## 1. Introduction

This is the second paper in a loose three-paper series concerned with assessing the relevance of the linearized upper-tropospheric vorticity equation in the low-frequency dynamics of the wintertime extratropical flow. The series was inspired by the study of Simmons et al. (1983; hereafter SWB). SWB placed particular emphasis on the climatological zonally and meridionally varying 300-mb flow as shaping the structure and evolution of extratropical low-frequency anomalies. Our interest is in determining to what extent the observed characteristics of low-frequency variability may be deduced from the properties of such a flow alone.

To address this issue, we consider unforced as well as stochastically forced versions of the barotropic vorticity equation linearized about a representative upper-tropospheric flow. The unforced calculations are intended to clarify whether the short-term evolution of low-frequency anomalies may be viewed as essentially free barotropic Rossby wave propapation and dispersion on such a flow. The stochastic forcing calculations are intended to determine whether extratropical low-frequency variability in general may be viewed as essentially randomly forced barotropic Rossby waves evolving on such a flow. The unforced calculations are presented in paper 1 (Borges and Sardeshmukh 1995) and paper 2 (this paper). The stochastic forcing problem is considered in paper 3 (Newman et al. 1997). These papers can be read independently of one another. However, some discussion will be borrowed from papers 1 and 3 here to maintain continuity.

Borges and Sardeshmukh studied the general characteristics of the unforced problem. A generalized stability analysis of 17 different representative upper-tropospheric flows was performed. The analysis consisted of determining not only the perturbation eigenfunctions (i.e., normal modes or “Rossby waves”), but also initial perturbations that maximize growth of global perturbation kinetic energy over finite time intervals (optimal perturbations). Growth from optimal initial perturbations is associated with the constructive interference of several nonorthogonal normal modes and is therefore fundamentally nonmodal. In particular, it can occur even when all the normal modes are stable, that is, decaying. Borges and Sardeshmukh found that a 5-day drag on the perturbations stabilizes all the normal modes of all their flows, but still allows optimal nonmodal growth for up to two weeks. Note that their analysis only reveals optimal nonmodal growth as a theoretical possibility. For it to be relevant in the atmosphere, an initial perturbation must be of substantially optimal form, as will be shown below.

We have two aims in this paper. One is to determine whether the *unforced* nondivergent barotropic vorticity equation linearized about the climatological 300-mb flow can explain the general short-term behavior of observed low-frequency anomalies. The other is to determine whether the theoretically possible optimal nonmodal growth revealed by the generalized stability analysis is actually realized in instances of observed anomaly growth.

Our observational dataset consists of daily 300-mb streamfunction anomalies during the eight northern winters of 1985–93, filtered in time to pass fluctuations with periods of greater than 10 days. By the “general short-term behavior” of these low-frequency anomalies, we mean the statistics of their evolution over ∼10 days. By “instances of observed anomaly growth,” we mean 21 cases in the eight-winter data record in which the global kinetic energy of these low-frequency anomalies increased monotonically over at least 7 days.

*d*

*ζ**dt*

**L**

*ζ***F**

**is a vector of perturbation vorticity coefficients in say a spherical harmonic basis,**

*ζ***L**

**F**represents forcing.

**L**

**includes advection of perturbation vorticity by the climatological 300-mb flow, advection of climatological 300-mb absolute vorticity by the perturbation flow, and dissipation terms. Note that (1) can always be arranged so that**

*ζ***B**

**is a real vector, containing the real and imaginary parts of the vorticity coefficients separately as components. Our aim in this paper is to determine whether the unforced equation**

*ζ**d*

*ζ**dt*

**L**

*ζ**t*to

*t*+

*τ*gives

*ζ**t*

*τ*

**L**

*τ*

*ζ**t*

**G**

*τ*

*ζ**t*

**(**

*ζ**t*) and taking an ensemble average gives

**C**

_{τ}

**G**

*τ*

**C**

_{0}

**C**

_{0}= 〈

**(**

*ζ**t*)

*ζ*^{T}(

*t*)〉 are the zero-lag and time-lag covariance matrices of

**, respectively, and angle brackets indicate an ensemble average that is often estimated as a time average. Approximating low-frequency dynamics as (2) therefore implies a definite relationship between**

*ζ***C**

_{0}and

**C**

_{τ}that can be checked against observations. We will specify the observed

**C**

_{0}in (4), compute

**C**

_{τ}, and compare it with the observed

**C**

_{τ}for

*τ*= 10 days.

*t*) may be written as the inner product

*ζ*^{T}(

*t*)

**D****(**

*ζ**t*), where

**D**

*t*

*τ*

*ζ*^{T}

*t*

*τ*

**D**

*ζ**t*

*τ*

*ζ*^{T}

*t*

**D**

*τ*

*ζ**t*

*τ*) =

**D**

^{−1}

**G**

^{T}(

*τ*)

**D**

**G**

*τ*). Now if at any particular time

*t*

_{a},

**(**

*ζ**t*

_{a}) happens to be an eigenfunction

**ϕ**

*(*

_{j}*τ*) of Γ(

*τ*) with eigenvalue

*λ*

*(*

_{j}*τ*), (5) gives

*t*

_{a}

*τ*

*λ*

_{j}*τ*

**ϕ**

_{j}^{T}

**D**

**ϕ**

_{j}*λ*

_{j}*τ*

*t*

_{a}

*λ*

*(*

_{j}*τ*) over the subsequent interval

*τ*. It is easy to show that if

**(**

*ζ**t*

_{a}) happens to be the eigenfunction

**ϕ**

_{1}(

*τ*) corresponding to the maximum eigenvalue

*λ*

_{1}(

*τ*), then one has the maximum possible energy growth (or minimum energy decay, if

*λ*

_{1}(

*τ*) < 1) over the interval

*τ*. This is the phenomenon of optimal nonmodal growth. The eigenfunctions

**ϕ**

_{1}(

*τ*) thus represent optimal initial perturbations for maximizing perturbation energy growth (or minimizing energy decay) over the interval

*τ*. The

**ϕ**

*(*

_{j}*τ*) are also called singular vectors in the meteorological literature.

Note that energy amplification can also occur for ** ζ**(

*t*

_{a}) ≠

**ϕ**

_{1}(

*τ*). However, a necessary condition for this is that

**(**

*ζ**t*

_{a}) have a nonzero projection on the subspace spanned by the

**ϕ**

*(*

_{j}*τ*) with eigenvalues

*λ*

*(*

_{j}*τ*) > 1. We will refer to this subspace as the subspace of growing optimals. Knowledge of the dimensionality of this subspace can be useful in anticipating the likely importance of nonmodal growth in a system. If

**(**

*ζ**t*

_{a}) has an isotropic probability distribution in total space [i.e., the space spanned by all the

**ϕ**

*(*

_{j}*τ*)], its likelihood of projecting on the subspace, and therefore the likelihood of nonmodal growth, will clearly depend upon the fractional dimension of the subspace. More generally, if

**(**

*ζ**t*

_{a}) has any stationary multinormal probability distribution, then that distribution as characterized by

**C**

_{0}may be used to derive an expectation value for barotropic nonmodal growth as described below in section 3. Note that for a system with stationary statistics the ensemble averages 〈E(

*t*+

*τ*)〉 and 〈E(

*t*)〉 are the same; that is, on average there is no actual energy amplification or decay.

The interest in nonmodal kinetic energy growth resulting from eddies tilted against the shear of a flow dates back to at least Orr (1907). Elegant discussions of the basic dynamics in idealized settings may be found in Boyd (1983) and Shepherd (1985). (We describe in appendix B a simple but relevant extension to their analysis that includes dissipative effects.) The interest in *optimal* nonmodal growth, in realistic settings, is more recent, in both meteorology (e.g., Farrell 1988, 1989a, 1990; Lacarra and Talagrand 1988; Borges and Hartmann 1992; Molteni and Palmer 1993; Buizza and Palmer 1995) and oceanography (e.g., Blumenthal 1991; Penland and Sardeshmukh 1995). Note that one may consider the optimal growth of any positive quantity, not just global perturbation kinetic energy, by suitably redefining **D***λ*_{1}(*τ*) as the maximum amplification (MA) curve. Borges and Sardeshmukh (1995) discuss the MA curve and the optimal perturbations **ϕ**_{1}(*τ*) for the barotropic model (2).

Our interest here is in determining if the observed state vectors ** ζ**(

*t*) project sufficiently often and sufficiently strongly on the subspace of growing optimals for barotropic nonmodal growth to be considered an important growth mechanism for low-frequency anomalies in the atmosphere. We will approach this question both theoretically, by determining the relative size of the subspace of growing optimals and the likelihood that the observed state vectors

**(**

*ζ**t*) lie in it, and experimentally, by examining how well the unforced model (2) performs with observed initial conditions in 21 observed cases of global anomaly growth sustained over at least 7 days.

The paper is organized as follows. Section 2 describes the model and data in greater detail than done here and assesses the general ability of (2) to describe low-frequency evolution over 10 days. Several statistics of the 10-day model integrations made with observed initial conditions are presented. The focus then shifts to nonmodal growth. Section 3a contains simple results of optimal versus expected growth, section 3b assesses to what extent the growth in the 21 observed cases can be attributed to nonmodal growth (considering both linear and nonlinear models), and section 3c follows with a discussion. Section 4 follows with a more general discussion and states our principal conclusions.

## 2. General performance of the linear barotropic model

The observational dataset of the daily 300-mb wind fields for the eight northern winters of 1985–93 was obtained from the National Meteorological Center (NMC). The winds were interpolated to a T21 Gaussian grid, their vorticity was calculated, and the eight-winter average removed to define vorticity anomalies at each grid point. A 61-point Lanczos filter passing fluctuations with periods longer than 10 days (see appendix A) was then applied to the daily anomaly time series to obtain a low-frequency anomaly time series. Unless stated otherwise, all “observations” of low-frequency 300-mb anomalies refer to this low-pass filtered anomaly time series.

*ζ*is perturbation vorticity,

**v**is the associated rotational wind, and

*ζ*

_{c}and

**v**

_{c}are the eight-winter average 300-mb absolute vorticity and rotational wind, respectively. The damping terms are modeled as a scale-independent drag with timescale

*τ*

_{D}and a scale-dependent biharmonic diffusion with coefficient

*K*

_{H}. Borges and Sardeshmukh (1995) stress that if (7) is interpreted as an equivalent barotropic vorticity equation applied at the 300-mb level, then the appropriate damping constants to specify in it should represent the damping on the vertically averaged flow and not the 300-mb flow. Studies of the vertically averaged horizontal momentum budget such as that of Klinker and Sardeshmukh (1992) suggest a value of 4–5 days for

*τ*

_{D}. We specify

*τ*

_{D}= 5 days and introduce a weak scale selectivity with an additional biharmonic diffusion with coefficient

*K*

_{H}= 2.34 × 10

^{16}m

^{4}s

^{−1}. The damping associated with the diffusion is very weak (with timescale > 250 days) at total wavenumber

*n*= 7, a representative wavenumber of observed low-frequency eddies.

Expanding all variables in a truncated series of spherical harmonics, the linearized barotropic vorticity equation (7) becomes the vector equation (2) for the vector of vorticity anomaly coefficients. As mentioned earlier, (2) can always be arranged so that **L**** ζ** is a real vector, containing the real and imaginary parts of the vorticity coefficients separately as components. At the truncation chosen (T21),

**L**

**C**

_{0}and

**C**

_{τ}in (4) are also real 483 × 483 matrices.

*ζ*_{j}

*t*

**a**

_{j}

*ω*

_{j}

*t*

**b**

_{j}

*ω*

_{j}

*t*

*σ*

_{j}

*t*

**a**

_{j}

^{T}

**D**

**b**

_{j}= 0,

**a**

_{j}

^{T}

**D**

**a**

_{j}= 1, and

**b**

_{j}

^{T}

**D**

**b**

_{j}≤ 1. Figure 1 shows the growth rate

*σ*

_{j}and circular frequency

*ω*

_{j}/2

*π*of each normal mode. These results are for no linear drag in the problem, that is,

*τ*

_{D}= ∞. The results for any finite value of

*τ*

_{D}may be obtained by shifting the entire plot downward by 1/[

*τ*

_{D}(in days)]. Thus there are 10 unstable modes if there is no linear drag, one unstable mode if

*τ*

_{D}= 20 days, and no unstable modes if

*τ*

_{D}= 12 days or shorter. In particular if

*τ*

_{D}= 5 days, all the normal modes are stable.

The general performance of the model (2) is depicted in Figs. 2 and 3, respectively. For each day in our eight-winter data record, we specify ** ζ**(

*t*) on the right-hand side of (3), predict

*ζ̂*(

*t*+ 10 days), and compare it with the observed

**(**

*ζ**t*+ 10 days). For ease of viewing we actually show the statistics of streamfunction instead of vorticity, transformed to grid space.

Figures 2a and 2b show the streamfunction variance 〈*ψ*(*t*)*ψ*(*t*)〉 and 10-day lag covariance 〈*ψ*(*t* + 10)*ψ*(*t*)〉 of the observed low-pass filtered anomalies during the eight northern winters of 1985–93. In effect these are maps of the diagonal elements of the observed vorticity covariance matrices **C**_{0} and **C**_{10}, discussed in (4), transformed to streamfunction and then grid space. The contour interval is 40 × 10^{12} m^{2} s^{−1} in both panels and negative values in Fig. 2b are indicated by both shading and dashed contours. The most prominent features in Fig. 2a are the by now familiar regional maxima over the northeast Pacific and Atlantic Oceans. Figure 2b shows, consistent with numerous observational studies (e.g., Dole 1986), that the eddies in these regions are also relatively more persistent.

Figure 2c shows the 10-day lag streamfunction covariance 〈*ψ̂*(*t* + 10)*ψ*(*t*)〉 predicted by the model in an identical format to that of Fig. 2b. These values are for *τ*_{D} = 15 days in (7). It is easy to see from (3) and (4) that the pattern in Fig. 2c will be identical for any other *τ*_{D}. The magnitudes however will change by a factor exp[(10/15)(1 − 15/*τ*_{D})], where *τ*_{D} is expressed in days.

In general, Fig. 2c compares poorly with Fig. 2b. The model correctly predicts the double maximum in the central North Pacific and the relative maxima and minima over North America, but generally it predicts a too rapid decay over the hemisphere. It performs particularly badly over the northeast Atlantic. It is important to note that this poor comparison of Fig. 2c with Fig. 2b cannot be improved by altering *τ*_{D}. As stated above, altering *τ*_{D} has no effect on the pattern in Fig. 2c; in particular one cannot change the negative values into positive values.

Figure 2d shows the local anomaly correlation of the predicted and observed day-10 streamfunction anomalies, 〈*ψ̂*(*t* + 10)*ψ*(*t* + 10)〉/[〈*ψ̂*(*t* + 10)*ψ̂*(*t* + 10)〉〈*ψ*(*t* + 10)*ψ*(*t* + 10)〉]^{1/2}. The contour interval is 0.15, negative values are indicated by dashed contours and light shading, and values greater than +0.3 by heavy shading. The pattern as well as the actual values on this plot are independent of *τ*_{D}, because both the numerator and the denominator in the definition of the anomaly correlation scale by the same exponential factor described above. The model has positive skill in the eastern hemisphere Tropics and in some isolated extratropical regions such as the northeast Pacific. However, if an anomaly correlation of at least +0.6 is considered useful, then the model has no useful skill anywhere on the hemisphere.

To determine whether the model is better at predicting spatial patterns than local values, the spatial pattern anomaly correlation corr(*ψ̂*(*t* + 10), *ψ*(*t* + 10)) of the predicted and observed streamfunction anomalies north of 20°N was computed for each 10-day forecast. The solid lines in Fig. 3 depict these values. As in Fig. 2d these are also independent of *τ*_{D}. For clarity only values greater than +0.4 are shown. Note that because there are no 10-day predictions verifying on the first 10 days of each winter, all the time series in the figure begin on 11 December.

Figure 3 apparently suggests that the model has encouraging 10-day forecast skill in some instances, particularly in late winter when the occurrence of blocking is relatively more common. The substantial skill throughout most of the winter of 1988/89 is also striking. Further analysis however shows this skill to be more apparent than real. The dashed curves in Fig. 3 are skill scores for 10-day persistence forecasts (i.e., a forecast that the anomaly field on day 10 is identical to that on day 0), computed and presented in exactly the same manner as for the model forecasts. In general, the model cannot be said to perform better than persistence. This is perhaps not a surprise: if blocking is defined as persistent anomalous ridging, then the persistence forecasts are guaranteed to perform well at least for the duration of a block. For example, as discussed by Hoskins and Sardeshmukh (1987), the tropospheric flow was blocked over western Europe throughout February 1986, but both our persistence and model forecasts start showing skill in Fig. 3 only in the middle of the month, when the block was already in place.

The high skill of the persistence forecasts during the winter of 1988/89 is also not surprising. This was a winter with persistently cold sea surface temperatures (SSTs) in the eastern equatorial Pacific Ocean associated with a cold ENSO event, and the troposphere in the Pacific–North American (PNA) sector was in a persistent (–) PNA state with a low over the subtropical eastern Pacific, a high over Alaska, and a low over Canada. This corresponds to an amplified quasi-stationary hemispheric wave pattern, and the model apparently performs well in this situation. What is more surprising in Fig. 3 is that neither the model nor persistence perform equally well in the winter of 1986/87, when the tropical SSTs were anomalously warm.

In sum, Figs. 2 and 3 show that the unforced model (2) has little general skill in predicting even the short-term (∼10 day) evolution of low-frequency 300-mb circulation anomalies. The 20-day forecasts (not shown) have even less skill. We stress again that these statements can be made regardless of the specification of the most uncertain parameter in the model, the linear drag timescale *τ*_{D}. Not only are the model forecasts poor (Figs. 2d and 3), but the model does not represent with reasonable fidelity even the statistical decorrelation of the eddies over 10 days (Figs. 2b and 2c).

Although these results are discouraging, the possibility remains that the free Rossby wave dynamics encapsulated in (2) are relevant in certain regions of the globe, such as the vicinity of jet streams, and at certain times, perhaps times of rapid kinetic energy growth. It is possible that this limited realm of the validity of (2) in both time and space is obscured in Figs. 2 and 3, Fig. 2 emphasizing global behavior in time at any particular location and Fig. 3 global behavior in space at any particular time. We consider this limited realm of validity in the following section.

## 3. Relevance of optimal nonmodal growth

### a. Optimal versus expected growth

*τ*) in (5):

**ΓΦ**

**ΦΛ**

**Λ**is the diagonal matrix of eigenvalues

*λ*

_{j}(arranged in decreasing order) and

**Φ**is the matrix of eigenvectors

**ϕ**

_{j}. All the

*λ*

_{j}are positive and the eigenvectors are orthonormal with respect to the norm

**D**

**Φ**

^{T}

**D**

**Φ**=

**I**. Expanding

**(**

*ζ**t*) in (5) in terms of these eigenvectors as

**(**

*ζ**t*) =

**Φ**

**α**(

*t*) gives E(

*t*) =

**α**

^{T}(

*t*)

**α**(

*t*), and

_{j}

*λ*

_{j}

**α**

^{2}

_{j}

*λ*

_{1}Σ

_{j}

**α**

^{2}

_{j}

*λ*

_{1}over the interval [

*t,*

*t*+

*τ*] is obtained for

**(**

*ζ**t*) =

*β*

**ϕ**

_{1}, where

*β*is an arbitrary constant. As stated earlier, we refer to a plot of

*λ*

_{1}(

*τ*) as the maximum amplification curve. Figure 4 shows the MA curve for our system (2), obtained using a 5-day linear drag. With such a drag, all the normal modes of the system are stable, yet the MA curve still allows the possibility of energy amplification from modal interference, of as much as a factor of 8 over 3.5 days. In fact it allows amplification for up to about 12 days (not shown). An MA curve similar to Fig. 4 obtained for a slightly different ambient flow is discussed in detail in Borges and Sardeshmukh (1995). For brevity that discussion will not be repeated here. It should be stressed that the shape of our MA curve is not determined solely by the ambient flow structure: the 5-day drag and biharmonic diffusion are also important. This point is explored further in appendix B in an idealized setting.

Figure 5a shows the streamfunction of the optimal initial perturbation **ϕ**_{1}(3.5) that gives the maximum energy growth of *λ*_{1} ∼ 8 over 3.5 days in our system. The perturbation has large amplitude equatorward and upstream of the African–Asian jet maximum shown in Fig. 5c and is strongly tilted against the shear of the jet. This structure and position are both crucial for the subsequent growth of perturbation kinetic energy (see also Farrell 1989b). For comparison the initial perturbation **ϕ**_{483}(3.5), which gives the fastest energy *decay* of *λ*_{483} = 5.5 × 10^{−3} over 3.5 days, is shown in Fig. 5b. This also has large amplitude in the vicinity of the African–Asian jet; however, it has a very different structure and is not tilted against the shear of the jet.

As stated earlier, energy amplification can also occur for ** ζ**(

*t*) ≠

*β*

**ϕ**

_{1}, but only if

**(**

*ζ**t*) has a nonzero projection on the subspace of the growing optimals, that is, the subspace spanned by the eigenvectors

**ϕ**

_{j}with eigenvalues

*λ*

_{j}greater than 1. Without knowledge of the system’s statistics, the dimension

*J*of this subspace compared with that of the full space,

*N*= 483, is useful as a crude indicator of the likelihood of

**(**

*ζ**t*) lying in this subspace. Figure 6 shows that

*J*decreases rapidly with

*τ*, from 80 for

*τ*= 0.5 days to 30 for

*τ*= 3.5 days, and is less than 5 for

*τ*longer than 7 days. Thus, the subspace of growing optimals is only about 14% of the total space for

*τ*= 0.5 days, 6% of the total space for

*τ*= 3.5 days, and less than 1% of the total space for

*τ*longer than 7 days. This suggests that in the absence of a special mechanism generating vorticity perturbations in these small subspaces, free barotropic nonmodal growth, although shown to be possible in Fig. 4, might be unlikely in the T21 representation of the atmosphere considered here.

*t*

*τ*

*t*

_{j}

*λ*

_{j}

**α**

^{2}

_{j}

_{j}

**α**

^{2}

_{j}

*λ*

**α**

^{2}

_{j}

**α**

^{2}

_{j}

*λ*} may be interpreted as the expected energy amplification over the interval [

*t,*

*t*+

*τ*]. For an ensemble average over all states of the system (often estimated as a time average), the actual value of 〈E(

*t*+

*τ*)〉 should be independent of

*τ*if the system has stationary statistics; for such an average, therefore, {

*λ*}(

*τ*) should be identically equal to 1. If the ensemble average is taken only over those initial perturbations that are followed by energy growth for some finite time, then {

*λ*}(

*τ*) should be greater than 1 for that finite time. We will consider both types of ensembles. In analogy with the MA curve, we will refer to a plot of {

*λ*}(

*τ*) as the expected amplification (EA) curve. To draw an EA curve, one needs to specify 〈

**α**

_{j}

^{2}

*V*of the system is distributed isotropically in optimal vector space, that is, 〈

**α**

^{2}

_{j}

*V*/

*N.*In that case {

*λ*} reduces to the

*average*eigenvalue

*λ̄*, of Γ, as opposed to the

*maximum*eigenvalue

*λ*

_{1}in the maximum amplification case.

**ϕ**

_{1}with somewhat greater probability. In other words, consider initial perturbations of the form

**(**

*ζ**t*) =

*β*

**ϕ**

_{1}+ (1 −

*β*

^{2})

^{1/2}

**r**, where

*β*is between 0 and 1 and

**r**is an arbitrary mix of the other optimals of unit energy. Then

**(**

*ζ**t*) is also of unit energy, and the expected amplification from (11) is

*t*

*τ*

*t*

*β*

^{2}

*λ*

_{1}

*β*

^{2}

*λ̃*

*λ̃*is now the average of all eigenvalues excluding

*λ*

_{1}. If the total dimension

*N*of the system is large, then

*λ̃*≈

*λ̄.*Now for

*β*= 1, energy amplification of

*λ*

_{1}is guaranteed; for smaller

*β*there will be smaller amplification. The question arises, what is the minimum value of

*β*below which there is no expected amplification? This is answered by equating (12) to unity and solving for

*β*=

*β*

_{min}= [(1 −

*λ̃*)/(

*λ*

_{1}−

*λ̃*)]

^{1/2}. Figure 7 shows these values of

*β*

_{min}for a range of intervals

*τ*. It is evident that

*β*

_{min}generally exceeds 0.3. This may be interpreted as implying that if the initial

**(**

*ζ**t*) consists of a mixture of the fastest growing optimal perturbation and a random perturbation, then it must have a projection of 0.3 or more on the former for one to see amplified energy at time

*t*+

*τ*. The expected projection of an arbitrary vector on any particular vector in a 483-dimensional space is (1/483)

^{1/2}∼ 0.05, so a projection of 0.3 must be considered substantial.

**α**

^{2}

_{j}

**α**

^{2}

_{j}

**α**(

*t*)

**α**

^{T}(

*t*)〉. Because

**(**

*ζ**t*) =

**Φ**

**α**(

*t*), this matrix is related to the observed vorticity covariance matrix

**C**

_{0}as

**α**

*t*

**α**

^{T}

*t*

**Φ**

^{T}

**D**

**C**

_{0}

**D**

**C**

_{0}.

Figure 8 shows five EA curves {*λ*}(*τ*); one, EA1, is estimated as*λ̄*(*τ*) in the manner described above and the other four by specifying four different **C**_{0} in (13). These are

〈

*ζ*_{unf}(*t*)*ζ*_{unf}^{T}(*t*)〉, where the subscript “unf” indicates that the covariance is that of the original unfiltered anomalies;〈

(*ζ**t*)*ζ*^{T}(*t*)〉, the covariance of the low-pass filtered anomalies;〈

*ζ*_{SE}(*t*)*ζ*_{SE}^{T}(*t*)〉, the covariance of the low-pass filtered anomalies in the south Asian region enclosed within the ellipse in Fig. 5c; and〈

*ζ*_{RG}(*t*)*ζ*_{RG}^{T}(*t*)〉, the covariance of the low-pass filtered anomalies on day 0 of the 21 cases of real global growth in our data record. The angle brackets 〈 〉 refer here to the 21-member ensemble average.

*τ*for at least 7 days. In Fig. 8, however, all five of the EA curves indicate a monotonic

*decay*with

*τ*. Also shown in Fig. 8 is the decay of global perturbation kinetic energy associated with the 5-day drag alone. It is interesting that the EA curves show only a slightly slower decay than this.

These results suggest that although growth of global perturbation kinetic energy by the constructive interference of decaying normal modes is possible (Fig. 4), it is not likely (Fig. 8). If (2) were a reasonable model of the evolution of low-frequency 300-mb anomalies, and if nonmodal growth were important, then one might at least have expected our EA curves not to show a monotonic decay. In light of Fig. 6, it is not a surprise that the EA1 curve [= *λI**τ*)] shows a monotonic decay. However, one might argue that the assumption of isotropically distributed variance in optimal vector space is a strong one, and therefore EA1 potentially misleading; EA2 improves on EA1 by considering the observed covariance structure, but of unfiltered anomalies, and one might again object that (2) is best thought of as a model of low-frequency anomalies. Also, EA3 considers the covariance of the observed low-frequency anomalies and still predicts a decay. However, one could now argue, as in the concluding paragraph of section 2, that optimal nonmodal growth might be relevant only in certain places and at certain times. EA4 and EA5 consider these possibilities. EA4 considers the covariance of anomalies only in the region of the fastest growing optimal perturbation (Fig. 5a) but still predicts a monotonic decay. EA5 considers the covariance at the initial times of observed global perturbation kinetic energy growth and also predicts a monotonic decay.

Figure 8 shows one more curve, the average evolution of global perturbation kinetic energy predicted by the linear model (2), starting with observed initial conditions in the 21 observed cases of energy growth. This curve is (and should be) identical to the EA5 curve, but was obtained by a completely independent numerical procedure. It is included here as a check on our computations.

In concluding this subsection, it should be remembered that the results in Figs. 4–8 are for the 5-day drag on vorticity. As in the previous section, however, these results can also be readily extended to other values of *τ*_{D}. The eigenvectors **ϕ**_{j} of **Γ** in (9) remain unchanged, but the eigenvalues *λ*_{j}(*τ*) are modified by a factor *ρ*(*τ*) = exp{−2[(*τ*/*τ*_{D}) − (*τ*/5)]}, where *τ* and *τ*_{D} are in days. The new MA curve is thus obtained by multiplying the curve in Fig. 4 by *ρ*(*τ*). The new EA curves are also obtained simply by multiplying those in Fig. 8 by *ρ*(*τ*). Figure 5 remains unchanged. The new values of *β*_{min} in Fig. 7 are now determined as *β* _{min} = [(*ρ*^{−1} −*λ̃*)/(*λ*_{1} − *λ̃*)]^{1/2}, where *λ̃* and *λ*_{1} are the *old* values. The changes to Fig. 6 cannot be described simply, as they depend upon the precise spectrum of singular values; but they are nevertheless also determined completely by *ρ*(*τ*). The modifications to Figs. 4–8 are thus all given in terms of the single function *ρ*(*τ*).

For a weaker drag (*τ*_{D} > 5 days), *ρ*(*τ*) rises monotonically with *τ* from *ρ*(0) = 1. Even for *τ*_{D} = 15 days, however, *ρ*(*τ* = 7 days) is only 6.46, that is, not large enough to arrest completely the decay of any of the curves in Fig. 8 over 7 days. This is interesting, because for *τ*_{D} = 15 days, there is now an unstable normal mode in the problem (see Fig. 1). The implication is that either the mode is not strongly represented in the observed mix of “initial” perturbations or that its growth is not rapid enough for it to dominate the perturbation evolution over the next 7 days. The simple analysis of Fig. 7 is again useful in anticipating this result: even with an unstable mode present for *τ*_{D} = 15 days, *β*_{min}(7) only decreases from 0.42 to 0.12, not to a vanishingly small value.

### b. Global growth in 21 real cases: Nonlinear runs

We now examine in greater detail the 21 observed cases of global energy growth. Figure 9 shows the time series of E, defined in (5) as the global kinetic energy of the low-pass filtered 300-mb circulation anomalies. The growth cases are identified as the times when E decreases for 2 days and then increases monotonically for the next 7 days. (The energy decrease for 2 days is stipulated to isolate distinct growth events, so as not to count, say, a single 14-day growth event as seven separate 7-day events.) Day 0 for each case is defined as the day on which E begins to increase and is indicated as a filled circle on the curve.

Figure 10 shows the average amplification of E over the next 7 days in these 21 cases. As shown in Fig. 8, the barotropic model (2) linearized about the eight-winter average 300-mb flow fails to capture this growth of E when run with the observed day 0 low-pass filtered anomalies as the initial condition. Indeed it does not capture the growth in a single case (not shown).

*ζ*

*t*

**v**

*ζ*

*K*

_{H}

^{4}

*ζ*

*ζ*

_{c}

*ζ*

*ζ*

_{c}

*τ*

_{D}

_{c}

Here, *ζ* = *f* + **k**·∇*x***v** is the total (not perturbation) absolute vorticity, **v** is the associated rotational wind, *ζ*_{c} is the absolute vorticity of the eight-winter average climatological flow, and F_{c} is a time-independent specified forcing. We specify F_{c} = **v**_{c}·∇*ζ*_{c}, so that the forcing maintains the climatological flow in a steady state. The constants *K*_{H} and *τ*_{D} are defined as in (7). With these specifications, the nonlinear model (14) differs from the linear model (7) only in that there is an additional term, −(**v** − **v**_{c})·∇(*ζ* − *ζ*_{c}), on the right-hand side. Unless stated otherwise, the model runs described here are made with *τ*_{D} = 5 days.

The model is run for 7 days in each one of the 21 cases, starting with the observed day 0 absolute vorticity *ζ*_{0}, defined as *ζ*_{c} plus the observed day 0 low-pass filtered vorticity anomaly. Figure 10 shows the average amplification of E in these 21 runs as the curve labeled “F_{c} runs.” The vertical hatching on the curve indicates the standard deviation of the values obtained in the 21 cases. As in the linear model runs, the energy “amplification” in these nonlinear runs is also actually a decay. Indeed the curve is very similar to the linear model curve in Fig. 8. Like the linear model, the nonlinear model also fails to capture the observed growth of E in a single case.

*ζ*

*t*

**v**

*ζ*

*K*

_{H}

^{4}

*ζ*

*ζ*

_{−}

*ζ*

*ζ*

_{−}

*τ*

_{D}

_{−}

_{−}=

**v**

_{−}·∇

*ζ*

_{−}, and

*ζ*

_{−}and

**v**

_{−}are the average absolute vorticity and rotational wind between day −10 and day 0. Thus, in each case, the constant forcing F

_{−}maintains the “instantaneous” ambient flow, defined as the average flow over 10 days prior to day 0, in a steady state. Like (14), the model (15) differs from a linear model (2) linearized about this instantaneous ambient flow only in the inclusion of the nonlinear advection term

**v**

**v**

_{−}

*ζ*

*ζ*

_{−}

*ζ*

_{0}as in (14). The curve labeled “F

_{−}runs” in Fig. 10 shows the average amplification of E in these 21 runs, in the same format as previously. The predicted amplification is again actually a decay. However, the decay is much less rapid than in the F

_{c}runs and also shows more variability from case to case. Indeed the energy evolution in individual cases, depicted by the dashed 7-day segments in Fig. 9, is not always a monotonic decay.

Figure 11a shows the squared streamfunction anomaly on day 0, *ψ*^{2}(0), averaged over the 21 cases. Figure 11b shows the same quantity observed 7 days later, *ψ*^{2}(7). The observed average global energy amplification depicted Fig. 10 is seen in Fig. 11b to be associated with increased anomaly amplitude over the North Pacific and Atlantic Oceans. Figures 11c and 11d show the average values of *ψ*^{2}(7) predicted by the F_{c} and F_{−} models, respectively. The global decay predicted by the F_{c} model in Fig. 10 is seen in Fig. 11c to be also local decay almost everywhere. Figure 11c clearly compares poorly with Fig. 11b. Like the F_{c} model, the F_{−} model also predicted global decay in Fig. 10. Locally, however, it predicts growth over most middle- and high-latitude areas as shown in Fig. 11d. Even so, the correspondence of Fig. 11d with Fig. 11b is poor.

Unlike the linear model results described in section 2, the nonlinear results cannot scale rigorously with the damping constant *τ*_{D} in a simple manner. Nevertheless, they do scale approximately. Figures 11e and 11f repeat the calculation of Figs. 11c and 11d with a weaker damping of *τ*_{D} = 15 days. The patterns are clearly similar to their *τ*_{D} = 5 day counterparts and the magnitudes are generally higher, as expected. As stated earlier, with the weaker damping, one of the modes of the climatological flow is unstable (see Fig. 1). The values in Fig. 11e are now comparable to those in Fig. 11a, but still not as high as in Fig. 11b. Using a weaker damping in the F_{−} runs actually results in global energy growth in some cases, as depicted by the solid 7-day segments in Fig. 9. Locally, the values in Fig. 11f are higher than those in Fig. 11a in most areas, indeed they are even higher than in Fig. 11b. The weakly damped F_{−} model is thus certainly capable of producing local as well as global growth. Nevertheless, the correspondence of Fig. 11f with Fig. 11b remains poor.

_{c}and F

_{−}runs. To a good approximation the nonlinear F

_{c}and F

_{−}models behave, at least in these 21 cases, as the linear model (2) linearized about the climatological and instantaneous ambient flows, respectively. However, it would be erroneous to attribute the generally slower energy decay and larger streamfunction amplitudes in the F

_{−}runs to the greater instability of the instantaneous flows. To see this, consider (15) rewritten as

_{−}and F

_{c}runs are made with the same initial conditions, the differences between the F

_{−}and F

_{c}predictions in Figs. 10 and 11 arise

*entirely*from the different steady forcing in the two models. Now (14) and (15) may also be written as the perturbation equations

_{c}or F

_{−}and whose linear terms are identical except that they are linearized about different ambient flows. One might therefore be tempted to link the differences in the F

_{c}and F

_{−}runs to the different stability properties of these flows. Note, however, that the perturbation

*ζ*′ is defined as

*ζ*−

*ζ*

_{c}in (14a) and

*ζ*−

*ζ*

_{−}in (15a). The model forecasts (14) and (15) are made with the same initial conditions

*ζ*

_{0}. To be exactly equivalent with (14) and (15), the forecasts made with (14a) and (15a) must be with different initial conditions,

*ζ*

_{0}′

*ζ*

_{0}−

*ζ*

_{c}in (14a) and

*ζ*

_{0}′

*ζ*

_{0}−

*ζ*

_{−}in (15a), respectively. The differences between the F

_{−}and F

_{c}runs in Figs. 10 and 11 might then indeed be described as arising from the different ambient flows as well as different initial conditions in the two runs, assuming that the nonlinear perturbation terms remain small. However, this would be a rather more convoluted “explanation” of the difference between the F

_{c}and F

_{−}runs than the simple and rigorously correct statement that the difference arises entirely from a difference in the forcing. [See also Andrews (1984) for a related discussion.]

Given this important role of forcing, it was decided not to make the nonlinear barotropic model even more “realistic” than (15) by specifying a time-dependent forcing F(*t*), which maintains an ambient flow in a specified trajectory *ζ*_{b}(*t*) over the 7 days of interest in our cases of growth. Quite apart from the difficulty of meaningfully defining an ambient flow evolving on the same time scale as the perturbation, our thinking was that even if such a “tangent-nonlinear” barotropic model were to reproduce the observed growth of E and *ψ*^{2} in Figs. 10 and 11, that growth would have to be attributed to F(*t*), and not to free nonmodal Rossby wave dynamics.

### c. Discussion

The results in section 3b show that barotropic nonmodal growth is not an important mechanism for the growth of low-frequency 300-mb anomalies. We could not explain the growth in a single one of 21 observed cases, regardless of whether our nonlinear model was perturbed about climatological or “instantaneous” ambient flows. We had anticipated this conclusion from the preliminary analysis of optimal versus expected nonmodal growth in section 3a. The optimal analysis revealed the theoretical possibility of substantial energy growth from modal interference, but the expected growth, given the statistical structure of observed initial conditions, was shown to be actually an energy *decay.*

It should be stressed that our conclusion concerning optimal nonmodal growth applies only to the *unforced, barotropic,* dynamics of the wintertime *low-frequency flow,* and as such may be difficult to generalize to other dynamical systems and phenomena. Indeed Penland and Sardeshmukh (1995) have unambiguously demonstrated the importance of optimal nonmodal growth in the evolution of tropical SST anomalies, Farrell and collaborators have made a good case for its relevance in extratropical synoptic-eddy development, and Palmer and collaborators for its relevance in the error dynamics of the ECMWF weather forecasts.

Nevertheless, our analysis highlights two aspects of optimal nonmodal growth theory that should be considered in any system. First, optimal nonmodal growth as revealed by an MA curve (Fig. 4) is not a “finite-time instability,” in the sense that an arbitrary initial perturbation is guaranteed to grow for a finite time before decaying. This is in contrast with “infinite-time” or normal mode instability, in which an arbitrary initial perturbation with *any* nonzero projection on the most unstable mode is guaranteed eventually to grow. In our system the normal modes are stable for a 5-day drag, so every initial perturbation is guaranteed eventually to decay. The MA curve shows that temporary growth is nevertheless possible, at least for the first optimal perturbation **ϕ**_{1}. Figure 7, however, makes the important additional point that an arbitrary initial perturbation must have a finite minimum projection on **ϕ**_{1}, not just any nonzero projection, for one to see that temporary growth.

In this context it is interesting to recall the simple analysis of Shepherd (1985). Boyd (1983) had discussed earlier how a plane wave tilted against the shear of a flow extracts energy from the flow even as the shear acts to rotate the wave. After its phase lines become perpendicular to the flow, the wave loses energy as it continues to rotate. Boyd suggested that this temporary energy amplification would also obtain in the case of a more general perturbation made up of a superposition of two plane waves of equal and opposite tilt. However, Shepherd, seeking to generalize this result, found that for an isotropic distribution of plane waves the energy would in fact remain constant, the amplification of some waves being precisely balanced by the decay of others. He concluded that while carefully chosen individual plane waves could amplify considerably before decaying, a general distribution of waves would tend to exhibit little or no amplification. Our own analysis suggests this conclusion to be rather more general.

This brings us to the second aspect of optimal nonmodal growth theory that we believe to be of general relevance. If the initial perturbations must be of substantially optimal form for one to see growth, that is, if they must project significantly on the subspace of growing optimals, then the relative size of that subspace becomes important. If the subspace is small, then optimal nonmodal growth may be unlikely. This point has recently also been emphasized by Ehrendorfer and Errico (1995). The subspace is small in our system, so it is not a surprise that optimal nonmodal growth is not important. Note, however, that we still needed the actual model runs (3) to demonstrate this, especially in the 21 cases of observed growth. The small dimension of our subspace notwithstanding, it was still always possible for the initial perturbations to have projected significantly onto it in those 21 cases, giving growth. This did not happen, however.

It should be borne in mind that the fractional dimension of the growing subspace is not the sole determinant of the importance of nonmodal growth in a dynamical system. In this context it is instructive to contrast our analysis with that of Penland and Sardeshmukh (1995). They consider a truncated 15-dimensional empirical–dynamical model of tropical sea surface temperature evolution, in which they find only one growing optimal structure. Like ours, their subspace of growing optimals is also small; and yet they find that the observed SST anomalies project frequently onto it, so optimal growth is important in their system. The dimension of the subspace by itself is therefore clearly not as reliable an indicator of the importance of optimal growth in a system as are indicators that also take its statistical structure into account. Without knowledge of those statistics, however, it may be the best available indicator. In our system, it turns out to be an excellent indicator.

With this caveat in mind, the fact that the dimension of the subspace of growing optimals is relatively large for short optimization times *τ* in Fig. 6 suggests that barotropic nonmodal growth may be more relevant in the very short range of say 1–2 days (see also Chang and Mak 1995). This is not inconsistent with the emphasis on horizontal flow deformation and dilatation characteristics placed by several investigators (e.g., Mak and Cai 1989; Farrell 1989b; Dole 1992).

## 4. Conclusions

Borges and Sardeshmukh (1995) found that including a realistic drag stabilizes all the normal modes of the nondivergent barotropic vorticity equation linearized about representative wintertime upper-tropospheric flows. Specifically, they determined that without forcing, any perturbation to the flow begins to decay in two weeks or less. This clearly limits the possibilities of explaining the general structure of atmospheric low-frequency variability with an unforced barotropic vorticity equation.

Borges and Sardeshmukh were however more optimistic as to the relevance of the unforced vorticity equation in describing the short-term evolution of the anomalies. They found that in the time range of two weeks not only could the decaying normal modes give local growth, but also global growth by interfering constructively with one another. Indeed one could optimally obtain a global perturbation energy increase of as much as a factor of 8 over 3.5 days from such interference. Thus, there was no restriction in principle on explaining the short-term evolution, decay or growth, local or global, of the observed low-frequency 300-mb anomalies with an unforced vorticity equation.

This paper put this hypothesis to test. We attempted to explain with the unforced 300-mb vorticity equation the observed evolution of low-frequency 300-mb anomalies over 10 days, given the anomalies on day 0, but had modest success at best. We then attempted to explain the global energy growth in 21 observed cases but found that we could not, in even a single case, when a realistic 5-day drag was specified. We should add that repeating all the model integrations at a higher numerical resolution of T42 gave very similar results (not shown) to the T21 results presented here.

Our results in sections 2 and 3 make it difficult to avoid the conclusion that one can explain little of even the short-term observed evolution of low-frequency anomalies with an *unforced* barotropic vorticity equation. “Forcing,” that is, the sum of the neglected terms in (7), is crucially important, and not just as a provider of initial perturbations in certain sensitive regions such as that equatorward of the African–Asian jet in Fig. 5a.

Chang and Mak (1995) have recently also investigated the importance of free barotropic modal and nonmodal growth in the dynamics of observed low-frequency disturbances. Their approach to the problem is different and they also consider a different vertical level (500 instead of 300 mb), but their results are nevertheless consistent with ours. In particular they imply that free barotropic nonmodal dynamics are not important beyond about 4 days and that some aspects of the forcing, especially synoptic-eddy forcing, must be taken into account.

Although we have demonstrated that free barotropic dynamics fail to explain the lag-covariance **C**_{τ} of the low-pass 300-mb eddies given **C**_{0}, one could ask if they are better at explaining certain parts of **C**_{τ}, say the part associated with westward propagating disturbances. In other words, is it possible that westward propagating low-frequency disturbances approximate free barotropic Rossby waves better than eastward propagating disturbances? We have not attempted such a stratification here. It would certainly be of interest to isolate that part of low-frequency variability that is explainable in terms of free barotropic dynamics.

Note that our analysis here has been concerned more with establishing the general importance of forcing than the relative importance of its constituent parts. The forcing in a barotropic model such as (1) represents the combined effects of diabatic heating, interactions with orography, extratropical synoptic-eddy feedbacks, and baroclinic dynamics. Various authors, too numerous to list here, have argued for the importance of one or more of these effects in the low-frequency dynamics of the wintertime extratropical flow. Of these the synoptic–eddy feedbacks are the most important and the subject of numerous observational and theoretical studies (e.g., Hoskins et. al 1983; Lau 1988; Robinson 1991; Branstator 1992). These are followed in importance perhaps by fluctuations in tropical diabatic heating (e.g., Sardeshmukh and Hoskins 1988; Lau and Phillips 1986; Ferranti et. al 1990). More recently, Black and Dole (1993) have emphasized the baroclinic aspects of persistent anomaly development over the North Pacific. Indeed they describe the development as a process similar to Type B synoptic-scale cyclogenesis, only occurring on a much larger spatial scale.

In view of the complicated nature of the forcing, it is remarkable that SWB’s most unstable normal mode bears any resemblance to the dominant observed spatial structures of low-frequency variability at all. For even if one could parameterize some of the forcing effects in terms of the low-frequency vorticity anomalies, this would change the **L** operator in (2) and lead to possibly very different normal mode structures. SWB themselves envisaged a somewhat different scenario, in which the different forcings could be viewed as effectively random, so the normal modes of **L** as defined in (2) would remain relevant for explaining at least the statistics if not individual realizations of the observed variability. Paper 3 in our series (Newman et al. 1997) will assess the relevance of this scenario. The forcing will be specified as stochastic noise, both with and without geographical and temporal coherence, and an attempt will be made to explain the general space–time structure (i.e., both the **C**_{0} and **C**_{τ}) of the observed variability.

This paper has been concerned with the unforced problem. Our main conclusion is that consideration of barotropic Rossby wave propagation and dispersion on a zonally and meridionally varying 300-mb flow is necessary, but far from sufficient, for understanding even the short-term (∼10 day) evolution of low-frequency anomalies in the wintertime extratropical troposphere. That the unforced barotropic model generally predicts a rapid energy decay in Fig. 8, when in fact on average the energy should remain the same, clearly signals the need to include at least some aspects of the forcing. Furthermore, it is only useful to retain (1) as a simple model for forecasting the evolution of observed low-frequency anomalies, and for explaining their statistical structure, if the forcing can be specified independently of the anomalies.

## Acknowledgments

Valuable discussions with our colleagues, especially Drs. Randall Dole, Cécile Penland, and Klaus Weickmann are gratefully acknowledged, as are constructive comments by all the anonymous referees. Catherine Smith assisted with the preparation of the observational data. This research was supported in part by a grant from the NOAA Office of Global Programs.

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## APPENDIX A

### The Lanczos Filter

*x*

_{i}, where

*i*is the time index, a bandpass-filtered time series

*x̃*

_{i}containing frequencies between

*f*

_{1}= 1/

*T*

_{1}and

*f*

_{2}= 1/

*T*

_{2}is obtained as

*N*required to define the filtered value

*x̃*

_{i}at any time

*i*is

*N*= 2

*K*+ 1. The higher the specified value of

*N,*the closer is the frequency response of the filter to unity in the frequency interval [

*f*

_{1},

*f*

_{2}] and to zero outside it.

For our problem, we obtain a low-pass filter by specifying *f*_{1} = 0, *f*_{2} = 1/(10 days), and *N* = 61. Our low-pass filtered values are thus weighted 61-day running means of the original anomalies. We extend our original time series by 30 days before 1 December and 30 days after 28 February to obtain a 90-day filtered series from 1 December to 28 February for each one of our eight winters.

## APPENDIX B

### Maximum Amplification of Damped PlaneWaves on a Linear Couette Flow

*∂ζ*

*∂t*

*∂*

*ζ*

*∂*

*x*

*β*

*∂*

*ψ*

*∂*

*x*

*rζ*

*K*

_{H}

^{4}

*ζ*

*r*=

*K*

_{H}= 0) is

*ζ*

_{k,l}

_{k,l}

*kx*

*k*

*t*

*y*

*ϕ*

*t*

*l*is the initial meridional wavenumber and Δ ≡

*l*/

*k*S. The quantity

*k*S(Δ −

*t*) may be viewed as a time-dependent meridional wavenumber

*l̂*(

*t*). Thus, Δ is the time at which (

*Î*(

*t*) = 0. Note that the only effect of

*β*in this problem is to introduce a phase shift

**ϕ**(

*t*) =

*β*tan

^{−1}(St)/S

*k.*

*A*cos(

*kx*+

*ly*) thus evolves as E =

*A*

^{2}

*D*

^{2}(

*t*)/[4

*k*

^{2}{1 +

*S*

^{2}(

*t*− Δ)

^{2}}]. In the absence of dissipation [

*D*(

*t*) ≡ 1], this reaches a maximum at

*t*= Δ. With dissipation included this is no longer true. In that case, setting

*d*E/

*dt*= 0 gives a sixth-order polynomial equation for the nondimensional time variable

*T*≡

*S*(

*t*− Δ):

*R*

*r*

*T*

^{6}

*R*

*r*

*T*

^{4}

*R*

*r*

*T*

^{2}

*ST*

*R*

*T*=

*T*

_{m}determine the time

*t*

_{m}= Δ +

*T*

_{m}/

*S*at which E reaches an extremum. Although six roots are possible, at least one corresponds to an energy maximum. The root corresponding to the largest maximum is relevant here.

*kx*+

*ly*). This is done as follows. For specified ambient shear

*S*and dissipation parameters

*r*and

*K*

_{H}, and fixed

*k,*we ask: what is the initial

*l*that leads to the greatest wave energy E

_{max}at a specified time

*τ*, and what is the energy amplification E

_{max}(

*τ*)/

*E*(0) at that time. The answers are obtained by first determining the relevant root

*T*

_{m}of (B4) for these specified values of

*S, r, K*

_{H}, and

*k.*The optimal initial meridional wavenumber is then determined as

*l*

*kS*

*k*

*S*

*τ*

*T*

_{m}

*l̂*(

*t*

_{m}) at time t

_{m}=

*τ*is not zero but −

*k*

*T*

_{m}. The energy amplification at that time is

_{max}

*τ*

*D*

^{2}

*τ*

*S*

*τ*

*T*

_{m}

^{2}

*T*

^{2}

_{m}

It is interesting to go through this exercise for the same values of *r* and *K*_{H} as used in the main body of the text, *r* = 1/*τ*_{D} = 1/(5 days) and *K*_{H} = 2.34 × 10^{16} m^{4} s^{−1}. For *S* we specify a typical observed value of the 300-mb east Asian jet, *S* = 1.5 day^{−1}. Three MA curves, for these values of *r, K*_{H}, and *S* and three values of *k* = (7, 8, 9) × 10^{−7} m^{−1}, are shown in Fig. B1. Also shown is the MA curve for these values of *r*, *S*, and *k* but no diffusion (*K*_{H} = 0). Note that in this case the peak amplification is considerably higher, and the MA curve does not depend upon *k.* Finally, the dotted line in Fig. B1 is the MA curve [1 + (*S**τ*)^{2}] obtained if there is no dissipation at all (*r* = *K*_{H} = 0).

The MA curves in Fig. B1 are for a much simpler problem than considered in the paper, yet they have the same overall characteristics as the MA curve of Fig. 4. In particular the peak amplification is also achieved in 3–5 days. It is evident that both damping and diffusion are important in determining this time and in limiting the maximum. The MA curve for the problem without dissipation has much higher values and no peak at finite *τ*.

The inclusion of dissipation also has a bearing on whether the T21 truncation employed in the main calculations of the paper is adequate to resolve the optimal initial perturbations. Again, the simple example considered here is instructive. Without dissipation, the initial meridional wavenumber of the maximally growing wave associated with the dotted MA curve in Fig. B1 is *l* = *kS*Δ = *kS**τ*. For sufficiently large values of *τ*, this becomes large enough that it cannot be resolved at T21 or any other finite resolution. With dissipation included, however, the MA curves in Fig. B1 dip below 1 beyond about *τ* = 5 days, so the larger values of *τ* become irrelevant. The value of *T*_{m} in (B5) is ∼−0.1, so for *τ* = 5 days, the optimal initial *l* is still ∼*kS**τ* = 7.5*k.* For the zonal wavenumbers *k* considered here, this meridional wavenumber is very well resolved at T21 truncation.