The Western Mediterranean summer variability and its feedbacks

Abstract

The anomalous climatic variability of the Western Mediterranean in summer, its relationships with the large scale climatic teleconnection modes and its feedbacks from some of these modes are the targets of this study. The most important trait of this variability is the recurrence of warm and cold episodes, that take place at 2–4 year intervals, and which are monitored in the Western Mediterranean Index. We find that the Western Mediterranean events are part of a basin scale mode, and are related to the previous spring atmospheric anomalies. These anomalies are related mainly to the Pacific North America teleconnection pattern and the North Atlantic Oscillation, but also to a number of other climatic modes, connected with the previous two, as the Southern Oscillation, the Indian Core Monsoon and the Scandinavian teleconnection pattern. We identify the main spatial and temporal traits of the Western Mediterranean summer variability, the physical mechanisms at play in the generation of the events and their impacts. Considering the Atlantic Ocean, the Mediterranean events influence the sea surface temperature in the southeastern part of the North Atlantic Gyre. Additionally, they are significantly related to summer precipitation anomalies of the opposite sign in the Baltic basin (Central Germany and Poland) and near the Black Sea. We then estimate the mutual influence that the anomalous previous state of the Western Mediterranean, of the Pacific North America teleconnection pattern and of the North Atlantic Oscillation have on their summer conditions using a simple stochastic model. As the summer Western Mediterranean events have an influence on a part of the Baltic basin, we propose a second stochastic model in order to investigate if thereafter the Baltic basin variability will feedback on the Western Mediterranean sea surface temperature anomalies. Among the variables included in the second model are, in addition to the Western Mediterranean previous state, that of the Baltic Sea and of the Scandinavian teleconnection pattern. From each of the feedback matrices, a linear statistical analysis extracts spatial patterns whose evolution in time exhibits predictive capabilities for the Western Mediterranean evolution in summer and autumn that are above those of persistence, and that could be improved.

Appendix

1.1 The principal oscillation pattern (POP) analysis

Let a system be described by a state vector (${\bf z} =\{ z_j,\; j=1,n\}$) where the z _j represent the centered values of one or several variables. Let (${\bf z}_{i},\; i=1,m$) represent m samples of the state vector z, obtained at m different times. Let the time evolution of the system be represented by a linear stochastic model

$$ \frac{d{\bf z}_j}{dt} = A{\bf z} + {\bf n} $$

(4)

where A is the dynamical or feedback matrix that represents the response of the system to changes produced by the forcing, and n represents the ’noise’, that we assume to be stationary, gaussian and delta correlated.

The finite difference equivalent of Eq. (4) is

$$ \frac{{\bf z}_{i+1}-{\bf z}_{i}}{\Updelta t} = A{\bf z}_{i} + {\bf n}_{i} $$

(5)

with $\Updelta t=1$. From this equation the dynamical matrix can be determined as

$$ A=\frac{<{\bf z}_{i+1}{\bf z}_{i}> -<{\bf z}_{i}{\bf z}_{i}>}{<{\bf z}_{i}{\bf z}_{i}>} $$

(6)

where the angular brackets denote averages to m samples. This can be written in terms of C ₁ and C ₀, covariance matrices at lag 1 and 0, respectively, as:

$$ A=\frac{C_1 - C_0}{C_0} $$

(7)

In this way, we take out of the feedbacks coefficients the part of the cross-covariance between each pair of variables that is determined by their own autocovariances, and that introduced by the common cross-covariance that they have with a third variable. Solutions to the deterministic part of Eq. (5) can be written as

$$ {\bf z}_i = \sum_{k=1}^n C_k e^{\mu_k \Updelta t_i} {\bf u}_k;\quad \Updelta t_i = t_i -t_0, i=1,k $$

(8)

where the characteristic exponents μ _k can be obtained from the eigenvalues λ _k of the dynamical matrix $A, \mu_k= \ln(\lambda_k)$, and the u _k are the corresponding eigenvectors.

As the dynamical matrix A is non-symmetric, its eigenvectors and eigenvalues will be in general complex. Also, due to the assumption of stationarity, the absolute values |λ _k | are lower than 1, and therefore the exponents μ _k are negative. The oscillations are always damped. If the number of complex eigenvalues is l _i , Eq. (8) can be written as

$$ z_i = \sum _{k=1}^{l_i} c_k e^{\mu_k\Updelta t_i}[ \hbox{Re}({\bf u}_k) + \imath\,\hbox{Im} ({\bf u}_k)] + \sum_{l_i+1}^n c_k e^{\mu_k \Updelta t_i} {\bf u}_k. $$

(9)

If we denote γ _k ≡Re(μ _k ) and ω _k ≡Im(μ _k ), it is easy to work out that for each pair of complex conjugated eigenvalues, μ _k and μ _k+1 we will have a pair of real patterns p _k  = Re(u _k ) and q _k  = Im(u _k ) associated to a frequency ω _k , a period T _k and a damping factor γ _k . These are the Principal Oscillation Patterns (POP) of the system at that frequency. For each pair of POP derived from a pair of complex eigenvectors, Eq. (9) prescribes the following evolution in time:

$$ {\bf p}_k \overset{T_k/4}\longrightarrow -{\bf q}_k \overset{T_k/4}\longrightarrow -{\bf p}_k \overset{T_k/4}\longrightarrow {\bf q}_k $$

(10)

The evolution in time for each pair can also be obtained empirically from the z vectors. If we collect the eigenvectors or patterns into a matrix W, with elements (w _j  = p _j , w _j+1 = q _j , j = 2k − 1, k = 1, l _i ) and (w _j  = Re(u _k ), k = l _i  + 1, n), then z _i  = W s _i and the time coefficients s _i can be obtained as

$$ {\bf s}_i = W^{-1}{\bf z}_i $$

(11)