Decaying singular vectors and their impact on analysis and forecast correction

The full set of kinetic energy singular values and singular vectors for the forward tangent propagator of a quasigeostrophic potential vorticity model is examined. In contrast to the fastest growing singular vectors, the fastest decaying vectors exhibit a downward and downscale transfer of energy and an eastward tilt with height. The near-neutral singular vectors resemble small-scale noise with no localized structure or coherence between levels. Post-time forecast and analysis correction techniques are examined as a function of the number of singular vectors included in the representation of the inverse of the forward tangent propagator. It is found that for the case when the forecast error is known exactly, the best corrections are obtained when using the full inverse, which includes all of the singular vectors. II is also found that the erroneous projection of the analysis uncertainty onto the fastest decaying singular vectors has a significant detrimental effect on the estimation of analysis error. Therefore, for the more realistic case where the forecast error is known imperfectly, use of the full inverse will result in an inaccurate estimate of analysis errors, and the best corrections are obtained when using an inverse composed only of the growing singular vectors. Running the tangent equations with a negative lime step is a very good approximation to using the full inverse of the forward tangent propagator.