Nonlinear relation of the Arctic oscillation with the quasi-biennial oscillation

Abstract

A nonlinear principal component analysis (NLPCA) is applied to a set of monthly mean time series from January 1956 to December 2007 consisting of the Arctic oscillation (AO) index derived from 1,000-hPa geopotential height anomalies poleward of 20°N latitude and the zonal winds observed at seven pressure levels between 10 and 70 hPa in the equatorial stratosphere to investigate the relation of the AO with the quasi-biennial oscillation (QBO). The NLPCA is conducted using a new, compact neural network model. The NLPCA modeling of the dataset of the AO index and QBO winds offers a clear picture of the relation between the two oscillations. In particular, the phase of covariation of the oscillations defined by the two nonlinear principal components of the dataset progresses with a predominant 28.4-month periodicity. This predominant cycle is modulated by an 11-year cycle. The variation of the AO index with the QBO phase also shows that the average AO index is positive when the westerly QBO phase descends past 30 hPa and, conversely, the average AO index is negative when the easterly QBO phase descends past 30 hPa. This relationship is evident during the boreal cold season from November to April but non-existent during the boreal warm season from May to October.

Appendix: Strategy of model optimization

Determination of appropriate values of the weights and biases of NLPCA neural network models relies on a procedure of optimization. The procedure involves minimizing a scalar objective function of the neural network, e.g., the mean square error of output to input signals used in this study. In this analysis, the minimization is performed using a hybrid procedure (Webb and Lowe 1988) consisting of the quasi-Newton method for nonlinear optimization and the least square method for linear optimization, since the neural network employs linear transfer functions for all the output neurons. The quasi-Newton method with a mixed quadratic and cubic line search and the Broyden–Fletcher–Goldfarb–Shanno formula for updating approximations of the Hessian matrix (Broyden 1970a, b; Fletcher 1970; Goldfarb 1970; Shanno 1970) is used to minimize the objective function with respect to weights and biases of the bottleneck and decoding neurons. Each time these weights and biases are updated, hence the decoding signals are changed, a one-step exact minimization of the objective function with respect to the output weights is obtained using the least square method.

The quasi-Newton method is a fast algorithm in training feed-forward neural networks (Bishop 1995), but stops at the first minimum of the objective function it finds. The topology of the mean square error surface of a nonlinear neural network model usually has many minima in addition to a global minimum. As an effort to search for the global minimum, multiple solutions of the model are determined from randomly initialized values of the weights and biases. Each set of the random initial values for a model run is normally distributed with a zero mean and a unit variance. To further specialize the search for the global minimum, the following is performed during each model run for a solution. After a minimum of the objective function is found, the current values of the weights and biases are adjusted by adding to them another set of random values with a zero mean and a variance of 0.01. Then a new minimum is searched. If the new minimum is lower, the newly optimized values will replace the current ones. Otherwise, the current values remain unchanged. The procedure repeats until no lower minimum has been found for several consecutive such adjustments. Then the solution of this model run is constructed using the current values. Among all runs of a model, the solution with the lowest minimum is taken as model solution.