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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union

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https://doi.org/10.5194/npg-2016-57
© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.
Research article
30 Sep 2016
Review status
A revision of this discussion paper was accepted for the journal Nonlinear Processes in Geophysics (NPG) and is expected to appear here in due course.
Controllability, not chaos, key criterion for ocean state estimation
Geoffrey Gebbie1 and Tsung-Lin Hsieh1,2,3 1Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, MA, USA
2Summer Student Fellow, Woods Hole Oceanographic Institution, Woods Hole, MA, USA
3Princeton 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 algorithm is hampered by multiple local minima. Although initialization is an important task for numerical weather prediction, ocean state estimation usually demands an additional task – 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 there is no fundamental obstacle to ocean state estimation with eddy-resolving, highly-nonlinear models, especially when using an improved first-guess trajectory.

Citation: Gebbie, G. and Hsieh, T.-L.: Controllability, not chaos, key criterion for ocean state estimation, Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npg-2016-57, in review, 2016.
Geoffrey Gebbie and Tsung-Lin Hsieh
Geoffrey Gebbie and Tsung-Lin Hsieh

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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.
The best reconstructions of the past ocean state involve the statistical combination of...
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