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Optimisation of simplified GCMs using circulation indices and maximum entropy production

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Abstract

Two kinds of objective functions for parameter optimisation in simplified general circulation models (SGCMs) are introduced and tested with an SGCM employing linear parameterisations for diabatic heating, surface friction and horizontal diffusion. (a) A set of circulation indices is introduced to characterise the zonal mean primary and secondary circulation and the global energetics. The objective function is then given by the distance between the modelled and a reference (e.g. observed) circulation in a state space spanned by these indices. (b) The global and time mean entropy production and kinetic energy dissipation are introduced as additional objective functions, following the maximum entropy production principle. It is found that both methods lead to optimal parameter values close to the standard configuration of the model, though the method of the second kind is restricted to those model parameters associated with internal processes such as heat and momentum fluxes.

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Acknowledgments

This work was funded by the Deutsche Forschungsgemeinschaft, project DFG FR 450 / 7-2. We thank Frank Lunkeit for helpful comments, Axel Kleidon for discussion of the MEP principle and also three anonymous reviewers for their valuable comments on the first draft of the manuscript.

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Correspondence to Torben Kunz.

Appendix

Appendix

The zonal mean meridional mass streamfunction ψ is defined by (Peixoto and Oort 1992):

$$ \left[{\bar{v}}\right] = \frac{g}{2\pi a\cos\phi} \frac{\partial {\psi}}{\partial {p}} $$
(10)
$$ \left[{\bar{\omega}}\right] = -\frac{g}{2\pi a^2\cos\phi} \frac{\partial {\psi}}{\partial {\phi}}, $$
(11)

with ψ = 0 at p = 0, and where \(\left[{\bar{v}}\right]\) and \(\left[\bar{\omega}\right]\) denote the time and zonal mean meridional and vertical velocity components, respectively, a the Earth’s radius, g the gravitational acceleration, ϕ is latitude and p pressure.

The energy conversions of the Lorenz energy cycle as outlined by Ulbrich and Speth (1991) and Arpe et al. (1986) are given by:

$$ \langle A Z\to A E\rangle = -\frac{\gamma} {g}\left([v^*T^*]\frac{1}{a}\frac{\partial {[T]}}{\partial {\phi}}+[\omega^*T^*]\left(\frac{\partial}{\partial {p}}([T]-\{T\}) - \frac{R}{c_p p}([T]-\{T\})\right)\right) $$
(12)
$$ \langle A E\to K E\rangle = -[\omega^*T_v^*]\frac{R}{g p} $$
(13)
$$ \langle K E\to K Z\rangle = \frac{1}{g}\left([u^*v^*]\frac{1}{a}\frac{\partial {[u]}} {\partial {\phi}}+[u^*v^*][u]\frac{\tan\phi}{a}+[v^*v^*]\frac{1} {a}\frac{\partial {[v]}}{\partial {\phi}} -[u^*u^*][v]\frac{\tan\phi}{a}+[\omega^*u^*]\frac{\partial {[u]}} {\partial {p}}+[\omega^*v^*]\frac{\partial {[v]}}{\partial {p}}\right) $$
(14)
$$ \langle A Z\to K Z\rangle = -([\omega]-\{\omega\})([T_v]-\{T_v\})\frac{R}{g p} $$
(15)

with virtual temperature T v , specific heat capacity at constant pressure c p , gas constant of dry air R, zonal wind u, zonal mean [], deviation from zonal mean * (eddy part), global horizontal mean {} and the stability parameter

$$ \gamma = -\frac{R}{p}\left(\frac{\partial {[T]}}{\partial {p}}-\frac{R}{c_p}\frac{[T]}{p}\right)^{-1}. $$
(16)

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Kunz, T., Fraedrich, K. & Kirk, E. Optimisation of simplified GCMs using circulation indices and maximum entropy production. Clim Dyn 30, 803–813 (2008). https://doi.org/10.1007/s00382-007-0325-y

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