## 1. Introduction

The leading principal component analysis (PCA) mode of extratropical zonal-mean zonal wind variability is known as the zonal index (e.g., Lorenz and Hartmann 2001). The spatial structure corresponding to this mode [i.e., the empirical orthogonal function (EOF)] is found to be a dipole in observations and in a range of models from randomly forced barotropic *β*-plane dynamics (e.g., Vallis et al. 2004) through dry dynamical cores (e.g., Fyfe and Lorenz 2005) to complex general circulation models (e.g., Fyfe et al. 1999) and is related to (but not identical with; cf. Monahan and Fyfe 2008, hereafter MF08) the leading mode of zonal-mean geopotential height variability (the annular mode). As noted in Wittman et al. (2005), the ubiquity of this dipolar structure suggests that it reflects some generic feature of variability of the extratropical atmosphere—in particular, the existence of a jet in zonal-mean zonal winds characterized by fluctuations in position. The numerical simulation results presented in Wittman et al. (2005) were confirmed analytically and extended in Monahan and Fyfe (2006, hereafter MF06) the central conclusions of which were as follows:

- A small number of basic shapes, corresponding to monopole, dipole, and tripole structures, contribute to the leading-order EOFs. As noted in MF08, these shapes are successive derivatives of the jet shape function. All of these basis functions and EOFs are either symmetric or antisymmetric around the jet axis. Symmetric and antisymmetric basis functions are mutually orthogonal, but the symmetric basis functions are not orthogonal to other symmetric basis functions. Similarly, antisymmetric basis functions are not mutually orthogonal.
- The leading EOF structures corresponding to kinematic degrees of freedom representing jet fluctuations in strength, position, or width individually can be computed and correspond respectively to monopole, dipole, and coupled monopole/tripole structures.
- If the jet fluctuates in more than one of these kinematic degrees of freedom, the dipole arises as a distinct EOF mode as a result of fluctuations in jet position (as the leading EOF if fluctuations in position are sufficiently large compared to those in strength and width). However, the associated principal component (PC) time series mixes together variability in strength, position, and width; thus, the “zonal index” mode cannot be uniquely associated with a single kinematic jet degree of freedom.
- The EOFs associated with the individual kinematic degrees of freedom are not generally orthogonal, and when more than one degree of freedom is active the EOFs other than the dipole will generally consist of a mixture of monopole, dipole, and tripole structures.

These conclusions were obtained through a perturbation analysis of the covariance structure of a jet in zonal-mean zonal wind with idealized spatial structure (Gaussian profile) and fluctuations in strength, position, and width (all Gaussian distributed). Although these are reasonable first-order approximations, the observed tropospheric zonal-mean jet is not exactly Gaussian in profile and the statistics of its fluctuations are not exactly Gaussian. The present study generalizes the results of MF06 for the case of a jet of arbitrary (sufficiently smooth) profile with fluctuations of arbitrary distribution (for which a sufficiently large number of moments exist). The fundamental conclusions of MF06 are recovered in generalized form, and new results associated with asymmetric jet shape and non-Gaussian fluctuations are obtained. In particular, the conditions under which a dipolelike structure arises as the leading EOF of the fluctuating jet are characterized. The generalized model is presented in section 2. Analytic computations of the EOFs of this model for a number of illustrative special cases are presented in sections 3 through 7 before the general case is considered in section 8. Conclusions follow in section 9. Zonal-mean zonal jets in observations and models fluctuate simultaneously in all of strength, position, and width, conserving angular momentum (to a first approximation). Most of the special cases of jet variability considered in this study do not resemble the actual variability of observed tropospheric jets (and may not conserve angular momentum), but are considered as illustrative limiting examples. We note that when all jet parameters are set to best-fit values from observations (such that angular momentum is in fact conserved), the first and second predicted EOFs are in excellent agreement with those of the observed extratropical tropospheric zonal-mean zonal wind (MF06; MF08).

## 2. Idealized jet model

*u*(

*x*,

*t*) of strength

*U*(

*t*), central position

*x*(

_{c}*t*), and width

*σ*(

*t*):where

*f*is a C

*(i.e.,*

^{K}*K*times continuously differentiable) and localized [i.e., with substantially nonzero values over only part of the domain

*x*

_{1},

*x*

_{2})] but otherwise arbitrary function. We assume that strength, position, and width are fluctuating quantities:where

*ξ*,

*λ*, and

*η*are random variables of mean zero and unit variance. Other than assuming that sufficiently many moments of these random variables exist for the following series expansions to be meaningful, their joint probability density function (pdf)

*p*(

*ξ*,

*λ*,

*η*) is arbitrary. Note that without any loss of generality we have defined our coordinate system so that the average jet central position 𝗘{

*x*} is zero and the average jet width 𝗘{

_{c}*σ*} is 1, where the expectation of any function

*q*(

*ξ*,

*λ*,

*η*) is defined asWe will assume that

*l*,

*h*, and

*υ*are all ≪ 1; for the Southern Hemisphere eddy-driven jet, MF06 obtained the estimates

*l*∼ 0.1,

*h*∼ 0.3, and

*υ*∼ 0.2. Other values for these parameters will be appropriate for other jets (e.g., in the Northern Hemisphere or the middle atmosphere); what is important for the following analysis is that these parameters are small.

*f*(

*x*) is sufficiently smooth that enough of the vectors |

*f*〉 and |

_{k}_{k}〉 exist for the series expansions in the following discussion to be meaningful. If the function

*f*(

*x*) is too irregular, or contains sharp jumps, then higher-order coefficients

*N*will be very large, invalidating the asymptotic analyses of the following sections. Note that if

_{j}*f*(

*x*) is a symmetric function, then the |

*f*〉 will be alternately odd and even functions, whereas |

_{j}_{j}〉 will all be even.

*f*(

*x*) is a “jet” characterized by a unique extremum, so its derivative changes sign only once in the domain and |

*f*

_{1}〉 is “dipolar” in structure. Furthermore, we assume that

*f*(

*x*) is sufficiently localized that the function and its derivatives vanish at the boundaries of the domain

*f*〉 are orthogonal:[where the inner product 〈

_{j}*f*|

_{j}*f*〉 is defined in Eq. (A1) of the appendix]. This result follows trivially by symmetry, for functions

_{k}*f*(

*x*) that are symmetric as |

*f*〉 are then alternately odd and even functions; it is important for the following discussion that Eq. (11) holds generally for localized jet shape functions

_{j}*f*(

*x*). No such selection rule exists among the vectors |

_{j}〉.

*E*

^{(k)}〉 of

*u*(

*x*,

*t*) are the eigenfunctions of the covariance “matrix” (technically, operator)That is,We now proceed to obtain expressions for the leading EOFs of

*u*(

*x*,

*t*), considering a set of illustrative special cases before addressing the general case.

## 3. Fluctuations in strength alone

*f*

_{0}〉 is an eigenfunction with eigenvalue

*U*

_{0}

^{2}

*N*

_{0}

^{2}

*l*

^{2}. A jet fluctuating in strength alone has a single EOF with nonzero variance, with the spatial pattern of the mean jet. This result is not surprising because the jet fluctuating in strength alone is a standing wave (i.e., separable into a product of functions of space and time alone). We also note that the structure of the EOF is independent of the pdf of

*ξ*. This result is in contrast to the EOFs of fluctuations in position alone, to which we now turn.

## 4. Fluctuations in position alone

*u*(

*x*,

*t*) as a Taylor series:or, in bra–ket notation,Withit follows that the covariance matrix is given byWith

*h*≪ 1, we determine to

*O*(

*h*

^{4}) thatwhere

*s*and

_{λ}*κ*are respectively the skewness and kurtosis of

_{λ}*λ*:

*s*= 𝗘{

_{λ}*λ*

^{3}} and

*κ*= 𝗘{

_{λ}*λ*

^{4}} − 3.

If the pdf of *λ* is symmetric so that *s _{λ}* = 0, it follows by the selection rule (11) both that |

*f*

_{2}〉 is an eigenfunction of 𝗖 with eigenvalue

*μ*

^{(2)}=

*N*

_{2}

^{2}(

*κ*+ 2)

_{λ}*U*

_{0}

^{2}

*h*

^{4}/4 and that to leading order in

*h*

^{2}, |

*f*

_{1}〉 is an eigenfunction with eigenvalue

*μ*

^{(1)}=

*N*

_{1}

^{2}

*U*

_{0}

^{2}

*h*

^{2}[the off-diagonal terms of the covariance matrix will result in

*O*(

*h*

^{4}) corrections to

*μ*

^{(1))}]. The relative variances of these two PCA modes will depend on the size of

*h*, the kurtosis of

*λ*, and the normalization factors

*N*

_{1}and

*N*

_{2}. Because

*h*is by assumption M 1, |

*f*

_{1}〉 will be the leading EOF unless the kurtosis

*κ*is very large.

_{λ}*L*are determined so that the jet has spatial mean zero, unit spatial variance, and spatial skewness

_{j}*L*

_{4}< 0 ensures the jet is spatially localized). Plots of

*f*(

*x*) for

*s*= 0 will bePlots of |

_{λ}*E*

^{(1)}〉 and |

*E*

^{(2)}〉 for

*h*= 0.5 (a relatively large value selected for illustrative purposes) as predicted from Eqs. (29) and (30) are presented in Fig. 2. Not surprisingly, the jet asymmetry results in EOF patterns that are themselves asymmetric, although |

*E*

^{(1)}〉 remains identifiably dipolar. The leading EOFs were also computed directly from a numerical simulation of the fluctuating jet; these EOFs (also presented in Fig. 2) agree very well with the theoretically predicted structures. The envelopes of the numerically simulated EOFs are somewhat wider than those of the theoretical EOFs, reflecting the fact that

*h*

^{2}= 0.25 is not strictly much less than 1. As

*h*is reduced, the simulated and predicted EOF patterns (not shown) become indistinguishable.

It can be seen from Eq. (26) that the effect of a skewed distribution of *λ* is to mix the vectors | *f*_{1}〉 and | *f*_{2}〉 in the EOFs of *u*(*x*, *t*), although this effect is only on the order of *O*(*h*). The effects of skewness in *λ* will be considered further in section 6a.

## 5. Fluctuations in width alone

*O*(

*υ*

^{4}),where

*s*and

_{η}*κ*are respectively the skewness and kurtosis of width fluctuations. To

_{η}*O*(

*υ*

^{2}) the leading EOF structure is |

_{1}〉, with associated eigenvalue

*μ*

^{(1)}=

*U*

_{0}

^{2}

_{1}

^{2}

*υ*

^{2}. In contrast to the case of fluctuations in position alone, we cannot conclude for the case

*s*= 0 that the second EOF of fluctuations in width alone is given by |

_{η}_{2}〉 because in general 〈

_{1}|

_{2}〉 ≠ 0. Instead, the second- and higher-order EOFs will necessarily be mixtures of the functions |

_{j}〉. Only for the leading EOF can the simple statement be made that (to leading order) it is given by |

_{1}〉. Note that for the symmetric Gaussian jet considered in MF06, |

_{1}〉 could be expressed as a linear combination of |

*f*

_{0}〉 and |

*f*

_{2}〉; such a decomposition of |

_{j}〉 into a small number of |

*f*〉 will not be possible in general.

_{k}*E*

^{(1)}〉 for

*υ*= 0.15 (both predicted and numerically simulated) are shown in Fig. 3 for

*υ*is reduced, the agreement between the predicted and simulated EOFs improves.

## 6. Fluctuations in both position and strength

*ξ*of a power higher than 2 do not affect the covariance structure (in contrast to those of

*λ*).

### a. Independent fluctuations in strength and position

*O*[(

*lh*)

^{4}], the covariance function isThe EOFs will then be given bywhere (

*α*,

*β*,

*γ*) is an eigenvector of the matrixwhereand the matrix (47) is not symmetric because the basis set is nonorthogonal.

*λ*is symmetric, so that

*s*= 0, the even and odd sectors of this covariance matrix decouple and, to leading order in

_{λ}*h*, |

*f*

_{1}〉 is an eigenfunction with eigenvalue

*μ*

^{(1)}=

*N*

_{1}

^{2}

*U*

_{0}

^{2}

*h*

^{2}(1 +

*l*

^{2}). The other two eigenfunctions will then take the formwhere (

*α*,

*γ*) is an eigenvector of the matrixThese other leading EOFs will therefore be hybrids of |

*f*

_{0}〉 and |

*f*

_{2}〉, with eigenvalueswhere(for a jet with Gaussian profile and Gaussian fluctuations,

*δ*= 3

*h*

^{4}/8

*l*

^{2}; cf. MF06; MF08). The degree of hybridization will depend on the inner product

*F*

_{02}and the size of the ratio

*δ*. Thus, the three leading EOFs will be given by the dipole |

*f*

_{1}〉 and the hybrids |

*E*

^{(±)}〉 of |

*f*

_{0}〉 and |

*f*

_{2}〉, with ordering (|

*f*

_{1}〉, |

*E*

^{(+)}〉, |

*E*

^{(−)}〉) or (|

*E*

^{(+)}〉, |

*f*

_{1}〉, |

*E*

^{(−)}〉), depending on

*δ*and

*F*

_{02}.

*λ*influences the EOF structure, consider a jet with symmetric profilewith independent fluctuations in position and strength. Fluctuations in

*λ*are assumed to be skewed, with a centered Weibull distributionof mean zero, unit variance, and skewness

*s*(this pdf is selected as an illustrative example of a skewed distribution). As discussed in Monahan (2006), the skewness of this pdf is a function of the shape parameter

_{λ}*b*alone; for

*b*l 3.6, the pdf is approximately Gaussian. For the jet profile Eq. (53) we haveand

*F*

_{02}= −1 /

*α*,

*β*,

*γ*) given by the eigenvectors of the matrixThe upper and lower panels of Fig. 4 show respectively the first and second EOFs obtained with

*h*= 0.3 and

*l*= 0.185 (values selected for illustrative purposes) and position fluctuation skewnesses

*s*= 0, 0.75, and 1.5. In all cases, the first and second EOFs are respectively a dipole and a monopole. As the skewness of

_{λ}*λ*increases, these structures become increasingly asymmetric around

*x*= 0 as |

*f*

_{1}〉 mixes with |

*f*

_{0}〉 and |

*f*

_{2}〉. In particular, one lobe of the dipole shrinks while the other strengthens, and the midpoints of both the dipole and monopole move away from

*x*= 0. As the skewness enters to third order in

*h*while the diagonal terms dominating the leading-order EOF are of second order in this small parameter, the effects of skewness in position fluctuation on the leading two EOFs are relatively weak.

In this example, the leading EOF remains recognizably dipolar; for the mixing of | *f*_{1}〉 with | *f*_{0}〉 and | *f*_{2}〉 to obscure the dipolar structure of the EOF, the skewness of *λ* would have to be very large.

### b. Dependent fluctuations in strength and position

*ξ*and

*λ*on the covariance structure of

*u*(

*x*,

*t*), consider the case in which strength and position fluctuations are both Gaussian and are perfectly correlated: i.e.,

*λ*=

*ξ*. Then, to

*O*(

*h*

^{2}):so the EOFs are vectors mixing |

*f*

_{0}〉 and |

*f*

_{1}〉, with eigenvaluesTo leading order, only one of the eigenvalues is nonzero; the associated eigenvector iswhereThe mixing of |

*f*

_{0}〉 and |

*f*

_{1}〉 in the EOFs will depend on the magnitude of E. When this ratio is very large or very small, either |

*f*

_{0}〉 or |

*f*

_{1}〉 will dominate |

*E*〉 and these basis vectors will not be strongly mixed. For intermediate values of E, the contribution of these two vectors to |

*E*〉 will be of the same order of magnitude and the mixing will be more pronounced. In the case of a jet which fluctuates in both strength and position, a mixing of the “dipole” structure |

*f*

_{1}〉 with other basis functions in the EOFs can be induced by either skewness in the position fluctuations or coupling of the strength and position fluctuations.

## 7. Fluctuations in both position and width

*h*and

*υ*,It follows thatwhere

*c*= 𝗘{

_{λη}*λη*} is the covariance of

*η*and

*λ*. Assuming for simplicity that fluctuations in position and width are independent,In general, the vectors |

_{1}〉 and |

*f*

_{1}〉 will not be orthogonal:It follows that EOFs will take the form |

*E*〉 =

*α*|

*f*

_{1}〉 +

*β*|

_{1}〉 where (

*α*,

*β*) is an eigenvector of the matrix:That is, the dipole |

*f*

_{1}〉 will not generally be an EOF of a jet that fluctuates in both position and width unless (i) the jet shape

*f*(

*x*) is symmetric about the jet axis, so by symmetry

*G*

_{11}= 0, or (ii) position fluctuations are much stronger than those of width

*h*≫

*υ*, so the dipole |

*f*

_{1}〉 emerges from the leading EOF whereas the second EOF is some orthogonal hybrid of |

*f*

_{1}〉 and |

_{1}〉.

As an example, consider the asymmetric jet given by Eq. (27) with spatial skewness S = 1. The leading EOFs obtained for *υ* = 0.2 and *h* = 0.1, 0.2, and 0.3 are presented in Fig. 5. For the smaller value of *h*, variability is dominated by width fluctuations and the leading EOF is |_{1}〉. As *h* increases, |_{1}〉 and | *f*_{1}〉 mix: the leading EOF for *h* = 0.2 is a hybrid of these two structures. As *h* increases further, position fluctuations dominate over width and the leading EOF becomes the dipole | *f*_{1}〉. Note once again that if the jet structure *f* (*x*) were symmetric then the width tripole and position dipole structures would not couple in the EOFs. For a jet with independent fluctuations in position and width, the degree of hybridization of |_{1}〉 and | *f*_{1}〉 depends on how strongly asymmetric the jet shape is and on the relative magnitude of strength and position fluctuations.

The special case of fluctuations in both strength and width could also be considered (as in MF06), but the leading EOFs will mix | *f*_{0}〉 and |_{1}〉 and not project strongly along the dipole | *f*_{1}〉. We thus now turn to the general case of fluctuations in all three kinematic degrees of freedom.

## 8. Fluctuations in strength, position, and width

For the general case of fluctuations in all of strength, position, and width, the covariance matrix can be computed as in the special cases considered above. Rather than present the full (very complicated) covariance matrix, the essential results of the analysis can be obtained from a qualitative discussion making use of the results of the previous sections. The basis vectors entering the state vector to leading order in the small parameters *l*, *h*, and υ will be | *f*_{0}〉, | *f*_{1}〉, | *f*_{2}〉, and |_{1}〉; note that | *f*_{1}〉 is orthogonal to the vectors | *f*_{0}〉 and | *f*_{2}〉, but it is not orthogonal in general to |_{1}〉. Building the covariance matrix from this leading-order expansion of the state vector demonstrates that the following factors influence the degree to which the “pure” dipole | *f*_{1}〉 arises as an EOF of the variability: (i) the skewness of jet position fluctuations, (ii) the dependence of position fluctuations on either width or strength fluctuations, and (iii) the relative strength of position and width fluctuations. This analysis also indicates that asymmetric dipole EOFs can arise either from asymmetries in the jet shape *f* (*x*) or through mixing of | *f*_{1}〉 with other basis functions in the EOFs. In particular, a symmetric jet can generate asymmetric dipole EOFs if fluctuations in position are skewed or coupled to fluctuations in strength or width.

## 9. Conclusions

This study generalizes the analysis of MF06, providing an analytic characterization of the leading EOFs of a localized jet of arbitrary (smooth and localized) structure *f* (*x*) with fluctuations in strength, position, and width of arbitrary distribution. The following generalizations of the central conclusions of MF06 listed in the introduction have been obtained:

- A small number of basic shapes contribute to the leading-order EOFs, corresponding to successive derivatives of the jet shape function
*d*/^{j}f*dx*and products^{j}*x*/^{j}d^{j}f*dx*. These basis functions and the EOFs are not generally either symmetric or antisymmetric around the jet axis. Basis functions produced by even and odd derivatives are orthogonal, but the even derivative basis functions are not mutually orthogonal (and similarly for the odd derivative basis functions). No simple orthogonality relationships exist among the functions^{j}*x*/^{j}d^{j}f*dx*.^{j} - The leading EOF structures corresponding to a jet fluctuating in one of strength, position, or width individually can be computed and for unskewed fluctuations are respectively
*f*(*x*),*f*′(*x*), and*xf*′(*x*). These EOF structures will be modified if the fluctuations in position or width are skewed, but they are insensitive to the shape of the pdf of strength fluctuations. - If the jet fluctuates in more than one kinematic degree of freedom, the dipole structure
*f*′(*x*) arises as a distinct EOF mode as a result of fluctuations in jet position (as the leading EOF if fluctuations in position are sufficiently large compared to those in strength and width), provided the position fluctuations are not strongly skewed or dependent on strength or width fluctuations. However, the associated principal component time series mixes together variability in strength, position, and width: the zonal index mode cannot be uniquely associated with a single kinematic jet degree of freedom. - The EOFs associated with individual degrees of freedom are not generally orthogonal and may be mixed when more than 1 degree of freedom is active.

Furthermore, it is clear that asymmetric jet EOFs can arise as a consequence of an asymmetric jet shape, skewed position or width fluctuations, or the coupling of position fluctuations with other kinematic degrees of freedom.

Returning to the question posed in the title, this analysis has demonstrated that—to the extent that a variable jet can be described as a smooth localized functional form *f* (*x*) with a single extremum [so *f* ′(*x*) changes sign only once] that fluctuates in strength, position, and width—the factors that influence the extent to which a dipole-like structure will arise as an EOF are (i) the skewness of position fluctuations, (ii) the dependence of position fluctuations on strength and width fluctuations, and (iii) the relative strength of the position and width fluctuations. In particular, the leading EOF will be a dipole if jet position fluctuations are not strongly skewed, not strongly dependent on position and width fluctuations, and sufficiently large relative to strength and width fluctuations. That these conditions appear to be characteristic of the tropospheric zonal-mean eddy-driven jets in observations and models (e.g., Fyfe and Lorenz 2005; MF06) explains the ubiquity of dipolar zonal-mean zonal wind EOFs in these systems.

This study demonstrates that an important factor in the dominance of the dipole EOF is that position fluctuations are (relatively speaking) stronger than those of either strength or width. What the present analysis cannot do is provide a mechanistic explanation of why it is that position fluctuations are observed to be dominant in the tropospheric eddy-driven jets. The model is kinematic and takes as input the jet shape and fluctuation parameters that are results of dynamical processes. For example, the coupling of jet strength and width fluctuations required to conserve angular momentum (MF06) is a tunable parameter. This flexibility is in fact a strength of the model, allowing it to be used in situations where angular momentum may not be conserved (e.g., a zonal sector of less than global extent or the middle atmosphere where breaking planetary waves impose a variable torque on the westerly jet). Gerber and Vallis (2005) demonstrate that the leading EOF of a model of zonal-mean zonal wind anomalies as a spatial random walk (a “Brownian bridge”) is dipolar; central to this conclusion was the requirement that the anomaly field be momentum-conserving. The results of the present study suggest that momentum conservation per se is not as important in the production of dipole EOFs as the relative ordering of the magnitudes of fluctuations in jet strength, position, and width (which, as a consequence of dynamical processes, will be influenced by momentum conservation or nonconservation). Although both the present analysis and that of Gerber and Vallis (2005) are able to account for the structure of the leading EOFs of zonal-mean zonal wind, we note that the present model predicts the structure of the EOFs of zonal-mean geopotential (MF08) with greater fidelity to observations than that of Gerber and Vallis (2005).

Because jets are generic features of flow on rotating spheres, to the extent that these jets can be characterized as a basic shape displaying fluctuations in strength, position, and width the results of this study are relevant to the characterization of variability in the middle atmosphere, the ocean, and the atmospheres of other planets. Furthermore, the present study reinforces in a more general context a central conclusion of MF06: in the troposphere, the dipole EOF arises because of variability in jet position, but its associated PC time series also carries information about strength and width fluctuations. The statistical analysis provides a picture of the jet dynamics, but a blurred one: as through a PCA, darkly.

## Acknowledgments

The authors thank Bill Merryfield, John Scinocca, and two anonymous reviewers for their very helpful 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

### Notation

*f*(

*x*) can be considered as a vector |

*f*〉 (denoted the “ket”) in a vector space

*f*110%| (denoted the “bra”). The inner product of the vectors |

*f*〉 and |

*g*〉 is given by the “bracket” (thus the terms

*bra*and

*ket*):The “dyadic product”

*A*= |

*f*〉〈

*g*| defines an operator acting on any vector |

*e*〉: