Figure 1
Spectra of a) a spherical harmonic coefficient simulated by an atmospheric model (James & James, 1989), b) zonally averaged SLP difference representing the Southern Annular Mode from the NCEP/NCAR reanalysis (solid black) and from models (gray) (Deser et al., 2012), c) the three components of the Lorenz’s 1963 model (von Storch, 2022), d) and e) current kinetic energy from instrumental records in the North Atlantic at 500 m and in the South Pacific at 1000 m (Ferrari & Wunsch, 2009). Using detrended time series (dashed black line in b) can be considered as a way to eliminate the influence from external forcings.
Figure 2
Scatter diagrams of Gτ,iτ against xiτ (dots) and the respective regression lines (black lines) for five values of τ (listed on the far left) and for the three Lorenz components (magenta, blue, green), as derived from n = 106 pairs of . , and are calculated following Eq. (A1) – Eq. (A4) in Appendix A. Numbers listed in each scatter diagram are values of and , where is the variance of and is the variance of . Points (xiτ,Gτ,iτ) are collected along a stationary Lorenz solution. A stationary Lorenz solution is obtained by first integrating the Lorenz model from an arbitrary initial state for a sufficiently long time. The integration is done using a Runge Kutta scheme with a time step of 0.01.
Figure 3
(top) and (bottom) for τ = 2 (left) and τ = 10000 (right) and for the three Lorenz components (magenta, blue and green) as functions of n, the number of pairs (xiτ,Gτ,iτ) used for their calculations. The calculation is carried out using an increment in n that equals one for 1 ≤ n ≤ 500 and equals 20 for 500 ≤ n ≤ 10000.
Figure 4
(top) and (bottom), with being set to the variance of each of the three Lorenz components (black lines) and to the variance of the solution of dx/dt = cos(2πt/P) with period P = 200 (orange line). The latter equals P2/(8π2) = 506.61. Colored dots are points (top) and points (bottom) with τ = 1,…,1000, each obtained using n = 106 pairs of (xiτ, Gτ,i) along a stationary Lorenz solution, with the colors (magenta, blue, and green) indicating the Lorenz components. Black dots are points with T = 1,2,…,P, obtained from (x(iT),GT(iT)) with i = 1,…, 5P. Both x(iT) and GT(iT) are calculated using the analytical expressions obtained from the cosine model. dT and are calculated using the regression defined in the same way as for the discrete solution.
Figure 5
and as functions of τ, derived using n = 105 pairs of (xiτ,Gτ,iτ) along a stationary Lorenz solution. and obtained from the first two Lorenz components (magenta, blue), which overlay each other, converge with increasing τ faster than those obtained from the third component (green). The calculation is done using an increment in τ that equals 10 for 1 ≤ τ ≤ 1001 and equals 200 for τ >1001.
Figure 6
Auto-correlation function of fluctuating component , defined as , for six values of τ, obtained for the three Lorenz components (magenta, blue, green) using n = 106 data points along a stationary Lorenz solution. is a function of k. The smallest non-zero time lag resolved by is obtained for k = 1, corresponding to a time lag of τ time steps.
Figure 7
Dissipation associated with integral forcing G1 (i.e. Gτ with τ = 1, solid lines) and damping amount due to differential forcing F (dashed lines) as functions of time increment Δt, for the three Lorenz components (magenta, blue, and green). The dissipation associated with G1 is quantified by . For a given value of Δt, is the regression slope obtained by regressing G1,i against xi using (xi, G1,i) with i = 1,…,106 along a Lorenz solution computed with this Δt. The damping amount due to F is quantified by ãΔt, where ã is the proportionality factor of the linear damping in the discretized Lorenz model. The values of ã differ slightly from a = –10, –1, –8/3 given in the Lorenz model. The difference results from the numerical scheme used, which is the fourth order Runge-Kutta scheme in this study. The two black lines are proportional to –Δt and –Δt2, respectively.
Figure 8
Same as Figure 2, but for the cosine model dx/dt = cos(2π t/P) with period P = 200 for six different values of T. Dots are the points (x(t), GT(t)) with t = iT, i = 0,1,… n, and n = 103. They overlap when the periods of (x(iT), GT(iT)), which vary with T, are shorter than n. Lines are regressions GT(iT) = cT+dT x(iT) obtained from the n points. Numbers listed are values of dT and with . Note that if T is a multiple of P/2, we have GT,iT = 0. Different from Figure 2, the symbol ^ is dropped, since for n that is a multiple of P, cT, dT, and do not change with increasing n.
Figure B1
A1 (top), A2 (middle), and A3 (bottom) as functions of n, derived for the three Lorenz components (magenta, blue and green) and for τ = 2 (left) and τ = 5000 (right). n is the number of consecutive data points along a stationary solution used to calculate A1, A2 and A3.