with the errors obtained (i) by an ensemble prediction and (ii) by initializing the model with the largest singular vector for several norms, in a wind-driven double-gyre using a 6-layer quasi-geostrophic code.