Articles | Volume 24, issue 3
Nonlin. Processes Geophys., 24, 351–366, 2017
Nonlin. Processes Geophys., 24, 351–366, 2017

Research article 19 Jul 2017

Research article | 19 Jul 2017

Controllability, not chaos, key criterion for ocean state estimation

Geoffrey Gebbie1 and Tsung-Lin Hsieh1,2,3 Geoffrey Gebbie and Tsung-Lin Hsieh
  • 1Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, MA, USA
  • 2Summer Student Fellow, Woods Hole Oceanographic Institution, Woods Hole, MA, USA
  • 3Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, NJ, USA

Abstract. The Lagrange multiplier method for combining observations and models (i.e., the adjoint method or 4D-VAR) has been avoided or approximated when the numerical model is highly nonlinear or chaotic. This approach has been adopted primarily due to difficulties in the initialization of low-dimensional chaotic models, where the search for optimal initial conditions by gradient-descent algorithms is hampered by multiple local minima. Although initialization is an important task for numerical weather prediction, ocean state estimation usually demands an additional task – a solution of the time-dependent surface boundary conditions that result from atmosphere–ocean interaction. Here, we apply the Lagrange multiplier method to an analogous boundary control problem, tracking the trajectory of the forced chaotic pendulum. Contrary to previous assertions, it is demonstrated that the Lagrange multiplier method can track multiple chaotic transitions through time, so long as the boundary conditions render the system controllable. Thus, the nonlinear timescale poses no limit to the time interval for successful Lagrange multiplier-based estimation. That the key criterion is controllability, not a pure measure of dynamical stability or chaos, illustrates the similarities between the Lagrange multiplier method and other state estimation methods. The results with the chaotic pendulum suggest that nonlinearity should not be a fundamental obstacle to ocean state estimation with eddy-resolving models, especially when using an improved first-guess trajectory.

Short summary
The best reconstructions of the past ocean state involve the statistical combination of numerical models and observations; however, the computationally efficient method that produces physically interpretable fields is thought to not be applicable to chaotic dynamical systems, such as ocean models with eddies. Here we use a model of the chaotic, forced pendulum to show that the most popular existing method is successful so long as there are enough uncertain boundary conditions through time.