## 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

*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 assumingwhere

*ξ*(

*t*),

*λ*(

*t*), and

*η*(

*t*) are individually Gaussian time series with mean zero, that is,(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.

*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:(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:

*U*

_{0}≈ 23.3 m s

^{−1},

*γ*≈ 2.7 m s

^{−1},

*ϕ*

_{0}≈ −47.5 degree (deg),

*w*≈ 2.7 deg,

*σ*

^{−1}

_{0}≈ 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.

*u*(

*ϕ*,

*t*) asand the spatial covariance function of

*u*(

*ϕ*,

*t*) aswhere 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*(

*ϕ*

_{1},

*ϕ*

_{2}) that follows it is useful to define the functionswhere

*H*

_{1}(

*x*) = 2

*x*and

*H*

_{2}(

*x*) = 4

*x*

^{2}− 2 are the Hermite polynomials of order 1 and 2 (Arfken 1985). The functions

*f*(

_{i}*ϕ*) are normalized to have unit square norm: ∫

^{∞}

_{–∞}

*f*

^{2}

_{i}(

*ϕ*)

*dϕ*= 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,

*f*

_{1}(

*ϕ*) is orthogonal to both

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*ϕ*), but

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*ϕ*) are not mutually orthogonal, that is, ∫

^{∞}

_{–∞}

*f*

_{0}(

*ϕ*)

*f*

_{2}(

*ϕ*)

*dϕ*= −1/

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

*γ*≠ 0,

*w*= 0,

*υ*= 0). The time-mean jet isIn this case the time-mean jet is identical to the instantaneous jet with time-mean strength. The covariance function isThus, 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

*f*

_{0}(

*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 monopoleNote that this provides an analytic demonstration of the result found numerically in Wittman et al. (2005).

## 4. Fluctuations in jet position alone

*γ*= 0,

*w*≠ 0,

*υ*= 0). The time-mean jet isIn 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 iswhere we have defined the new coordinatesDefining the parameter

*h*=

*w*/

*σ*

_{0}the covariance function can be expressedThis covariance function is not obviously separable for general values of

*h*. However, in the limit of small fluctuations in position (

*h*

^{2}≪ 1; the 500-hPa SH estimates give

*h*

^{2}= 0.066), expanding the quantity in square brackets in (22) in powers of

*h*

^{2}yieldssoThe covariance function is manifestly separable in terms of the functionswhere the

*O*(

*h*

^{2}) terms are corrections to the width of the Gaussian envelope, which for

*h*

^{2}≪ 1 alter the width but not the shape of the functions

*Q*

_{1}(

*x*) and

*Q*

_{2}(

*x*). Changing coordinates back to

*ϕ*

_{1},

*ϕ*

_{2}, the covariance function can be writtenAs discussed in appendix A, because the functions

*f*

_{1}(

*ϕ*) and

*f*

_{2}(

*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:*[where the

*O*(

*h*

^{2}) 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

*f*

_{1}(

*ϕ*) and the second EOF is the tripole

*f*

_{2}(

*ϕ*), precisely in accordance with the numerical results of Wittman et al. (2005).

*α*

^{(1)}(

*t*) and

*α*

^{(2)}(

*t*) are clearly

*not*independent: their joint distribution is parabolic. Statistical independence of two variables

*X*

_{1}and

*X*

_{2}requires that the conditional distribution

*p*(

*X*

_{2}|

*X*

_{1}) is equal to the marginal distribution

*p*(

*X*

_{2}) alone; that is, that knowledge of

*X*

_{1}has no effect on the distribution of

*X*

_{2}(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

*γ*= 0,

*w*= 0,

*υ*≠ 0). The time-mean jet isAs was the case with fluctuations in position, the mean jet is not equal to the instantaneous jet at mean width. The covariance function iswhereAs was the case with fluctuations in position alone, this covariance function is not obviously separable for general values of

*υ*. However, for

*υ*

^{2}≪ 1 (the 500-hPa SH estimates give

*υ*

^{2}= 0.027),so to

*O*(

*υ*

^{4}),The leading EOF for pure fluctuations in jet width is therefore[again up to

*O*(

*υ*

^{2}) 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 *f*_{0}(*ϕ*), *f*_{1}(*ϕ*), and *f*_{2}(*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

*γ*≠ 0,

*w*≠ 0,

*υ*= 0), and where, as suggested by the observations, the fluctuations in strength and position are independent, that is,Then the time-mean jet isNot 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 iswhere

*l*=

*γ*/

*U*

_{0}. 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*:where we have also assumed that

*l*is “small.” Transforming back to the original coordinates we obtain(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

*f*(

_{j}*ϕ*),

*j*= 0, 1, 2. As noted earlier, while

*f*

_{1}(

*ϕ*) is orthogonal to both

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*ϕ*),

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*ϕ*) are not mutually orthogonal. Thus, while

*f*

_{1}(

*ϕ*) is one of the three leading EOFs of the covariance function (44), the other two will be orthogonal linear combinations of

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*ϕ*),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 3

*h*

^{4}/8

*l*

^{2}. For small values of the ratio, that is, for

*h*

^{4}≪

*l*

^{2},

*β*

^{(+)}

_{0}≃ 1 and

*β*

^{(+)}

_{2}≃ 0. The EOF

*E*

^{(+)}(

*ϕ*) is therefore approximately equal to

*f*

_{0}(

*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

*h*

^{4}≫

*l*

^{2},

*β*

^{(+)}

_{2}≃ 1 and

*β*

^{(+)}

_{0}≃ 0. In this limit,

*E*

^{(+)}(

*ϕ*) ≃

*f*

_{2}(

*ϕ*), the second EOF for pure fluctuations in jet position. However, for intermediate values of the ratio,

*E*

^{(+)}(

*ϕ*) is necessarily a mixture of

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*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 3

*h*

^{4}/8

*l*

^{2},

*E*

^{(−)}(

*ϕ*) is also a hybrid of the vectors

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*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.

*E*

^{(−)}(

*ϕ*) to that of the eigenvector

*f*

_{1}(

*ϕ*) is (at most) of order

*h*

^{2}, so

*f*

_{1}(

*ϕ*) is always a higher-order EOF than

*E*

^{(−)}(

*ϕ*). The ordering of

*E*

^{(+)}(

*ϕ*) relative to

*f*

_{1}(

*ϕ*) depends on the relative magnitudes of the variance of the PC corresponding to

*E*

^{(+)}(

*ϕ*):(appendix A), and that of the PC corresponding to

*f*

_{1}(

*ϕ*),

*π*

*σ*

_{0}

*U*

^{2}

_{0}

*h*

^{2}/2: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.

*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).

*α*

^{(2)}(

*t*), is even more complicated because

*E*

^{(2)}(

*ϕ*) is a hybrid of

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*ϕ*):Regardless of the degree of alignment of

*E*

^{(2)}(

*ϕ*) along either

*f*

_{0}(

*ϕ*) or

*f*

_{2}(

*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

*l*≫

*h*

^{2}and

*l*≪

*h*

^{2}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, 10^{4} realizations of the field (1) with *h* = *w*/*σ*_{0} = 0.26 were made for a range of values of *l* = *γ*/*U*_{0} [selected such that *f*_{1}(*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

*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,The time-mean jet isAs before, the mean jet is distinct from the instantaneous jet with mean width. Calculating the covariance function, we find thatwhere

*x*and

*y*are defined as in Eqs. (35) and (36). This function is not manifestly separable for general values of

*υ*, but for

*υ*

^{2}≪ 1, we havesowhereis the leading EOF for pure correlated fluctuations in jet strength and inverse width [up to

*O*(

*υ*

^{2}) correction terms associated with the width of the Gaussian envelope].

The mixing of the functions *f*_{0}(*ϕ*) and *f*_{2}(*ϕ*) in *E*^{(1)}(*ϕ*) depends on the ratio *ρ*/*U*_{0}; in particular, when this ratio is equal to 1, *E*^{(1)}(*ϕ*) is aligned along *f*_{2}(*ϕ*) (i.e., it is a tripole) and when the ratio is very large, *E*^{(1)}(*ϕ*) is approximately aligned along *f*_{0}(*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*/*U*_{0}.

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

*f*

_{0}(

*ϕ*),

*f*

_{1}(

*ϕ*), and

*f*

_{2}(

*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

*f*

_{0}(

*ϕ*),

*f*

_{1}(

*ϕ*), and

*f*

_{2}(

*ϕ*) alone,We then consider the statistical structure of the zonal mean zonal wind projected into this three-dimensional subspace.

*u*(

*ϕ*,

*t*) on the basis vectors

*f*(

_{i}*ϕ*),direct integration gives the following explicit forms:whereBecause the vectors in the basis [

*f*

_{1}(

*ϕ*),

*f*

_{2}(

*ϕ*),

*f*

_{3}(

*ϕ*)] are not mutually orthogonal, the projection coefficients

*p*(

_{i}*t*) will not equal the components

*a*(

_{i}*t*) in Eq. (56). In fact, the vector of components

**a**(

*t*) is related to the vector of projections

*p*(

*t*) bywheresoDenoting by

*a*′

_{i}(

*t*) the anomalies of the components

*a*′

_{i}(

*t*) =

*a*(

_{i}*t*) − 〈

*a*(

_{i}*t*)〉, we can write the covariance matrix aswhere we have assumed that position fluctuations are uncorrelated with fluctuations in strength and inverse width (as suggested by the observations), so 〈

*a*′

_{0}

*a*′

_{1}〉 = 〈

*a*′

_{1}

*a*′

_{2}〉 = 0. Writing the eigenvector

*E*(

*ϕ*) over the basis set

*f*(

_{i}*ϕ*),the integral equation for the eigenfunctions (A1) can be expressed aswhereand

**b**= (

*b*

_{0},

*b*

_{1},

*b*

_{2}). From this, we can read off the fact that

*f*

_{1}(

*ϕ*) is an EOF with PC time series

*a*

_{1}(

*t*) =

*p*

_{1}(

*t*) (as 〈

*a*

_{1}〉 = 0 for symmetric fluctuations in

*λ*). From Eqs. (59) and (61), we obtain the rough approximation (assuming

*η*≪ 1 and

*λ*/

*σ*

_{0}≪ 1)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*/

*U*

_{0}≈ 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

^{2}+ 12

^{2}

*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.

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*ϕ*), with components given by the solutions to the reduced eigenvalue problemThe 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

*f*

_{1}(

*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,
*f*_{0}(*ϕ*),*f*_{1}(*ϕ*),*f*_{2}(*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

*C*(

*ϕ*

_{1},

*ϕ*

_{2}), as defined in Eq. (10). The EOFs

*E*

^{(j)}(

*ϕ*) of

*u*(

*ϕ*) are then defined as solutions of the integral equation,If the covariance matrix is separable, that is, if it can be expressed in the formfor some linearly independent functions

*g*(

_{i}*ϕ*),

*i*= 1, . . . ,

*N*, then the integral equation can be reduced to the matrix equation,where

*a*

_{i}^{(j)}are the components of

*E*

^{(j)}(

*ϕ*) over the basis

*g*(

_{i}*ϕ*):andThus, 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

*g*(

_{i}*ϕ*) are mutually orthogonal, that is, if 𝗚

*=*

_{jk}*ν*, then the functions

_{j}δ_{ij}*g*(

_{i}*ϕ*) are identical to the EOFs (up to a normalization factor).

*f*

_{1}(

*ϕ*) is orthogonal to both

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*ϕ*), it is an eigenvector of

*C*(

*ϕ*

_{1},

*ϕ*

_{2}); the EOF problem is thus reduced to a 2 × 2 problem for the eigenvectors in the space spanned by

*f*

_{0}(

*ϕ*) and

*f*

_{2}(

*ϕ*),for which the eigenvalues (corresponding to the EOF variances) areThe components of the associated eigenvectors satisfywith normalization constraintIn the limit that

*l*

^{2}≪

*h*

^{4},and |

*β*

^{(+)}

_{0}| ≪ |

*β*

^{(+)}

_{2}|, while |

*β*

^{(−)}

_{0}| and |

*β*

^{(−)}

_{2}| are of the same order of magnitude. Conversely, in the limit that

*l*

^{2}≫

*h*

^{4},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 *f*_{0}(*ϕ*) and *f*_{2}(*ϕ*) corresponds to the second EOF of the case with fluctuations in jet position alone, lying almost parallel to *f*_{2}(*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 *f*_{0}(*ϕ*) and *f*_{2}(*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 *f*_{0}(*ϕ*) and *f*_{2}(*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

*u*(

*ϕ*) > 0] to obtainOptimizing the parameters

*a*

_{1},

*a*

_{2}, and

*a*

_{3}to minimize the squared misfit,is a simple nonlinear regression problem that can be expressed aswhere 〈·〉 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**= (

*a*

_{1},

*a*

_{2},

*a*

_{3}) as

**X**=

*M*

^{−1}

**Y**. From this, we obtain

List of symbols used in this study.