A robust numerical method for the potential vorticity based control variable transform in variational data assimilation

• Published in: Quarterly journal of the Royal Meteorological Society Vol. 137; no. 657; pp. 1083 - 1094
• Main Authors: Buckeridge, S., Cullen, M.J.P., Scheichl, R., Wlasak, M.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.04.2011; Wiley
• Subjects: Data assimilation; Earth, ocean, space; Exact sciences and technology; External geophysics; finite volume; Mathematical analysis; Mathematical models; Meteorology; multigrid; Offices; parallelisation; Partial differential equations; Physics of the high neutral atmosphere; potential vorticity; preconditioner; solver; Three dimensional; Tolerances; Vorticity; Linear system; Multigrid; Vorticity; Potential vorticity; Variational calculus; finite volume; parallelisation; preconditioner; Partial differential equation; Data assimilation; Convergence; solver
• DOI: 10.1002/qj.826
• ISSN: 0035-9009; 1477-870X

The potential vorticity based control variable transformation for variational data assimilation, proposed in Cullen (2003), is a promising alternative to the currently more common vorticity based transformation. It leads to a better decorrelation of the control variables, but it involves solving a highly ill‐conditioned elliptic partial differential equation (PDE), with a constraint. This PDE has so far been impossible to solve to any reasonable accuracy for realistic grid resolutions in finite difference formulations. Following on from the work in Buckeridge and Scheichl (2010) we propose a numerical method for it based on a Krylov subspace method with a multigrid preconditioner. The problem of interest includes a constraint in the form of two‐dimensional elliptic solves embedded within the main three‐dimensional problem. Thus the discretised problem cannot be formulated as a simple linear equation system with a sparse system matrix (as usual in elliptic PDEs). Therefore, in order to precondition the system we apply the multigrid method in Buckeridge and Scheichl (2010) to a simplified form of the three‐dimensional operator (without the embedded two‐dimensional problems) leading to an asymptotically optimal convergence of the preconditioned Krylov subspace method. The solvers used at the Met Office typically take over 100 iterations to converge to a residual tolerance of 0.1 and fail to converge to a tolerance of 10−2. The method proposed in this paper, in contrast, can converge to a tolerance of 10−2 within 15 iterations on all typical grid resolutions used at the Met Office, and is convergent to a tolerance of 10−6. In addition, the method demonstrates almost optimal parallel scalability. Copyright © 2011 Royal Meteorological Society and British Crown Copyright, the Met Office