Linear Amplification and Error Growth in the 2D Eady Problem with Uniform Potential Vorticity

Abstract

The concept of a singular mode underlies optimal linear amplification theories. This concept is studied in the frame of the two-dimensional, quasigeostrophic Eady problem with uniform potential vorticity. Analytical solutions are produced for the relevant physical norms. Exact relations are also derived for the amplifications, which give the lower and upper bounds to any linear development. Results show significant differences in the structure of the singular modes, as well as in the associated amplifications, when the horizontal wavenumber is varied or the inner product is changed. It is found that the singular modes can depart significantly from the normal modes, though the dynamics of the problem are very simple. Comparisons with previous works are also performed. Finally, the derived equations are used to present the linear evolution of error growth within the Eady problem, as predicted by a Kalman filter. Considerations on the spectral space error covariance matrix are made, and a particular case of error dynamics in the 2D physical space is shown. The derivation of the general algebraic solutions is included in the appendix.