Reducing stochasticity in the North Atlantic Oscillation index with coupled Langevin equations

We present a critical investigation of the functional relationship between the two pressure time series routinely used to define the index characterizing the North Atlantic Oscillation (NAO), well known to regulate global climate variability and change. First, by a standard Markov analysis we show that the standard NAO index based on the pressure difference is not optimal in the sense of producing sufficiently reliable forecasts because it contains a dominating stochastic term in the corresponding Langevin equation. Then, we introduce a variationally optimized Markov analysis involving two coupled Langevin equations tailored to produce a NAO quasi-index having the desired minimum possible stochasticity. The variationally optimized Markov analysis is very general and can be applied in other physical situations involving two or more time series.

Figure 1

Time series of the monthly NAO index $\mathcal{N}$ as defined in Eq. (1), computed from measured data [11].

Figure 2

The correlation function $C\left(t\right)$ and mutual information $I\left(t\right)$ for the standard monthly NAO index $\mathcal{N}$ as defined in Eq. (1). As one sees, correlations may be neglected for $t>1$ (see text).

Figure 3

The first and second conditional moments, $M^{\left(1\right)}$ and $M^{\left(2\right)}$ for the time series $\mathcal{N}$ shown in Fig. 1. Here $t_{\mathit{ref}}=32$ and ${\mathcal{N}}_{2t}=\overline{\mathcal{N}}$ (top), ${\mathcal{N}}_{2t}=\overline{\mathcal{N}}-\sigma$ (center), and ${\mathcal{N}}_{2t}=\overline{\mathcal{N}}+\sigma$ (bottom). Fitting the curves beyond the Markov length ${\mathcal{l}}_M=16$ and intersecting the fits with the vertical axis $\Delta t=0$ yields the corresponding Kramers-Moyal coefficient (see text).

Figure 4

The drift and diffusion coefficients, $D^{\left(1\right)}$ and $D^{\left(2\right)}$, obtained from linear fits of the corresponding conditional moments (see dashed lines in Fig. 3). Here ${\mathcal{N}}_{\tau }$ is given in the same units as the NAO index and varies within the range $[-6,6]$ for $t_{\mathit{ref}}=32$ (top), $t_{\mathit{ref}}=64$ (center), and $t_{\mathit{ref}}=96$ (bottom).

Figure 5

Comparison between the probability density functions of the auxiliary time series for NAO (circles) and the integration of the Fokker-Planck equation (solid lines), starting with an initial PDF at $t_{\mathit{ref}}=32$. The integration begins at $t=t_{\mathit{ref}}$ $(\Delta \tau =0)$ and PDFs are plotted at time steps $t=32,16,8,4,2$, i.e., at $\Delta \tau ={\log }_2(t_{\mathit{ref}}∕t)=0,1,2,3,4$, respectively. For each time step the PDF is plotted in linear (left) and logarithmic (right) scales. The good agreement seen indicates the Markov length ${\mathcal{l}}_M=16$ used in the simulation to be a good choice and the process to be almost Gaussian in the range $[-2,2]$ (see text).

Figure 6

(Color online) The Kramers-Moyal coefficients for the two coupled Langevin equations in (9): (a) $D_1^{\left(1\right)}$, (b) $D_{11}^{\left(2\right)}$, (c) $D_{12}^{\left(2\right)}$, (d) $D_2^{\left(1\right)}$, (e) $D_{21}^{\left(2\right)}$, and (f) $D_{22}^{\left(2\right)}$. Each plot shows a surface obtained from Eqs. (C1) in Appendix and the corresponding fitted surface shifted for better inspection.

Figure 7

Function $L=\mathcal{A}-\mathcal{B}-\mathcal{F}$ [see Eqs. (20)] as a function of $P_1$ and $P_2$ in the range $[-1,1]\times [-1,1]$. The contour line indicates $L=0$ for which Eq. (19) holds. Anywhere else the differential equation is not satisfied, meaning that the standard NAO index is not the one with minimal stochasticity (see text).

Figure 8

Solution $N_1(P_1,P_2)$ of Eq. (19) with linear boundary conditions (see text), minimizing the variational in Eq. (14). In (a) and (b) the surface is shown in two different views, to emphasize its curvature, showing that $N_1$ is not linear in $P_1$ and $P_2$ as the standard NAO index. In (c) one sees the surface projected in the $(P_1,P_2)$ plane with contour lines, emphasizing the asymmetry around the secondary diagonal $P_1=-P_2$.

Figure 9

Kramers-Moyal coefficients, as a function of the NAO index, for the optimized index (diamonds) compared with the standard index (circles): (a) Drift coefficient $D^{\left(1\right)}$ and (b) diffusion coefficient $D^{\left(2\right)}$. Here, the reference is $t_{\mathit{ref}}=32$.