Fluctuation analysis of the atmospheric energy cycle

The atmosphere gains available potential energy by solar radiation and dissipates kinetic energy mainly in the atmospheric boundary layer. We analyze the fluctuations of the global mean energy cycle defined by Lorenz in a simulation with a simplified hydrostatic model. The energy current densities are well approximated by the generalized Gumbel distribution and the Generalized Extreme Value (GEV) distribution. In an attempt to assess the fluctuation relation of Evans, Cohen, and Morriss we define entropy production by the injected power and use the GEV location parameter as a reference state. The fluctuation ratio reveals a linear behavior in a finite range.

Figure 1

Northern hemispheric relative vorticity in the midtroposphere (500 hPa) $[{10}^{-5}$/s].

Figure 2

Lorenz energy cycle with energy compartments (boxes) and energy currents (arrows). Available potential energy is $P$, kinetic energy is $K$, forcing is $R$, and dissipation is $D$; the zonal means are denoted by $m$ and eddies by $e$. Internal conversions are $C(P_m,P_e),C(P_e,K_e),C(K_e,K_m)$, and $C(K_m,P_m)$. Intense currents are thick, moderate thin, and weak dotted.

Figure 3

Normalized histograms of energy input and dissipation: (a) zonal mean forcing (injected power) $R_m$, (b) negative eddy forcing $-R_e$, (c) zonal mean dissipation $D_m$, (d) eddy dissipation $D_e$. The solid (red) line is a GEV fit, and the dashed (blue) line a generalized Gumbel (GG) fit. The vertical lines indicate the GEV-location parameters $\mu$ (solid, black) and the means $m$ (dotted, black).

Figure 4

Normalized histograms of internal currents (conversions): (a) zonal mean to eddy available potential energy $C(P_m,P_e)$, (b) eddy available potential energy to eddy kinetic energy $C(P_e,K_e)$, (c) eddy kinetic energy to zonal mean kinetic energy $C(K_e,K_m)$, (d) zonal mean kinetic energy to zonal mean potential energy $C(K_m,P_m)$. The fits are as in Fig. 3.

Figure 5

Fluctuation ratio for the shift to the location parameters: (a) injected power $\sigma =R_m$ (defined as the entropy production), (b) eddy dissipation $D_e$, (c) conversion of zonally averaged potential to eddy potential energy $C(P_m,P_e)$, (d) i.i.d. GEV random variates $r$ with the distribution of $R_m$.

Figure 6

Fluctuation ratio for the shift to the means of the current distributions: (a) injected power $\sigma =R_m$, (b) eddy dissipation $D_e$, (c) zonally averaged potential to zonally averaged kinetic energy conversion $C(P_m,P_e)$, (d) i.i.d. GEV random variates $r$ with the distribution of $R_m$.