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
3. Fluctuations in strength alone
4. Fluctuations in position alone
If the pdf of λ is symmetric so that sλ = 0, it follows by the selection rule (11) both that | f2〉 is an eigenfunction of 𝗖 with eigenvalue μ(2) = N22(κλ + 2) U02h4/4 and that to leading order in h2, | f1〉 is an eigenfunction with eigenvalue μ(1) = N12U02h2 [the off-diagonal terms of the covariance matrix will result in O(h4) 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 N1 and N2. Because h is by assumption M 1, | f1〉 will be the leading EOF unless the kurtosis κλ is very large.
It can be seen from Eq. (26) that the effect of a skewed distribution of λ is to mix the vectors | f1〉 and | f2〉 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
6. Fluctuations in both position and strength
a. Independent fluctuations in strength and position
In this example, the leading EOF remains recognizably dipolar; for the mixing of | f1〉 with | f0〉 and | f2〉 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
7. Fluctuations in both position and width
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 |
The special case of fluctuations in both strength and width could also be considered (as in MF06), but the leading EOFs will mix | f0〉 and |
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 | f0〉, | f1〉, | f2〉, and |
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 djf/dxj and products xjdjf/dxj. 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 xjdjf/dxj.
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
Jet shape function f (x) for spatial skewness [Eq. (28)]
Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2814.1
EOFs of a jet with Gaussian fluctuations in position alone (gray curves: predicted; black curves: numerically simulated) with h = 0.5 for spatial skewness
Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2814.1
Leading EOF |E(1)〉 of a jet with Gaussian fluctuations in width alone (gray curves: predicted; black curves: numerically simulated) with υ = 0.15 for spatial skewness (a)
Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2814.1
EOFs of a symmetric jet with Gaussian strength fluctuations (l = 0.185) and skewed position fluctuations (h = 0.3) (gray curves: predicted; black curves: numerically simulated) for skew(λ) = 0, 0.75, and 1.5. In the second and third columns, the dashed line is the simulated EOF pattern for skew(λ) = 0. (top row) first EOF |E(1)〉; (bottom row) second EOF | E(2)〉.
Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2814.1
Leading EOF |E(1)〉 of an asymmetric jet (with spatial skewness
Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2814.1