An attempt to deconstruct the Atlantic Multidecadal Oscillation

Abstract

The link between the Atlantic Multidecadal Oscillation (AMO) and low-frequency changes of the Atlantic Meridional Overturning Circulation (AMOC) is investigated in three historical and five control simulations with different climate models. An AMOC intensification is followed by a positive AMO phase in each case, but the time lag and the strength of the AMO–AMOC link depend on the model and the type of simulation. In historical simulations, the link is sensitive to the method used to remove the influence of external and anthropogenic forcing from the sea surface temperature (SST) before defining the AMO. Subtracting the regression onto the global mean SST leads to better correlations between the AMO and the AMOC than linear or quadratic detrending, or removing the global mean SST, but a dynamical filter based on linear inverse modeling (LIM) yields even slightly higher correlations. The LIM filter, which decomposes the SST field into non-orthogonal normal modes that may have a physical interpretation, allows investigating whether removing Pacific links from SST improves the AMOC–AMO correlation. In several cases, there is a small improvement when removing the links to the El Niño Southern Oscillation, but the correlation becomes weaker in one historical simulation, so no firm conclusion can be drawn. Additionally removing the modes associated with the Pacific decadal variability strongly degrades the representation of AMOC changes by the AMO in one model, and it tends to reduce the AMOC–AMO correlation in most others, reflecting the strong relation between the Pacific and the Atlantic at decadal scales. The LIM-based filter is finally applied to observed SSTs, confirming that the AMO amplitude is smaller and its recent positive phase weaker than when the global effects are represented by a linear trend. When the global signal is removed, the observed AMO leads the Pacific Decadal Oscillation, but does not significantly lag it, as suggested earlier, stressing the need to carefully remove global changes when investigating low-frequency interbasin connections.

Appendices

Appendix 1: Testing the validity of the LIM hypothesis

If LIM is a good approximation of the system, the matrix B should not depend on the lag τ used to determine it, ie determining B from $\varvec{C}\left( {\tau_{1} } \right) = e^{{\tau_{1} \varvec{B}}}$ or $\varvec{C}\left( {\tau_{2} } \right) = e^{{\tau_{2} \varvec{B}}}$, with τ ₁ ≠ τ ₂ and C(l) the sample covariance matrix of x (in EOF space), should lead to the same results. Hence, a simple test of the accuracy of the LIM model is to compare B estimated at different lags (tau test, Penland and Sardeshmukh 1995). However, as discussed by Newman (2007), if B can only be determined at one lag because of the Nyquist problem, an alternate test is to verify that the empirically derived matrix B reproduces the sample covariance C at a longer lag l (Winkler et al. 2001). If the model is valid, the covariance matrix at lag l derived from the estimated value of B (C _lim  = e ^lB C(0)) should not substantially differ from the sample covariance matrix C(l). This can be evaluated by computing (with tr being the trace):

$$d\left( l \right) = \frac{{tr(\varvec{C}_{lim} \left( l \right)) - tr(\varvec{C}\left( l \right))}}{{tr(\varvec{C}\left( l \right))}} $$

(5)

As it has long been shown that tropical SST anomalies are well represented by LIM (e.g. Penland and Sardeshmukh 1995), we simply compared in two cases d(l) with a distance d _ref (l) obtained by applying LIM only in the tropical strip, retaining the same number of tropical EOF than in the “global” B. This was done for the observations and the IPSLCM5 control simulation, using l = 1 to 30. In the observations, d ranged between −0.8 and 0.1 and d _ref (l) between −1 and 0.3, depending on the lag. In the IPSLCM5 control simulation, d was in the interval [−1.3,1] and d _ref (l) in [−1.5,1.5]. Thus, in both cases, d and d _ref (l) were of the same order of magnitude, broadly supporting the validity of LIM in the global domain.

Appendix 2: Removing ENSO using LIM

The non-orthogonality of the B eigenmodes allows a constructive interference of several modes, leading to a transient amplification of the system. Penland and Sardeshmukh (1995) showed that the optimal SST perturbations leading to the maximum amplification of SST anomaly variance in the tropical strip evolved into a structure strongly reminiscent of an ENSO event after 7 months. ENSO could then be filtered out by selecting the empirical eigenmodes that most significantly contributed to the growth (Penland and Matrosova 2006). Here, the LIM modes are defined in a broader domain extending from 20°S to 70°N, but we only maximize the amplification of the SST variance in the Tropics (20°S–20°N), where ENSO is strongest. The tropical SST variance can be measured by the norm vector y _N = y ^T Ny when N is a positive definite hermitian form (normalized to have unit determinant) defined by N = (WE)^T(WE), where the matrix E of size (n, p) contains the patterns of the p retained EOF restricted to the n grid points of the Tropics, and W is a matrix of area weighting, which takes into account the earth curvature. The amplification μ of the system over a time τ is then defined by

$$\mu \left( \tau \right) = \frac{{\|\varvec{x}(\tau )_{N}^{2}\| }}{{\|\varvec{x}(0)_{N}^{2}\| }} = \frac{{\varvec{x}\left( \tau \right)^{T} \varvec{Nx}(\tau )}}{{\varvec{x}\left( 0 \right)^{T} \varvec{x}(0)}} = \frac{{\varvec{x}\left( 0 \right)^{T} \varvec{G}\left(\varvec{\tau}\right)^{\varvec{T}} \varvec{NG}(\varvec{\tau})\varvec{x}(0)}}{{\varvec{x}\left( 0 \right)^{T} \varvec{x}(0)}} $$

(6)

We thus search for the initial structure with small SST anomaly variance in the entire domain leading to the largest SST anomaly variance in the tropical strip.

The optimal initial condition x(0) that maximizes $\|\varvec{x}(\tau )\|_{N}^{2}$ subject to x(0) having unit norm under the L ₂ norm (ie x(0)^T x(0) = 1) is given by

$$\mathop {\hbox{max} }\limits_{{\varvec{x}(0)}} \left\{ {\varvec{x}\left( 0 \right)^{T} \varvec{G}\left(\varvec{\tau}\right)^{\varvec{T}} \varvec{NG}\left(\varvec{\tau}\right)\varvec{x}\left( 0 \right) + \gamma \varvec{x}\left( 0 \right)^{T} \varvec{x}(0)} \right\} $$

(7)

which leads to the generalized eigenvalue problem:

$$\varvec{G}\left(\varvec{\tau}\right)^{\varvec{T}} \varvec{NG}\left(\varvec{\tau}\right)\varvec{x}\left( 0 \right) = \gamma \varvec{x}(0) $$

(8)

with the optimal initial structure normalized such as x(0)^T x(0) = 1. The eigenvector φ1(τ) of (8) associated with the largest eigenvalue β ₁(τ) is the initial spatial structure leading to the maximum amplification after a time τ. The function β ₁(τ), called the maximum amplification curve, quantifies the maximum growth possible over an interval τ in the absence of forcing. The optimal initial structure is given by x(0) = φ ₁(τ _m ) with τ _m  = argmax(φ ₁(τ)), leading to the maximum amplification x(τ _m) = G(τ _m )x(0) = G(τ _m )φ ₁(τ _m ) in τ _m  months.

In all models, the optimal initial structure shows strong SST anomalies in the eastern equatorial Pacific with a maximum generally found between 110° and 130°W, as in the observations. This is illustrated for HadCM3 in Fig. 13, where hints of the seasonal footprinting mechanism (Vimont et al. 2003) can be seen in the eastern subtropical Pacific. The optimal growth occurs after 6 or 9 months depending on the model, showing very large SST anomalies spanning the tropical Pacific and a horseshoe pattern in the North Pacific, reminiscent of a mature ENSO event.

Fig. 13

Maximum Amplification curve (top left) and projection of normal modes onto the optimal initial structure (top right) in HadCM3. Optimal initial structure (middle) leading to maximum amplification (bottom) of SST variance in the tropical strip

The contribution from each eigenmode u _i of B to the optimal initial structure can be estimated by the magnitude of its projection onto the corresponding modal adjoint, the eigenvector v _i of B ^T, which has the same eigenvalue as u _i . Eigenmodes corresponding to global SST changes and Pacific decadal changes may contribute to the optimal initial structure, but cannot be considered as ENSO modes. Hence, modes having a larger period than 8 yr or a decay time larger than 24 months were not classified as ENSO modes. As discussed in Penland and Matrosova (2006), the identification of ENSO modes is somewhat subjective, and this holds for the Pacific decadal modes. Choosing too many modes may add noise in the North Atlantic SSTs and include signals that are involved in the link between the AMO and the AMOC, while choosing too few modes may miss a significant part of the ENSO signal.

In all simulations and in the observations, the reconstructed ENSO indices correlated well with the traditional Nino 1.2 and Nino 3.4 indices, r ranging between about 0.7 and 0.9, and the associated SST patterns were very similar, although their amplitude was generally overestimated (see Marini 2011 for more details). Adding more oscillatory modes fitting our criteria to the ENSO subset leads to realistic amplitudes, but more noisy patterns, so that the link between the ENSO-unrelated AMO and the AMOC was deteriorated. This supports our choice of a limited number of ENSO modes.