Articles | Volume 25, issue 3
Nonlin. Processes Geophys., 25, 481–495, 2018

Special issue: Numerical modeling, predictability and data assimilation in...

Nonlin. Processes Geophys., 25, 481–495, 2018

Research article 09 Jul 2018

Research article | 09 Jul 2018

Parametric covariance dynamics for the nonlinear diffusive Burgers equation

Olivier Pannekoucke1, Marc Bocquet2, and Richard Ménard3 Olivier Pannekoucke et al.
  • 1INPT-ENM, CNRM UMR 3589, Météo-France/CNRS, CERFACS, Toulouse, France
  • 2CEREA, Joint Laboratory École des Ponts ParisTech and EDF R&D, Université Paris-Est, Champs-sur-Marne, France
  • 3ARQI/Air Quality Research Division, Environment and Climate Change Canada, Dorval (Québec), Canada

Abstract. The parametric Kalman filter (PKF) is a computationally efficient alternative method to the ensemble Kalman filter. The PKF relies on an approximation of the error covariance matrix by a covariance model with a space–time evolving set of parameters. This study extends the PKF to nonlinear dynamics using the diffusive Burgers equation as an application, focusing on the forecast step of the assimilation cycle. The covariance model considered is based on the diffusion equation, with the diffusion tensor and the error variance as evolving parameters. An analytical derivation of the parameter dynamics highlights a closure issue. Therefore, a closure model is proposed based on the kurtosis of the local correlation functions. Numerical experiments compare the PKF forecast with the statistics obtained from a large ensemble of nonlinear forecasts. These experiments strengthen the closure model and demonstrate the ability of the PKF to reproduce the tangent linear covariance dynamics, at a low numerical cost.

Short summary
The forecast of weather prediction uncertainty is a real challenge and is crucial for risk management. However, uncertainty prediction is beyond the capacity of supercomputers, and improvements of the technology may not solve this issue. A new uncertainty prediction method is introduced which takes advantage of fluid equations to predict simple quantities which approximate real uncertainty but at a low numerical cost. A proof of concept is shown by an academic model derived from fluid dynamics.