Temporal growth and vertical propagation of perturbations in the winter atmosphere

Transcription

1 Q. J. R. Meteorol. Soc. (2003), 129, pp doi: /qj Temporal growth and vertical propagation of perturbations in the winter atmosphere By BO CHRISTIANSEN Danish Meteorological Institute, Denmark (Received 3 September 2001; revised 17 July 2002) SUMMARY We present a general-circulation model study of the temporal growth and vertical propagation of perturbations following transient, vertically con ned forcings. Five ensembles of perturbation experiments are performed, each with eight different initial conditions and with the perturbations enforced at ten different vertical levels from the lower troposphere to the upper stratosphere. The motivation for the study is the recent recognition of downward propagation of anomalies from the stratosphere to the troposphere and its implications both for medium-range forecasts and for a possible physical mechanism for stratospheric impacts on weather and climate. The perturbations grow in an initial period after which they saturate to a level comparable to the natural variability. In the initial period the growth of the perturbations is quadratic in time. This power-law growth suggests a predictability time which converges towards a nite value in the limit of small perturbations. In the troposphere the predictability time, de ned as when perturbations reach a size of 50% of the natural variability, is days. In the stratosphere the response is delayed and a predictability time of days is observed. In the initial period of growth, the vertical spread of perturbations seems decoupled from the background vacillations in the sense that no vertical propagation of perturbations with the time-scale of the vacillations is observed. Forcings con ned to the stratosphere result instantaneously in perturbations of the troposphere and vice versa. Thus, stratospheric vacillations seem to have no active role in the propagation of perturbations from the stratosphere to the troposphere: stratospheric vacillations are not a vehicle for transient perturbations in the stratosphere to be transported to the troposphere. The disconnected nature of the stratospheric vacillations and the growth of perturbations is further substantiated in experiments with a time-independent troposphere. In these experiments the periodic stratospheric vacillations are shown not to support the growth of perturbations. KEYWORDS: Arctic Oscillation Downward propagation Error growth General-circulation model Stratosphere troposphere coupling 1. INTRODUCTION Downward propagation from the stratosphere to the troposphere in the northernhemisphere cold season has been reported in both observations (Kodera et al. 1990, 2000; Baldwin and Dunkerton 1999) and models (Christiansen 2000b). Zonal-mean zonal-wind anomalies seem to be born in the mesosphere and propagate down through the stratosphere and into the troposphere on a time-scale of 2 3 weeks. When anomalies reach the lower troposphere, the Arctic Oscillation (AO) (Thompson and Wallace 1998) is predominantly in its positive phase. As the AO controls much of the weather in the northern hemisphere (Hurrell 1995; Thompson and Wallace 2001) the downward propagation opens obvious possibilities for medium-range forecasting (Thompson et al. 2002). The downward propagation might also offer a mechanism for changes in the upper atmosphere to affect the tropospheric climate. Here we are thinking of changes in trace gases such as ozone (Hartmann et al. 2000; Kindem and Christiansen 2001) and in modulations related to the 11-year solar cycle (Geller and Alpert 1980; Haigh 1999; Shindell et al. 1999). On the shorter time-scales the downward propagation could perhaps provide a mechanism (Hines 1974; Arnold and Robinson 1998, 2001) for the asserted effects of variations in the solar activity on weather (Taylor 1986; Bochn cek et al. 1999). However, such a mechanism requires that the stratosphere actually exercises control over the troposphere so that transient or stationary perturbations in the stratosphere are Corresponding address: Danish Meteorological Institute, Lyngbyvej 100, DK-2100, Denmark. boc@dmi.dk c Royal Meteorological Society,

2 1590 B. CHRISTIANSEN transported downwards. The description of the downward propagation in the stratosphere also known as stratospheric vacillations involves both zonal-mean and wave quantities and also both the troposphere and the stratosphere. As shown in Christiansen (2001) the mechanism of the downward propagation includes the interaction of planetary waves with the zonal-mean ow. The planetary waves are generated in the troposphere due to orography and land sea contrasts and propagate vertically into the stratosphere and mesosphere. The vertical level where the planetary waves eventually break and transfer momentum to the zonal-mean ow depends partly on the structure of the zonal-mean ow and partly on the characteristics of the planetary waves themselves. As the structure of the zonal-mean ow is modulated by the downward propagation the possibility exists that the downward propagation is fully determined by the planetary waves and that the vacillations are insensitive to perturbations in the stratosphere. The present paper investigates this possibility by studying the development of vertically con- ned transient disturbances in a general-circulation model (GCM). We note that even if the stratosphere is responding passively to the wave forcing from the troposphere, the downward propagation may still be a source for enhanced predictability of near-surface weather. It should also be noted that the mechanism described above is not fully understood, e.g. the downward propagation of a critical line has been suggested (Christiansen 1999) but it has not yet been established. Also, the downward propagation from the tropopause to the surface may be caused by a different mechanism to the downward propagation in the stratosphere. A related subject is the temporal growth of perturbations and the limitation of predictability. Lorenz (1969) suggested that in systems with many length-scales, perturbations will cascade from smaller to larger scales resulting in a power-law growth of the size of the perturbations. As a consequence the predictability time will be nite: for any time there is a limit below which the error cannot be reduced, no matter how small the initial error is made. These ideas have been tested in several models of fully developed turbulence (Aurell et al. 1996), but to our knowledge not yet investigated in detailed models of the atmosphere. In this paper the temporal growth and vertical propagation of perturbations in the northern-hemisphere winter will be studied with a GCM. Transient, vertically con ned forcings in the zonal wind, U, and the temperature will be imposed at different heights in the model. The forcings will be imposed on an ensemble of experiments in order to identify a possible statistically robust response. As the mechanism of the downward propagation is only partly understood we have chosen to study highly idealized forcings in this paper. However, even with this limitation we believe that our study will provide useful information on the possibility of the stratospheric vacillations transporting perturbations from the stratosphere to the troposphere. Following a description of the model and the experiments in section 2 we present a few individual members of the ensemble experiments in section 3. In section 4 we consider the temporal growth of the perturbations and the predictability time. In section 5 we take a closer look at the possible downward propagation of the perturbations and the connection to the AO. Section 6 describes the temporal growth of perturbations in an additional ensemble experiment with a time-independent troposphere. Finally, we close with conclusions in section MODEL AND EXPERIMENTS We use the Action de Recherche Petite Echelle et Grande Echelle (ARPEGE) general-circulation model, cycle 14 (Déqué et al. 1994). The model has 41 levels in

3 GROWTH AND SPREAD OF PERTURBATIONS 1591 Figure 1. Zonal-mean zonal-wind anomaly (m s 1 ) at 60 ± N as a function of time and pressure for a 300-day interval of the control experiment. Contours are separated by 5 m s 1. A smoothed plot of the whole simulation is presented in Christiansen (2000b). the vertical and spans the atmosphere from the surface to 0.1 hpa. All experiments are performed in perpetual-january mode, at T21 horizontal resolution, and with sea-surface temperatures and sea-ice extent prescribed from climatology. Outputs from the model are stored as daily averages. A 3650-day simulation of the unperturbed atmosphere was described in Christiansen (2000b). This experiment will be used as the control experiment upon which the forcings will be imposed. The downward propagation in the control experiment of the zonal-mean zonal-wind anomaly at 60 ± N is demonstrated in Fig. 1 for a time interval of 300 days. Here, data have not been smoothed in contrast to Fig. 1 of Christiansen (2000b) which shows the downward propagation for the whole simulation. With the annual cycle included the modelled stratospheric vacillations and their coupling to the troposphere are in excellent agreement with the National Centers for Environmental Prediction (NCEP) re-analyses (Christiansen 2001). Both the reanalyses and the model show that downward propagation dominates the variability in the northern-hemisphere cold season. From the control experiment we choose N D 8 days suf ciently far apart to be considered independent and with both positive and negative zonal-wind anomalies. These days are used as starting dates for the perturbation experiments. For each starting date M D 10 perturbation experiments are performed each with the forcing con ned to a single vertical layer. The vertical layers are 1.5, 6.0, 13.2, 25.4, 46.5, 140.2, 355.0, 619.5, 857.2, and hpa. The forcings are imposed on all latitudes north of 20 ± N. The perturbation experiments last 100 days but the forcing is transient and only imposed for at most 10 days. Five different types of forcings were considered by adding an additional term to either the temperature tendency equation (two types) or the zonal-wind tendency equation (three types). The forcings on the zonal wind have the form of a relaxation towards its long-term zonal-mean average with a relaxation time of 1 day. The forcings are imposed for 10 days, 5 days or 30 minutes (one time step), respectively. The forcings on the temperature are accomplished by imposing an extra heating rate of 0.5 K day 1 for 10 days and 1 day, respectively. The forcing on the temperature is zonal symmetric. The forcing on the zonal wind has a longitudinal dependence through the initial eld, although both the relaxation time and the eld relaxed towards have no longitudinal dependence. The ve ensembles are denoted U10d, U5d, U30m (forcing on the zonal wind), H10d, and H1d (forcing on the temperature). Thus, altogether our ensembles contain 5 N M D 400 experiments each lasting 100 days. The forcings are purposely idealized, but for comparison a heating rate of 0.5 K day 1 is of the order of magnitude

4 1592 B. CHRISTIANSEN of the radiative heating in the middle stratosphere in midlatitude winter following a 50% change in ozone. The forcing in the U30m ensemble is effectively an error on the initial conditions. We denote the zonal-mean zonal wind of the perturbation experiments with U r;l, where the subscripts r D 1 : : : N and l D 1 : : : M indicate the initial date and the forcing level. The control experiment has l D 0 and the zonal-wind response is calculated as ±U r;l D U r;l U r;0. Ensemble averages are calculated as the root mean square, e.g. [±U] D f. P r;l ±U 2 r;l /=.NM/g1=2, where the average is taken over all initial dates and the selected forcing levels. Sometimes ensemble averages are taken over experiments with forcings at all levels, sometimes only over experiments with forcing at a speci c level, in the troposphere or in the stratosphere. 3. A FIRST LOOK AT THE RESPONSES In this section we rst describe the response in a few individual experiments before we proceed to the statistical properties of the whole ensemble in the next sections. The rst three paragraphs in this section describe the vertical propagation which is taken up again in section 5. The last paragraph in this section gives a rst impression of the temporal growth which is also the subject of section 4. Figure 2 shows the response for three different experiments with the forcing situated in the stratosphere, and one experiment with the forcing in the troposphere. The three experiments in panels (b) (d) have the same initial conditions. The responses are shown for the rst 100 days in panel (a) and for the rst 40 days in panels (b) (d). First we mention the rather trivial observation that perturbations in the stratosphere eventually affect the troposphere and vice versa. For the experiments with the forcing on U at 6 hpa (Figs. 2(a) and (b)) the response is localized to the region near the forcing for the 10-day time interval when the forcing is active. The response reaches 1.5 m s 1 at the forced level at day 10. In the next 5 days the response seems to decay in the stratosphere, while a response of opposing sign is found in the troposphere. Following the decay, the strength of the perturbation starts to grow after day 15. Similar behaviour is observed in the experiments in U10d and U5d with forcing in the stratosphere. When the control experiment has strong or weak zonal wind at the initial date, a negative (as in Fig. 2(a)) or positive (as in Fig. 2(b)) response will evolve during the period when the forcing is active. This response will then decay when the forcing is terminated. For the experiments with the forcing on the heating rate at 6 and 355 hpa, respectively, a weaker response is found in both the troposphere and the stratosphere. Figure 2(e) shows the root-mean-square response of the eight experiments with forcing at 6 hpa in the H10d ensemble. We see that the strength of the perturbations grows in the rst days after which it seems to saturate. We also note that in the rst days where the response builds up no downward or upward propagation is observed either in the four individual experiments in Figs. 2(a) (d) or in the root-meansquare response presented in Fig. 2(e). Under the assumptions that the model atmosphere is chaotic and that the forcing does not drive the atmosphere into a different regime, the fully developed response should have the same vertical and temporal structure as the control experiment. This is indeed the case in the second half of the 100-day period as is exempli ed by the experiment shown in Fig. 2(a). In particular, here the response shows a downward propagation similar to that in the control experiment (Fig. 1). This observation holds both for experiments with forcings in the stratosphere and in the troposphere.

5 GROWTH AND SPREAD OF PERTURBATIONS 1593 Figure 2. The zonal-mean zonal-wind response at 60 ± N (m s 1 ) as a function of time and pressure for four experiments. (a) An experiment from the U10d ensemble with forcing at 6 hpa, (b) another experiment from U10d with forcing at 6 hpa, (c) an experiment from H10d with forcing at 6 hpa, (d) an experiment from H10d with forcing at 355 hpa, (e) the root-mean-square error of the eight experiments in H10d with the forcing at 6 hpa. The black bar shows the vertical position and extent, and the duration of the forcing. Experiments in (b), (c) and (d) have the same initial conditions. In (a) contours are separated by 5 m s 1. In (b), (c) and (d) the contours are : : : 2; 1; 0:5; 0; 0:5; 1; 2; : : :, in (e) 0:01; 0:025; 0:05; 0:075; 0:1; 0:25; 0:5; 0:75; 1; 2; 3 : : :. Note that the time interval in (a) is 100 days, while it is 40 days in the other panels.

6 1594 B. CHRISTIANSEN Figure 3. The zonal-mean zonal-wind response at 60 ± N and 10 hpa (a) and 550 hpa (b) for all 80 perturbation experiments in the U30m ensemble. Blue and red curves have the forcing in the stratosphere and troposphere, respectively. The responses have been divided by p 2¾ U.p/, where ¾ U.p/ is the standard deviation of the control experiment at the appropriate level. In Fig. 3 the responses at 10 hpa and 550 hpa of all 80 perturbation experiments of the U30m ensemble are shown. The natural variability is much larger in the stratosphere than in the troposphere. To facilitate the comparison of the two levels the responses have, therefore, been divided by p 2¾ U.p/, where ¾ U.p/ is the standard deviation of U at the pressure level p, under consideration. The standard deviation is obtained from the control experiment and is shown in Fig. 4(a) of Christiansen (1999). Experiments with the forcing in the troposphere and the stratosphere are shown with red and blue curves, respectively. Perturbations grow faster in the troposphere than in the stratosphere. Even in the experiments with forcing in the stratosphere the response develops faster in the troposphere. The transition between the initial period of fast growth and the later period

7 GROWTH AND SPREAD OF PERTURBATIONS 1595 Figure 4. The strength of the average perturbation at 60 ± N and 10 hpa (blue curves) and 550 hpa (red curves) as functions of time. The perturbations are averaged over experiments with forcing in the troposphere (dashed curves) and in the stratosphere (solid curves). (a) is semi-log, (b) log log. The two black curves represent an exponential growth with a time-constant of 6 days and a growth with time squared. Data are from the U30m ensemble. dominated by a saturated response of strength comparable to the natural variability in the model, takes place after approximately 25 and 35 days in the troposphere and stratosphere, respectively. An almost equal number of positive and negative responses are found. There seems to be no pattern in the sign of the responses except for the one described above for the experiments in U10d and U5d with forcings imposed in the stratosphere. Otherwise, no simple relation was found connecting the initial condition or forcing level to the sign of the response. 4. TEMPORAL GROWTH AND PREDICTABILITY TIME The normalized ensemble-average response [±U]=f p 2¾ U.p/g, at 10 hpa (red curves) and 550 hpa (blue curves) is shown in Figs. 4(a) and (b) in double and single logarithmic coordinate systems for the U30m ensemble. The responses to forcings in the troposphere (dashed curves) and stratosphere (solid curves) are almost identical.

8 1596 B. CHRISTIANSEN Figure 5. The strength of the average perturbation at 60 ± N and 550 hpa as a function of time. The perturbations are averaged over all experiments in the ve ensembles: U10d (blue), U5d (red), U30m (yellow), H10d (cyan), and H1d (green). The thin grey curves are the responses of the individual experiments in the U30m ensemble. The straight line represents a growth with time squared. Figure 6. The amplitude of the zonal waves of the surface pressure (hpa) at 60 ± N. Shown is the ensemble average over the H1d ensemble. Wave numbers are 1, 3, 6, 8, 11, 13, 16, 18. The straight line represents a growth with time squared. The two black curves are an exponential growth with a time constant of 6 days, U D U 0 exp t= (U 0 D 10 2, D 6), and a growth with the square of time U D.t=t 0 / 2 C U 0 (U 0 D 0, t 0 D 50). The response grows almost perfectly with the square of time until it saturates after approximately days. In the log log plot the end of the initial period of growth is well de ned. This behaviour is found for all our ensembles independently of the strength and duration of the forcing, as seen in Fig. 5. Only the U10d and the U5d ensembles show a less steep increase in the rst 10 days where the forcing is active. The quadratic growth is not restricted to the zonal wind or zonal-mean variables but is also found in zonal-mean and single-cell values of temperature, geopotential

9 GROWTH AND SPREAD OF PERTURBATIONS 1597 Figure 7. The time (days) for the average perturbation to grow to ¾ U.p/=2 as function of latitude and pressure. Panel (a) is averaged over the experiments with the forcing in the stratosphere. Panel (b) is averaged over the experiments with the forcing in the troposphere. Data are from the U30m ensemble. and surface pressure. In Fig. 6 the growth of perturbations in the surface pressure is shown expanded in zonal wave numbers. The quadratic growth is observed for all wave numbers. There is a tendency that high wave numbers saturate faster than low wave numbers and that the growth is less steep for high wave numbers before day 5. However, the rst few days could be biased as the analysis is based on daily averages. This behaviour is very different from the exponential growth of in nitesimal perturbations related to deterministic chaos in low-dimensional systems. However, systems with many degrees of freedom can show power-law growth of perturbations as shown by Lorenz (1969). Scaling theories for homogeneous, isotropic turbulence suggest that perturbations grow with time following a power law with the power depending sensitively on the power n of the kinetic-energy spectrum E.k/ k n. In In the inertial range we have by dimensional analysis the eddy turn-over time (e.g. Bohr et al. 1998).k/ D fk 3 E.k/g 1=2, where E.k/ k n is the energy spectrum and k is the wave number. From dt= D dk=k we nd k t 2=.n 3/. Now dimensional analysis gives ±U 1=fk.k/g k.1 n/=2 t, with D.1 n/=.n 3/. With the Kolmogorov n D 5=3 law we have ±U t 1=2, but grows fast with n and diverges at n D 3, which characterizes two-dimensional turbulence. For n D 7=3 we have D 2.

10 1598 B. CHRISTIANSEN the Kolmogorov theory n D 5=3 and perturbations grow with the square root of time, ±U p t (Bohr et al. 1998). The power-law behaviour of the growth of perturbations has profound consequences for the predictability of the system. Whereas an exponential growth implies that the system is inde nitely predictable, because the perturbation at any time can be made as small as possible by reducing the initial perturbation, this is not the case for the power-law growth. De ning the predictability time T as the time it takes for the perturbation to grow to an arbitrarily chosen level U p we get T D t 0 p Up U 0 and T D ln.u p =U 0 / for the power-law and exponential growth, respectively. Thus, in the limit of small initial perturbations, U 0, the predictability time converges for the powerlaw growth and diverges for the exponential growth. The predictability time, de ned as the time it takes for the response to grow to ¾ U.p/=2, is shown in Fig. 7 as function of latitude and pressure for the H30m ensemble. The values shown in the upper and lower panels are the averages over the experiments with the forcing in the stratosphere and troposphere, respectively. Values in the troposphere are days while they are around days in the stratosphere, except near the equator where the stratospheric values are up to 60 days. The gure is remarkably symmetric with respect to the equator when we recall the large asymmetry in ¾ U.p/, which in the stratosphere is up to a factor of ten larger in the northern hemisphere than in the southern hemisphere. The delayed response in the stratosphere is thus found for all latitudes. No signi cant differences are found between experiments with the forcing in the stratosphere and the troposphere. Almost identical predictability times are found for the other ensembles. 5. THE VERTICAL PROPAGATION AND THE ARCTIC OSCILLATION In section 3 a lack of systematic upward or downward propagation of the responses in the rst days was postulated based on a visual inspection of the perturbation experiments. In subsection (a) we present statistical evidence for this postulate. We do this by calculating ensemble means of time-lagged correlations of the zonal-mean zonalwind response. In subsection (b) we consider the temporal growth of the AO index and show that it follows the same power law as was demonstrated for the zonal-mean zonal wind in the last section. The AO is of particular interest as this mode is the dominant pattern of variability in the northern-hemisphere surface pressure and as its dynamics have been shown to be related to the stratospheric vacillations. (a) Statistics of the vertical propagation Statistical evidence for the downward propagation of zonal-mean zonal-wind anomalies in the last 42 years of the NCEP re-analyses and for GCM experiments of comparable length has previously been provided by calculating the correlations between U at 10 hpa and U at other levels as a function of time lag (Baldwin and Dunkerton 1999; Christiansen 2000b, 2001). It is more dif cult to quantify the vertical propagation in an ensemble of independent time series of length comparable to the time-scale of the vertical propagation itself. As an attempt we calculated the average of the correlations of the responses over the ensemble for each level and time lag. This was done for the 0 30-day period, where the responses build up, and for the day period, where the responses are fully developed. This analysis (Fig. 8) shows downward propagation from the stratosphere to the troposphere in the latter period and instantaneous connection in the former period. The difference between the two 30-day periods is also sustained if only stratospheric or tropospheric perturbations are included. Also, there seems to be

11 GROWTH AND SPREAD OF PERTURBATIONS 1599 Figure 8. Correlations of the zonal-mean zonal wind at 60 ± N with that at 10 hpa as a function of pressure and time lag. The correlations were averaged over all members of the U30m ensemble. (a) and (b) are restricted to the periods 1 30 days and days, respectively. no dependence on the vertical propagation of the phase of the underlying vacillation at the time when the forcing is imposed. We note that the time-scale of the downward propagation is distorted in Fig. 8(b) due to boundary effects. Using the same averaging procedure after chopping the control experiment into 30-day periods has the same effect. An additional complication arises from the growth of the responses in the initial period and the related increase of the contribution to the correlations. We have, therefore, repeated the analysis with the zonalmean zonal-wind response scaled (by dividing ±U r;l.p; t/ by r;l.t/, where r;l.t/ 2 is the mean of.±u r;l / 2.p; t/ over all model layers) to have the same strength for the whole 30-day period, and we found that the scaling does not have any signi cant effect on the correlations. In summary we believe that the statistical analysis presented here provides strong evidence for the lack of vertical propagation of responses in the rst 30 days. (b) The Arctic Oscillation When the downward propagation of westerly zonal-mean zonal-wind events reaches the troposphere the phase of the AO will have a signi cant tendency to be positive. In the control experiment the AO, de ned as the leading empirical orthogonal function (EOF) of surface pressure north of 20 ± N, explains 27% of the total variance. To investigate to what extent the AO is triggered by the forcing we calculate the spatial covariance between the normalized leading EOF (Ã. ; Á/, with Ã. ; Á/ 2 D 1) from the control

12 1600 B. CHRISTIANSEN Figure 9. As a function of time are shown (a) the strength of the surface-pressure perturbation (hpa) averaged (root mean square) over latitudes north of 20 ± N, (b) the perturbation of the leading principal component (hpa), i.e. the Arctic Oscillation, (c) the fraction of the surface-pressure perturbation explained by the Arctic Oscillation. The perturbations are averaged over all experiments in the ve ensembles: U10d (blue), U5d (red), U30m (yellow), H10d (cyan), and H1d (green). The thin grey curves are the responses of the individual experiments in the U30m ensemble. The straight line represents a growth with time squared.

13 GROWTH AND SPREAD OF PERTURBATIONS 1601 experiment and the surface-pressure response a r;l.t/ D ±p r;l. ; Á; t/ã. ; Á/, where the overline denotes the spatial average over the region north of 20 ± N. This quantity is a measure of how the AO responds to the forcing. It should be compared with the averaged perturbation north of 20 ± N, 1p r;l.t/ D f±p r;l. ; Á; t/ 2 g 1=2. The squared ratio, ar;l 2 =1p2 r;l, measures how much of the total surface-pressure response can be explained by the AO. In Fig. 9, 1p r;l.t/, ja r;l.t/j and the squared ratio are plotted as a function of time. The grey clouds are the 80 individual members of the H1d ensemble and the thick curves are the ensemble means [1p].t/, [a].t/ and [ar;l 2 =1p2 r;l.t/]. Both [1p].t/ and [a].t/ grow approximately as time squared for the rst 30 days. However, [a].t/ grows faster, as can be seen from the ratio which increases by more than a factor of 10 from 0.01 to 0.3 in U30m, H1d and H10d and by a factor of 0.1 to 0.3 in U5d and U10d. Also, [a].t/ continues to grow after [1p].t/ has saturated after approximately 30 days. Less than 1% of the initial response in the surface pressure can be explained by changes in the AO in the U30m, H1d and H10d ensembles. At the end of the 100-day period the AO explains about 20% of the response which is comparable to the 27% of the variance the AO explains in the control experiment. These numbers refer to the ensemble means. In particular, ar;l 2 =1p2 r;l varies strongly among the members of the ensembles. 6. A STRATOSPHERE-ONLY EXPERIMENT In Christiansen (1999) we showed that the GCM can also sustain stratospheric vacillations in the absence of temporal variability in the troposphere. Here we present an ensemble of perturbation experiments with a constant troposphere to study the role of the stratospheric vacillations in the growth of perturbations. As in Christiansen (1999) the state of the troposphere, de ned as the levels with pressure larger than 200 hpa, was obtained as a 5-day mean, here starting from day 1745 of the control experiment, and the stationary waves were enhanced by a factor D 1:5 to obtain conditions for periodic vacillations. A control experiment and an ensemble experiment were performed. The control experiment lasted for 1000 days and N D 4 starting dates were chosen, for which M D 6 experiments were performed with forcing in the upper six of the layers listed in section 2. The forcings were of the same type as in U5d. The zonal-mean zonal wind in the control experiment at 60 ± N is shown in Fig. 10 as a function of pressure and time for 200 days. The state is periodic with a period of 52 days. It shows a weak downward propagation with a time-scale of approximately 10 days. The growth of the perturbations is shown in Fig. 11. As in Fig. 5, the perturbations are normalized with p 2¾ U.p/, where ¾ U.p/ now refers to the control experiment with constant troposphere. The perturbations grow while the forcing is active, but the further development of the perturbations is very different from the growth found in the full system (Fig. 5). In the experiment with a constant troposphere the strength of the perturbations does not grow beyond the level they had when the forcing terminated. Thus, the experiments with constant troposphere demonstrate that the model atmosphere can sustain stratospheric vacillations without the growth of perturbations. This is actually trivial given the existence of periodic vacillations in numerical experiments as these are obviously stable. Increasing the strength of the stationary waves in the troposphere above a threshold a quiescent state is replaced by a periodic vacillating state in agreement with the Hopf bifurcation observed in a low-dimensional model of

14 1602 B. CHRISTIANSEN Figure 10. Zonal-mean zonal wind (m s 1 ) at 60 ± N for the control experiment with constant troposphere. The stratosphere is in a periodic state with a weak downward propagation. Figure 11. The strength of the perturbation at 10 hpa, 60 ± N averaged over the ensemble (thick black curve) with constant troposphere and periodic background variability in the stratosphere. The thin curves are the responses in the 24 individual perturbation experiments. stratospheric vacillations (Yoden 1987). In this model the increase of the strength of the stationary waves results in secondary bifurcations that make the vacillating state irregular (Christiansen 2000a). The period of the regular vacillations is determined by the rst Hopf bifurcation and this period is also carried over as the dominant time-scale in the irregular vacillations. The growth of the perturbations on the other hand, is related to time-scales associated with the secondary bifurcations. 7. CONCLUSIONS We studied the growth of perturbations following transient forcings in a perpetual- January GCM. Ensemble experiments were performed with ve types of forcing, eight different initial conditions, and the forcings imposed at ten different vertical levels. All forcings were applied from 20 ± N to the north pole. The results from the different types of forcings agree in general and only small differences are found in the periods during which the forcings are active. The model atmosphere is chaotic and shows perturbation growth no matter which level is forced. The perturbations grow to a size comparable to the variability of the

15 GROWTH AND SPREAD OF PERTURBATIONS 1603 unperturbed atmosphere on a time-scale of days in the troposphere and days in the stratosphere. After the initial period of growth the perturbations have in a statistical sense the same vertical and temporal structure as the unperturbed atmosphere as shown Fig. 8(b). Although the forcing is restricted to the northern hemisphere, the perturbations encompass the whole atmosphere and develop on the same time-scale in both hemispheres. In the unperturbed model atmosphere the stratospheric variability has the form of downward-propagating stratospheric vacillations as shown in Fig. 1 and in more detail in Christiansen (2000b). However, in the initial period of growth the perturbations do not propagate downward and seem, in general, uncoupled from the background vacillations. This suggests that the downward propagation is a robust feature determined more by the processes in the troposphere than the state of the stratosphere, at least for the restricted types of perturbations studied. The perturbation experiments presented in this paper all had forcings in the region north of 20 ± N and one could argue that these perturbations are too extended to trigger changes in the stratospheric vacillations as they would affect both the polar vortex and its surroundings. However, additional experiments with the forcing imposed only north of 60 ± N con rm our conclusions. The forcings we studied were all con ned to a single model level and all had a degree of zonal symmetry and, therefore, we cannot totally eliminate the possibility that forcings with a spatial pattern matching the natural variability (Volodin and Galin 1998) could trigger such changes in the stratospheric vacillations. More effective forcings can probably be designed when a deeper understanding of the stratospheric vacillations has been obtained. We note that any perturbation in any atmospheric layer will almost immediately be communicated through the vertical extent of the atmosphere through, for example, hydrostatic adjustments. In our experiments this seems to be the mechanism that determines the vertical spread of perturbations. Perturbations are developed faster in the troposphere than in the stratosphere even when the forcing is imposed in the stratosphere. At the surface the in uence of the perturbations on the AO increases gradually in the initial period. This in uence also seems to increase, although somewhat more slowly, after the initial period. The absence of a slow downward propagation of transient perturbations in the stratosphere does not rule out a stratospheric in uence on the tropospheric weather, e.g. related to variations in the solar activity. As mentioned, perturbations in the stratosphere eventually affect the troposphere and vice versa. However, the seemingly chaotic response found in both the stratosphere and troposphere (Fig. 3) will make such an in uence very hard if not impossible to detect. Perturbations grow with time squared both when zonal-mean and single-cell values are considered. Such a power-law growth suggests the existence of a nite predictability time which is independent of the initial perturbation as long as it is small. Consistently, all ve ensembles show comparable predictability times ranging from days in the troposphere to days in the stratosphere. Power-law growth of perturbations and nite predictability time are expected in some systems with many length-scales such as fully developed three-dimensional, isotropic and homogeneous turbulence. The atmosphere and the GCM do not ful ll any of these requirements so there is no reason to believe that detailed results from turbulence theory should directly carry over. Indeed, in the Kolmogorov approach to turbulence the perturbations grow with the square root of time. We are not aware of other GCM studies reporting on the temporal growth of nitesize perturbations. A theoretical understanding of the growth with the square of time will be needed if our results are con rmed by other models. The power-law growth is related to the way perturbations are transferred from smaller to larger length-scales and

16 1604 B. CHRISTIANSEN is not related to the maximum Luapunov exponent. Thus, the traditional methods used to analyse predictability, which are based on the experience gained from low-dimensional systems, will not be useful. This could have practical consequences if the power-law growth holds for GCMs used for weather forecasting. The predictability time and the time-scale of the downward propagation are physically unrelated although they are of the same order of magnitude. This is clearly seen from model experiments with time-independent wave forcing at the tropopause level. Perfectly periodic stratospheric vacillations can be obtained when the strength of the wave forcing exceeds a critical threshold and the quiescent state becomes unstable. Imposing transient forcings on the periodic state results in perturbations that do not grow but remain of the same magnitude as they had when the forcing was terminated. The error growth seen in the full system is, therefore, not connected to the bifurcation that created the vacillations, but rather to the secondary bifurcations that make the vacillations irregular. As mentioned in the introduction, the predictive potential of the downward propagation is unrelated to the extent of the stratospheric control. The predictability times for the forcings considered here are long enough to preserve some predictive power of the vacillations. Forcings of the strength and duration studied in this paper will not seriously change the downward propagation of an observed zonal-mean zonal-wind anomaly. We close with a comment on the model dependence of our results. The stratospheric vacillations and their connection to the troposphere seem to be very well represented both in the ARPEGE GCM (Christiansen 2001) and in other GCMs (Yamazaki and Shinya 1999). This is not surprising as the mechanism behind the vacillations involves the interaction between the mean ow and the planetary waves. These large scales are well simulated in the models. Therefore, we do not believe that the absence of downward propagation of the perturbations is a model artefact. The growth of perturbation with the square of time needs to be con rmed by other models. This growth presumably depends on eddies of all scales and is, therefore, much more sensitive to model differences and de ciencies. ACKNOWLEDGEMENT This work was supported by the Danish Climate Center. REFERENCES Arnold, N. F. and Robinson, T. R Solar cycle changes to planetary wave propagation and their in u- ence on the middle atmospheric circulation. Ann. Geophys., 16, Solar magnetic ux in uences on the dynamics of the winter middle atmosphere. Geophys. Res. Lett., 28, Aurell, E., Bofetta, G., Crisanti, A., Paladin, G. and Vulpiani, A Predictability in systems with many characteristic times: The case of turbulence. Phys. Rev. E, 53, Baldwin, M. P. and Dunkerton, T. J Propagation of the Arctic Oscillation from the stratosphere to the troposphere. J. Geophys. Res., 104, Bochn cek, J., Hejda, P., Bucha, V. and Pýcha, J Possible geomagnetic activity effects on weather. Ann. Geophys., 17, Bohr, T., Jensen, M. H., Paladin, G. and Vulpiani, A Dynamical systems approach to turbulence. Cambridge University Press Christiansen, B Stratospheric vacillations in a general circulation model. J. Atmos. Sci., 56, a Chaos, quasi-periodicity and interannual variability: Studies of a stratospheric vacillation model. J. Atmos. Sci., 57,

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