Chaos, Quasiperiodicity, and Interannual Variability: Studies of a

15 SEPTEMBER 2000

CHRISTIANSEN

3161

Chaos, Quasiperiodicity, and Interannual Variability: Studies of a Stratospheric Vacillation Model BO CHRISTIANSEN Danish Meteorological Institute, Copenhagen, Denmark (Manuscript received 21 January 1999, in final form 22 November 1999) ABSTRACT The parameter space of a stratospheric vacillation model is investigated numerically. The model is the quasigeostrophic b-plane model introduced by Holton and Mass with time-independent forcing at the lower boundary. Chaotic and quasiperiodic states, diagnosed by the values of the largest Lyapunov exponents, are found for realistic values of the parameters. The quasiperiodic state branches off the periodic vacillations observed in previous studies through a Hopf bifurcation, and chaotic states appear after additional bifurcations. Defining an appropriate variable, a circle map behavior is revealed and the route to chaos identified as an example of the Ruelle–Takens–Newhouse scenario. The existence of additional bifurcations with constant forcing indicates that the explanation of the stratospheric variability in the Northern Hemisphere does not need to include external factors such as tropospheric variations or variations in the equatorial quasi-biennial oscillation. The bifurcations produce low-frequency components with timescales long enough to interfere with the annual cycle.

1. Introduction Although the first observations of stratospheric sudden warmings were reported almost 50 years ago (Scherhag 1952) and the first successful theoretical insight goes almost 30 years back (Matsuno 1971), a thorough understanding of its mechanism is still lacking. Together with the equatorial quasi-biennial oscillation, the stratospheric sudden warmings provide a unique opportunity to study the nonlinear dynamics in general and the wave–mean interactions of the large-scale flow in particular. In addition to the theoretical interest, stratospheric sudden warmings are also a subject of much practical attention related to the transport properties of the Northern Hemisphere winter. A major warming that is characterized by a heating of the polar region and reversal of the temperature gradients and the zonal wind has a typical lifetime of a few weeks. Such warmings are observed in average every second year but with irregular intervals between one and five years (see, e.g., Andrews et al. 1987). This dramatic variability in the Northern Hemispheric stratosphere has previously been attributed to interannual variability in one or more of the stratospheric boundary conditions. A different possibility is that the variability is created internally within the extratropical strato-

Corresponding author address: Dr. Bo Christiansen, Danish Meteorological Institute, Lyngbyvej 100, DK-2100 Copenhagen, Denmark. E-mail: [email protected]

q 2000 American Meteorological Society

sphere. This possibility will be considered in this paper by numerical analysis of a simple model. The dynamics in the stratosphere are known to be driven by planetary waves entering the stratosphere through the lower boundary. The stationary planetary waves are generated by zonal asymmetries in the orography and in the land–sea distribution, while transient planetary waves are generated by instabilities in the troposphere related to synoptic-scale weather systems. In the Northern Hemisphere winter, the stationary and transient components are of comparable strength at the tropopause level. Interannual variations in the tropospheric state might thus be a natural source of the interannual variability in the stratosphere, and correlations do exist between stratospheric variability and, for example, the North Atlantic oscillation (Baldwin et al. 1994). The equatorial boundary is appropriately determined by the position of the critical line with vanishing mean zonal flow. The critical line influences the propagation of the planetary waves in the stratosphere by working as an impenetrable boundary. Interannual variability in the equatorial quasi-biennial oscillation change the position of the critical line and may thus change the wave driving of the extratropical stratosphere (Holton and Tan 1982; Kodera 1995). Other tropospheric factors such as volcanos (Labitzke and McCormick 1992) and the El Nino–Southern Oscillation (van Loon and Labitzke 1987) have been suggested. Also the influence of the solar cycle has been studied (Labitzke and van Loon 1995; Hood et al. 1993). Less attention has been paid to the possibility that the

3162

JOURNAL OF THE ATMOSPHERIC SCIENCES

variability is of internal nature. However, studies with a one-dimensional quasigeostrophic model formulated by Holton and Mass (1976) have shown the existence of a Hopf bifurcation when the system is forced from below with a time-independent forcing (Yoden 1987a). When the strength of the forcing exceeds a critical threshold, a stable steady state loses stability and a periodic vacillating state emerges. The vacillations strongly resemble stratospheric warmings and result from wave–mean flow interactions. Christiansen (1999) suggested a conceptual picture explaining the vacillations in terms of a downward moving critical line as a consequence of the competing effects of the relaxation toward the radiative equilibrium profile and the wave– mean interactions. These concepts are closely related to the theory of the quasi-biennial oscillation suggested by Plumb (1977, 1984). In the present study the Holton–Mass model will be studied with the emphasis on internal variability. The disadvantage of a simple model like this is the coarse or even absent representation of physical processes. In the Holton–Mass model the main simplifications are the absence of horizontal wave flux and the severe truncation leaving only one zonal and meridional mode. The advantage is that the parameter space can be investigated in detail allowing a deeper understanding of the physical processes that are included. Several results indicate that the Holton–Mass model offers a reasonable trade-off. First, the transition from a quiescent state to a vacillating state has been observed both in a general circulation model run in perpetual January mode (Christiansen 1999) and in a primitive-equation model of the middle atmosphere (Scaife and James 2000). Christiansen (1999) also indicates that the vertical wave flux drives the vacillations and that horizontal wave flux is less important. Second, Pierce and Fairlie (1993) and Pawson and Kubitz (1996) found observational evidence for preferred flow regimes in the Northern Hemisphere winter similar to the regimes found in the Holton–Mass model for an intermediate range of forcings. Recently, Kodera and Kuroda (1999a, manuscript submitted to J. Geophys. Res.) analyzed the National Centers for Environmental Prediction dataset and identified slowly downward propagating zonal wind anomalies in the Northern Hemisphere winters. They showed (Kodera and Kuroda 1999b, manuscript submitted to J. Geophys. Res.) that the observed features can be simulated qualitatively by an extended version of the Holton–Mass model including two meridional modes. Previously, the only time-dependent state observed in the Holton and Mass model has been a periodic vacillation. While the existence of periodic vacillations might suggest that internal stratospheric dynamics play a major part in the mechanism of sudden warmings, it does not explain the irregular occurrence of the sudden warmings in the real stratosphere. In this paper, we report the existence of quasiperiodic and chaotic states for realistic values of the wave forcing at the tropopause,

VOLUME 57

and the radiative equilibrium profile. The slow modes present in the nonperiodic states generate interannual variability when a periodic annual cycle is included. This result suggests, to the extent it carries over to the real stratosphere, that at least part of the interannual variability might be explained as a result of internal stratospheric instabilities. The paper is organized as follows. In section 2 the Holton–Mass model is briefly summarized and the numerical approach is described. The critical lines separating regions of qualitative different solutions in the parameter space are discussed in section 3. Periodic, quasiperiodic, and chaotic states are identified from the spectrum of Lyapunov exponents in section 4, where also the route to chaos is discussed. The interannual variability occurring when the model is forced with a periodic annual cycle is discussed in section 5. The paper is closed with conclusions in section 6. 2. The model The model was introduced by Holton and Mass (1976) and later studied more intensively by other authors (Holton and Dunkerton 1978; Chao 1985; Yoden 1987a,b; Yoden 1990). It is a quasigeostrophic, b-plane model truncated to include only one zonal wavenumber and one meridional mode. The model includes two independent variables, time and height, and three vertical fields of dependent variables, the zonal mean state and the amplitude and phase of the eddies. Here we briefly summarize the model and our numerical procedures. Expanding the geostrophic streamfunction for a wave c9(x, y, z, t) 5 R[C(z, t) exp(ikx)] exp[z/(2H)] sin(ly) and the mean zonal wind u(y, z, t) 5 U(z, t) sin(ly) the linearized quasigeostrophic potential vorticity equation becomes

5

(]t 1 ikeU ) 2(k 2 1 l 2 ) 1 1 b9ikC 1 e

6

f 02 [] 2 (2H )22 ] C N 2 zz

f 02 [] 2 (2H )21 ]{a []z 1 (2H )21 ]C} N2 z

5 0,

(1)

where

b9e 5 b 1 el 2 U 2

f 02 e (] U 2 H 21]z U ). N 2 zz

(2)

The mean flow equation becomes

[

]t 2l 2 U 1 52 1

]

f 02 (] U 2 H 21]z U ) N 2 zz

f exp(z /H )]z [a exp(2z /H )]z (U 2 UR )] N2 2 0

l 2 ke f 02 exp(z /H )I(C]zzC*). 2 N2

(3)

15 SEPTEMBER 2000

CHRISTIANSEN

The notation is the same as in Holton and Mass (1976) and it is summarized in the appendix. The radiative equilibrium profile U R is chosen as a linear function of height, U R 5 U R,0 1 gz. The physical interpretation of the terms can be found in Yoden (1987b); in particular we note that the two terms on the right-hand side of the mean flow equation Eq. (3) can readily be identified as the radiative relaxation and the wave–mean interaction in the form of the divergence of the Eliassen–Palm flux, respectively. We choose the same boundary conditions as in Yoden (1987a), that is, U(0, t) 5 U R (0) and C(0, t) 5 gh/ f 0 at the bottom z 5 0 (the tropopause level), and C(ztop , t) 5 0 and ] z U(ztop , t) 5 ] z U R (ztop ) at the upper boundary z 5 ztop . As in Yoden (1987a) we proceed adopting the centered differencing scheme ] z X| z5j dz 5 (X j11 2 X j21 )/(2dz) and ] zz X| z5j dz 5 (X j11 1 X j21 2 2X j )/dz 2 , where j 5 1, 2, . . . , N denotes the N (N 5 27 if not explicitly stated otherwise) equidistantly spaced vertical levels, that is, X j 5 X(z 5 jdz). Defining the state vector f by concatenation of R(F), I(F), and U the resulting system of 3N coupled ordinary differential equations can be written [Eq. (2.13) in Yoden (1987a)]: Mi, j

df j 5 Ci 1 L i, j f j 1 Ni, j,k f j f k . dt

(4)

Linearizing gives Mi, j

ddf j 5 L i, j df j 1 Ni,j,k f j df k 1 Ni, j,k f k df j . dt

(5)

The arrays M i,j , L i,j , and N i,j,k are highly sparse. In particular, we note that M i,j is tridiagonal and Eq. (4) is therefore easily solved for df i /dt using the Gauss elimination scheme. Likewise, Eq. (5) is readily transformed to the form ddf i /dt 5 J i,j df j , where J is the Jacobian matrix. We use the standard fourth-order Runge–Kutta method to integrate Eqs. (4) and (5), rather than the scheme originally introduced by Holton and Mass (1976) and used in previous studies of the model. As in Yoden (1987a), steady states are found as the roots of the righthand side of Eq. (4). We use a globally convergent Broyden method (see, e.g., Kelley 1995) to solve the set of 3N nonlinear equations. The stability of a steady state is determined by the sign of the largest real part of the eigenvalues of the Jacobian matrix. At a bifurcation point from a steady state, where the largest real part of the eigenvalues is zero, the bifurcation is identified as a saddle-node or Hopf bifurcation if the corresponding imaginary part is zero or nonzero, respectively. More generally, the type of the solution is determined by the values of the largest Lyapunov exponents. If the largest Lyapunov exponent is negative or positive, then the state is steady or chaotic, respectively. If the largest Lyapunov exponent is zero and the second largest Lyapunov exponent is negative or zero, then the solution is periodic or quasiperiodic, respectively (see, e.g., Berge´ et

3163

al. 1986). The numerical technique for calculating the Lyapunov exponents originates from Shimada and Nagashima (1979) and Benettin et al. (1980) but is adopted here from Ott (1993). To calculate the k largest exponents, an initial set of k orthonormal vectors j m , m 5 1, . . . , k is chosen arbitrarily. The k vectors are then evolved in time according to dj mi /dt 5 J i,jj mj . These equations have to be solved simultaneously with Eq. (4) as the Jacobian matrix depends on the basic state f. This gives a set of 3N(k 1 1) ordinary differential equations. At regular intervals pt , p 5 1, 2, . . . , P, the vectors are orthonormalized by the Gram–Schmidt procedure and their lengths b m,p before normalization recorded. In the limit of large P the exponents l m are given in decreasing order by l m 5 (1/Pt ) S Pp51 lnb m,p . 3. The parameter space In this section we explore the two-dimensional parameter space spanned by the forcing at the lower boundary h and the vertical slope g of the radiative equilibrium. The radiative equilibrium at the lower boundary U R,0 is fixed at 10 m s21 . The results presented here extend the previous analysis by Yoden (1987a, 1990) by considering the full two-dimensional parameter space in greater detail. Figure 1 shows bifurcation diagrams for several values of g with U R,0 5 10 m s21 . The values of the zonal mean wind U at 28.5 km for stable and unstable steady states are plotted as a function of h. For g 5 2 m s21 km21 we recover the general picture described by Yoden, although a few details are different. For low forcing, the only solution is a stable steady state close to the radiative equilibrium, that is, a ‘‘cold’’ state only slightly influenced by the waves. The cold state is stable for a large range of forcings, but loses stability at hcold 5 188.2 m, indicated with a thick arrow in Fig. 1, through a saddle-node bifurcation. For an intermediate range of forcings, [hwarm , hHopf ] 5 [37.3, 68.5] m, a ‘‘warm’’ stable steady state coexists with the cold state. The label ‘‘cold’’ (‘‘warm’’) refers to a strong (weak) mean zonal wind that through the thermal wind relationship is related to a strong (weak) latitudinal temperature gradient with a cold (warm) polar region. The warm stable steady state and an unstable steady state are borne through a saddle-node bifurcation at hwarm , and the warm stable steady state undergoes a Hopf bifurcation at hHopf to an oscillating state. In Fig. 1 these bifurcations are indicated with a thin and a solid arrow, respectively. In a region close to hwarm , which was poorly resolved in Yoden (1987a), we find the coexistence of the two stable states and three unstable states. For h 5 61.1 m, two of the unstable steady states merge and disappear, while the coldest of the two remaining unstable states merges with the cold stable state at hcold . The vertical structures of the five coexistent states for h 5 50 m are shown in Fig 2. The two new unstable states differ from the previously described states in par-

3164


VOLUME 57

FIG. 1. The zonal mean wind U at 28.5 km for the steady solutions as a function of the strength h of the forcing on the tropopause for different values of the vertical slope g of the radiative equilibrium profile. Stable and unstable solutions are shown with solid and dashed curves, respectively. The Hopf bifurcation at hHopf is indicated with solid arrows, and the two saddle-node bifurcations at hwarm and hcold with thin and thick arrows, respectively. Here, U R,0 5 10 m s21 .

15 SEPTEMBER 2000

CHRISTIANSEN

3165

FIG. 2. The five steady solutions for g 5 2 m s21 km21 , U R,0 5 10 m s21 and h 5 50 m. (a) The zonal wind U (m s21 ), (b) the wave amplitude |C| f 0 /g (m), (c) the wave phase, (d) the wave forcing on the zonal wind = · F/r (m s21 day21 ). The dotted curve and the short dashed curve are the stable states.

ticular by having weak zonal mean flow above 40 km. One of these states has a critical line at approximately 40 km, whereas the warm stable steady state has a critical line near 20 km. Just above the critical levels the phase of the waves changes drastically, while it is almost constant elsewhere. The wave forcing on the mean flow (Fig. 2d), which for the steady states is exactly balanced by the radiative relaxation, shows large values at the critical lines. This is in agreement with theory, which predicts that the group velocity vanishes and the refractive index becomes infinite on the critical line. Returning to Fig. 1, we see that there exists an upper threshold in g above which the warm stable state has ceased to exist and has been replaced by a warm unstable state. For g 5 2.5 and g 5 3 m s21 km21 , five steady states coexist for an intermediate range of forcings just

as for g 5 2 m s21 km21 . The situation is different for g below 2. For g 5 1.5, 1.25, and 1 m s21 km21 , three steady states, two stable and one unstable, exist for the intermediate range of forcings. For g less than 1, hwarm exceeds hcold and a stable section appears on the otherwise unstable branch connecting the bifurcation points at hwarm and hcold . For g 5 0.75, hcold 5 6.0 m is only visible in Fig. 1b as a bend, while for g 5 0.5 it has vanished totally. The new section of stable steady states is borne near hcold by a saddle-node bifurcation and disappears near hwarm in a Hopf bifurcation. For g less than 1, periodic vacillations have been observed near hwarm . Figure 3 shows the phase diagram spanned by g and h for U R,0 5 10 m s21 . The critical forcing hcold , where the cold stable state undergoes a saddle-node bifurcation (the dotted curve), increases fast with increasing g. The

3166


FIG. 3. The phase diagram for U R,0 5 10 m s21 . The dotted line, the solid line, and the slashed line show the positions of hcold , hHopf , and hwarm , respectively. The hatched region is where quasiperiodic and chaotic states are found.

increase is faster than linear and follows the power law hcold ; (g 2 0.69)1.27 to high precision. The thick solid line shows the forcing hHopf for which the warm stable state loses stability and the periodic vacillation appears. The critical forcing for the Hopf bifurcation has only a weak dependence on g and is near 70 m for the whole range of g studied. The critical forcing hwarm for which the warm stable state and an unstable state are created through a saddle-node bifurcation (dashed line) shows an overall increase with g, although some minor wriggles are observed. For g less than 1, the additional stable steady state described in the previous paragraph is found in the crosshatched region between hcold and hwarm . For g equal to approximately 2.7 m s21 km21 the two critical forcings hwarm and hHopf meet at a value of around 60 m. As mentioned in the last paragraph the warm stable state

VOLUME 57

does not exist for g above this threshold. However, there still exists a transition to the vacillating state. For g larger than 2.7 m s21 km21 the vacillating state is found above the dot-dashed line, which continues hwarm . This line is obtained as the values of h where the vacillation disappears when h is decreased slowly enough for the system to adjust to the new attractor. The hatched region in Fig. 3 is where nonperiodic vacillations are found. The lower edge of this region is taken as the lowest value of h where nonperiodic vacillations are observed when h is decreased slowly. Periodic and nonperiodic solutions are distinguished by inspection of the power spectrum. It is not everywhere in this region that nonperiodic states exist. As is commonly found in dynamical systems the supercritical region contains both periodic, quasiperiodic, and chaotic states mixed together in a complicated fashion. This is consistent with Yoden (1987a), who reports periodic solutions in two experiments inside the hatched region. Only periodic solutions have been found for g below 1.5 m s21 km21 and h below 250 m, which is the maximum forcing applied in this study. The increase of the lower edge with increasing g does not continue; in an experiment with g 5 4 m s21 km21 the edge is found below 100 m. All previously published studies of the Holton–Mass model have used 27 vertical levels, tacitly assuming that the results obtained with this resolution are representative for the continuum limit, N → `. To test this assumption, we have reproduced the bifurcation diagram for g 5 2 m s21 km21 with N 5 13 and N 5 54. The result shown in Fig. 4 should be compared with Fig. 1f. The general picture is identical for all three values of

FIG. 4. Bifurcation diagram for g 5 2 m s21 km21 and U R,0 5 10 m s21 with N 5 13 and N 5 54.

15 SEPTEMBER 2000

CHRISTIANSEN

FIG. 5. Bifurcation diagram for h 5 200 m and U R,0 5 10 m s21 .

N but a few details are different. For N 5 13 the warm state is unstable for an interval around h 5 70 m and, furthermore, three unstable states coexist in a short interval between 119.75 and 125.5 m. Two of these unstable states meet in a cusp for h 5 125.55 m. Such effects of the finite resolution are not seen for N 5 27 and N 5 54. The main difference between N 5 27 and N 5 54 is the value of hcold , which is 232, 189, and 172 m for N513, 24, and 54, respectively. The values of hHopf are 88.5, 68.6, and 61.7 m, respectively. 4. Chaos and quasiperiodicity In this section we study the nonperiodic states and their transitions in more detail. We also provide quantitative evidence—in the form of Lyapunov exponents— for the existence of both two- and three-frequency quasiperiodicity and chaos. A bifurcation diagram for h 5 200 m and g in the interval between 1.8 and 2.3 m s21 km21 is shown in Fig. 5. The diagram is compiled from a series of 4000day-long simulations with g 5 1.800, 1.805, . . . , 2.300 m s21 km21 . For each simulation a map u*i , i 5 1, 2, . . . , is constructed as the values of U at 28.5 km with local maxima. This map corresponds to a single component of the Poincare´ map with dU (28.5 km)/dt 5 0 as the surface of section. The values of u*i are then plotted as the ordinate in Fig. 5, with the abscissa set to the particular value of g. The first 10 periods, that is, u*i , i 5 1, 2, . . . 10, have been discarded to avoid the transients. In Fig. 6 the time evolution of the zonal mean wind at 28.5 km is shown together with its power spectrum and the first return map, that is, u*i11 as function of u*i , for g 5 1.825, 1.875, 1.910, 1.940, 2.000, 2.075, 2.150, and 2.235 m s21 km21 . The plots are based on experiments lasting for 16 000 days. Going one step further in the analysis, we construct the map Q i11 5 f (Q i ), with tanQ i 5 y i /x i , where y i 5 u*i11 2 u*, x i 5 u*i 2 u*, and u* is the time average of u*i . The variable Q i is the polar angle of the point (u i , u i11 ) measured with respect to a coordinate system whose origin is the center (u*, u*) of the first return map. A similar transformation

3167

was introduced by Brandstater and Swinney (1987) in a study of Taylor–Couette flow. As we will see below, this approach reveals circle map–type dynamics in the Holton–Mass model. The circle map Q i11 5 Q i 1 V 1 K sinQ i (mod 2p) can be viewed as a Poincare´ map of a flow on a two-dimensional torus, and it is a celebrated toy model for studying the quasiperiodic route to chaos (Jensen et al. 1983, 1984). The map is invertible for K , 1, and in this region only quasiperiodic and periodic solutions exist. For K . 1 the map is noninvertible and shows periodic and chaotic solutions. For a review of the circle map see, for example, Ott (1993). The bifurcation diagram (Fig. 5) shows the existence of several different types of solutions. We first describe the solutions and transitions qualitatively, based on the bifurcation diagram, the power spectra, and the first return map. We then provide quantitative evidence in the form of Lyapunov exponents for the existence of twoand three-frequency quasiperiodicity and chaos. Finally we describe the map Q i11 5 f (Q i ) and its relation to the circle map. For g below 1.850 m s21 km21 the attractor is a limit cycle (periodic state) resulting in a single point in both the bifurcation diagram and the first return map. For g 5 1.825 m s21 km21 the power spectrum consists of a set of well-separated individual peaks corresponding to the basic frequency f 1 5 0.0249 day21 and its harmonics. At g 5 1.850 m s21 km21 a qualitative change takes place. For g above 1.850 m s21 km21 , u* now continuously fills a line segment in the bifurcation diagram, and the first return map consists of a closed loop corresponding to the attractor, dense filling a two-dimensional toroidal surface. The power spectrum now contains two basic frequencies, f 2 5 0.0056 day21 and f 1 5 0.0253 day21 for g 5 1.875 m s21 km21 , and their harmonics. The two frequencies are incommensurable and the state is quasiperiodic. The width of the line segment as well as the size of the closed loop in the first return map increases from zero at g 5 1.850 m s21 km21 , where the low-frequency component f 2 is born through a secondary Hopf bifurcation. At g 5 1.910 m s21 km21 sharp wriggles have developed in the return map and in the power spectrum a third frequency f 3 , lower than both f 1 and f 2 , has appeared. For g 5 1.910 m s21 km21 the three frequencies, f 1 5 0.0253, f 2 5 0.0057, and f 3 5 0.0030 day21 , are incommensurable and the attractor is three-frequency quasiperiodic. The incommensurability of the three frequencies and the necessity of the third frequency to explain the spectrum are seen by comparing linear combinations nf 1 1 mf 2 , nf 1 1 mf 2 1 pf 3 of two and three frequencies, respectively, with the spectrum. Above g 5 1.910 m s21 km21 the first return map becomes plane filling, indicating the breakdown of the two-dimensional torus, and broadband noise appears in the power spectrum. These features are characteristic of chaotic attractors. For g larger than 1.980 m s21 km21 the attractor is again periodic. The three basic frequencies are commensurable and locked

3168


VOLUME 57

FIG. 6. The attractor for g 5 1.825 (top row), 1.875, 1.910, and 1.940 (bottom row) m s 21 km21 , with h 5 200 m and U R,0 5 10 m s21 . (a) The zonal wind U at 28.5 km as a function of time; (b) power spectrum of U at 28.5 km; (c) first return map, i.e., u*(i 1 1) vs u*(i); (d) Q i11 vs Q i .

to each other on the ratios of f 1 : f 2 : f 3 5 9:2:1 in a finite interval of g. This means that the period is 9/ f 1 . The frequency locking continues until g exceeds 2.050 m s21 km21 . For larger values of g, two more locked regions are found separated by regions with quasiperiodicity and chaos. In the region between g 5 2.105 and g 5 2.210 m s21 km21 the frequencies fulfill the ratios of f 1 : f 2 : f 3 5 8:2:1 and the period is 8/ f 1 . The locked region above g 5 2.255 m s21 km21 involves only the two largest basic frequencies with f 1 : f 2 5 3:1, and the corresponding period is 3/ f 1 . Table 1 shows the three largest Lyapunov exponents for the eight attractors shown in Fig. 6 calculated by the algorithm described in section 2. The length of the integrations were 16 000 days, and an interval t of 400 days between the orthonormalizations was chosen. The standard deviation for the Lyapunov exponents is around 30 3 1025 day21 . Additional calculations with t 5 800, 1600, and 2000 days, summarized in Table 2 for g 5 1.940 m s21 km21 , show only small deviations from the values in Table 1. It is also worth noting that the algorithm was able to produce the correct eigenvalues when applied to a stable steady state.

We find from Table 1 that all eight attractors have at least one vanishing exponent as required for nonsteady continuous dynamical systems. The attractors are characterized by the signs of the exponents. For g 5 1.825 m s21 km21 the signs are (0, 2, 2) and the attractor is a limit cycle. For g 5 1.875 m s21 km21 we have (0, 0, 2) indicating two-frequency quasiperiodicity densely filling a two-dimensional toroidal surface. For g 5 1.910 m s21 km21 we have three vanishing exponents, (0, 0, 0), and therefore three-frequency quasiperiodicity. The attractor is now densely filling a threedimensional toroidal surface in the phase space. The largest exponent for g 5 1.940 m s21 km21 is positive while the two next are zero, (1, 0, 0), providing evidence for chaotic dynamics on a strange attractor developed through the breakdown of the three-dimensional toroidal surface. For g 5 2.000 m s21 km21 the attractor is a limit cycle, (0, 2, 2), and for g 5 2.075 and 2.235 m s21 km21 we again encounter chaotic solutions, (1, 0, 2). We now discuss the map Q i11 5 f (Q i ) shown in the last column of Fig. 6. For g immediately above the secondary Hopf bifurcation the map consists of two

15 SEPTEMBER 2000

3169

CHRISTIANSEN

FIG. 6. (Continued ) The attractor for g 5 2.000 (top row), 2.075, 2.150, and 2.235 (bottom row) m s 21 km21 .

almost linear pieces. This mimics the circle map with K 5 0 and corresponds to a rigid rotation of the circle. Increasing g corresponds to increasing the nonlinear parameter K in the circle map: the map deviates increasingly from the pure rotation. For g 5 1.910 m s21 km21 the map f develops a noninvertibility resembling the circle map for K 5 1. For g 5 1.940 m s21 km21 the map is noninvertible and multivalued reflecting the breakdown of the torus. For higher values of g the torus

is still apparent in the (Q i11 , Q i ) plots although now superposed with a layered or fractal structure. 5. The interannual variability The introduction of low-frequency components in the Holton–Mass model has an interesting implication. Yoden (1990) forced the model with a periodic annual forcing and concluded that such a forcing is not likely to introduce interannual variability in the model. This is because the internal timescales, that is, the damping time

TABLE 1. The three largest Lyapunov exponents for the attractors shown in Fig. 6.

g

(m s

21

21

km )

1.825 1.875 1.910 1.940 2.000 2.075 2.150 2.235

l1 3 105 (day21)

l2 3 105 (day21)

l3 3 105 (day21)

1.1 8.5 17.4 277.6 2.8 193.3 22.7 482.4

2180.1 22.5 5.3 66.9 2289.6 225.8 2519.1 1.5

2184.8 2532.5 222.2 3.6 2983.9 2952.3 2678.4 21099.7

TABLE 2. The three largest Lyapunov exponents and their uncertainty level (in parentheses) for different values of the interval t between orthonormalization. The parameters are g 5 1.940 m s21 km21 , h 5 200 m, and U R,0 5 10 m s21 .

t (day) 400 800 1600 2000

l1 3 105 (day21) 277.6 272.4 269.6 257.1

(51.5) (28.9) (33.2) (28.7)

l2 3 105 (day21) 66.9 60.0 58.3 92.5

(45.0) (39.7) (24.8) (22.6)

l3 3 105 (day21) 3.59 2.05 21.1 21.5

(33.9) (28.0) (16.2) (12.9)

3170


VOLUME 57

FIG. 7. The zonal wind U at 28.5 km and its power spectrum for the system with annual forcing on the vertical equilibrium slope, g 5 g 0 1 G sin2pt/T, with g 0 5 1.700 (a), (b); 1.825 (c), (d); 1.875 (e), (f ); and 2.000 m s 21 km21 (g), (h). The other parameters are G 5 2.5 m s21 km21 , h 5 200 m, U R,0 5 10 m s21 , and T 5 365 days.

and the vacillation period, both are considerably shorter than a year. However, we have in the previous section seen that secondary Hopf bifurcations may introduce timescales comparable to a year and, thus, provide the attractor with a memory long enough to interfere with the annual cycle. To investigate this point further we added a periodic varying component to the vertical slope of the radiative equilibrium, that is, g 5 g 0 1 G sin2pt/T. In a series of simulations with T 5 365 day, h 5 200 m, G 5 2.5 m s21 km21 , and g 0 5 1.700, 1.825, 1.875, 2.000 m s21 km21 , we can follow the effect of increasingly longer internal timescales. These experiments differ from those in Yoden (1990) only by the

values of g 0 . Yoden used g 0 5 0.75 m s21 km21 , which is well outside the hatched region in Fig. 3. The simulations are 50 years long, and the last 30 years have been used to produce the time series and the power spectra in Fig. 7. For g 0 5 1.700 m s21 km21 no interannual variability is seen and the spectrum consists of separated peaks. The internal variability, as obtained for the autonomous system, that is, with G 5 0, is strictly periodic with a period of 40.8 days. This internal variability has been effectively suppressed, although remains are present in the form of relatively high power around 0.02 and 0.04 day21 . For g 0 5 1.825 m s21 km21 a period doubling has taken place and the system shows

15 SEPTEMBER 2000

3171

CHRISTIANSEN

a biannual oscillation. The internal period is 40.1 days, but the autonomous system is very close to the first bifurcation point as can be seen from Fig. 6a. For g 0 5 1.875 m s21 km21 the autonomous system is quasiperiodic with a secondary timescale of 177 days, and the nonautonomous (G 5 2.5) system shows a pronounced nonperiodic interannual variability. For g 0 5 2.000 m s21 km21 the autonomous system is frequency locked with a largest period of 357 days, and the nonautonomous system now shows a very large interannual variability. As expected, for all the simulations both the interannual and intra-annual variability is largest in the winter season (day 365/4 in Fig. 7 corresponds to midwinter) when g is large and positive. Note that even for g 0 5 2.000 m s21 km21 the variability is very weak in summer and spring.

stratosphere might be internally driven. However, recent work by Kinnersley (1998) emphasizes the importance of tropospheric variability by reproducing the observed interannual variability in a stratospheric model forced with observed waves on the tropopause. Thus, a full understanding of the variability probably requires inclusion of both internal chaotic dynamics and transient wave forcing from the troposphere. Simulations with more comprehensive models such as general circulation models will be needed to learn to what extent the insight obtained with the Holton–Mass model carries over to more realistic conditions. Acknowledgments. This work was supported by the Commission of the European Communities (Contract EV5V-CT94-492).

6. Conclusions The parameter space of a simple model of stratospheric dynamics introduced by Holton and Mass (1976) has been studied. The critical lines marking the appearance of the ‘‘warm’’ stable steady state, the appearance of the vacillating state, and the disappearance of the ‘‘cold’’ stable steady state have been traced out in (g, h)-space. In addition to the previously described steady and periodic vacillating states, we observe both quasiperiodic and chaotic states. Quasiperiodic states with both two and three frequencies are found. Regions with quasiperiodic and chaotic states are separated by regions of periodic, frequency-locked states. The twofrequency quasiperiodic state is reached through a secondary Hopf bifurcation from the periodic state. A third Hopf bifurcation introduces a third frequency after which chaotic states are found. The transition to chaos is related to the development of a noninvertibility in the return map. In fact, defining an appropriate angle variable Q i , the circle map behavior becomes evident. We suggest that the transition to chaos can be described as an example of the Ruelle– Takens–Newhouse scenario (Ruelle and Takens 1971; Newhouse et al. 1978), which, in crude words, says that a chaotic attractor is likely to appear after three successive Hopf bifurcations [see, e.g., the discussion in Ott (1993)]. When the system is forced with a periodic annual cycle, strong interannual variability may develop as a consequence of the long timescales introduced to the system by the secondary Hopf bifurcations. Although the variability in Fig. 7 is clearly unrealistic, the results presented here suggest that the extratropical interannual stratospheric variability to some extent can be explained by internal stratospheric dynamics. Pawson and Kubitz (1996) found no relation between the interannual variability and the possible forcing mechanisms such as the quasi-biennial oscillation or tropospheric variability. This supports the suggestion that much of the interannual variability in the Northern Hemisphere

APPENDIX The Notation and the Discretized Equations The notation follows previous papers and is summarized below.

f z ztop g a s k l e N dz H N2 V f0 b a g U R,0 UR h u c U C

Latitude of channel, 608N The height relative to the tropopause Upper boundary, 70 km (above tropopause) Acceleration of gravity, 9.82 m s22 Radius of Earth, 6370 km Zonal wave number Inverse zonal wavelength, s/(a cosf ) Inverse meridional wave length, 3/a Expansion parameter, 8/(3p) Number of vertical levels Level spacing Scale height, 7000 m Brunt–Va¨isa¨la¨ frequency, 4 3 1024 s22 Frequency of Earth, 1/24/60/60 s21 Coriolis parameter, 2V sinf Meridional derivative of Coriolis parameter, 2V cosf /a Radiative damping time, {1.5 1 tanh[(z/1000 2 25)/7]}/10 6 s21 Vertical slope of radiative equilibrium profile Radiative equilibrium mean wind at tropopause Radiative equilibrium profile, gz 1 U R,0 Geopotential height wave forcing at tropopause Zonal mean wind Eddy geostrophic streamfunction Expanded zonal mean wind Expanded eddy geostrophic streamfunction

Because of the boundary conditions the coefficients of Eq. (4) are most conveniently presented by giving the full equations. In the following formulas the coefficients are to be evaluated at level k:

3172


VOLUME 57

A˙ k11 2 [2 1 dz 2 /(4H 2 ) 1 (k 2 1 l 2 )d 2 ]A˙ k 1 A˙ k21 5 2(a 1 dza z /2)A k11 1 [2a 1 adz 2 /(4H 2 ) 2 a z dz 2 /(2H )]A k 2 (a 2 dza z /2)A k21 1 bkd 2 Bk 2 e k[1 2 dz /(2H )]Uk11 Bk 2 e k[1 1 dz /(2H )]Uk21 Bk 2 e k[k 2d 2 1 dz 2 /(4H 2 )]Uk Bk 1 e kUk Bk11 1 e kUk Bk21,

(A1)

B˙ k11 2 [2 1 dz 2 /(4H 2 ) 1 (k 2 1 l 2 )d 2 ]B˙ k 1 B˙ k21 5 2 (a 1 dza z /2)Bk11 1 [2a 1 adz 2 /(4H 2 ) 2 a z dz 2 /(2H )]Bk 2 (a 2 dza z /2)Bk21 2 bkd 2 A k 1 e k[1 2 dz /(2H )]Uk11 A k 1 e k[1 1 dz /(2H )]Uk21 A k 1 e k[k 2d 2 1 dz 2 /(4H 2 )]Uk A k 2 e kUk A k11 2 e kUk A k21,

(A2)

˙ k11 1 (2 1 l d )U ˙ k 2 [1 1 dz /(2H )]U ˙ k21 2[1 2 dz /(2H )]U 2

2

5 1[a (UR ) zz 1 (a /H 2 a z )(UR ) z ]dz 2 1 [a 1 dz(a z 2 a /H )/2]Uk11 2 2aUk 1 [a 2 dz(a z 2 a /H )/2]Uk21 1 e kl 2 exp(z /H )/2(A k Bk11 1 A k Bk21 2 A k11 Bk 2 A k21 Bk ).

Here d 2 5 (dzN/ f ) 2 . These equations are directly valid for k 5 2, . . . , N 2 1 and can be extended to k 5 1 and k 5 N by the boundary conditions U 0 5 u R,0 , A 0 5 hg/ f, B 0 5 0, and U N11 5 2dzg 1 U N21 , A N11 5 0, B N11 5 0. REFERENCES Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. Academic Press, 489 pp. Baldwin, M. P., X. Cheng, and T. J. Dunkerton, 1994: Observed correlations between winter-mean tropospheric and stratospheric circulation anomalies. Geophys. Res. Lett., 21, 1141–1144. Benettin, G., L, Galgan, A. Giorgilli, and J.-M. Strelcyn, 1980: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part 2: Numerical application. Meccanica, 15, 21– 30. Berge´, B., Y. Pomeau, and C. Vidal, 1986: Order within Chaos. Wiley, 329 pp. Brandstater, A., and H. L. Swinney, 1987: Strange attractors and weakly turbulent Couette–Taylor flow. Phys. Rev. A, 35, 2207– 2220. Chao, W. C., 1985: Sudden stratospheric warmings as catastrophes. J. Atmos. Sci., 42, 1631–1646. Christiansen, B., 1999: Stratospheric vacillations in a general circulation model. J. Atmos. Sci., 56, 1858–1872. Holton, J. R., and C. Mass, 1976: Stratospheric vacillation cycles. J. Atmos. Sci., 33, 2218–2225. , and T. Dunkerton, 1978: On the role of wave transience and dissipation in stratospheric mean flow vacillations. J. Atmos. Sci., 35, 740–744. , and H.-C. Tan, 1982: The quasi-biennial oscillation in the Northern Hemisphere lower stratosphere. J. Meteor. Soc. Japan, 60, 140–147. Hood, L. L., J. L. Jirikowic, and J. P. McCormack, 1993: Quasidecadal variability of the stratosphere: Influence of long-term solar ultraviolet variations. J. Atmos. Sci., 50, 3941–3958. Jensen, M. H., P. Bak, and T. Bohr, 1983: Complete Devil’s Staircase, fractal dimension and universality of mode-locking structure in the circle map. Phys. Rev. Lett., 50, 1637–1639. , , and , 1984: Transition to chaos by interactions of

(A3)

resonances in dissipative systems. I. Circle maps. Phys. Rev. A, 30, 1960–1969. Kelley, C. T., 1995: Iterative Methods for Linear and Nonlinear Equations. Vol. 16, Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics, 165 pp. Kinnersley, J. S., 1998: Interannual variability of stratospheric zonal wind forced by the northern lower-stratospheric large-scale waves. J. Atmos. Sci., 55, 2270–2283. Kodera, K., 1995: On the origin and nature of the interannual variability of the winter stratospheric circulation in the Northern Hemisphere. J. Geophys. Res., 100, 14 077–14 087. Labitzke, K., and M. P. McCormick, 1992: Stratospheric temperature increases due to Pinatubo aerosols. Geophys. Res. Lett., 19, 207– 210. , and H. van Loon, 1995: Connections between the troposphere and stratosphere on a decadal scale. Tellus, 47A, 275–286. Matsuno, T., 1971: A dynamical model of the stratospheric sudden warming. J. Atmos. Sci., 28, 1479–1494. Newhouse, S., D. Ruelle, and F. Takens, 1978: Occurrence of strange axiom A attractors near quasiperiodic flows on T m (m $ 3). Commun. Math. Phys., 64, 35–40. Ott, E., 1993: Chaos in Dynamical Systems. Cambridge University Press, 385 pp. Pawson, S., and T. Kubitz, 1996: Climatology of planetary waves in the northern stratosphere. J. Geophys. Res., 101, 16 987–16 986. Pierce, R. B., and T. D. A. Fairlie, 1993: Observational evidence of preferred flow regimes in the Northern Hemisphere winter stratosphere. J. Atmos. Sci., 50, 1936–1949. Plumb, R. A., 1977: The interaction of two internal waves with the mean flow: Implications for the theory of the quasi-biennial oscillation. J. Atmos. Sci., 34, 1847–1858. , 1984: The quasi-biennial oscillation. Dynamics of the Middle Atmosphere, J. R. Holton and T. Matsuno, Eds., Terra Scientific, 217–251. Ruelle, D., and F. Takens, 1971: On the nature of turbulence. Commun. Math. Phys., 20, 167–192. Scaife, A. A., and I. N. James, 2000: Response of the stratosphere to interannual variability of tropospheric planetary waves. Quart. J. Roy. Meteor. Soc., 126, 275–297. Scherhag, R., 1952: Die explosionsartigen Stratospha¨ renwa¨rmungen des Spa¨twinters 1952. Ber. Dtsch. Wetterdienstes (US zone), 38, 51–63. Shimada, I., and T. Nagashima, 1979: A numerical approach to ergodic problem of dissipative dynamical systems. Prog. Theor. Phys., 61, 1605–1616.

15 SEPTEMBER 2000

CHRISTIANSEN

van Loon, H., and K. Labitzke, 1987: The Southern Oscillation. Part V: The anomalies in the lower stratosphere of the Northern Hemisphere in winter and a comparison with the quasi-biennial oscillation. Mon. Wea. Rev., 115, 357–369. Yoden, S., 1987a: Bifurcation properties of a stratospheric vacillation model. J. Atmos. Sci., 44, 1723–1733.

3173

, 1987b: Dynamical aspects of stratospheric vacillations in a highly truncated model. J. Atmos. Sci., 44, 3683–3695. , 1990: An illustrative model of seasonal and interannual variations of the stratospheric circulation. J. Atmos. Sci., 47, 1845–1853.