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    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).

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    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.

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    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.

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    Leading EOF of pure Gaussian fluctuations in jet width [Eq. (39)]. Normalization as in Fig. 2.

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    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.

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    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(−).

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    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).

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    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.

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    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).

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On the Nature of Zonal Jet EOFs

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  • 1 School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, and Earth System Evolution Program, Canadian Institute for Advanced Research, Toronto, Ontario, Canada
  • | 2 Canadian Centre for Climate Modelling and Analysis, Meteorological Service of Canada, University of Victoria, Victoria, British Columbia, Canada
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Abstract

Analytic results are obtained for the mean and covariance structure of an idealized zonal jet that fluctuates in strength, position, and width. Through a systematic perturbation analysis, the leading empirical orthogonal functions (EOFs) and principal component (PC) time series are obtained. These EOFs are built of linear combinations of basic patterns corresponding to monopole, dipole, and tripole structures. The analytic results demonstrate that in general the individual EOF modes cannot be interpreted in terms of individual physical processes. In particular, while the dipole EOF (similar to the leading EOF of the midlatitude zonal mean zonal wind) describes fluctuations in jet position to leading order, its time series also contains contributions from fluctuations in strength and width. No simple interpretations of the other EOFs in terms of strength, position, or width fluctuations are possible. Implications of these results for the use of EOF analysis to diagnose physical processes of variability are discussed.

Corresponding author address: Adam H. Monahan, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3055, STN CSC, Victoria BC V8W 3P6, Canada. Email: monahana@uvic.ca

Abstract

Analytic results are obtained for the mean and covariance structure of an idealized zonal jet that fluctuates in strength, position, and width. Through a systematic perturbation analysis, the leading empirical orthogonal functions (EOFs) and principal component (PC) time series are obtained. These EOFs are built of linear combinations of basic patterns corresponding to monopole, dipole, and tripole structures. The analytic results demonstrate that in general the individual EOF modes cannot be interpreted in terms of individual physical processes. In particular, while the dipole EOF (similar to the leading EOF of the midlatitude zonal mean zonal wind) describes fluctuations in jet position to leading order, its time series also contains contributions from fluctuations in strength and width. No simple interpretations of the other EOFs in terms of strength, position, or width fluctuations are possible. Implications of these results for the use of EOF analysis to diagnose physical processes of variability are discussed.

Corresponding author address: Adam H. Monahan, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3055, STN CSC, Victoria BC V8W 3P6, Canada. Email: monahana@uvic.ca

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

Following Fyfe (2003), Fyfe and Lorenz (2005), and Wittman et al. (2005), we consider our fundamental dynamical object to be a jet in the zonal mean zonal wind with a simple Gaussian profile,
i1520-0442-19-24-6409-e1
where the U(t), Φ(t), and σ(t) are the jet strength, position, and width, respectively. We proceed to investigate the statistical structure of u(ϕ, t) by assuming
i1520-0442-19-24-6409-e2
i1520-0442-19-24-6409-e3
i1520-0442-19-24-6409-e4
where ξ(t), λ(t), and η(t) are individually Gaussian time series with mean zero, that is,
i1520-0442-19-24-6409-e5
i1520-0442-19-24-6409-e6
i1520-0442-19-24-6409-e7
(Table 1 summarizes these and other symbols used in this study.) We do not assume that the joint distribution p(ξ, λ, η) can be factored as the product of the individual distributions p(ξ)p(λ)p(η): in other words, we allow in general for dependence between these variables. For the purposes of calculating means and covariances of the zonal wind, while cross correlations between ξ(t), λ(t), and η(t) are important, the temporal autocorrelation structures of these time series are irrelevant. Note that the expression for the inverse width [Eq. (4)] allows σ to become negative, which is of course unphysical; in practice, the standard deviation of η is sufficiently small that the probability of negative values of the jet width is negligible.
Observational justification for these approximations is provided in Fig. 1, the top panel of which shows the leading EOFs of daily National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996) Southern Hemisphere winter (May–September) 500-hPa zonal mean zonal wind after fitting to the profile (1) (following the direct procedure in appendix B) and then reconstructing λ(t) and η(t) to be individually Gaussian with observed variances. Additionally, in order to preserve the observed correlation between strength and inverse width, we set ξ(t) = ρη(t) + ε(t) [where ε(t) is a small residual that will be neglected]. As discussed in Fyfe and Lorenz (2005), the observed correlation between strength and inverse width reflects conservation of momentum:
i1520-0442-19-24-6409-e8
(where the sphericity of the domain has been neglected). We note that there is no manifest dependence between either strength and position or width and position. Of course, angular momentum conservation on a spherical domain implies that poleward (equatorward) displacements of the jet should be associated with increased (decreased) jet strength; the lack of correlation between U(t) and Φ(t) indicates that this relationship is weaker than that between U(t) and σ(t). Under these assumptions the system is completely described in terms of the following best-fit parameters: U0 ≈ 23.3 m s−1, γ ≈ 2.7 m s−1, ϕ0 ≈ −47.5 degree (deg), w ≈ 2.7 deg, σ−10 ≈ 0.095 deg−1, υ ≈ 0.165, and ρ ≈ 13.1 m s−1. We now compare the leading EOFs obtained in this way with the leading EOFs of the unfit zonal mean zonal winds (Fig. 1, bottom). We see that despite all the rather stringent approximations imposed by (1)(7) the two sets of leading EOFs compare very well: the idealized model does a good job of capturing the mean and covariance structure of the original data.
Having specified the statistical structure of the fluctuations in jet strength, position, and width, we can calculate the time mean of u(ϕ, t) as
i1520-0442-19-24-6409-e9
and the spatial covariance function of u(ϕ, t) as
i1520-0442-19-24-6409-e10
where the angle brackets 〈·〉 denote the expectation (or averaging) operator [and we use the simplified notation 〈u(ϕ)〉 = 〈u(ϕ, t)〉]. Once the covariance function has been calculated, the eigenvalue problem for the EOFs can be posed as an integral equation as described in appendix A. For the analysis of C(ϕ1, ϕ2) that follows it is useful to define the functions
i1520-0442-19-24-6409-e11
i1520-0442-19-24-6409-e12
i1520-0442-19-24-6409-e13
where H1(x) = 2x and H2(x) = 4x2 − 2 are the Hermite polynomials of order 1 and 2 (Arfken 1985). The functions fi(ϕ) are normalized to have unit square norm: ∫–∞ f2i(ϕ) = 1, i = 0, 1, 2. Plots of these functions are given in Fig. 2. It is worth noting that despite their resemblance to parabolic cylinder functions (e.g., Gill 1982), these functions are not mutually orthogonal. By symmetry, f1(ϕ) is orthogonal to both f0(ϕ) and f2(ϕ), but f0(ϕ) and f2(ϕ) are not mutually orthogonal, that is, ∫–∞ f0(ϕ) f2(ϕ) = −1/3. Figure 3 provides a geometric representation of the vector space spanned by these functions.

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

Consider first the case in which only fluctuations in strength are nonzero (i.e., γ ≠ 0, w = 0, υ = 0). The time-mean jet is
i1520-0442-19-24-6409-e14
In this case the time-mean jet is identical to the instantaneous jet with time-mean strength. The covariance function is
i1520-0442-19-24-6409-e15
i1520-0442-19-24-6409-e16
Thus, the integral equation defining the eigenvalue problem for the EOFs [Eq. (A1) in the appendix] is characterized by a kernel that is trivially separable in the function f0(ϕ), which is therefore the only eigenfunction associated with a nonzero eigenvalue (Arfken 1985). In the presence of Gaussian fluctuations in the jet strength alone, the leading (and only) EOF is the monopole
i1520-0442-19-24-6409-e17
Note that this provides an analytic demonstration of the result found numerically in Wittman et al. (2005).

4. Fluctuations in jet position alone

Now consider the case in which only fluctuations in jet position are nonzero (i.e., γ = 0, w ≠ 0, υ = 0). The time-mean jet is
i1520-0442-19-24-6409-e18
In this case the time-mean jet differs from the instantaneous jet at the time-mean position. In particular, the jet wobbling around its mean position produces a mean jet that is weaker and wider than the instantaneous jet at the mean position. The covariance function is
i1520-0442-19-24-6409-e19
where we have defined the new coordinates
i1520-0442-19-24-6409-e20
i1520-0442-19-24-6409-e21
Defining the parameter h = w/σ0 the covariance function can be expressed
i1520-0442-19-24-6409-e22
This covariance function is not obviously separable for general values of h. However, in the limit of small fluctuations in position (h2 ≪ 1; the 500-hPa SH estimates give h2 = 0.066), expanding the quantity in square brackets in (22) in powers of h2 yields
i1520-0442-19-24-6409-e23
so
i1520-0442-19-24-6409-e24
The covariance function is manifestly separable in terms of the functions
i1520-0442-19-24-6409-e25
i1520-0442-19-24-6409-e26
where the O(h2) terms are corrections to the width of the Gaussian envelope, which for h2 ≪ 1 alter the width but not the shape of the functions Q1(x) and Q2(x). Changing coordinates back to ϕ1, ϕ2, the covariance function can be written
i1520-0442-19-24-6409-e27
As discussed in appendix A, because the functions f1(ϕ) and f2(ϕ) are mutually orthogonal, it follows that these functions are also EOFs. Furthermore, the ordering of the EOFs is clear from the expansion in powers of h:
i1520-0442-19-24-6409-e28
i1520-0442-19-24-6409-e29
[where the O(h2) correction terms arising from the width of the Gaussian envelope have been neglected]. For small fluctuations in jet position (relative to jet width), the first EOF is the dipole f1(ϕ) and the second EOF is the tripole f2(ϕ), precisely in accordance with the numerical results of Wittman et al. (2005).
Having obtained analytic expressions for the EOFs, it is possible to calculate the PC time series as
i1520-0442-19-24-6409-e30
To leading order, these are respectively
i1520-0442-19-24-6409-e31
i1520-0442-19-24-6409-e32
While the PC time series are uncorrelated by construction, α(1)(t) and α(2)(t) are clearly not independent: their joint distribution is parabolic. Statistical independence of two variables X1 and X2 requires that the conditional distribution p(X2|X1) is equal to the marginal distribution p(X2) alone; that is, that knowledge of X1 has no effect on the distribution of X2 (and vice versa). This is clearly not the case for α(1)(t) and α(2)(t). For the problem under consideration, at every time t, the wind field is specified by the single number λ(t). Because variations in the jet project on a spectrum of EOFs, the PC of all of these EOFs must be determined uniquely by the scalar time series λ(t), and cannot be mutually independent. This result was first obtained by Fyfe and Lorenz (2005) using a Taylor series expansion of the wobbling jet; the above analysis formalizes the argument.

5. Fluctuations in jet width alone

Consider the case in which only fluctuations in jet width are nonzero (i.e., γ = 0, w = 0, υ ≠ 0). The time-mean jet is
i1520-0442-19-24-6409-e33
As was the case with fluctuations in position, the mean jet is not equal to the instantaneous jet at mean width. The covariance function is
i1520-0442-19-24-6409-e34
where
i1520-0442-19-24-6409-e35
i1520-0442-19-24-6409-e36
As was the case with fluctuations in position alone, this covariance function is not obviously separable for general values of υ. However, for υ2 ≪ 1 (the 500-hPa SH estimates give υ2 = 0.027),
i1520-0442-19-24-6409-e37
so to O(υ4),
i1520-0442-19-24-6409-e38
The leading EOF for pure fluctuations in jet width is therefore
i1520-0442-19-24-6409-e39
[again up to O(υ2) correction terms associated with the width of the Gaussian envelope]. This EOF, illustrated in Fig. 4, is in excellent agreement with the leading EOF of pure fluctuations in jet width obtained numerically in Wittman et al. (2005).

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

We now turn to the case in which only fluctuations in width are zero (i.e., γ ≠ 0, w ≠ 0, υ = 0), and where, as suggested by the observations, the fluctuations in strength and position are independent, that is,
i1520-0442-19-24-6409-e40
Then the time-mean jet is
i1520-0442-19-24-6409-e41
Not surprisingly, the mean jet depends on fluctuations in jet position but not in jet strength. Defining the coordinates x and y as in Eqs. (20) and (21), the covariance function is
i1520-0442-19-24-6409-e42
where l = γ/U0. As was the case of fluctuations in jet position alone, this covariance function becomes manifestly separable when we assume that h is small and expand in powers of h:
i1520-0442-19-24-6409-e43
where we have also assumed that l is “small.” Transforming back to the original coordinates we obtain
i1520-0442-19-24-6409-e44
(where the higher-order terms include small corrections resulting from the width of the Gaussian envelope, which will be neglected). The covariance function is clearly separable in fj(ϕ), j = 0, 1, 2. As noted earlier, while f1(ϕ) is orthogonal to both f0(ϕ) and f2(ϕ), f0(ϕ) and f2(ϕ) are not mutually orthogonal. Thus, while f1(ϕ) is one of the three leading EOFs of the covariance function (44), the other two will be orthogonal linear combinations of f0(ϕ) and f2(ϕ),
i1520-0442-19-24-6409-e45
where the plus (minus) superscript labels the EOF associated with the larger (smaller) variance. The EOF problem can then be recast as a two-dimensional eigenvalue problem, details of which are given in appendix A. Figure 6 displays plots of β(±)0 and β(±)2 as functions of the ratio 3h4/8l2. For small values of the ratio, that is, for h4l2, β(+)0 ≃ 1 and β(+)2 ≃ 0. The EOF E(+)(ϕ) is therefore approximately equal to f0(ϕ), the leading EOF for pure fluctuations in jet strength (up to a sign, always arbitrary for EOFs). For large values of the ratio, that is, for h4l2, β(+)2 ≃ 1 and β(+)0 ≃ 0. In this limit, E(+)(ϕ) ≃ f2(ϕ), the second EOF for pure fluctuations in jet position. However, for intermediate values of the ratio, E(+)(ϕ) is necessarily a mixture of f0(ϕ) and f2(ϕ), that is, a hybrid structure that does not correspond to an EOF found in either of the cases of pure jet position or strength fluctuations (Fig. 7). For all values of the ratio 3h4/8l2, E(−)(ϕ) is also a hybrid of the vectors f0(ϕ) and f2(ϕ). Fluctuations in both strength and position are combined inextricably in the structure of the EOFs. The structure of the dipole EOF of jet position variations, on the other hand, is unaffected by fluctuations in the jet strength.
Note that these calculations allow us not only to identify the leading EOFs, but their ordering as well. It is straightforward to show [appendix A, Eq. (A7)] that the ratio of the variance of the PC time series associated with the eigenvector E(−)(ϕ) to that of the eigenvector f1(ϕ) is (at most) of order h2, so f1(ϕ) is always a higher-order EOF than E(−)(ϕ). The ordering of E(+)(ϕ) relative to f1(ϕ) depends on the relative magnitudes of the variance of the PC corresponding to E(+)(ϕ):
i1520-0442-19-24-6409-e46
(appendix A), and that of the PC corresponding to f1(ϕ), πσ0U20h2/2:
i1520-0442-19-24-6409-e47
As it is the first of these two cases that is relevant to the midlatitude tropospheric jets, we will assume this ordering for the remainder of this section.
While the spatial pattern of the dipole EOF arising in the case of pure jet position fluctuations is unaffected by the addition of jet strength fluctuations, the associated principal component time series couples both forms of variability:
i1520-0442-19-24-6409-e48
Variations of ξ(t) change the overall amplitude of u(ϕ, t), so they must project on the dipole EOF. Clearly, the time series of the dipole EOF cannot be simply interpreted as reflecting variability in jet position alone (although for relatively weak fluctuations in jet strength the differences will be small).
The interpretation of the second principal component time series, α(2)(t), is even more complicated because E(2)(ϕ) is a hybrid of f0(ϕ) and f2(ϕ):
i1520-0442-19-24-6409-e49
Regardless of the degree of alignment of E(2)(ϕ) along either f0(ϕ) or f2(ϕ), the PC time series α(2) is an inextricable mixture of variability in both jet strength and position. Only in the limiting cases of lh2 and lh2 can this time series reasonably be interpreted as reflecting (to leading order) fluctuations in strength or position, respectively.

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

Consider now the case in which only fluctuations in position are zero (i.e., γ ≠ 0, w ≠ 0, υ ≠ 0), and where, as suggested by the observations, the fluctuations in strength and inverse width are correlated, that is, ξ(t) = ρη(t). Then,
i1520-0442-19-24-6409-e50
The time-mean jet is
i1520-0442-19-24-6409-e51
As before, the mean jet is distinct from the instantaneous jet with mean width. Calculating the covariance function, we find that
i1520-0442-19-24-6409-e52
where x and y are defined as in Eqs. (35) and (36). This function is not manifestly separable for general values of υ, but for υ2 ≪ 1, we have
i1520-0442-19-24-6409-e53
so
i1520-0442-19-24-6409-e54
where
i1520-0442-19-24-6409-e55
is the leading EOF for pure correlated fluctuations in jet strength and inverse width [up to O(υ2) correction terms associated with the width of the Gaussian envelope].

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

In considering the cases of fluctuations in both strength and width, or all three of strength, position, and width, direct calculation of the covariance matrix is mathematically intractable. In the limiting cases considered in sections 3 through 7, the leading EOFs lie in the three-dimensional space spanned by f0(ϕ), f1(ϕ), and f2(ϕ). Motivated by these results, we proceed by assuming that (to a first approximation) the zonal mean zonal wind u(ϕ, t) can be expressed in terms of the basis vectors f0(ϕ), f1(ϕ), and f2(ϕ) alone,
i1520-0442-19-24-6409-e56
We then consider the statistical structure of the zonal mean zonal wind projected into this three-dimensional subspace.
Defining the projections of u(ϕ, t) on the basis vectors fi(ϕ),
i1520-0442-19-24-6409-e57
direct integration gives the following explicit forms:
i1520-0442-19-24-6409-e58
i1520-0442-19-24-6409-e59
i1520-0442-19-24-6409-e60
where
i1520-0442-19-24-6409-e61
Because the vectors in the basis [ f1(ϕ), f2(ϕ), f3(ϕ)] are not mutually orthogonal, the projection coefficients pi(t) will not equal the components ai(t) in Eq. (56). In fact, the vector of components a(t) is related to the vector of projections p(t) by
i1520-0442-19-24-6409-e62
where
i1520-0442-19-24-6409-e63
so
i1520-0442-19-24-6409-e64
Denoting by ai(t) the anomalies of the components ai(t) = ai(t) − 〈ai(t)〉, we can write the covariance matrix as
i1520-0442-19-24-6409-e65
where we have assumed that position fluctuations are uncorrelated with fluctuations in strength and inverse width (as suggested by the observations), so 〈a0a1〉 = 〈a1a2〉 = 0. Writing the eigenvector E(ϕ) over the basis set fi(ϕ),
i1520-0442-19-24-6409-e66
the integral equation for the eigenfunctions (A1) can be expressed as
i1520-0442-19-24-6409-e67
where
i1520-0442-19-24-6409-e68
and b = (b0, b1, b2). From this, we can read off the fact that f1(ϕ) is an EOF with PC time series a1(t) = p1(t) (as 〈a1〉 = 0 for symmetric fluctuations in λ). From Eqs. (59) and (61), we obtain the rough approximation (assuming η ≪ 1 and λ/σ0 ≪ 1)
i1520-0442-19-24-6409-e69
for the time series of the dipole EOF. Given the parameter estimates for the SH 500-hPa jet presented in section 2 (υ ≈ 0.165, γ/U0 ≈ 0.12), variations in jet width and strength respectively contribute approximately 8% and 12% corrections to the amplitude of the dipole PC. The combined correction would be approximately 82 + 122 =14% and 8 + 12 = 20% for uncorrelated and correlated fluctuations; given the strong correlation between η(t) and ξ(t) characteristic of the SH zonal mean jet, the value of correction is expected to be closer to the larger of these two numbers. By symmetry, neither fluctuations in jet strength nor width (alone or in combination) will project on the dipole EOF in the absence of fluctuations in jet position [as was noted in Wittman et al. (2005)]. Fluctuations in jet position break this symmetry, allowing these fluctuations to project on the dipole: strength fluctuations by changing its overall amplitude, and width fluctuations by changing the width of the perturbed jet relative to the fixed-width dipole.
The other two EOFs are hybrids of f0(ϕ) and f2(ϕ), with components given by the solutions to the reduced eigenvalue problem
i1520-0442-19-24-6409-e70
The ordering of these EOFs will depend on the details of the covariance structure of λ(t), η(t), and ξ(t). When restricted to this three-dimensional subspace, the dipole arises as an EOF of u(ϕ, t) even in the presence of simultaneous fluctuations in jet strength, width, and position. As was the case in the absence of fluctuations in width, while the PC time series associated with the dipole EOF represents fluctuations in jet position to leading order, there are higher-order correction terms involving fluctuations in jet strength and width. Furthermore, while the PC time series will be uncorrelated, they will not be independent, as each will involve the time series of fluctuations in jet strength, position, and width. The dipole f1(ϕ) clearly emerges as a privileged pattern in the set of leading EOFs, but one without a direct physical interpretation.

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

Consider the covariance function C(ϕ1, ϕ2), as defined in Eq. (10). The EOFs E(j)(ϕ) of u(ϕ) are then defined as solutions of the integral equation,
i1520-0442-19-24-6409-ea1
If the covariance matrix is separable, that is, if it can be expressed in the form
i1520-0442-19-24-6409-ea2
for some linearly independent functions gi(ϕ), i = 1, . . . , N, then the integral equation can be reduced to the matrix equation,
i1520-0442-19-24-6409-ea3
where ai(j) are the components of E(j)(ϕ) over the basis gi(ϕ):
i1520-0442-19-24-6409-ea4
and
i1520-0442-19-24-6409-ea5
Thus, if the covariance matrix is separable, the infinite dimensional integral equation for the EOFs can be reduced to a finite dimensional eigenvalue problem. In particular, if the functions gi(ϕ) are mutually orthogonal, that is, if 𝗚jk = νjδij, then the functions gi(ϕ) are identical to the EOFs (up to a normalization factor).
For the covariance function (44) of section 6, the integral Eq. (A1) reduces to the problem of diagonalizing a 3 × 3 matrix. As the vector f1(ϕ) is orthogonal to both f0(ϕ) and f2(ϕ), it is an eigenvector of C(ϕ1, ϕ2); the EOF problem is thus reduced to a 2 × 2 problem for the eigenvectors in the space spanned by f0(ϕ) and f2(ϕ),
i1520-0442-19-24-6409-ea6
for which the eigenvalues (corresponding to the EOF variances) are
i1520-0442-19-24-6409-ea7
The components of the associated eigenvectors satisfy
i1520-0442-19-24-6409-ea8
with normalization constraint
i1520-0442-19-24-6409-ea9
In the limit that l2h4,
i1520-0442-19-24-6409-ea10
i1520-0442-19-24-6409-ea11
and |β(+)0| ≪ |β(+)2|, while |β(−)0| and |β(−)2| are of the same order of magnitude. Conversely, in the limit that l2h4,
i1520-0442-19-24-6409-ea12
i1520-0442-19-24-6409-ea13
and |β(+)0| ≫ |β(+)2|, while |β(−)0| and |β(−)2| are of the same order of magnitude.

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

Given the jet structure
i1520-0442-19-24-6409-eb1
we can take logarithms of both sides [over the latitude range where u(ϕ) > 0] to obtain
i1520-0442-19-24-6409-eb2
Optimizing the parameters a1, a2, and a3 to minimize the squared misfit,
i1520-0442-19-24-6409-eb3
is a simple nonlinear regression problem that can be expressed as
i1520-0442-19-24-6409-eb4
where 〈·〉 denotes the spatial average over the range of latitudes used for the fitting. Writing this equation as Y = M X we solve for the vector X = (a1, a2, a3) as X = M−1Y. From this, we obtain
i1520-0442-19-24-6409-eb5
i1520-0442-19-24-6409-eb6
i1520-0442-19-24-6409-eb7

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

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

Fig. 3.
Fig. 3.

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

Fig. 4.
Fig. 4.

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

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

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

Fig. 8.
Fig. 8.

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

Fig. 9.
Fig. 9.

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

Table 1.

List of symbols used in this study.

Table 1.
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