Abstract
Conditional nonlinear optimal perturbation (CNOP) is an extension of the linear singular vector technique in the nonlinear regime. It represents the initial perturbation that is subjected to a given physical constraint, and results in the largest nonlinear evolution at the prediction time. CNOP-type errors play an important role in the predictability of weather and climate. Generally, when calculating CNOP in a complicated numerical model, we need the gradient of the objective function with respect to the initial perturbations to provide the descent direction for searching the phase space. The adjoint technique is widely used to calculate the gradient of the objective function. However, it is difficult and cumbersome to construct the adjoint model of a complicated numerical model, which imposes a limitation on the application of CNOP. Based on previous research, this study proposes a new ensemble projection algorithm based on singular vector decomposition (SVD). The new algorithm avoids the localization procedure of previous ensemble projection algorithms, and overcomes the uncertainty caused by choosing the localization radius empirically. The new algorithm is applied to calculate the CNOP in an intermediate forecasting model. The results show that the CNOP obtained by the new ensemble-based algorithm can effectively approximate that calculated by the adjoint algorithm, and retains the general spatial characteristics of the latter. Hence, the new SVD-based ensemble projection algorithm proposed in this study is an effective method of approximating the CNOP.
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References
Birgin E G, Martínez J M, Raydan M. 2000. Nonmonotone spectral projected gradient methods on convex sets. SIAM J Opt, 10: 1196–1211
Blumenthal M B. 1991. Predictability of a coupled ocean-atmosphere model. J Clim, 4: 766–784
Buizza R, Montani A. 1999. Targeting observations using singular vectors. J Atmos Sci, 56: 2965–2985
Cai M, Kalnay E, Toth Z. 2003. Bred vectors of the Zebiak-Cane model and their potential application to ENSO predictions. J Clim, 16: 40–56
Duan W S, Mu M, Wang B. 2004. Conditional nonlinear optimal perturbation as the optimal precursors for El Niño-Southern Oscillation events. J Geophys Res, 109: 1984–2012
Duan W S, Xue F, Mu M. 2009. Investigating a nonlinear characteristic of ENSO events by conditional nonlinear optimal perturbation. Atmos Res, 94: 10–18
Duan W S, Yu Y S, Xu H, et al. 2012. Behaviors of nonlinearities modulating El Niño events induced by optimal precursory disturbance. Climate Dyn, 40: 1399–1413
Foias C, Teman R. 1997. Structure of the set of stationary solution of the Novier-Stokes equations. Commun Pure Appl Math, 30: 149–164
Houtekamer P L, Mitchell H L. 2001. A sequential ensemble Kalman filter for atmospheric data assimilation. Mon Weather Rev, 129: 123–137
Lorenz E N. 1965. A study of the predictability of a 28-variable atmospheric model. Tellus, 17: 321–333
Mantua N J, Battisti D S. 1995. Aperiodic variability in the Zebiak-Cane coupled ocean-atmosphere model: Air-sea interactions in the western equatorial Pacific. J Clim, 8: 2897–2927
Moore A M, Kleeman R. 1996. The dynamics of error growth and predictability in a coupled model of ENSO. Quart J Roy Meteor Soc, 122: 1405–1446
Mu M, Duan W S, Wang B. 2003. Conditional nonlinear optimal perturbation and its applications. Nonlinear Process Geophys, 10: 493–501
Mu M, Duan W S. 2003. A new approach to studying ENSO predictability: Conditional nonlinear optimal perturbation. Chin Sci Bull, 48: 1045–1047
Mu M, Xu H, Duan W S. 2007. A kind of initial errors related to “spring predictability barrier” for El Niño events in Zebiak-Cane model. Geophys Res Lett, 34: L03709, doi: 10.1029/2006GL027412
Osborne AR, Pastorello A. 1993. Simultaneous occurrence of low-dimensional chaos and colored random noise in nonlinear physical systems. Phys Lett A, 181: 159–171
Palmer T N, Gelaro R, Barkmeijer J, et al. 1998. Singular vectors, metrics, and adaptive observations. J Atmos Sci, 55: 633–653
Qin X H, Duan W S, Mu M. 2013. Conditions under which CNOP sensitivity is valid for tropical cyclone adaptive observations. Quart J Roy Meteor Soc, 139: 1544–1554
Qin X H, Mu M. 2011. Influence of conditional nonlinear optimal perturbations sensitivity on typhoon track forecasts. Quart J Roy Meteor Soc, 138: 185–197
Sun G D, Mu M. 2011. Nonlinearly combined impacts of initial perturbation from human activities and parameter perturbation from climate change on the grassland ecosystem. Nonlinear Process Geophys, 18: 883–893
Teman R. 1991. Approximation of attractors, large eddy simulations and multiscale methods. Proc R Soc Lond A, 434: 23–29
Thompson C J, Battisti D S. 1995. A linear stochastic dynamical model of ENSO. Part I: Model development. J Clim, 8: 2897–2927
Thompson C J. 1998. Initial conditions for optimal growth in a coupled ocean-atmosphere model of ENSO. J Atmos Sci, 55: 537–557
Wang B, Tan X W. 2010. Conditional nonlinear optimal perturbations: Adjoint-free calculation method and preliminary test. Mon Weather Rev, 138: 1043–1049
Yu Y S, Duan W S, Xu H. 2009. Dynamics of nonlinear error growth and season-dependent predictability of El Niño events in the Zebiak-Cane model. Quart J Roy Meteor Soc, 135: 2146–2160
Yu Y S, Mu M, Duan W S, et al. 2012. Contribution of the location and spatial pattern of initial error to uncertainties in El Nino predictions. J Geophys Res, 117: 1–13
Zebiak S E, Cane M A. 1987. A model El Niño-Southern oscillation. Mon Weather Rev, 115: 2262–2278
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Chen, L., Duan, W. & Xu, H. A SVD-based ensemble projection algorithm for calculating the conditional nonlinear optimal perturbation. Sci. China Earth Sci. 58, 385–394 (2015). https://doi.org/10.1007/s11430-014-4991-4
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DOI: https://doi.org/10.1007/s11430-014-4991-4