1. Introduction
Empirical orthogonal function (EOF) analysis, also known as principal component analysis (PCA), is a standard technique for decomposing an observed geophysical field into a set of orthogonal spatial patterns with associated temporally uncorrelated time series. These spatial patterns (denoted the EOFs) are obtained as the eigenvectors of the covariance matrix (or sometimes the correlation matrix) of the field, while the time series (denoted the principal components, or PCs) arise as the projection coefficients of the corresponding EOF pattern on the original field. It is common to interpret individual EOF/PC pairs (together referred to as a mode) as corresponding to distinct physical processes, where the term physical process is used in this study to denote a degree of freedom of the system with a clear physical interpretation. It was emphasized by North (1984), however, that individual EOF modes correspond to individual physical modes only in a very limited class of physical systems (those governed by linear dynamics for which the linear operator commutes with its adjoint). In general, observed geophysical flows do not belong to this class of systems (e.g., Farrell and Ioannou 1996; Penland 1996; Palmer 1999). In particular, if the underlying physical processes are localized, nonstationary, not mutually orthogonal, or nonlinearly coupled, they will generally be spread across a number of EOF modes (e.g., Ambaum et al. 2001; Dommenget and Latif 2002; Fyfe 2003; Monahan et al. 2003; Fyfe and Lorenz 2005). Individual EOF modes cannot in general be expected to correspond to individual physical processes.
In particular, EOF analysis has been used to study the low-frequency (10–100 days) variability of the extratropical atmosphere (e.g., Barnston and Livezey 1987; Thompson and Wallace 2000). In both hemispheres, throughout the troposphere, it is found that the meridional spatial structure of the dominant EOF mode of the zonal mean zonal wind is a dipole centered at approximately the latitude of the core of the time-mean jet. This structure is generally interpreted as representing meridional displacements of the eddy-driven jet (the so-called zonal index), while higher-order EOFs (when they are considered) are interpreted as reflecting changes in jet strength or width (e.g., Feldstein and Lee 1998; Feldstein 2000; DeWeaver and Nigam 2000; Codron 2005; Vallis et al. 2004). Wittman et al. (2005) consider numerical simulations of the EOF structure of an idealized midlatitude zonal jet [as in Fyfe (2003) and Fyfe and Lorenz (2005)] characterized by Gaussian fluctuations in strength, position, and width (denoted, respectively, as pulsing, wobbling, and bulging). It is shown that a meridional dipole arises as the leading EOF of pure wobbling motion, and that neither pulsing nor bulging (both of which are symmetric about the jet axis) produces dipole EOF patterns (which are asymmetric about the jet axis). A meridional dipole was also found in the study of Gerber and Vallis (2005) as the leading EOF of a one-dimensional spatially stochastic process that conserves momentum.
The present study takes as its starting point the idealized midlatitude jet considered in Wittman et al. (2005), and obtains analytic expressions for the (covariance based) EOFs and PCs in terms of the fluctuations in jet strength, position, and width. These analytic results allow unambiguous diagnoses of the relationships between the EOF modes and the underlying physical processes. It will be shown that while the leading EOFs are made up of a small number of basic spatial patterns, and are therefore simple in structure, the associated time series inextricably couple the underlying processes of jet variability. In particular, a dipole is shown to arise as an EOF of the fluctuating jet under quite general conditions, but the associated PC time series mixes fluctuations in jet strength, position, and width. Furthermore, because the EOFs associated with one physical process are not orthogonal to those associated with another, these EOFs will be seen to be mixed when both processes are present simultaneously. It will be shown that in this idealized (but physically motivated) system, while some EOF modes may be associated with individual physical processes to a leading-order approximation, this association cannot generally be made. In the physically motivated context of the fluctuating zonal jet, the present study reinforces the conclusions of earlier studies that demonstrated difficulties in associating individual physical processes with individual EOF modes (e.g., Ambaum et al. 2001; Dommenget and Latif 2002).
Section 2 describes the idealized fluctuating midlatitude jet considered in this study. The EOFs of the jet in the case of pure fluctuations in strength, position, and width are considered respectively in sections 3, 4, and 5. Section 6 describes the covariance structure in the presence of simultaneous fluctuations in both strength and position, while the case of correlated fluctuations in strength and width is considered in section 7. The EOF structure for simultaneous fluctuations in strength, position, and width is discussed in section 8. A discussion and conclusions are presented in section 9.
2. The idealized Gaussian jet
We now proceed to develop analytic expressions for the covariance function and EOFs of the fluctuating jet for progressively complex forms of variability: first, individual fluctuations in strength, position, and width; and second, simultaneous fluctuations in strength and position, and in strength and width. While these individual examples do not describe the full covariance structure of the fully variable jet, they represent important limiting cases that can be used to understand the more complex case. In this analysis, the sphericity of the earth will be neglected: the jet will be taken to exist on an infinite domain.
3. Fluctuations in jet strength alone
4. Fluctuations in jet position alone
5. Fluctuations in jet width alone
The orientations in the space spanned by f0(ϕ), f1(ϕ), and f2(ϕ) of the leading EOFs for the cases of pure fluctuations in jet strength, position, and width are presented in Fig. 5.
6. Independent fluctuations in strength and position
To illustrate this coupling of strength and position fluctuations in the PC time series, 104 realizations of the field (1) with h = w/σ0 = 0.26 were made for a range of values of l = γ/U0 [selected such that f1(ϕ) remains the leading EOF], and the time series α(1) (t) and α(2) (t) were calculated. Scatterplots of these time series are plotted in Fig. 8, conditioned on the sign of ξ(t) (dark for ξ < 0, light for ξ > 0). It is evident in Fig. 8 that, for sufficiently weak fluctuations in jet strength, the distribution clusters around the parabolic curve associated with the projection of position fluctuations on both E(1)(ϕ) and E(2)(ϕ). It is also clear that both fluctuations in strength and position generally project along E(2)(ϕ), precluding its interpretation in terms of either individually.
7. Dependent fluctuations in strength and width
The mixing of the functions f0(ϕ) and f2(ϕ) in E(1)(ϕ) depends on the ratio ρ/U0; in particular, when this ratio is equal to 1, E(1)(ϕ) is aligned along f2(ϕ) (i.e., it is a tripole) and when the ratio is very large, E(1)(ϕ) is approximately aligned along f0(ϕ) (i.e., it is a monopole). Figure 9 displays plots of E(1)(ϕ) for a representative range of the values of the ratio ρ/U0.
8. Fluctuations in strength, position, and width
9. Discussion and conclusions
This study has considered analytic calculations of the statistical structure of an idealized (yet physically motivated) midlatitude jet characterized by Gaussian fluctuations in its strength, position, and width. The following results were obtained.
In general, the time-mean jet does not equal the jet with time-mean strength, width, and position. This difference implies an ambiguity in the definition of the jet climatology, although the difference is small for the parameters characteristic of the midlatitude troposphere.
The leading (covariance based) EOF patterns of the fluctuating jet found in Wittman et al. (2005) were obtained analytically. To a first approximation, these EOFs can be described in terms of three elementary functions, f0(ϕ), f1(ϕ), f2(ϕ) [Eqs. (11)–(13) and Figs. 2 and 3], representing monopole, dipole, and tripole patterns, respectively. The dipole is spatially orthogonal to the monopole and the tripole, so it arises naturally as an EOF if fluctuations in jet position are uncorrelated with fluctuations in strength and inverse width. The monopole and tripole, however, are not orthogonal, so the remaining EOFs arise as linear combinations of these two structures. In particular, the EOF patterns for fluctuations in jet position or strength alone are not retained in the presence of fluctuations in both strength and position; these structures are mixed, or hybridized. Caution must therefore be exercised in interpreting a particular EOF pattern in terms of the EOFs produced by individual physical processes.
The PC time series inextricably couple together the time series associated with the individual underlying physical processes, defined in this study as degrees of freedom of the system with clear physical interpretations. Individual EOF modes therefore cannot be associated in general with individual physical processes. The dipole EOF, which only occurs in the presence of fluctuations in jet position, has a corresponding PC time series that combines fluctuations in jet strength, position, and width. Although the strength and width fluctuations only come in as first-order corrections to the PC time series (with an estimated ∼20% effect on the PC amplitude for the SH 500-hPa jet), the dipole EOF mode does not simply correspond to fluctuations in jet position. The higher-order EOFs cannot be simply interpreted (even to leading order) in terms of fluctuations in strength, width, or position alone. Note that the distinct physical processes in the PC time series cannot be decoupled through the selection of another basis set, such as would follow from a rotated EOF analysis. Furthermore, the dependence of each of the PC time series on all of the underlying processes implies that while these time series are uncorrelated, they are not independent (as is often assumed).
The dipole pattern does arise naturally as an EOF of the fluctuating jet (although one without a straightforward physical interpretation), requiring only that fluctuations in jet position be uncorrelated with those in strength and inverse width. Momentum conservation, which was used in the model of Gerber and Vallis (2005), is not required to produce this structure (although it will in general have an effect on the ordering of EOFs). In fact, momentum conservation (and the consequent correlation between ξ and η) is not expected for a sectorial (as opposed to a zonal average) jet. On the other hand, Gerber and Vallis (2005) obtained a dipole EOF without assuming a mean jet structure. The results of the present study and that of Gerber and Vallis (2005) emphasize that dipole EOFs arise for a broad class of systems, further compounding the difficulty of interpreting these structures.
The fluctuating jet considered in this study was idealized both in its meridional structure, and in the statistical structure of its fluctuations. In fact, observed jets are generally asymmetric about their peak and fluctuations in strength, position, and inverse width are generally non-Gaussian. A goal of future work is to investigate the effects on the jet covariance structure of jet asymmetries and non-Gaussian fluctuations.
The present study has therefore demonstrated, in the context of a fluctuating idealized zonal jet, that individual EOF modes cannot generally be associated with individual physical processes. There may be cases in which such an identification may be made to leading order (such as the association of the dipole EOF with fluctuations in jet position), but this connection will not be exact. This result, which is not surprising in light of the general study of North (1984), has consequences for the interpretation of EOF modes of the zonal wind as characterized in earlier studies (e.g., Feldstein and Lee 1998; Feldstein 2000; DeWeaver and Nigam 2000; Vallis et al. 2004; Codron 2005; Wittman et al. 2005), and therefore for the study of atmospheric extratropical low-frequency variability. Empirical orthogonal function analysis is a powerful tool for dimensionality reduction in multivariate datasets, but it is a purely statistical operation. At times, it may be possible to interpret individual EOF modes in terms of underlying physical processes, but such interpretations should be approached with the utmost caution.
Acknowledgments
The authors thank Steven Feldstein and two anonymous referees, as well as Slava Kharin and John Scinocca, for their thoughtful comments on this manuscript. Adam Monahan acknowledges support from the Natural Sciences and Engineering Research Council of Canada and from the Canadian Institute for Advanced Research Earth System Evolution Program.
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APPENDIX A
Calculating the EOFs
Thus, when jet strength fluctuations are relatively weak, the larger of the two EOFs spanned by f0(ϕ) and f2(ϕ) corresponds to the second EOF of the case with fluctuations in jet position alone, lying almost parallel to f2(ϕ) and explaining exactly the correct amount of variance [cf. Eq. (27)]. When jet strength fluctuations are relatively strong, the larger of the two EOFs spanned by f0(ϕ) and f2(ϕ) corresponds to the leading EOF of the situation in which there are fluctuations in jet strength alone, also with the correct variance [cf. Eq. (16)]. In both limiting cases, the smaller of the two EOFs spanned by f0(ϕ) and f2(ϕ) is a mixture of both vectors. To leading order, in the limits of both weak and strong fluctuations in jet strength, the respective cases of fluctuations in jet position alone is recovered and in jet strength alone is recovered.
APPENDIX B
Gaussian Jet Fitting Procedure
Leading EOFs of daily Southern Hemisphere winter (May–September) 500-hPa zonal mean zonal wind (1958–2003). (top) Following the fitting procedure in accord with Eqs. (l)–(7). (bottom) Not following the fitting procedure. Solid curves: E(1). Dashed curves: E(2).
Citation: Journal of Climate 19, 24; 10.1175/JCLI3960.1
Plots of the functions f0(ϕ), f1(ϕ), and f2(ϕ) [Eqs. (11)–(13)] from which the leading EOFs are constructed, rescaled to be of unit norm.
Citation: Journal of Climate 19, 24; 10.1175/JCLI3960.1
Geometric illustration of the vectors (in function space) f0(ϕ), f1(ϕ), and f2(ϕ). Because f0(ϕ) and f2(ϕ) are not orthogonal, they cannot simultaneously be eigenvectors of a symmetric function such as the covariance. If both of these vectors contribute to the leading EOFs, these EOFs must be orthogonal linear combinations of these vectors.
Citation: Journal of Climate 19, 24; 10.1175/JCLI3960.1
Leading EOF of pure Gaussian fluctuations in jet width [Eq. (39)]. Normalization as in Fig. 2.
Citation: Journal of Climate 19, 24; 10.1175/JCLI3960.1
The orientations in the space spanned by f0(ϕ), f1(ϕ), and f2(ϕ) of the leading EOFs for the cases of pure fluctuations in jet strength, position, and width.
Citation: Journal of Climate 19, 24; 10.1175/JCLI3960.1
Components of the EOFs spanned by f0(ϕ) and f2(ϕ) for fluctuations in both jet strength and position, for (a) the EOF with the larger variance μ(+) and (b) the EOF with the smaller variance μ(−).
Citation: Journal of Climate 19, 24; 10.1175/JCLI3960.1
Hybrid EOFs E(+) of the covariance function (44) for values of the ratio 3h4/8l2 equal to 0.3 (thin solid curve), 1 (thick curve), and 3 (dashed curve).
Citation: Journal of Climate 19, 24; 10.1175/JCLI3960.1
Scatterplots of numerically calculated α(1)(t)/U0 vs α(2)(t)/U0 for h = 0.26 and 3h4/8l2 = 0.1, 0.25, 1, 2.5, and 10. Dark dots denote those points for which ξ < 0; light dots denote those points for which ξ > 0.
Citation: Journal of Climate 19, 24; 10.1175/JCLI3960.1
Leading EOF in case of correlated strength and inverse width fluctuations, for ρ/U0 = 0.01 (thin solid line), ρ/U0 = 1 (thick solid line), and ρ/U0 = 100 (dashed line).
Citation: Journal of Climate 19, 24; 10.1175/JCLI3960.1
List of symbols used in this study.