Computationally efficient methods for climate model inversion

Abstract

The dynamical equations of a model are used to obtain the `forcing function', which is a representation of climate change drivers, from an observed climatic anomaly. This inversion problem is mathematically difficult because of the two-way interaction between the mean field and transient eddies; this is known as the turbulence closure problem. The first method that we explore for overcoming the closure problem involves iteratively nudging a climate simulation towards the observed climate. We demonstrate how this method is used to successfully calculate the climatic forcing function. The second method that we explore involves finding approximations to the turbulence closure problem. In this method, the transient eddy feedback term in the mean field equation is represented as a linear combination of the mean fields and a constant term. We demonstrate that the closure method yields a good approximation to the climatic forcing function. This forcing function is then used as an improved first estimate in the iterative method, thereby yielding a scheme that converges very quickly to the correct solution in only a few iteration steps. References G. C. Hegerl, T. R. Karl, M. Allen, N. L. Bindoff, N. Gillett, D. Karoly, X. Zhang and F. Zwiers. Climate change detection and attribution: beyond mean temperature signals. J. Clim. , 19 :5058--5077, 2006. doi:10.1175/JCLI3900.1 S. Corti, A. Giannini, S. Tibaldi and F. Molteni. Patterns of low-frequency variability in a three-level quasi-geostrophic model. Clim. Dyn. , 13 :883--904, 1997. doi:10.1007/s003820050203 C. E. Leith. Climate response and fluctuation dissipation. J. Atmos. Sci. , 32 :2022--2026, 1975. 2.0.CO;2>doi:10.1175/1520-0469(1975)032 2.0.CO;2 T. L. Bell. Climate sensitivity from fluctuation dissipation: some simple model results. J. Atmos. Sci. , 37 :1700--1707, 1980. 2.0.CO;2>doi:10.1175/1520-0469(1980)037 2.0.CO;2 A. S. Gritsun. Fluctuation-dissipation theorem on attractors of atmospheric models. Russ. J. Numer. Anal. Math. Modelling , 16 :115--133, 2001. A. Gritsun and G. Branstator. Climate response using a three-dimensional operator based on the fluctuation-dissipation theorem. J. Atmos. Sci. , 64 :2558--2575, 2007. doi:10.1175/JAS3943.1 J. S. Frederiksen. Subgrid-scale parametrisations of eddy-topographic force, eddy viscosity, and stochastic backscatter for flow over topography. J. Atmos. Sci. , 56 :1481--1494, 1999. 2.0.CO;2>doi:10.1175/1520-0469(1999)056 2.0.CO;2 J. S. Frederiksen. Statistical dynamical closures and subgrid modeling for inhomogeneous QG and 3D turbulence. Entropy , 14 :32--57, 2012. doi:10.3390/e14010032 J. S. Frederiksen. Self-energy closure for inhomogeneous turbulent flows and subgrid modeling. Entropy , 14 :769--799, 2012. doi:10.3390/e14040769 R. Salmon. Lectures on Geophysical Fluid Dynamics , 1998. Oxford University Press. M. J. Zidikheri and J. S. Frederiksen. Inverse method for attribution of climate change. ANZIAM J. , 52 :C823--C836, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3930 . I. M. Held and M. J. Suarez. A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Amer. Meteor, Soc. , 75 :1825--1830, 1994. 2.0.CO;2>doi:10.1175/1520-0477(1994)075 2.0.CO;2 J. Namias, X. Yuan and D. R. Cayan. Persistence of North Pacific sea surface temperature and atmospheric flow patterns. J. Clim. , 1 :682--703, 1988. 2.0.CO;2>doi:10.1175/1520-0442(1988)001 2.0.CO;2