a. The γ = 0.1280 limit cycle

A numerical exploration of (2.1)–(2.3) has been conducted by Pedlosky and Frenzen (1980). The present study focuses on a set of solutions with m = 1 and K² = 2π² (k = π), corresponding to a wave with equal zonal and meridional scales. For these values of m and K, the results of Pedlosky and Frenzen (1980, Fig. 3) suggest that solutions of (2.1)–(2.3) undergo a period-doubling transition to chaos near γ = 0.13. Unless otherwise noted, the results described here were obtained with the simple truncation at J = 6 in (2.2). In some cases, slightly different results were obtained for other truncations (see Fig. 3, below) but the main points made here were not affected. Differences between the present numerics and those of Pedlosky and Frenzen (1980) also lead to small but inessential differences in the solutions and their dependence on γ. A brief summary of other numerical issues is given in the appendix.

For γ = 0.1280 (and for a range of adjacent γ; see Fig. 3 below), the numerical solutions approach a limit cycle (Figs. 1, 2a). Let the period of this limit cycle (and its unstable continuation; see below), which depends weakly on γ, be denoted by T; for γ = 0.1280, T ≈ 24.176 (Fig. 1). Note that, since the Eqs (2.1)–(2.3) are unchanged by the transformation (A, B) → (−A, −B), an asymmetric cycle is accompanied by a twin of opposite parity, corresponding to an arbitrary along-channel phase shift of a half-wavelength. For simplicity, attention is restricted here to the cycle with parity such that the maximum value of |A| occurs for A > 0. Corresponding results for the twin cycle may be obtained by changing the appropriate signs or phases.

In the wave–mean oscillation corresponding to this limit cycle, there is a periodic reduction in zonal shear driven by the growing wave, followed by saturation of wave growth as the source of instability is removed, and subsequent decay of the wave amplitude and resurgence of the zonal shear (Fig. 1; the origin of time is fixed here and below so that t = 0 at the point of the cycle where B = 0 and A > 0). Coriolis forces that arise from the wave-induced meridional circulation drive the changes in zonal flow. The wave amplitude A oscillates between a positive and a negative maximum, each time remaining small for substantial times as it changes sign (Fig. 1). The mean-flow correction reduces the vertical shear everywhere, and is nearly in phase with the squared wave amplitude, consistent with a dominant balance between mean-flow acceleration and potential vorticity fluxes due to secular changes in the wave amplitude, as might be anticipated for small values of friction γ. Similarly, since γ is small, the weakly nonlinear phase shift B ≈ dA/dt over most of the cycle, as for the linear modes of instability of the steady zonal flow. The cycle is asymmetric and not sinusoidal. The positive maximum for A is slightly larger than the negative minimum, and the transition from negative minimum to positive maximum takes longer than the reverse. The system evolves relatively slowly near the unstable equilibrium A = B = U_j = 0, and |B| is largest for intermediate values of |A| (Figs. 1, 2a).

b. Unstable and higher-order cycles (γ = 0.1315)

As γ increases past 0.1280, the system undergoes a sequence of period-doubling bifurcations. Near γ = 0.1292, the period-T limit cycle loses stability, and a stable period-2T cycle appears. (Away from the bifurcation point, the ratio of the periods of these two cycles is only approximately 2, but it is customary and useful to continue to identify it as a period-2T cycle, and similarly for the higher-order cycles.) Near γ = 0.1305, the period-2T cycle loses stability, and a stable period-4T cycle appears. Chaos appears to ensue for γ greater than about 0.1309 (Fig. 2b). This sequence is conveniently visualized in a Feigenbaum diagram based on a Poincaré section, in which the value of A when B = 0 and A > 0 is plotted against γ for a set of numerical solutions (Fig. 3a). Near γ = 0.1316, there is an abrupt change in the solution structure; additional analysis suggests that this change marks the merger of a pair of antisymmetric attractors. More accurate numerical solutions show a similar bifurcation sequence, shifted slightly toward smaller γ (Figs. 3b,c).

The periodic orbits that lose their stability in this period-doubling sequence persist as unstable periodic orbits (e.g., Fig. 2a). For γ less than about 0.1309, the stable state is a finite-period limit cycle, and the unstable periodic orbits are not directly relevant to the long-time dynamics. In the chaotic regime, however, the set of unstable periodic orbits can be related to the structure of the attractor and the long-time dynamics in a specific, quantitative manner, and may be used to compute statistical averages over the attractor (e.g., Artuso et al. 1990a,b). This relation is only briefly illustrated here, and will be pursued in more detail elsewhere.

For γ = 0.1315, in the chaotic regime, the evolution may be usefully represented by a one-dimensional map (Fig. 4). From the spline representation of this map, unstable period orbits may be determined systematically in the standard way, by associating the symbols 0 and 1 with the intervals to the left and right, respectively, of the point where the map achieves its maximum, generating a set of binary symbol sequences, and finding the corresponding unique orbit points by inverse iteration. These points may then be used as first guesses for periodic points of the differential equations, which may be improved by a shooting technique that uses Newton’s method.

The corresponding unstable cycles for the repeated symbol sequences 1, 10, 1110, 11010, 11110, and 111111111010 are shown in Fig. 5. Because of the geometry of the map, many symbol sequences do not have corresponding cycles. No cycle corresponds to the repeated sequence 0, nor to any sequence containing 00, since all points in the left-hand interval are mapped to points on the right-hand interval (Fig. 4). The unstable cycles “fill out” (more precisely, and under certain conditions that may or may not strictly hold here, are dense on) the attractor (Figs. 5, 2b); it is this property that makes them useful for computing averages.

a. Formulation

Small disturbances to the periodic cycles described in the previous section will satisfy a linearized form of (2.1)–(2.3), in which the given periodic cycle is the time-dependent basic state about which the equations are linearized. The solutions of the corresponding linearized equations may be computed by standard techniques for linear differential systems with periodic coefficients (e.g., Coddington and Levinson 1955), often known as Floquet theory. For a periodic orbit solution P(t) = [A(t), B(t), U₁(t), U₂(t), . . . , U_J(t)], any linear disturbance may be written as a fixed sum of the J + 2 Floquet eigenvectors {ϕ_j(t), j = 1, 2, . . . , J + 2}, where (for nondegenerate systems) each vector has the form ϕ_j = Φ_j(t) exp(λ_jt), for a (J + 2)-component vector function Φ_j(t), with Φ_j(t + T) = Φ_j(t) [or possibly Φ_j(t + T) = −Φ_j(t)]. Each ϕ_j is a solution of the linearized equations around the given cycle.

Here, attention is restricted to the shortest cycles, the unstable period-T cycle for γ = 0.1315 and the stable period-T cycle for γ = 0.1280. These two cycles are very similar (Fig. 2a; see also Figs. 1 and 9a, below), with nearly indistinguishable spatial and temporal wave and mean-flow structure, the main difference being only that one cycle is stable and the other slightly unstable. The corresponding periods T are T = 24.176 for γ = 0.1280 and T = 24.479 for γ = 0.1315. The Floquet problems were solved numerically, as described in the appendix.

b. Results

The Floquet characteristic exponents λ_j for these two cycles are shown in Fig. 6. Positive exponents indicate instability, and negative indicate stability. The solutions are obtained for the truncation J = 6, so there are eight Floquet eigenvectors. Some of the corresponding structure functions Φ_j for γ = 0.1280 are shown in Fig. 7. The Floquet vectors essentially divide into two types. The neutral vector (ϕ₁ for γ = 0.1280, or ϕ₂ for γ = 0.1315) is proportional at each point of the cycle to the rate of change along the cycle, and so describes wave-dynamical processes, as does the oscillation itself. The least and most stable nonneutral Floquet vectors (ϕ₂ and ϕ₈ for γ = 0.1280, or ϕ₁ and ϕ₈ for γ = 0.1315) are also of this type, as they each differ in structure from the neutral vector essentially only by a change in phase at minima of |A| and an exaggerated asymmetry between the maxima in the unstable mode, relative to the neutral mode (Figs. 7a,b,d). In contrast, ϕ₄–ϕ₇ have mean-flow structure dominated by higher meridional harmonics (Fig. 7c). These modes have decay rates nearly equal to γ, and evidently describe the frictional decay of the higher mean-flow harmonics. The vector ϕ₃, which has intermediate stability, appears to be a mixture of these two types.

The decaying wave-dynamical mode ϕ₈ distinguishes itself from the corresponding least stable nonneutral wave-dynamical mode (ϕ₂ or ϕ₁) by its larger amplitude during the decay phase that follows each wave maximum along the basic-state wave–mean cycle. This distinction can be seen by comparing the wave or mean-flow disturbances in Figs. 7b and 7d immediately before and after each wave amplitude maximum in Fig. 1. The sustained disturbance growth or decay described by the wave-dynamical Floquet vectors is accomplished by this asymmetry in the disturbance oscillations.

The dynamical splitting into two types of modes is evident also in the disturbance heat (or potential vorticity) flux associated with these cycles. An appropriate heat flux quantity is the cycle average of the product Φ_AΦ_B, the first two components of Φ_j, for each j. For both cycles, this quantity is of order 10⁻² for ϕ₁–ϕ₃; −10⁻² for ϕ₈; and 10⁻⁵, 10⁻⁶, 10⁻⁷, 10⁻⁸ for ϕ₄–ϕ₇, respectively. Thus, the wave-dynamical modes have much larger heat fluxes than the decaying zonal flow modes; the heat flux is large but countergradient for the rapidly decaying wave-dynamical mode ϕ₈.

This dynamical splitting persists for the higher-order γ = 0.1315 cycles. Characteristic exponents for 138 of these cycles, ordered by symbol sequence, exhibit substantial variations for the least and most unstable modes, and for the third mode, and very little variation for modes 4 through 7 (Fig. 8). The variations in the exponents compensate, as they must, since the sum of the exponents equals the trace of the Jacobian matrix; numerically these sums were equal and constant to 10⁻⁷ or less, while the neutral eigenvalue was numerically of order 10⁻⁶ or less. The eigenvectors for γ = 0.1280 and γ = 0.1315 are very similar, as can be anticipated from the similarity of the cycles (Figs. 1, 2a, and 9a). The unstable vector (ϕ₁) for γ = 0.1315 is similar to the least stable nonneutral vector (ϕ₂) for γ = 0.1280, with a modest change in the relative magnitude of the wave amplitude peaks (Figs. 7a, 9b); the bifurcation to instability of the cycle as γ varies is associated with the change in sign of the Floquet exponent for this vector.

c. The bred growing mode analogy

The Floquet vectors are the simplest analog of the bred growing modes (BGMs; Toth and Kalnay 1997) used for ensemble generation in operational forecast models. As noted above, bred growing modes are computed by an iterated breeding cycle, in which the differences between forecast ensemble members and a control forecast is rescaled and added to the analysis at each analysis cycle to initialize a new ensemble. In the simplest idealization, in which only linear disturbances are permitted and no additional model of the forecast–analysis cycle is included, the Floquet vectors (or suitably orthogonalized combinations thereof) may be viewed as analogs of the modes that one may expect eventually to emerge from a BGM iteration for a control forecast corresponding to the periodic basic state. (For a nonperiodic basic state, the idealized analogs of BGMs are Lyapunov vectors.)

For the stable (γ = 0.1280) cycle, all disturbances decay except those that are tangent to the cycle. The resulting BGM-analog disturbance is a time-dependent phase shift along the cycle, which is described in the linear approximation by the neutral Floquet vector. A corresponding BGM-analog ensemble would itself oscillate periodically, spreading exponentially along the cycle during the exponential growth phase of the cycle, and contracting exponentially during the decay phase. Thus, in spite of its overall long-term stability, the time-dependent flow will have a nontrivial BGM analog that is associated with the cyclic growth and decay of the baroclinic wave. The local oscillation of disturbance amplitude that is described by the neutral Floquet vector is due to precisely the same process that causes the oscillation of the wave amplitude itself, not an additional or secondary oscillation or instability. The growth of disturbances is a direct consequence of the growth of the time-dependent basic state. It can be inferred that any time-dependent state will induce a similar type of disturbance growth during its periods of growth.

For the unstable (γ = 0.1315) cycle, one of the Floquet exponents in positive, and the corresponding disturbance will grow exponentially. This mode supplies a second BGM analog, in addition to the neutral mode. However, the positive Floquet exponent is roughly 0.025, 40 times smaller than the growth rate of the baroclinic instability and the inverse timescale of the basic baroclinic oscillation. Thus, the disturbance growth processes that are relevant on the baroclinic wave timescale are still primarily associated with the growth and decay phases of the oscillation itself, rather than with this additional weak time-dependent instability.

If attention is extended to the nonperiodic attractor indicated by the γ = 0.1315 numerical solutions, the unstable Floquet vector of the period-T cycle provides additional information, as the large-scale structure of the attractor tends to be tangent to this unstable eigenvector (Fig. 10). Since the attractor is in general a fractal object (Fig. 10), the precise meaning of the phrase “tangent to the attractor’s large-scale structure” is not easily understood, but in practice it is useful if the large-scale geometry is sufficiently simple. For example, in Fig. 10, both the large-scale structure and the disturbance along ϕ₁ (and, in this case, also those along ϕ₂ and ϕ₈) lie on the diagonal from upper left to lower right. Thus, in this case, the BGM analog can give an indication of the distribution of dynamically accessible states near the unstable periodic orbit that would be relevant to ensemble construction for this idealized model.

a. Formulation

The Floquet vectors give a complete description of the behavior of linear disturbances, and identify the disturbances of fixed spatiotemporal structure that will grow or decay at fixed exponential rates over the course of a period. Alternatively, it is possible to inquire which linear disturbances are maximally amplified after a fixed interval of time (Lorenz 1965; Farrell 1989). Such optimal disturbances, referred to here as singular vectors (SVs), generally exhibit transient growth, and do not have fixed modal structure.

Since the Floquet vectors span the phase space, the disturbance ξ(t) may be written as a sum of Floquet vectors,

View Expanded

where Z(t) = [ϕ₁(t), ϕ₂(t), . . . , ϕ_J+2(t)] is the matrix whose columns are the Floquet vectors ϕ_j. The expansion coefficient vectors a = [a₁, a₂, . . . , a_J+2] are independent of time, so if the Φ_j and λ_j are known, it suffices to determine the a at the initial time t₁ in order to compute ξ(t) for any t. Define the “linear propagator” L(t₁, t) = Z(t)Z(t₁)⁻¹, in terms of which ξ(t) = L(t₁, t)ξ(t₁). Then the SV maximization yields the generalized eigenvalue problem

Lt₁t₂^TN₂Lt₁t₂ξt₁μ²N₁ξt₁(5.3)

for the SVs in the usual way (e.g., Buizza et al. 1993). The matrices Z and L may be calculated directly from the Floquet vectors ϕ_j at the initial (t₁) and final (t₂) times. The eigenvalue problem (5.3) may then be solved by standard methods for the eigenvector–eigenvalue pairs {(ξ^l, μ²_l), l = 1, . . . , J + 2}. From (5.2), the Floquet decompositions of the SVs are then a^l = Z⁻¹(t₁)ξ^l. The square root μ_l of each eigenvalue is the corresponding singular value.

For a given periodic orbit p, the SVs depend on the initial and optimization times t₁ and t₂, and on the norms defined by the matrices N₁ and N₂. Since the periodic orbits have been determined numerically, analytical solutions for the SVs are not available, and the SVs must be calculated numerically, which requires that specific choices for these quantities be made. Attention is restricted here to two simple choices for N₁ and N₂, while the dependence of the corresponding SVs on t₁ and t₂ is explored more completely.

The two choices of norm are referred to here as the“orbit-relative” (OR) and “amplitude energy” (AE) norms. For convenience, the matrices N₁ and N₂ are taken equal in each case. Since the initial and final norms are then equivalent, the singular values μ_l are amplification factors. This restriction is not necessary, but simplifies the interpretation. In both cases, N₁ and N₂ are also taken to be diagonal matrices. For the AE norm, N₁ and N₂ are the identity matrix, so the norm is simply the sum of squares of the disturbance amplitudes ξ_j. In terms of the physical variables, this is closely related to the usual energy, except that terms of first and second order in the small-amplitude parameter ε are weighted equally. A similar norm would be obtained by computing the usual energy and formally setting ε = 1 in (2.7)–(2.8), so the AE norm is essentially a physical energy norm based on extrapolation of the small-amplitude results to finite amplitude. For the OR norm, the jth diagonal element in N₁ and N₂ is the inverse square of the peak-to-peak amplitude excursion of the jth component of the periodic orbit solution. That is, the linear disturbance of each component is measured relative to the magnitude of the variation of the corresponding component along the periodic orbit. Because higher mean-flow components U_j have relatively small variation along the periodic orbit, the OR norm gives them greater weight than does the AE norm.

b. Results

Singular vectors are computed for the stable cycle at γ = 0.1280 and for the period-T unstable cycle at γ = 0.1315. The dependence of the SVs on initialization time t₁ and final time t₂ is investigated, for both fixed-length (variable t₁, constant τ = t₂ − t₁) and variable-length (constant t₁, variable τ) optimization intervals. As above, the calculations are done for the truncation J = 6, so each yields 8 SVs. Although there are quantitative differences between the results for the two norms, qualitatively the results are similar.

For γ = 0.1280, the OR-norm SVs for t₁ = 0 (Fig. 11) divide into two classes, as did the Floquet vectors. The first and last SVs (in order of decreasing singular value) rapidly amplify and decay, respectively, in each case more rapidly than the corresponding Floquet vector. The singular value for the first SV, μ₁, depends strongly on the optimization interval τ = t₂ − t₁ = t₂, and is relatively large where the wave amplitude of the periodic cycle is large, and relatively small where the wave amplitude is small (Fig. 11). The maximum value of μ₁ for 0 < t₂ < T is approximately 800, near t₂ = 23. On longer timescales, μ₁ continues to oscillate with t₂, in phase with the wave amplitude, but never exceeds 800. This bounded amplification is consistent with the stability of the γ = 0.1280 periodic orbit. The fluctuations in μ₂ appear to grow with time, at least during the period 0 < t₂ < 5T. The intermediate SVs have amplification factors that correspond closely to the decay rates of the intermediate Floquet vectors. Thus, the dynamical splitting of the Floquet vectors into wave-dynamical modes and decaying zonal-flow modes is also reflected in the SVs.

The first SV is primarily a combination of the two wave-dynamical Floquet vectors, ϕ₂ and ϕ₈, plus a contribution from the neutral vector ϕ₁. Despite the strong dependence of μ₁ on t₂ (or, equivalently, on the optimization interval τ) for t₁ = 0, the Floquet decomposition a¹ of the first SV is nearly independent of t₂, as the ratios |a¹_j|/|a¹| for the dominant components a¹_j of the first SV change by only a few percent for 0 < t₂ < T. The Floquet vectors themselves are strongly time-dependent, so the final physical structure of this SV depends strongly on t₂, and only the Floquet decomposition is nearly independent of t₂.

When t₁ varies (for fixed optimization interval τ), the singular values μ_l continue to split in the same way, and μ₁ oscillates in the same way, with maxima where t₂ corresponds to maxima in the wave amplitude for the basic cycle (Fig. 12a). The Floquet decomposition of the first SV depends more strongly on t₁ with fixed τ (Fig. 12b) than on t₂(τ) with fixed t₁. The wave-dynamical and neutral Floquet vectors still dominate the first SV, but there are rapid oscillations with t₁ in the relative proportion of these contributions during periods where the wave amplitude is small (t₁ ≈ 5, t₁ ≈ 20). Elsewhere in the cycle, however, the decomposition in terms of Floquet vectors is often independent of t₁ for substantial periods. During these periods, the corresponding physical structure of the first SV can change rapidly and dramatically. For example, during the interval 6 < t₁ < 13, the Floquet decomposition is nearly fixed (Fig. 12b), but the Floquet vectors change markedly (Fig. 7), as the A and B components sequentially pass through zero crossings, implying rapid changes in the physical structure of the first SV.

Additional SV solutions for other values of t₁ and t₂, for fixed-length intervals with different τ and for variable-length intervals with different t₁, confirm these general tendencies: the amplification factor μ₁ oscillates with t₂, in phase with the wave amplitude of the cycle;and the Floquet decomposition a¹ primarily involves the wave-dynamical and neutral vectors, and is relatively independent of t₂ when t₁ is near the phase of maximum wave amplitude (or when t₂ is large), and strongly dependent on t₁ and t₂ when t₁ is near the phase of minimum wave amplitude. For the AE norm, the dependence of the amplification factors on t₁ and t₂ is qualitatively similar, though quantitatively different (Figs. 13, 14a). The decomposition a¹ is still dominated by the wave-dynamical and neutral Floquet vectors, but the dependence of the decomposition on t₁ is relatively stronger (Fig. 14b).

For the unstable cycle at γ = 0.1315, the results are much the same. This can be anticipated from the similarity of the cycles themselves, and of the Floquet vectors and exponents, and the relatively long timescale of the instability (1/σ ≈ 40) relative to the baroclinic wave timescale. The major difference is that, on long timescales, the leading SV amplification factor μ₁ tends to increase at an exponential rate equal to the leading Floquet exponent, as should be expected (Fig. 11). The Floquet vector decomposition of the leading SV has the same structure as for γ = 0.1280, with contributions primarily from the wave-dynamical and neutral vectors.

Some singular vectors were computed for a third, scaled physical energy norm, which was equivalent to the AE norm except that the second-order quantities B and U_j were scaled by ε = 0.01, corresponding roughly to a physical energy for a small but finite-amplitude disturbance. The qualitative results were again generally similar (Fig. 13, dashed line).

As noted above, the leading Floquet vector of an unstable cycle in a chaotic regime tends to be tangent, in some large-scale sense, to the attractor. Because of the contributions from other vectors, the leading SV does not generally share this property. This suggests that, to the extent that prediction errors are due solely to errors in estimating the initial conditions on the attractor, as they will be if the model attractor accurately describes the dynamically accessible states, the leading SV will in general be irrelevant to the predictability of the nonlinear system, since the disturbances it describes will evidently be dynamically inaccessible. However, if the attractor is examined on smaller scales, its fractal structure becomes apparent, and a leading SV disturbance may apparently still lie on the attractor (Fig. 10). This illustrates that inferences about the accessibility of states described by Floquet vectors (or BGMs) and SVs in an idealized model such as this one rely on the simple geometric structure of the model attractor, and may not easily generalize to more complex models. If model error is such that the dynamically accessible states are not accurately described by the attractor, then of course such inferences also do not follow.