## 1. Introduction

Linear stability theory addresses a set of problems in the dynamics of physical systems that include the origin, energetics, structure, and growth to finite amplitude of perturbations, and the conceptually distinct problem of error growth, which involves the rate of divergence of initially nearby trajectories in state space. In the context of the midlatitude atmosphere these address, respectively, the cyclogenesis and the predictability problems. While traditional stability analysis confined attention to determining the *t* â†’ âˆž asymptotic of perturbations to autonomous dynamical operators, more recent generalizations of stability theory have examined the more physically relevant finite-time stability of both autonomous and nonautonomous systems (Farrell and Ioannou 1996a, 1996b). In the case of an autonomous operator the temporal asymptotic perturbation or error growth occurs at the rate of the most unstable eigenvalue of the linear dynamical operator and takes the form of the associated most unstable normal mode. The analogous structure and growth rate in the case of a nonautonomous operator are given by the time-dependent first Lyapunov vector and first Lyapunov exponent, respectively (Oseledec 1968). Asymptotic structure and growth rate are readily obtained for autonomous systems through eigenanalysis of the dynamical operator and well-known theorems constrain autonomous system stability (Rayleigh 1880). Analogous results are not available for nonautonomous systems and integral bounds on growth in time-dependent systems are relatively loose (Farrell and Ioannou 1996b, hereafter F&I). Moreover, while eigenanalyses of canonical autonomous systems provide familiar examples of growth rate and structures for time-independent systems, growth rates and structures of Lyapunov vectors are not commonly available and intuition concerning such issues as the Lyapunov structure, its variability with time, and the nature of the energetics producing Lyapunov growth lacks example. Given that all physical systems are to a greater or lesser extent time dependent, and that the atmosphere in particular is highly time dependent, it is of more than strictly theoretical interest to better understand the nature of asymptotic stability of nonautonomous systems. Indeed, forecast accuracy is likely to be ultimately limited by asymptotic error growth, which is governed by the time-dependent tangent linear system linearized about the forecast trajectory until very near the end of the period of error growth when nonlinear effects become important. The approach of an initial perturbation to this asymptotic limit is of interest as is characterizing the universal structure assumed by the disturbance in this limit, which is the Lyapunov vector. (Hereafter, the terms Lyapunov vector and Lyapunov exponent refer to the first Lyapunov vector and the associated first Lyapunov exponent.)

We begin by examining destabilization of a barotropic model in which the flow and effective *Î²* are allowed to be time dependent. The results obtained are then interpreted from the point of view of nonnormal dynamics using simple time-dependent model systems.

## 2. The time-dependent stability problem

*U*(

*y, t*), in a channel with

*y*the northward direction and

*x*the zonal direction. Choosing the inverse of the mean shear and the channel width as characteristic time- and space scales, the nondimensional barotropic vorticity equation for the meridionally and temporally varying component of the streamfunction Î¨(

*x, y, t*) =

*Ïˆ*(

*y, t*)

*e*

^{ikx}is given by

*k*is the zonal wavenumber. The operator âˆ‡

^{2n}is defined as

*R*

_{n}is chosen appropriately for dissipation of order

*n*; for

*n*= 0 the dissipation models Ekman damping of the equivalent barotropic atmosphere with time constant 1/

*R*

_{0}; for

*n*= 1 the dissipation constant

*R*

_{1}is an inverse Reynolds number. The boundary conditions at the channel walls are

*Ïˆ*(Â±1,

*t*) = 0 and for

*n*= 1 the nonslip condition

*Ïˆ*

_{y}(Â±1,

*t*) = 0 is also imposed. The assumption of zonal homogeneity is severe and will be relaxed in future investigations, but for now the simplicity of this zonally homogeneous flow will be exploited to facilitate the exposition.

The perturbation barotropic vorticity equation can be interpreted as governing perturbations to a time-dependent flow in which the time dependence is produced by external forcing such as a barotropic stratospheric flow forced by tropospheric planetary waves, or as the evolution equation of errors on a time-dependent zonal trajectory *U*(*y, t*) obtained as the free solution of the nonlinear barotropic equation.

*N*grid points so that (1) takes the matrix form:

**is the column vector of streamfunction values at the discretization points, and**

*Ïˆ***A**

^{âˆ’2}has been rendered invertible by imposition of the boundary conditions. Convergence of the discrete approximate barotropic Eq. (3) to its continuous counterpart (1) for the examples considered was verified by doubling resolution.

*m*advances of the state of the system by the infinitesimal propagators

*e*

^{A(Î½Ï„)Ï„}at times

*Î½Ï„,*where

*m*and

*Ï„*satisfy the relation

*t*=

*mÏ„.*

It is well known that **A***t*) can have a stable spectrum at each instant and yet the time-dependent system (3) can be asymptotically unstable; this is the regime in which the midlatitude atmosphere typically operates. To understand the mechanism of this destabilization, we select for analysis operators that have stable spectrum at each time instant.

The Lyapunov exponent for time-independent flows reduces to the maximum over the real parts of the spectrum of the operator **A****A****A****A***t*) has a stable spectrum at each time instant, then for the time-dependent flow to be asymptotically unstable (i.e., *Î»* > 0) **A***t*) must be nonnormal in all inner products and in addition exhibit transient growth in all inner products. At first sight it is surprising that the Lyapunov exponent is independent of the norm considering that the nonnormality of an operator depends on the norm (cf. appendix A). The reason that nonnormality in all inner products produces the same asymptotic growth stems from the fundamental fact that time-dependent operators with noncommuting instantaneous realizations cannot be simultaneously rendered normal by choosing a time-independent inner product (while this can easily be done for autonomous operators or time-dependent but serially commuting operators). For this reason nonautonomous operators that do not serially commute and therefore do not have the same eigenvectors are fundamentally and irreducibly nonnormal and we will show that instability of time-dependent flows with stable mean operators is a consequence of this requisite nonnormality of the operators.

## 3. Necessary conditions for existence of positive Lyapunov exponents in flows with stable mean operators

### a. The time-dependent operator must be nonnormal and its time realizations must not commute

**A**

*t*

**A**

**N**

*t*

**N**

**N**

*t*) to be a stochastic operator (matrix) with zero mean (

*N*

_{ij}

*N*

^{2}

_{ij}

*i, j*). We choose the operator

**N**

*t*) to be a sum of red noise processes:

*Ïµ*

_{i}is a red noise process and

**B**

_{i}a time-independent matrix. The number of independent processes

*m*is at most

*n*

^{2}where

*n*is the dimension of

**N**

**N**

*t*), respectively.

^{1}between the mean operator and the deviation operator vanishes for all time: [

**N**

*t*)] = 0. In that case the propagator can be written:

^{t}

_{0}

*ds*

**N**

*s*) grows at most as

*t*

^{1/2}. This shows that when the commutator [

**A**

*t*

_{1}),

**A**

*t*

_{2})] of the operator matrix vanishes for all times

*t*

_{1},

*t*

_{2}the asymptotic stability of the flow is determined by the stability of the mean operator

*f*plane with Rayleigh friction of timescale 1/

*R*

_{0}and a mean flow consisting of a temporally varying constant shear, that is,

*U*(

*y, t*) =

*Î±y*+

*Ïµ*(

*t*)

*y*with

*Ïµ*

*Ïµ*

^{2}

*Î¶*= âˆ‡

^{2}

*Ïˆ*from (1) is

**N**

^{2}This argument also applies to a baroclinic flow on an

*f*-plane with time-dependent shear and this explains the stability of the Eady model with temporally varying shear (Hart 1971). Note that inclusion of either

*Î²*or any dissipation other than Rayleigh friction or Ekman damping (which in this problem is equivalent to Rayleigh friction) will lead to noncommuting operators in (12) in which case no statement can then be made about the stability of the time-dependent flow from the above argument.

*u*(

*y*,

*t*), in an atmosphere with Rayleigh damping and no vorticity gradient. Although this flow requires cancelation of the flow curvature term

*U*

_{yy}in the effective

*Î²*term,

*Î²*âˆ’

*U*

_{yy}, by a temporally and spatially varying

*Î²*(

*y, t*), it is instructive as an example. The equation for perturbation vorticity is

*U*(

*y, t*) =

*U*

_{0}with time varying

*Î².*The equation for the vorticity takes the form:

^{âˆ’2}commutes with âˆ‚/âˆ‚

*x*and âˆ‡

^{2n}. This proves that time variation of a meridionally uniform

*Î²*cannot destabilize a constant mean flow.

^{3}

We conclude that destabilization of a stable mean operator by a time-dependent deviation operator requires that the operators not commute.

### b. Effect of breaking the commutativity between the mean and deviation operator on the structure of the Lyapunov vector

The Lyapunov vector is the time-dependent structure to which all initial perturbations to a linear time-dependent system converge after sufficient time (cf. F&I for its properties).

*Î», Î¼, Î½*) assumed to have negative real part. We wish to find those deviation matrices of the form (9) generated by a single noise process, which will destabilize (17). Noise matrices

**N**

**N**

*Î±, Î²,*and

*Î³,*the commutator [

**N**

**N**

*t*) unless this deviation operator mixes the eigenmodes, that is, does not commute with the mean operator. We wish to know if there are other requirements on the structure of a noise matrix in order that it destabilize a mean operator. Consider the deviation operator produced by the noise matrix

**B**

*Î²*=

*Î³*= 0 to which has been applied a rotation

**R**

**N**

*Ïµ*(

*t*)

**RB**

**N**

*Ïµ.*The reason is that although

**N**

**RB**

**R**

**B**

*Î±*and

*Î²*nonzero and

*Î³*= 0. In this case the system is unstable for

*Ïµ*

^{2}

*y*=

**U**

^{âˆ’1}

*Ïˆ,*with

**U**

**Î›**

*Ïµ*has been increased by as much as the condition number â€–

**U**

^{âˆ’1}â€–â€–

**U**

**U**

^{âˆ’1}

**BU**

**U**

^{âˆ’1}â€–â€–

**B**

**U**

**Î›**

*Î¸*between the two nonorthogonal eigenvectors. Consider a noise matrix

**N**

**R**

_{Ï€/4}

**B**

**B**

*Î±*= 0 and

*Î²*=

*Î³*= 1 and

**R**

_{Ï€/4}is a rotation about the first eigenvector by an angle

*Ï€*/4. This noise matrix mixes the two nonorthogonal eigenvectors and as argued above for fixed magnitude noise process

*Ïµ*(

*t*) the degree of instability is expected to increase with the degree of nonnormality of the mean operator

*Î¸.*Indeed this is verified in Fig. 1, which shows the Lyapunov exponent as a function of

*Î¸.*

### c. On Rayleighâ€™s theorem in time-dependent flows

For inviscid zonally homogeneous time-independent flows, Rayleighâ€™s theorem provides a necessary condition for the existence of exponential instability, which is rooted in momentum conservation. We inquire whether a similar theorem constrains the possibility of positive Lyapunov exponents in inviscid time-dependent zonally homogeneous flows.

*M*=

*d*[

*uÏ…*]/

*dy,*where the bracket denotes the zonal average of

*u, Ï…,*which are, respectively, the zonal and meridional velocity. If the boundary conditions require vanishing of the perturbation momentum stress at

*y*= Â±1, then the perturbation cannot alter the integrated momentum of the mean flow because

*Q*

_{y}(

*y, t*) the time-dependent vorticity gradient of the background flow. Integrating over the channel width, we obtain in the absence of dissipation (

*Q*

_{y}is time independent, we obtain from (26) as a necessary condition for instability that it change sign (Rayleigh 1880). However, this result does not generalize to time-dependent flows because episodic decrease in [

*Î¶*

^{2}] can coincide with small values of

*Q*

_{y}so as to maintain constraint (26) while

*Î¶*

^{2}]/âˆ‚

*t*

*Q*

_{y}(

*y, t*) is illustrated with a simple schematic example in appendix B; however, in order to verify that the Rayleigh theorem does not constrain the stability of dynamical systems arising from time-dependent flows it suffices to consider a time periodic flow produced by periodic discontinuous alteration between two mean velocity profiles. We choose nondimensional

*Î²*= 10 in order to ensure that the associated mean vorticity gradient is of one sign. The two velocity profiles and the associated mean vorticity gradients are shown in Fig. 2. Each flow state is maintained for a time interval

*T.*The Lyapunov exponent is readily calculated using Floquet analysis. The time-dependent flow is unstable and the growth rate as a function of the switching period

*T*is shown in Fig. 3 for zonal wavenumber

*k*= 1.

*U*(

*y, t*) =

*Î±*(

*t*)

*y,*with time-varying

*Î²*(

*t*) of one sign. For this case the constraint (25) becomes

^{1}

_{âˆ’1}

*Î¶*

^{2}]

*dy*from which it follows that perturbations cannot grow in vorticity and therefore cannot have positive Lyapunov exponent, as discussed in section 3a.

## 4. The mechanism producing instability in time-dependent flows with stable mean operators

We will show that temporal variation of **A****A****A**

**A**

*t*) is approximated by a piecewise constant sequence of operators,

**A**

_{i}, each of which represents the continuous operator for a finite time interval

*Ï„.*The initial state

*Ïˆ*

_{0}evolves to the following state at time

*nÏ„*:

*G*

_{i}, over the interval

*Ï„*:

*G*

_{i}approaches the Lyapunov exponent in (6) as

*nÏ„*â†’ âˆž. The total perturbation growth over

*nÏ„*can be expressed alternatively in terms of the projection of the state vector on the optimal (or right singular) vectors of the time-varying propagator:

*Î±*

_{k}are the projection coefficients of the unit vector lying in the direction of the state vector on the optimal (or right singular) vectors of the incremental propagator

*e*

^{AiÏ„}

*Ïƒ*

_{k}are the associated optimal growths (singular values).

^{n}

_{k=1}

*Ïƒ*

_{k}= 1. If we assume that the state vectors are uniformly distributed over the state space hypersphere, then the Lyapunov exponent would be the averaged growth rate over the hypersphere:

^{n}

_{i=1}

*Î±*

^{2}

_{i}

*Ïƒ*

_{i}(so long as at least one

*Ïƒ*

_{i}> 1), the Lyapunov exponent

*Î»*> 0, so that the dynamical system is asymptotically unstable despite the fact that at each time step the system is neutral (appendix C). The fundamental reason for this generic instability can be traced to the convexity of the logarithmic function. It is remarkable that growth is inevitable even when the state vector projects uniformly on the optimal vectors of the instantaneous operator. However, in practice for atmospheric applications, expression (33) gives significantly more accurate estimates for the Lyapunov exponent if the observed statistical distribution of the state vector on the optimal vectors is taken into account by assigning the

*Î±*

_{i}â€™s in (33) their observed statistical properties of variance and temporal correlation.

The magnitude of growth calculated using (33) can be shown to depend primarily on the variance of the incremental growths over the characteristic interval *Ï„,* which can be conveniently measured by the standard deviation (std) of the optimal growths about their mean, which for nondissipative systems is unity. For nondissipative systems if the std of the optimal growths about unity is zero, the Lyapunov exponent is also zero, but as we have seen, for any other value of the std of the optimal growths of the incremental propagator the Lyapunov exponent is positive. As the variance increases the Lyapunov exponent approaches asymptotically the logarithm of the std of the optimal growths of the incremental propagators.

We illustrate the above by Monte Carlo evaluation of (33). The dependence of the Lyapunov exponent on the logarithm of the std of the optimal growths can be seen in Fig. 4 an example with *n* = 2 degrees of freedom, and in Fig. 5 an example with *n* = 20 degrees of freedom. An example with dissipation is also included in these graphs. While for dissipative systems arbitrarily small optimal growth stdâ€™s do not necessarily result in positive Lyapunov exponent, even for dissipative systems the exponent soon asymptotes to that obtained in the inviscid case as can be seen in the example in Fig. 4, demonstrating that this generic mechanism leading to destabilization of time-dependent systems is robust.

## 5. A Floquet approximation to the Lyapunov exponent

*U*(

*y, t*) governed by the barotropic operator (4) restricted for simplicity to be dissipationless. Consider random flow states chosen discontinuously every time interval

*Ï„*and obtain a further simplification by assuming that the sequence of these random but piecewise steady flow states is repeated with period

*T*=

*nÏ„.*The assumption of periodicity allows us to easily obtain an approximation to the Lyapunov exponent by Floquet analysis. Because of the periodicity the Lyapunov exponent is given by eigenanalysis of the propagator, Î¦

_{T}, that advances the state of the system a time interval equal to the period,

*T,*of the time-dependent flow. The Lyapunov exponent is given by

*Î»*

_{p}is the eigenvalue of the propagator with greatest absolute value.

It is an assumption easily verified that as the period *nÏ„* is increased the growth rate of the first eigenmode of the Floquet approximation approaches the growth rate of the Lyapunov vector of the aperiodic flow, that is, the growth rate obtained in the case the states never repeat. We consider recurrence intervals *n* = 2, 10, 50, 100 with each flow state constrained to be neutral by imposition of a large nondimensional *Î²* = 10, which renders the mean vorticity gradient one-signed for the velocity states chosen, assuring stability of the individual states by the Rayleigh theorem. We have verified that the Lyapunov exponent for *n* large enough approaches the Lyapunov exponent of an aperiodic system with the same fluctuation statistics of *U*(*y, t*) and the approach to this limit is instructive. In the two-flow state case (*n* = 2) (upper left Fig. 6), we obtain islands of instability that are characteristic of parametric instability of time-dependent systems with strictly periodic variation of parameters. An example of such a system is the harmonic oscillator with periodically varying restoring force, and analysis of this system reveals the islands of instability of the familiar Mathieu equation. The islands of instability in our example gradually blend into a continuum as the number of states increases ultimately producing the universal instability for all *Ï„* on [0, âˆž] (cf. the lower-right graph of Fig. 6 for *n* = 100). It can be verified that the Lyapunov exponent vanishes on approach to both limits of this interval *Ï„* â†’ 0 and *Ï„* â†’ âˆž and that the maximum Lyapunov exponent occurs at an intermediate value of *Ï„.*

The asymptotic behavior of the Lyapunov exponent as *Ï„* â†’ 0 and *Ï„* â†’ âˆž found in the above example is generally valid. It is shown in appendix D that for a fixed number of independent piecewise constant realizations if fluctuations of the operator are bounded, uncorrelated, and rapid, then in the limit *Ï„* â†’ 0 the Lyapunov exponent vanishes. In this limit the stability properties of the time-dependent system approach the stability properties of the mean autonomous system. In the large *Ï„* limit *Î»* approaches the mean of the decay rates of the least-damped modes of the incremental operators. Given that for an operator that has a stable mean and stable realizations at each time the Lyapunov exponent vanishes for *Ï„* â†’ 0 and *Ï„* â†’ âˆž, we anticipate that if destabilized by noise the Lyapunov exponent reaches a maximum at an intermediate *Ï„* and that this *Ï„* for which the maximum Lyapunov exponent is obtained depends on the timescale of transient growth. However, if the instantaneous operators are not restricted to be stable, then the Lyapunov exponent does not necessarily vanish as *Ï„* â†’ âˆž but rather asymptotes to the average of the maximum growth rates of the instantaneous states and there may not be an intermediate maximum.

In order to sharpen the correspondence between our general analysis of instability of time-dependent operators and the specific instability of the barotropic atmospheric model, we need to more closely relate the parameters in the analysis to the model. While the atmospheric flow does not evolve in piecewise constant steps, nevertheless the time variation of the flow state is characterized by a finite decorrelation time of the order of a few days, which we take to be the *Ï„* appropriate for our analysis. In our example we assume a timescale of jet vacillation and a mean state of the jet as well as the spatial spectrum of variance about the mean state. Such a model consists of a flow state decomposed into a mean part and a time-dependent stochastic part, which is adequately modeled for our purposes as a red noise process with the observed spatial structure, variance, and decorrelation time.

## 6. Error growth in a barotropic atmosphere

*U*

*y,*on which are superimposed time-dependent wind components produced by modulation of the six gravest latitudinal harmonics

*Ïµ*(

*t*) are red noise processes, with identical variance and decorrelation time,

*T*

_{c}, and zero mean. The magnitude of the velocity fluctuation is characterized by its nondimensional rms average. Typical velocity profile realizations for rms velocity fluctuations of magnitude 0.3 nondimensionalized by the mean shear and the half-channel width are shown in Fig. 7.

Consider the viscously damped barotropic vorticity Eq. (1), made nondimensional by spatial scale of the half-channel width, *L,* and timescale of the inverse shear 1/*Î±.* In these variables the Reynolds number is Re = *Î±*/*Î½* where *Î½* is the coefficient of viscosity, the nondimensional magnitude of the noise process is related to its dimensional magnitude, *ÏµÌƒ,**ÏµÌƒ**Î±L,* and the nondimensional *Î²* parameter is related to its dimensional value, *Î²Ìƒ,**Î²ÌƒL**Î±.* Note that in this nondimensionalization as the dimensional value of the shear *Î±* â†’ âˆž, the nondimensional value of the rms fluctuations *Ïµ* â†’ 0 while Re â†’ âˆž. In the calculations that follow we do not require that each realization be stable.

*U*

*y,*

Forward integration yields the Lyapunov exponent. We have verified that the same Lyapunov exponent is obtained if the flow is approximated by a sequence of steady states each with time duration *Ï„* equal to the decorrelation time of the red noise process. The dependence of the Lyapunov exponent on the rms value of the velocity fluctuations for various decorrelation times of the noise, *T*_{c}, is shown in Fig. 8 for zonal wavenumber *k* = 1, Re = 200, and *Î²* = 0. As the decorrelation time decreases the Lyapunov exponent approaches the decay rate of the stable mean operator (36).^{4} As the decorrelation time increases the Lyapunov exponent asymptotes to the average of the maximum growth rates of the instantaneous realizations. Because the individual flows have not been restricted to be stable as *T*_{c} â†’ âˆž, the Lyapunov exponent approaches a positive limit. This contrasts with the *T*_{c} â†’ âˆž limit shown in Fig. 6 in which case the individual realizations were chosen to be stable and therefore the Lyapunov exponent tended to zero. The relevant regime for atmospheric examples is the case in which each realization is stable.

A contour plot of the resulting Lyapunov exponent as a function of nondimensional rms velocity fluctuations and Reynolds number is shown in Fig. 9 for zonal wavenumber *k* = 1 and a red noise process with decorrelation time *T*_{c} = 1. As expected the growth increases with increasing Reynolds number, and with increasing velocity fluctuations.

We expect that as *Î²* increases the Lyapunov exponent will decrease because the nonnormality of the operator is suppressed by *Î².* The typical dependence is shown in Fig. 10, which shows the Lyapunov exponent as a function of *Î²* for Re = 200, wavenumber *k* = 1, rms velocity fluctuations 0.4, and decorrelation time *T*_{c} = 1. We also expect the Lyapunov exponent to be suppressed at high zonal wavenumbers again because nonnormality is suppressed at large *k.* We verify this by considering the flow damped with Rayleigh friction (Ekman damping) to avoid confusing this effect with the decrease of the Lyapunov exponent with wavenumber arising due to the wavenumber dependence of viscous damping. Figure 11 shows the typical dependence of Lyapunov exponent on zonal wavenumber.

In general, if the mean operator *T*_{c} â†’ 0 the Lyapunov exponent and structure approach the growth rate and structure of the most unstable mode of *T*_{c} â†’ âˆž, the growth rate becomes the average of the growth rates of the individual realizations and the structure assumes the form of the most unstable mode of the operator at a given time. For intermediate *T*_{c} assuming that the modes of *T*_{c} â†’ 0 and *T*_{c} â†’ âˆž limit is as in the case of unstable *T*_{c} â†’ 0 it is the structure of the least damped mode, for *T*_{c} â†’ âˆž it assumes the form of the maximally growing (alternatively least damped) structure of each flow realization. This brings up the question of how to most economically characterize the structure of the Lyapunov vector for intermediate *T*_{c} and we consider the typical case of *T*_{c} = 1. A snapshot of the time-varying Lyapunov vector at four consecutive time instants, each a decorrelation time interval apart, is shown in Fig. 12. The zonal wavenumber is *k* = 2, the Reynolds number is Re = 800, *Î²* = 0, the rms velocity fluctuation is 0.16, and the associated Lyapunov exponent is *Î»* = 0.2.

*y*=

**U**

^{âˆ’1}

*Ïˆ,*where

**U**

*k*= 2,

*Î²*= 0, and Re = 800. The mean coefficient of projection of the normalized Lyapunov vector on the eigenvectors of the mean operator and its standard deviation is shown in Fig. 14. The Lyapunov vector is found to be primarily a superposition of the eigenvectors associated with the numbered eigenvalues shown in Fig. 13. These eigenvectors form the eigenvector subspace of the operator, which is most nearly linearly dependent. As a result their excitation leads to robust growth, which through time dependence, can lead in the manner discussed in section 3b, to a positive Lyapunov exponent. Perhaps the most direct way to measure the contribution of an eigenvector,

**e**, to the nonnormality of

**b**is the biorthogonal

^{5}of

**e**and the inner product in the denominator of (38) is the dot product of the two vectors. For real eigenvectors we recognize that (38) is the secant of the angle between the eigenvector and its biorthogonal. We also recognize that

*Î½*(

**e**) gives the gain in initial projection obtained when an eigenvector is optimally excited by introducing at the initial time its biorthogonal rather than the eigenvector itself (cf. Farrell and Ioannou 1996a). Figure 15 shows the contribution of the eigenvectors of the mean operator to the nonnormality of

*Î½.*Comparison of Figs. 14 and Fig. 15 confirms that the Lyapunov vector results from destabilization of the dominant nonnormal subspace of the mean operator. The probability distribution of the three dominant eigenvector components of the Lyapunov vector is shown in Fig. 16. It should be noticed that the probability distribution is not normal, but similar to the distribution expected if the Lyapunov vector were the sum of a dominant vector (the common structure of the nonorthogonal vectors) and a vector in a random direction (see appendix C).

The nonorthogonal eigenvectors provide insight into the mechanism of time-dependent instabilities but because of their nonorthogonality the eigenvectors do not provide an economical basis for investigating the structure of the Lyapunov vector. We inquire now how Lyapunov vectors project on the orthogonal set of the optimal vectors and evolved optimal vectors (the right and left singular vectors). Projections of the Lyapunov vector on the optimal vectors and on the time-evolved optimal vectors of the mean flow propagator over a period *T*_{c} in the energy inner product are shown in Fig. 17. We observe that the Lyapunov vector does not project uniformly on the optimal vectors. In the energy inner product the Lyapunov vector projects primarily on the top 10 optimal vectors. It follows that we can obtain an improved upper bound on the Lyapunov exponent if we take account in the calculation (33) of the probability distribution of the projection of the Lyapunov vector on these optimal vectors of the propagator over a decorrelation time. This suggests that it is possible to obtain a good upper-bound estimate for the Lyapunov exponent of the large-scale atmospheric flow using the routinely collected analyses of optimal growth distributions. We observe that the Lyapunov vector projects strongly on the top evolved optimal vectors, and consequently the Lyapunov vector can be usefully characterized statistically as having the spatial structure of the top optimal or time-evolved optimal vectors over the decorrelation time *T*_{c}.

## 7. Discussion and conclusions

It is useful to distinguish between two problems addressed by analysis of the linear stability of time-dependent operators: the growth of errors and the growth of perturbations. Calculation of error growth involves the tangent linear equations in which the linearization has been performed about a known time-dependent trajectory and the perturbation is regarded as a small error in specification of the initial conditions. The result of the calculation is the difference between the perturbed and the unperturbed trajectories and for an initial error of specified magnitude this trajectory difference is valid until nonlinear effects become important. If a positive Lyapunov exponent exists then an arbitrarily small perturbation to initial conditions assumes after a period of adjustment the form of the time-dependent first Lyapunov vector after which it proceeds to grow at the mean rate of the first Lyapunov exponent. In this case the asymptotic stability calculation is interpreted as constraining the predictability of the system and the Lyapunov exponent, the structure of the Lyapunov vector, and the time interval required for it to be established as an asymptotic are of practical importance for forecast.

The growth of perturbation problems by contrast envisions development to finite amplitude of a disturbance on a time-dependent flow such as the stratospheric flow forced from below by the temporally varying tropospheric flow and the result is interpreted in this context as a generalization of the transient growth to finite amplitude of perturbations on stationary flows. Another example of this type of problem would be an instability calculation studied as a model for cyclogenesis. Such instability problems have almost always been examined assuming that the underlying operator is autonomous. This assumption is seldom critically examined and it would be of interest, for example, to find that the midlatitude jet while stable if its time mean were analyzed were found to be unstable if realistic variation of the jet with time were included in its specification. It could be that stable mean jets support unstable perturbations in the form of the Lyapunov vector when their temporal variability is included. The nature of these perturbations and their relationship to cyclogenesis and the maintenance of waves of synoptic and planetary scale would be of great interest.

In a more general context, given that all physical problems are to a greater or lesser extent time dependent, the stability of realistic flows whether interpreted as an analysis of error or as the development of a perturbation must, if it is to be comprehensive, be assessed taking into account the time dependence of the flow.

For short enough time intervals, the time dependence of forecast model tangent linear trajectories may be ignored without unacceptable effect on the growth of errors in synoptic forecast (Vukicevic 1991; Errico et al. 1993). In addition, the dominant energetics of the midlatitude atmosphere appear to be associated with growth on synoptic timescales so that the statistics of the atmosphere can be accurately modeled using autonomous time-mean operators (Farrell and Ioannou 1994, 1995; Whitaker and Sardeshmukh 1998). For longer time periods, such as are associated with medium and longer range forecast, explicit account of time dependence needs to be taken. In addition, time-dependent instabilities while not dominant in terms of variance and fluxes at synoptic and planetary scale may be important episodically in the growth of some perturbations to finite amplitude.â€ƒFor the purpose of understanding these phenomena the methods of nonnormal dynamics can be extended from the study of perturbation growth over finite time in autonomous systems to address the perturbation growth over finite time in nonautonomous systems (F&I). In the limit of long time the analog of the most rapidly growing normal mode, which asymptotically dominates in the autonomous system, is the first Lyapunov vector that asymptotically dominates in the nonautonomous system. This asymptotic growth in the nonautonomous system can also be analyzed through the nonnormal dynamics of the underlying time-dependent dynamical operator. But this raises an issue concerning methods in the study of nonautonomous operators. While a stationary state can be easily specified by a deterministic function, a time-dependent system such as the atmosphere can be examined only in generality through its statistical properties. We have turned this to advantage to obtain results that transcend the particularities of a realization of the system but depend rather on the general statistical properties of the systemâ€™s time dependence. We have studied this problem by using stochastic perturbation of the mean operator with the perturbations chosen to model statistically the observed deviations from the mean. We have found that the asymptotic stability properties of time-dependent systems with stable time-mean operators can be understood in general terms through considerations employed by Zelâ€™dovich et al. (1984) in his analysis of vector growth in random unitary systems; that is instability in time-dependent systems results because transient growth given by the propagator over a time interval can dominate over decay even if the system is stable at each instant, a mechanism that can be traced to the general property of the convexity of the logarithm. We have seen that this general mechanism of destabilization in time-dependent systems can be modeled by extension of Floquet analysis to approximate aperiodic systems.

Using these methods and with knowledge of the statistical properties of the time dependence of the operator, the asymptotic stability properties and the nature of the associated Lyapunov vector can be obtained. This allows us to evaluate circumstances under which destabilization of the system due to time dependence occurs and to understand the nature of the resulting growing structures. For the example of the barotropic jet we have found that increasing the amplitude of jet vacillation is destabilizing provided the vacillation does not have the same functional form as the mean flow. We found that increasing *Î²* is stabilizing because nonnormality of the mean operator decreases with increased *Î²,* although no generalization of the Rayleigh theorem exists: examples demonstrate that it is possible for a time-dependent system that satisfies necessary conditions for stability at each instant of time to be unstable. Sufficient spatial and temporal variation of effective *Î²* was found to be destabilizing and this is identified as the mechanism by which variation of jet structure with time produces instability. In the case of a stable mean operator, asymptotic stability is obtained when the temporal correlation of the time dependence of the jet vacillation is either too long or too short, while the optimal correlation time for growth is at an intermediate value.

Identification of the mechanism by which stable mean operators are destabilized by mode mixing induced by time dependence of stable mean operators shows that the Lyapunov vector is not in general the least stable mode of the mean operator; rather mode mixing in the nonorthogonal subspace of the operator produces the unstable Lyapunov vector, which includes structural characteristics from the entire excited subspace.

If the atmosphere is asymptotically unstable due to time dependence of the jet^{6} those instabilities may explain the existence of structures that are not easily explained by appeal to the modes of the mean jetâ€”the origin of short waves being one particular example. A more detailed analysis of the asymptotic stability of realistic flows is needed to answer this and other questions concerning the asymptotic stability of the atmosphere.

## Acknowledgments

This work was supported by NSF ATM-9623539 and by ONR N00014-99-1-0018.

## REFERENCES

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*Tellus,***45A,**462â€“477.Farrell, B. F., and P. J. Ioannou, 1994: A theory for the statistical equilibrium energy spectrum and heat flux produced by transient baroclinic waves.

*J. Atmos. Sci.,***51,**2685â€“2698.â€”â€”, and â€”â€”, 1995: Stochastic dynamics of the midlatitude atmospheric jet.

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*J. Atmos. Sci.,***53,**2041â€“2053.Hart, J. E., 1971: A note on the baroclinic instability of general time-dependent basic fields of the Eady type.

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*Proc. London Math. Soc.,***11,**57â€“70.Vukicevic, T., 1991: Nonlinear and linear evolution of initial forecast errors.

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## APPENDIX A

### The Lyapunov Exponent is Independent of the Norm

*p*is given by

*x*(

*t*) is the state at time

*t.*We show that the Lyapunov exponent does not depend on the norm.

*Î±*and

*Î²*for which

*t*â†’ âˆž, we have

*Î»*

_{1}â©½

*Î»*

_{2}â©½

*Î»*

_{1}and the Lyapunov exponent is independent of norm.

## APPENDIX B

### Schematic Example of Violation of the Momentum Integral Constraint in Time-Dependent Flows

*Q*

_{y}(

*y*) does not depend on time, the time derivative can be moved out of the integrand in (25). We thus obtain that

*Î¶*(

*t, y*) satisfies

*Q*

_{y}(

*y*) has no zero in the flow, then for all times

*Q*

^{âˆ’}

_{y}

*Q*

_{y}(

*y*)| and

*Q*

^{+}

_{y}

*Q*

_{y}(

*y*)|. This proves Rayleighâ€™s theorem: if

*Q*

_{y}(

*y*) â‰ 0, perturbations with nonvanishing vorticity cannot grow without bound in the integral sense of (B2).

*Q*

_{y}(

*y, t*) > 0 everywhere over the interval [

*a,â€‰b*] and

*Q*

^{âˆ’}

_{y}

*Q*

_{y}(

*y, t*) â©½

*Q*

^{+}

_{y}

*f*(

*y, t*) is everywhere positive. For example, consider

*f*(

*y*) = âˆ’

*y*sin

*y*and

*Q*

_{y}(

*y*) =

*y*on the interval [

*Ï€*/2, 3

*Ï€*/2], then

^{3Ï€/2}

_{Ï€/2}

*f*(

*y*)/

*G*(

*y*) = 0 but

^{3Ï€/2}

_{Ï€/2}

*f*(

*y*) > 0, contradicting the inequality. Consequently in time-dependent flows episodic decrease of [

*Î¶*

^{2}] at locations with small values of

*Q*

_{y}can lead to exponential growth of perturbations while maintaining the constraint (25) obtained from momentum conservation.

*A*(

*t, y*) = 1/

*Q*

_{y}(

*t,â€‰y*) takes the following discrete values:

*A*is 1 at low latitudes and 2 at high latitudes, while during the other half

*A*is 2 at low latitudes and 1 at high latitudes. Observe that

*Q*

_{y}has been chosen to be always positive.

*Q*

_{y}for each of the time period vanishes. An enstrophy growth

*B*(

*t, y*) = âˆ‚[

*Î¶*

^{2}]/âˆ‚

*t*that satisfies the above condition is

*A*(1,â€‰1)

*B*(1,â€‰1) +

*A*(1,â€‰2)

*B*(1,â€‰2) = 1 âˆ’ 1 = 0, and during the other half we also have

*A*(2,â€‰1)

*B*(2,â€‰1) +

*A*(2,â€‰2)

*B*(2,â€‰2) = âˆ’1 + 1 = 0. In this example we have enstrophy growth in the time mean at each latitude, that is,

*B*(1,â€‰1) +

*B*(2,â€‰1) = 1 âˆ’ Â½ > 0 and

*B*(1,â€‰2) +

*B*(2,â€‰2) = âˆ’Â½ + 1 > 0. Clearly, positivity of the mean vorticity gradient in time-dependent systems does not rule out the existence of mean growth at every latitude.

While this example demonstrates that it is impossible to exclude the possibility of growth when the potential vorticity is of one sign in time-dependent flows, it does not prove that exponential growth is possible in a dynamical system corresponding to a fluid stability problem. A constructive example of asymptotic exponential growth in a two-state periodic system is given in section 3c.

## APPENDIX C

### Mechanism of Universal Destabilization

Consider singular value decomposition of the propagator for a finite interval of time *Ï„.* The action of the propagator on a unit sphere is to distort the sphere in the direction of the optimal vectors producing an ellipse with semiaxes given by the corresponding optimal growths *Ïƒ*_{i}. We order the optimal growths in descending order: *Ïƒ*_{1} â‰¥ *Ïƒ*_{2} â‰¥ Â·â€‰Â·â€‰Â· â‰¥ 0.

*Ï„*is given by

*Î±*

_{i}are the projections of a state vector on the coordinate axes of the optimal vectors,

*Ïƒ*

_{i}are the optimal growths, and Î£ is the surface of the unit hypersphere of area |Î£|.

^{n}

_{i=1}

*Ïƒ*

_{i}=

*p*only

*n*âˆ’ 1 of the optimal growths are independent. Without loss of generality, we choose to eliminate

*Ïƒ*

_{2}. Then

This demonstrates that *Î»**Ïƒ*_{1}, the greatest optimal growth. (Note that a decrease in *Ïƒ*_{3} also leads to an increase of the growth rate, as then *Ïƒ*_{2} must increase to keep the product constant.)

It is immediate from (C2) that the minimum growth rate is attained when all the optimal growths are equal to *p*^{1/n}. The minimum is Î»_{min} = 1n*p*^{1/N}/Ï„.

For nondissipative dynamics (*p* = 1) these results demonstrate that when all the optimal growths are equal to unity the growth rate is zero, and that the slightest distribution of the optimal growths around unity guarantees growth.

*n*dimensions is for all

*i*:

*P,*of the magnitude,

*x,*of the projection on a coordinate axis of a vector uniformly distributed on the sphere obeys the following distribution on [âˆ’1, 1]:

*n*is the dimension of the space (

*n*â‰¥ 2). The probability distribution is shown in Fig. 18. It is the ratio of the area of the projection of a coordinate axis increment

*dx*onto the shell of a hypersphere to the total area of the hypersphere.

It is remarkable that while in two dimensions it is more probable for a random unit vector to project along the coordinate axes, as the dimension increases the probability distribution becomes highly concentrated near zero indicating that projection on a coordinate axis is increasingly unlikely (in three dimensions the distribution is uniform).

This probability distribution is useful in assessing the randomness of a given perturbation distribution. If the mean projections of the normalized perturbations to any orthogonal basis are distributed as in (C6), then the perturbations are uniformly distributed on the unit sphere. In practice uniform distributions over the unit hypersphere surface in *n*-dimensions can be obtained by choosing *n* uniformly distributed components on [âˆ’1,â€‰1] and normalizing them to unity.

## APPENDIX D

### An Approximate Expression for the Propagator

*Ï„*â†’ 0 can be written as

*n*produces the following small

*Ï„*approximation of the propagator that advances a system state over the interval

*nÏ„*:

*n*the propagator of the time-dependent flow in the limit

*Ï„*â†’ 0 approaches the propagator of the mean flow with operator

^{n}

_{i=1}

**A**

_{i}/

*n.*Consequently if the flow is periodic with period

*nÏ„,*with

*n*fixed, and

*Ï„*â†’ 0 the stability of the periodic flow will be determined from the stability of the mean flow, and consequently if the mean flow is stable the periodic flow will also be stable. Expression (D2) also shows that the next-order correction to the stability of the time-dependent flow involves the commutators among the operators of the flows.

The propagator of time-dependent flows on an arbitrary time interval with uncorrelated time-dependent deviations also approaches the propagator of the mean as the correlation time of the derivations *Ï„* â†’ 0. This can be seen by considering a subdivision of a finite time interval *T* in *n* subintervals of duration *Ï„* and assuming that as *Ï„* â†’ 0 the operators **A**_{i} remain uncorrelated. The reason is that as *n* â†’ âˆž the *O*(*n*^{2}) terms in the sum of commutators in the second exponential of (D2) add to a sum that by the central limit theorem is *O*(*n*) while the *Ï„*^{2} term multiplying this sum decreases as *O*(1/*n*^{2}) making the second exponential of (D2) inferior to the first. This shows that the stability of a time-dependent state consisting of a sum of a mean operator and uncorrelated time-dependent operators is governed by the stability of the mean operator.

(left panels) The mean flow velocity as a function of latitude for the Rayleigh stable example. (right panels) The associated mean vorticity gradient with *Î²* = 10.

Citation: Journal of the Atmospheric Sciences 56, 21; 10.1175/1520-0469(1999)056<3622:PGASIT>2.0.CO;2

(left panels) The mean flow velocity as a function of latitude for the Rayleigh stable example. (right panels) The associated mean vorticity gradient with *Î²* = 10.

Citation: Journal of the Atmospheric Sciences 56, 21; 10.1175/1520-0469(1999)056<3622:PGASIT>2.0.CO;2

(left panels) The mean flow velocity as a function of latitude for the Rayleigh stable example. (right panels) The associated mean vorticity gradient with *Î²* = 10.

Citation: Journal of the Atmospheric Sciences 56, 21; 10.1175/1520-0469(1999)056<3622:PGASIT>2.0.CO;2

The Lyapunov exponent as a function of the switching period *T* for the Rayleigh stable example, consisting of the velocity profiles in Fig. 2. The time-dependent flow results from periodic switching every *T* time units of the flows shown in Fig. 2. The case shown is for zonal wavenumber *k* = 1 and *Î²* = 10.

The Lyapunov exponent as a function of the switching period *T* for the Rayleigh stable example, consisting of the velocity profiles in Fig. 2. The time-dependent flow results from periodic switching every *T* time units of the flows shown in Fig. 2. The case shown is for zonal wavenumber *k* = 1 and *Î²* = 10.

The Lyapunov exponent as a function of the switching period *T* for the Rayleigh stable example, consisting of the velocity profiles in Fig. 2. The time-dependent flow results from periodic switching every *T* time units of the flows shown in Fig. 2. The case shown is for zonal wavenumber *k* = 1 and *Î²* = 10.

The Lyapunov exponent as a function of the standard deviation of the optimal growths for a two-dimensional dissipationless (curve 1) and a representative dissipative system (curve 2).

The Lyapunov exponent as a function of the standard deviation of the optimal growths for a two-dimensional dissipationless (curve 1) and a representative dissipative system (curve 2).

The Lyapunov exponent as a function of the standard deviation of the optimal growths for a two-dimensional dissipationless (curve 1) and a representative dissipative system (curve 2).

The Lyapunov exponent as a function of the standard deviation of the optimal growths of a 20D dissipationless (curve 1) and a representative dissipative system (curve 2). The spread in the values is due to the fact that the optimal growths in a high-dimensional system are not characterized completely by their standard deviation. The continuous line provide a quadratic fit to the Lyapunov exponents.

The Lyapunov exponent as a function of the standard deviation of the optimal growths of a 20D dissipationless (curve 1) and a representative dissipative system (curve 2). The spread in the values is due to the fact that the optimal growths in a high-dimensional system are not characterized completely by their standard deviation. The continuous line provide a quadratic fit to the Lyapunov exponents.

The Lyapunov exponent as a function of the standard deviation of the optimal growths of a 20D dissipationless (curve 1) and a representative dissipative system (curve 2). The spread in the values is due to the fact that the optimal growths in a high-dimensional system are not characterized completely by their standard deviation. The continuous line provide a quadratic fit to the Lyapunov exponents.

The Lyapunov exponent as a function of the time step *Ï„* in an inviscid barotropic flow consisting of a periodic recurrence of (upper-left corner) two-flow states, (upper-right corner) ten-flow states, (lower-left corner) 50 flow states, and (lower-right corner) 100 states. Although each state is neutrally stable and obeys the Rayleigh criterion for stability, the resulting time-dependent system is unstable, approaching neutral stability in the limit *Ï„* â†’ 0 and *Ï„* â†’ âˆž.

The Lyapunov exponent as a function of the time step *Ï„* in an inviscid barotropic flow consisting of a periodic recurrence of (upper-left corner) two-flow states, (upper-right corner) ten-flow states, (lower-left corner) 50 flow states, and (lower-right corner) 100 states. Although each state is neutrally stable and obeys the Rayleigh criterion for stability, the resulting time-dependent system is unstable, approaching neutral stability in the limit *Ï„* â†’ 0 and *Ï„* â†’ âˆž.

The Lyapunov exponent as a function of the time step *Ï„* in an inviscid barotropic flow consisting of a periodic recurrence of (upper-left corner) two-flow states, (upper-right corner) ten-flow states, (lower-left corner) 50 flow states, and (lower-right corner) 100 states. Although each state is neutrally stable and obeys the Rayleigh criterion for stability, the resulting time-dependent system is unstable, approaching neutral stability in the limit *Ï„* â†’ 0 and *Ï„* â†’ âˆž.

The velocity profile that results from modulation of the five gravest harmonics in the barotropic flow example. The rms velocity fluctuation is 0.3.

The velocity profile that results from modulation of the five gravest harmonics in the barotropic flow example. The rms velocity fluctuation is 0.3.

The velocity profile that results from modulation of the five gravest harmonics in the barotropic flow example. The rms velocity fluctuation is 0.3.

The Lyapunov exponent as a function of rms velocity fluctuation for different decorrelation times of the noise process. The decorrelation times are marked on the curves. The Reynolds number is Re = 200 and the zonal wavenumber *k* = 1 and *Î²* = 0. For *T*_{c} = 100 the Lyapunov exponent has approached its asymptotic *Ï„* â†’ âˆž limit, which is equal to the mean over the maximum real parts of the eigenvalues of the instantaneous operators (marked with crosses).

The Lyapunov exponent as a function of rms velocity fluctuation for different decorrelation times of the noise process. The decorrelation times are marked on the curves. The Reynolds number is Re = 200 and the zonal wavenumber *k* = 1 and *Î²* = 0. For *T*_{c} = 100 the Lyapunov exponent has approached its asymptotic *Ï„* â†’ âˆž limit, which is equal to the mean over the maximum real parts of the eigenvalues of the instantaneous operators (marked with crosses).

The Lyapunov exponent as a function of rms velocity fluctuation for different decorrelation times of the noise process. The decorrelation times are marked on the curves. The Reynolds number is Re = 200 and the zonal wavenumber *k* = 1 and *Î²* = 0. For *T*_{c} = 100 the Lyapunov exponent has approached its asymptotic *Ï„* â†’ âˆž limit, which is equal to the mean over the maximum real parts of the eigenvalues of the instantaneous operators (marked with crosses).

The Lyapunov exponent as a function of Reynolds number and rms fluctuation about a mean Couette velocity profile. The case is for zonal wavenumber *k* = 1, *Î²* = 0, and noise decorrelation time *T*_{c} = 1.

The Lyapunov exponent as a function of Reynolds number and rms fluctuation about a mean Couette velocity profile. The case is for zonal wavenumber *k* = 1, *Î²* = 0, and noise decorrelation time *T*_{c} = 1.

The Lyapunov exponent as a function of Reynolds number and rms fluctuation about a mean Couette velocity profile. The case is for zonal wavenumber *k* = 1, *Î²* = 0, and noise decorrelation time *T*_{c} = 1.

The Lyapunov exponent as a function of *Î².* The case is for rms velocity fluctuation 0.4 about a constant shear flow, Reynolds number Re = 200, zonal wavenumber *k* = 1, and noise decorrelation time *T*_{c} = 1.

The Lyapunov exponent as a function of *Î².* The case is for rms velocity fluctuation 0.4 about a constant shear flow, Reynolds number Re = 200, zonal wavenumber *k* = 1, and noise decorrelation time *T*_{c} = 1.

The Lyapunov exponent as a function of *Î².* The case is for rms velocity fluctuation 0.4 about a constant shear flow, Reynolds number Re = 200, zonal wavenumber *k* = 1, and noise decorrelation time *T*_{c} = 1.

The Lyapunov exponent as a function of zonal wavenumber *k.* The case is for rms velocity fluctuation 0.4 about a constant shear flow, *Î²* = 0, and noise decorrelation time, *T*_{c} = 1. The dissipation here is Ekman damping with coefficient *R*_{1} = 0.1.

The Lyapunov exponent as a function of zonal wavenumber *k.* The case is for rms velocity fluctuation 0.4 about a constant shear flow, *Î²* = 0, and noise decorrelation time, *T*_{c} = 1. The dissipation here is Ekman damping with coefficient *R*_{1} = 0.1.

The Lyapunov exponent as a function of zonal wavenumber *k.* The case is for rms velocity fluctuation 0.4 about a constant shear flow, *Î²* = 0, and noise decorrelation time, *T*_{c} = 1. The dissipation here is Ekman damping with coefficient *R*_{1} = 0.1.

The structure of the Lyapunov vector in the zonal (x), meridional (y) plane at four consecutive times separated by *T*_{c}. The rms velocity fluctuation is 0.16 and the noise decorrelation time is *T*_{c} = 1. The zonal wavenumber is *k* = 2, *Î²* = 0, and the Reynolds number is Re = 800. The Lyapunov exponent is *Î»* = 0.2. At first (top panel) the Lyapunov vector is configured to grow producing an increase over *T*_{c} of 1.7, in the next period the Lyapunov vector has assumed a decay configuration (second panel from top) and suffers a decrease of 0.7, subsequently (third panel from top) it enjoys a slight growth of 1.1, and finally (bottom panel) a growth by 1.8.

The structure of the Lyapunov vector in the zonal (x), meridional (y) plane at four consecutive times separated by *T*_{c}. The rms velocity fluctuation is 0.16 and the noise decorrelation time is *T*_{c} = 1. The zonal wavenumber is *k* = 2, *Î²* = 0, and the Reynolds number is Re = 800. The Lyapunov exponent is *Î»* = 0.2. At first (top panel) the Lyapunov vector is configured to grow producing an increase over *T*_{c} of 1.7, in the next period the Lyapunov vector has assumed a decay configuration (second panel from top) and suffers a decrease of 0.7, subsequently (third panel from top) it enjoys a slight growth of 1.1, and finally (bottom panel) a growth by 1.8.

The structure of the Lyapunov vector in the zonal (x), meridional (y) plane at four consecutive times separated by *T*_{c}. The rms velocity fluctuation is 0.16 and the noise decorrelation time is *T*_{c} = 1. The zonal wavenumber is *k* = 2, *Î²* = 0, and the Reynolds number is Re = 800. The Lyapunov exponent is *Î»* = 0.2. At first (top panel) the Lyapunov vector is configured to grow producing an increase over *T*_{c} of 1.7, in the next period the Lyapunov vector has assumed a decay configuration (second panel from top) and suffers a decrease of 0.7, subsequently (third panel from top) it enjoys a slight growth of 1.1, and finally (bottom panel) a growth by 1.8.

Distribution of eigenvalues for the Couette flow at *k* = 2 and Re = 800. Only the least decaying eigenvalues are shown (dots). The remaining eigenvalues are highly damped diffusive modes with zero phase speed (*c*_{r} = 0). The Lyapunov vector for the time-dependent flow with velocity fluctuating about the constant shear profile as in (35) is predominantly a superposition of the numbered eigenvectors (cf. Fig. 14). These numbered eigenvalues denote the most nonorthogonal eigenvectors (cf. Fig. 15).

Distribution of eigenvalues for the Couette flow at *k* = 2 and Re = 800. Only the least decaying eigenvalues are shown (dots). The remaining eigenvalues are highly damped diffusive modes with zero phase speed (*c*_{r} = 0). The Lyapunov vector for the time-dependent flow with velocity fluctuating about the constant shear profile as in (35) is predominantly a superposition of the numbered eigenvectors (cf. Fig. 14). These numbered eigenvalues denote the most nonorthogonal eigenvectors (cf. Fig. 15).

Distribution of eigenvalues for the Couette flow at *k* = 2 and Re = 800. Only the least decaying eigenvalues are shown (dots). The remaining eigenvalues are highly damped diffusive modes with zero phase speed (*c*_{r} = 0). The Lyapunov vector for the time-dependent flow with velocity fluctuating about the constant shear profile as in (35) is predominantly a superposition of the numbered eigenvectors (cf. Fig. 14). These numbered eigenvalues denote the most nonorthogonal eigenvectors (cf. Fig. 15).

Mean projection of the Lyapunov vector on the eigenvectors of the mean operator (36). The eigenvectors are indexed in ascending order of their decay rates. The error bars show the standard deviation of the projection coefficients. The Reynolds number is Re = 800, the zonal wavenumber is *k* = 2, *Î²* = 0, and the rms velocity fluctuation about the constant shear is 0.16. Note that the Lyapunov vector consists primarily of superposition of eigenvectors with eigenvalues denoted by their order number in Fig. 13.

Mean projection of the Lyapunov vector on the eigenvectors of the mean operator (36). The eigenvectors are indexed in ascending order of their decay rates. The error bars show the standard deviation of the projection coefficients. The Reynolds number is Re = 800, the zonal wavenumber is *k* = 2, *Î²* = 0, and the rms velocity fluctuation about the constant shear is 0.16. Note that the Lyapunov vector consists primarily of superposition of eigenvectors with eigenvalues denoted by their order number in Fig. 13.

Mean projection of the Lyapunov vector on the eigenvectors of the mean operator (36). The eigenvectors are indexed in ascending order of their decay rates. The error bars show the standard deviation of the projection coefficients. The Reynolds number is Re = 800, the zonal wavenumber is *k* = 2, *Î²* = 0, and the rms velocity fluctuation about the constant shear is 0.16. Note that the Lyapunov vector consists primarily of superposition of eigenvectors with eigenvalues denoted by their order number in Fig. 13.

Degree of nonorthogonality as defined in (38) for the first 40 eigenvectors (ordered in ascending order of their decay rate) of the mean Couette flow at *k* = 2 and Re = 800. The most nonnormal eigenvectors are those that correspond to eigenvalues denoted by their number in Fig. 13.

Degree of nonorthogonality as defined in (38) for the first 40 eigenvectors (ordered in ascending order of their decay rate) of the mean Couette flow at *k* = 2 and Re = 800. The most nonnormal eigenvectors are those that correspond to eigenvalues denoted by their number in Fig. 13.

Degree of nonorthogonality as defined in (38) for the first 40 eigenvectors (ordered in ascending order of their decay rate) of the mean Couette flow at *k* = 2 and Re = 800. The most nonnormal eigenvectors are those that correspond to eigenvalues denoted by their number in Fig. 13.

Probability distribution of the percentage projection of Lyapunov vector on its three dominant eigenvector components: modes with index 21, 22, 20. The example is that of Figs. 13, 14, and 15.

Probability distribution of the percentage projection of Lyapunov vector on its three dominant eigenvector components: modes with index 21, 22, 20. The example is that of Figs. 13, 14, and 15.

Probability distribution of the percentage projection of Lyapunov vector on its three dominant eigenvector components: modes with index 21, 22, 20. The example is that of Figs. 13, 14, and 15.

(top) Mean projection and standard deviation of the Lyapunov vector on the optimal vectors of the mean flow calculated for a time interval equal to *T*_{c} in the energy norm. (bottom) The mean projection and standard deviation of the Lyapunov vector on the *T*_{c} evolved optimal vectors of the mean flow in the energy norm.

(top) Mean projection and standard deviation of the Lyapunov vector on the optimal vectors of the mean flow calculated for a time interval equal to *T*_{c} in the energy norm. (bottom) The mean projection and standard deviation of the Lyapunov vector on the *T*_{c} evolved optimal vectors of the mean flow in the energy norm.

(top) Mean projection and standard deviation of the Lyapunov vector on the optimal vectors of the mean flow calculated for a time interval equal to *T*_{c} in the energy norm. (bottom) The mean projection and standard deviation of the Lyapunov vector on the *T*_{c} evolved optimal vectors of the mean flow in the energy norm.

Probability distribution of the magnitude of the projection on a coordinate axis of a vector in *n*-space distributed uniformly on the surface of the unit hypersphere. The case *n* = 3 yields a uniform distribution. Cases *n* = 2 and *n* = 20 are also shown.

Probability distribution of the magnitude of the projection on a coordinate axis of a vector in *n*-space distributed uniformly on the surface of the unit hypersphere. The case *n* = 3 yields a uniform distribution. Cases *n* = 2 and *n* = 20 are also shown.

Probability distribution of the magnitude of the projection on a coordinate axis of a vector in *n*-space distributed uniformly on the surface of the unit hypersphere. The case *n* = 3 yields a uniform distribution. Cases *n* = 2 and *n* = 20 are also shown.

^{1}

The commutator of two linear operators **A****B****A****B****AB****BA**

^{2}

Equation (12) implies that the eigenmodes of the mean operator and of the deviation operator are the same and therefore there is a variable (the projection on the modes) in which the modes are *Î´* functions and the operators therefore commute and destabilization cannot occur.

^{3}

Inclusion in (1) of temporal variation in *Î²* with the form *Î²*(*t*) in addition to variation in shear of the form *U*(*y, t*) = (*Î±* + *Ïµ*(*t*))*y* can be shown not to produce instability using simple integral arguments although the mean and deviation matrices do not commute in this case. However, a term of the form *Î²*(*y, t*) may be destabilizing.

^{4}

It should be noted that the limit *T*_{c} â†’ 0 is not the white noise limit. The white noise limit requires also that the fluctuations increased in amplitude as *T*^{âˆ’1/2}_{c}*T*_{c} â†’ 0 the noise approaches unit delta correlation in time. This white noise limit implies divergent velocity fluctuations that are unphysical. In the white noise limit the Lyapunov exponent does not vanish, as shown in F&I.

^{5}

If **U****b** of the eigenvector **u**, corresponding to a given column of **U****U**^{âˆ’1})^{â€ } (â€ denotes the Hermitian transpose). The biorthogonal of **u** is orthogonal to all other eigenvectors.

^{6}

It is not necessary for a turbulent flow to have a positive Lyapunov exponent. Consider a forced but highly damped flow; the forcing can produce an arbitrarily complex flow field that is completely determined by the, in principle, known forcing. Meanwhile the damping can be sufficiently great that no positive Lyapunov exponent exists. While this would mean that an arbitrarily small perturbation would not result in a completely different state after sufficient time has passed, it would not mean that the state would be predictable given that the forcing is unknown. It may be of interest to remark in this connection that even unforced turbulence has not been established to exist on a chaotic attractor and therefore to have a positive Lyapunov exponent (Brosa 1989).