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Ensemble Dynamics and Bred Vectors

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  • 1 Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota
  • | 2 Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania
  • | 3 Department of Mathematics and Department of Physics, The University of Arizona, Tucson, Arizona
  • | 4 School of Mathematics, University of Minnesota, Minneapolis, Minnesota
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Abstract

The new concept of an ensemble bred vector (EBV) algorithm is introduced to assess the sensitivity of model outputs to changes in initial conditions for weather forecasting. The new algorithm is based on collective dynamics in essential ways. As such, it keeps important geometric features that are lost in the earlier bred vector (BV) algorithm. By construction, the EBV algorithm produces one or more dominant vectors and is less prone to spurious results than the BV algorithm. It retains the attractive features of the BV algorithm with regard to being able to handle legacy codes, with minimal additional coding.

The performance of the EBV algorithm is investigated by comparing it to the BV algorithm as well as the finite-time Lyapunov vectors. With the help of a continuous-time adaptation of these algorithms, a theoretical justification is given to the observed fact that the vectors produced by the BV, EBV algorithms, and the finite-time Lyapunov vectors are similar for small amplitudes. The continuum theory establishes the relationship between the two algorithms and general directional derivatives.

Numerical comparisons of BV and EBV for the three-equation Lorenz model and for a forced, dissipative partial differential equation of Cahn–Hilliard type that arises in modeling the thermohaline circulation demonstrate that the EBV yields a size-ordered description of the perturbation field and is more robust than the BV in the higher nonlinear regime. The EBV yields insight into the fractal structure of the Lorenz attractor and of the inertial manifold for the Cahn–Hilliard-type partial differential equation.

Current affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana.

Corresponding author address: Juan M. Restrepo, Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ 85721. E-mail: restrepo@math.arizona.edu

Abstract

The new concept of an ensemble bred vector (EBV) algorithm is introduced to assess the sensitivity of model outputs to changes in initial conditions for weather forecasting. The new algorithm is based on collective dynamics in essential ways. As such, it keeps important geometric features that are lost in the earlier bred vector (BV) algorithm. By construction, the EBV algorithm produces one or more dominant vectors and is less prone to spurious results than the BV algorithm. It retains the attractive features of the BV algorithm with regard to being able to handle legacy codes, with minimal additional coding.

The performance of the EBV algorithm is investigated by comparing it to the BV algorithm as well as the finite-time Lyapunov vectors. With the help of a continuous-time adaptation of these algorithms, a theoretical justification is given to the observed fact that the vectors produced by the BV, EBV algorithms, and the finite-time Lyapunov vectors are similar for small amplitudes. The continuum theory establishes the relationship between the two algorithms and general directional derivatives.

Numerical comparisons of BV and EBV for the three-equation Lorenz model and for a forced, dissipative partial differential equation of Cahn–Hilliard type that arises in modeling the thermohaline circulation demonstrate that the EBV yields a size-ordered description of the perturbation field and is more robust than the BV in the higher nonlinear regime. The EBV yields insight into the fractal structure of the Lorenz attractor and of the inertial manifold for the Cahn–Hilliard-type partial differential equation.

Current affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana.

Corresponding author address: Juan M. Restrepo, Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ 85721. E-mail: restrepo@math.arizona.edu
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