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Dynamics of Atmospheres and Oceans 27 (1997) 485-503

Characteristics of annulus baroclinic flow structure during amplitude vacillation Huei-Iin Lu a,*, Timothy L. Miller b a Institute for Global Change Research and Education, 977 Explorer Blvd., Huntsville, AL 35806, USA b Marshall Space Flight Center, Huntsville, AL, USA

Received 30 August 1995; revised 8 August 1996; accepted 20 August 1996

Abstract An investigation is made of the mechanics of amplitude vacillation in a numerically simulated rotating annulus flow system. Amplitude vacillation is characterized by a periodic change of vertical wave structure in concert with growth and decay of wave amplitude and phase speed. The temperature wave amplitude profile for the dominant component consists of three local maxima: (1) lower boundary layer, (2) upper half layer and (3) lower half layer. The lower layer waves lead the time-dependent structural variation during vacillation. Two types of amplitude vacillation found in the experimental measurements (Buzyna et al., 1989: J. Atmos. Sci. 46, 2716-2729) can be distinguished in the temperature wave by whether the lower layer waves split from and travel behind the upper layer waves by one wave period in each cycle of vacillation. Linear eigenvalue analyses with respect to the instantaneous axisymmetric state at various points in time are performed to elucidate the simple interaction between the dominant wave and the zonal mean state. During the vacillation cycle, the zonal mean state is modified by the wave, which causes a change in growth rate and vertical structure of the linearly most unstable eigenmode. This, in turn, forces the actual changes of the nonlinear solutions. © 1997 Elsevier Science B.V.

1. Introduction The phenomenon of wave amplitude vacillation observed in the rotating differentially heated annulus flow system (e.g., Pfeffer et al., 1980a; Hignett, 1985; Tamaki and Ukaji,

* Corresponding author. Tel.: + 1 205 9225831; fax: + 1 205 9225788; e-mail: [email protected] 0377-0265/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0377-0265(97)00027-4

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H.-L Lu. T.L. Miller / Dynamics q[A tmospheres and Oceans 27 (1997) 485-503

1985) has long been regarded as a possible analogy to the zonal index cycle of the atmospheric general circulation (e.g., Webster and Keller, 1975; McGuirk and Reiter, 1976; McGuirk, 1982; Kidson, 1988). Despite the controversial evidence as to whether the notion of index cycle can be interpreted in terms of amplitude vacillation (Wallace and Hsu, 1985), the phenomenon is one of the purest forms of periodic behavior of baroclinic waves, and study of its underlying dynamics may provide new ideas about the life cycle of the extratropical cyclone, including its effects on the zonal mean flow. It is commonly known that the amplitude vacillation phenomenon, as normally seen from a time series of horizontal stream function (or temperature) field, is accompanied by significant variations in the azimuthal phase speed but no noticeable changes in the 'north-south' phase tilt with time (e.g., Pfeffer et al., 1974). However, the relationship between phase speed and wave amplitude of the dominant wave varies among different studies. It has been reported in some experiments (Pfeffer et al., 1974) that phase speed increases and in other experiments (Hignett et al., 1985; Buzyna et al., 1989) decreases when the wave amplitude is low. Equally controversial is how the vertical wave tilt between a 'top' and a 'bottom' level varies with time. Jonas (1981) found that the most rapid wave growth (at the bottom level) occurs at the time of maximum vertical tilt, while most of the experiments of Pfeffer et al. (1980b) show that the vertical tilt reaches its maximum when the wave is approaching its minimum amplitude (see Buzyna et al., 1989). The atmospheric observations (McGuirk, 1982; see also, Wang and Barcilon, 1986) also indicate that the largest tilt normally occurs at the minimum wave amplitude, rather than when the wave attains its maximum growth rate. Buzyna et al. (1989) analyzed the kinematic properties of amplitude vacillation in terms of two superimposed waves (or modes) with the same zonal wavenumber but different vertical structures that travel with different phase speeds around the annulus. The correlation between growth rate and vertical phase tilt has been identified in two types of dependence between the synthesized pair of wave amplitude functions, i.e., whether they intersect in the vertical. The intersection type, or the 'X'-type, represents the case in which one mode dominates in amplitude at the upper level, and another, slower traveling mode, is larger at lower levels. Because of the vertical differential in the mean drift speeds of the superimposed waves, a phase splitting, or usually called 'structural separation' in this paper, between the upper and lower parts of the wave can be seen in the phase progression plots. Most of the weakly nonlinear wave theories developed to date yield a single-mode baroclinic wave structure in the vertical direction (Pedlosky, 1979; Chou and Loesch, 1986; Wang and Barcilon, 1986). The time rate of change of wave amplitude has been found to be in phase with the westward tilt of the stream function perturbation, which is apparently at odds with the observational results cited previously. The single-mode wave theory fails to explain the phase splitting phenomena. To account for the presence of two waves (modes) represented in the kinematic analyses, the theory of dual-mode instability (Lindzen et ai., 1982; Barcilon and Drazin, 1984) has been developed based upon the weakly nonlinear dynamics of quasi-geostrophic perturbations on a basic state differing slightly from Eady's model. Although the occurrence of two weakly unstable modes and their equilibration with zonal flow may be possible for the basic states and points in the parameter space chosen by the theoretical model, the assumption of a weakly perturbed

H.-I. Lu, T.L Miller~Dynamics of Atmospheres and Oceans 27 (1997) 485-503

487

basic state configuration and weak nonlinearity has never been validated with experimental measurements (Barcilon and Drazin, 1984; Buzyna et al., 1989). Ukaji and Tamaki (1994) recently performed a 'two-mode' spectral analysis using the pressure data (not the temperature as the laboratory measurements usually provide) obtained from their numerical model. Their results do not indicate the existence of two dominant traveling modes. However, as it will become obvious later in this paper, the 'X'-type of wave structure may never exist in the pressure wave. Recent numerical simulations (Lu et al., 1994) of the laboratory experiments using a high resolution fully nonlinear model, by traversing the parameter space defined in Pfeffer et al. (1980b), have generated a number of amplitude vacillation cases. The purpose of this paper is to examine the baroclinic flow structure obtained from the numerical integrations, seeking the key features that are unique to the vacillation phenomena. The two types of amplitude vacillation found in the laboratory measurements (Buzyna et al., 1989) will be illustrated in the analyses. A new view on the mechanics of amplitude vacillation will be presented. 2. The numerical model The GEOSIM model (Miller et al., 1992) used for the present study is a numerical approximation of the Navier-Stokes equations, discretized by finite differences in the meridional and vertical directions and with a spectral representation in the azimuthal direction. The resolution of the model used here consists of 25 fully nonlinear waves, 25 and 45 grid points in the radial and vertical directions, respectively, with smoothly-varying grid compressions in the sidewall and lower boundary layers. The physical parameters for the present model are chosen to be the same as those in the laboratory experiments series B described in Pfeffer et al. (1980b), hereafter referred to as PBK. The annulus is rotating at a constant rotation rate ( O ) with no lid on the top. The centrifugal force term is included in the interior but not in affecting the shape of the upper surface (see Miller and Butler, 1991). Note that Ukaji and Tamaki (1994) omitted the centrifugal force term, while Hignett et al. (1985) included the term along with a non-slip boundary condition. In the present study, more than 40 cases with various /2 values ranging from 0.8 to 3.3 s -~ are conducted, including the five PBK cases. The quiescent-start initial condition (Lu et al., 1994) used by the model is different from the 'sudden-start' procedure used in the PBK experiments in which vigorous spin-up occurs initially. James et al. (1981) used a steady state axisymmetrical solution with a small perturbation as initial condition. Hignett et al. (1985) and Ukaji and Tamaki (1994) employed a procedure of decrementing 12 to achieve the vacillation phenomena. In all of the cases run, the equilibrated integrations were obtained by running the model for at least 3100 s. Three kinds of equilibrated time-dependent behavior are obtained from this series of numerical experiments: amplitude vacillation, tilted-trough vacillation and side-banded wave dispersion. The locations in parameter space where such behaviors occur were described in Lu et al. (1994). Data samples are taken at the rate of 1 per 3 s over the last few vacillation periods. The spatial locations denoted in this paper are normalized by the annulus geometrical parameters d and H. We will emphasize the analyses of two cases of different rotation rates, /2 = 1.30 and 1.72 s- 1.

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H.-1. Lu, T.L. Miller~Dynamics of Atmospheres and Oceans 27 (1997) 485-503

3. Results 3.1. Time variations o f the dominant wat'e

Fig. 1 shows the time variations of typical amplitudes and phase angles of the temperature wave for the dominant wavenumber, here wave 4, at the 3 / 4 , 1 / 2 and 1 / 4 heights at mid-radius. The two cases shown differ mainly in the behavior of phase speed, i.e., the time derivative of phase angle, for the lower level ( H / 4 ) wave when the wave amplitude reaches a minimum value. In the lower O case, the lower level wave travels at a mean drift speed slower than the upper and middle level waves due to a phase lag of 27r per vacillation cycle. This corresponds to the 'X'-type vacillation obtained in the laboratory analyses of Buzyna et al. (1989). In contrast, the higher case only shows that the phase difference between the upper and lower levels increases to a maximum shortly before the lower level wave reaches its minimum amplitude, and that the mean drift speed is constant with height. The amplitude functions show that growth/decay of the upper level waves lags behind the lower level waves for about two rotations (or days) in the high amplitude stage and about 15 days in the low amplitude stage. Hignett et al. (1985) reported that minima of wave amplitude are closely in phase (in time) at 'all' heights, but maxima near mid-level lead maxima in the top and bottom of the annulus. Such a lagging in the vertical wave development, though relatively unnoticeable in the laboratory measurements of Buzyna et al. (1989), suggests that there is a considerable variation in vertical structure during the vacillation cycle, Fig. 2 shows the progression of pressure wave phase angles at the same elevations for the two cases noted before. The main difference between the pressure and temperature waves is that in the pressure wave there is no vertical difference in the mean drift speeds, hence no phase splitting nor 'X'-type vacillation. In fact, the variations of vertical tilt are relatively difficult to detect for the higher $2 case. In the lower/2 case, however, the vertical tilt in the pressure wave decreases sharply near the minimum amplitude, due primarily to the decreased wave speed in the lower level. This would contradict the increase of vertical tilt in the temperature wave described previously if the quasi-geostrophic assumption were made. The apparent contradiction to the behavior of temperature wave is due to the poor representation of vertical tilt based upon the phase angle information at the three arbitrarily chosen elevations. The vertical wave structure is time-dependent and multi-modal, and only through careful examinations of the vertical profiles can we understand the actual flow behavior involved in the amplitude vacillation. 3.2. Vertical wave and mean f l o w structures

To give an accurate view of the vertical wave structure and its periodic changes, we present a series of plots depicting the vertical profiles of wave amplitude and phase angle in one cycle of vacillation, T. We denote ( 0 / 8 ) T and ( 8 / 8 ) T as the time when the temperature wave at z = 0.75 and y = 0.5 reaches the peak value. Fig. 3 depicts the time variations of temperature wave vertical structures for the two cases defined earlier. It reveals that each amplitude profile typically has three local

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maxima, and that the vertical amplitude structure undergoes significant changes during the vacillation cycle. We identify that the temperature wave consists of three local maxima which are heuristically called modes in this paper: the upper mode centers near z = 0.75; the lower mode centers near ~ = 0.25 and the boundary layer mode forms below z = 0.1. Consistent with previous discussion of the amplitude functions (Fig. 1), there is a lag of wave development in the upper mode (and boundary layer mode) behind the lower mode. Note that the usual interpretation of the term 'mode' is that of a set of mutually orthogonal structures which superpose a n d / o r interact to produce a resultant flow, Here we adapt the term to describe the unique features of wave structure/amplitude vacillations associated with these three local maxima. We will later show that these 'phases' of wave development are related to the characteristics of the zonal mean states in terms of the most unstable eigenmodes. The lower panels of Fig. 3 show the corresponding phase profiles as a function of time. Note that to show the phase angles in procession using phase diagrams (Figs. 1-4) requires recovering the branches when the individual wave functions are computed. Since the data sampling rate is sufficiently high, we assume that the difference between two successive phase angles, say ~2 - q~, never exceed _+ ~-. If q~2 is greater/smaller than ~ by + ~-, then 2~" is subtracted f r o m / a d d e d to q~2. A similar procedure is applied to ensure that no branch-cut discontinuity exists in the vertical at the wave 'incipient' time (0/8)T. The relative phase angle profiles given in Fig. 3 and Fig. 4 represent the vertical-dependent phase functions tot the 8 time frames with respect to the dominant temperature wave at the reference location (at z = 0.75 and y = 0.5) at the same sampling times. Structural separation (or lack thereof) was also verified by viewing detailed computer animations in an azimuthal cross-section. It is evident that below the z = 0.25 level, which includes the boundary mode and half of the lower mode, the vertical tilt of the temperature wave is westward (Fig. 3). The major eastward tilt is between ,- = 0.25 and ; = 0.5 which increases as the lower wave amplitude decreases, attaining a maximum tilt near the weakest wave stage. In the higher ,Q case, the tilt varies within 50 to 120 degrees, while in the lower ~2 case, it exceeds 180 degrees. The latter case further indicates that as the tilt exceeds 180 degrees, the 'normal' 50-degree tilt is reestablished by quickly merging the lower layer waves with the next coming upper layer waves which travel 'eastward' at about the reference wave speed. As in Fig. I. the mean drift speeds for the lower layer waves are slower than those of the upper and middle layer waves due to a phase lag of 27r per vacillation cycle, resulting in a structural separation during the lower amplitude state. The first harmonic component was found to be the second dominant wave in most of the amplitude vacillation cases. Generally the harmonic waves travel at the same mean wave speed as the primary waves around the annulus. In the case of structural separation, the harmonic waves in most part of the lower layer also travel at a 27r lag per vacillation cycle behind the upper layer waves. Typically the harmonic wave consists of a well defined upper mode and a lower mode, both generally concur with the corresponding modes of the dominant wave. The ratio of wave amplitudes between the harmonic wave and primary wave are about 1:5. Despite this seemingly large contribution from the harmonic wave, we have found that amplitude vacillation phenomena can occur in a simple model which includes only

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wave-mean flow interactions in the nonlinear advective terms. Wave-wave interactions are only essential for predicting the wavenumber regimes and details of the vacillation behavior. As expected, the pressure amplitude functions consist of three large vertical gradients in correspondence to the three-mode structure seen in the temperature waves. However, if the vertical tilt of the pressure wave (Fig. 4) is defined based upon the phase angle difference between z = 0.25 and 0.75, then the largest tilt appears to occur at the time of maximum growth, and smaller tilts occur during weak and decaying stages. Our result would be then consistent with the results of both theoretical models (e.g., Pedlosky, 1970; Wang and Barcilon, 1986) and some laboratory measurements (Douglas et al., 1972; Jonas, 1981). However, such comparison is misleading because the vertical tilt is a function of the layer chosen for computing the phase difference. If the vertical tilt is computed locally, then it is the middle layer where the maximum vertical tilt is located and where the largest temporal variability occurs. In the lower ~ case, a sharp pressure gradient changes its sign briefly near t = (3/8)T. This corresponds to the time when the temperature wave just begins to undergo a structural separation. By virtue of hydrostatic equilibrium, the pressure field maintains a smooth continuity in the vertical without splitting. We have verified that splitting does occur for the vertical derivatives of the pressure field, In the middle level of both cases, that increase of ' westward' phase tilt of the pressure wave is found to be consistent with the increase of 'eastward' phase tilt in the temperature wave when the amplitude is decreased to a minimum. The pressure waves lie to the east of the temperature waves, about 10 degrees in the upper layer and 130 degrees in the lower layer. This phase difference accounts for a factor of two larger magnitude of horizontal eddy heat flux in the lower half of the fluid, despite the fact that the temperature wave amplitude there is generally smaller. It is interesting to point out that long before the upper layer reaches its peak amplitude state, the baroclinic wave has begun to reduce its pressure-density solenoids. An equivalent barotropic configuration, that is when the isotherms and isobars are parallel, is reached in the upper layer at t = ( 2 / 8 ) T when the temperature wave amplitude has decreased to about half of the peak value. The zonal velocity profile consists of a positive vertical shear in the interior, and either positive or negative in the Ekman boundary layer (Fig. 5a). The sign of vertical shear in the Ekman layer depends upon whether the zonal flow in the lower layers is westerly or easterly, which is a function of wave intensity. Easterly surface flow occurs during the weak wave stage in which the meridional circulation is dominated by a weak Hadley cell across the annulus channel, while westerly surface flow occurs when the baroclinic wave is strong and the Ferrel cell is developed in the middle of the annulus. In the interior, the increase of vertical shear begins when the wave has decayed to half of its peak value. There is a sharp transition for the center of maximum vertical shear lifting from z = 0.3 up to z = 0.55 during the time the lower layer waves are undergoing an intensive growth. This raising of the interior baroclinic zone could be responsible for the subsequent growth of baroclinic waves in the upper layers. As the wave amplitude has just passed its peak at about t = (1/8)T, the baroclinicity drastically decreases toward its minimal value - - a well known characteristic of amplitude vacillation (Pfeffer et al., 1974).

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The static stability parameter, N 2 = g [ 3 ( d T / d z ) , was computed based upon the area-mean temperature away from the sidewall boundary layers. Fig. 5b shows that the profiles can be described as a two-step exponential function in the vertical, with a steeper rate of decrease in the lower layer than in the upper layer. The stability in the mid-lower layer (0.3 < z < 0.5) is relatively uniform with height. In contrast to the vertical shear, the static stability varies very little with time, except in the mid-lower layer. The presence of a stepwise stability profile could be dynamically interrelated with the formation of upper and lower modes of the temperature waves.

3.3. Relationship with the most unstable eigenmode One of the most striking characteristics of amplitude vacillation shown here is that the vertical structure of the baroclinic wave vacillates in concert with the growth and

496

H.-I. Lu, T.L. Miller/Dynamic~ of Atmospheres and Oceans 27 (1997) 485-503

decay of the wave, from the lower layers up during growth phase. With the above information on the time evolution of wave and zonally symmetric states, we hypothesize that the annulus baroclinic flow is capable of supporting at least two modes, which are unstable/stable at different times during the vacillation cycle. To show that such instability/stability sources exist we compute the most unstable linear waves with respect to the instantaneous zonal mean state. The method of computing the eigenmode was discussed in Miller et al. (1992). In these calculations, all the linear terms in the governing equations are kept, and the numerical procedures are identical with the nonlinear integration. The zonal mean state is fixed, while the wave is allowed to grow. The obtained most unstable eigenmode can be viewed as a characterization of the zonal mean state's instability at that point in time, which is instantaneously forcing the growth of wave if the eigenvalue is positive. If the eigenvalue of the most unstable mode is negative, which actually happens during part of the vacillation cycle, then the wave with that eigenmode structure would have the highest retaining (or lowest decay) rate over the other modal structures having even larger negative eigenvalues. As long as the eigenfunction differs from the existing waves and has a growth (or decay) rate significantly different from the other eigenmodes, there is a forcing of structural change. Fig. 6 and Fig. 7 show time sequences of longitudinal-height cross-sections of two temperature fields each covering the first quarter of the annulus ring at mid-radius. 'Nonlinear' refers to the actual integration of the fully nonlinear model, while 'Linear' is the most unstable eigenfunction. Notice that the calculations of linear solutions involve an arbitrary multiplying factor for amplitudes and an arbitrary adding constant for phase angles. The choices of these constants are self-determined by the corresponding nonlinear solutions which are used as the initial guess for the iteration scheme. For the lower $2 case (Fig. 6), the vertical tilt of the nonlinear wave increases as wave amplitudes decrease, attaining a maximum value near t = ( 3 / 8 ) T when the lower layer wave amplitudes reach a minimum and the upper layer waves are still decaying. A splitting of these two layer waves occurs as the tilt exceeds 180 degrees, followed by a rapid reattachment near t = (4/8)T. The contour lines show a 'scar' of waves being recently separated. As the upper layer waves decay to a minimum state near t = ( 5 / 8 ) T , the lower layer waves penetrate upward, developing into a single mid-layer mode structure before reestablishing a upper-heavy straight-up configuration at the mature stage. The computed linear wave consists of two transitions of modal structure during the decaying stage, i.e., ( I / 8 ) T < t < (3/8)T. It is evident that the transition to the lower heavy mode occurs before the nonlinear solutions take the corresponding form. The transition to a single mode structure during the growing stage, t > ( 5 / 8 ) T , is coincident with the nonlinear solutions. For the higher ~ case, the time variations of the nonlinear wave structure are somewhat simpler (Fig. 7), revealing that the lower layer waves are always connected with the upper layer waves in the same vertical configuration. Only a 'wiggling' wave movement associated with changes of vertical tilt is discernible, which attains a maximum when the lower layer waves are weakest. The corresponding linear solutions show similar alternations between upper and lower modes. In particular, the lower mode becomes most unstable at the time the upper layer waves of the nonlinear solution are still undergoing decay, i.e., ( 2 / 8 ) T < t < (4/8)T.

H.-L Lu, T.L. Miller~Dynamics of Atmospheres and Oceans 27 (1997) 485-503

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498

H.-I. Lu, T.L. Miller / Dynamic~ of Atmospheres and Oceans 27 (1997) 485 503

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H.-I. Lu, T,L Miller/Dynamics of Atmospheres and Oceans 27 (1997 485-503

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The extent to which amplitude vacillation can be accounted for by the time-dependent most unstable linear wave is explored by projecting the nonlinear solutions onto the eigenfunction space. Fig. 8 and Fig. 9 show the time variations of projection error normalized by the variance of temperature field N(T) 2, mean growth rates, and frequencies obtained from the linear and nonlinear solutions. It is important to note that the numerical scheme for solving the linearly most unstable mode fails to converge when more than one solution with similar growth rates coexist. These show up as a 'spike' in the curves for the errors of the linear projections. For this reason, we shall only look for a longer trend of modal selections, ignoring the exact time when the transitions actually take place. In general, the linearly most unstable mode undergoes three stages of structural change in one vacillation cycle. In the wave growing to saturation stage, which takes about 1 / 3 of the vacillation period, the shifting from a lower mode to a upper mode accounts for as much as 90% of the actual tendency of temperature wave. In the first half of the wave decaying stage, the most unstable eigenmode (with negative growth rate) accounts for about 70% of the actual rate of change. In the second half of the decaying stage, the existing nonlinear solutions are of a fast moving upper mode, while the linear solutions are characterized by a positive growth rate and a considerably slower phase speed. Apparently, it is this growth of the slowly moving lower mode that causes the increase of vertical tilt. As expected, the difference of mean wave speeds between nonlinear and linear solutions at this time stage is considerably larger for the case with structural separation (Fig. 8) than for the case without separation (Fig. 9).

4. Discussion and conclusions Amplitude vacillation is characterized by a periodic change of vertical wave structure in concert with the growth and decay of the wave amplitude. Corresponding to the local maxima in the temperature amplitude profile, the baroclinic wave consists of a boundary layer mode, a lower mode, and an upper mode. These three modes evolve periodically with the lower mode leading the vacillation cycle. Because of the presence of three modes, the relationship between the vertical tilt and growth rate of baroclinic waves depends upon the choice of two elevations that define the vertical tilt. We believe that some of the controversies in the earlier studies are partially due to poor presentations of the vertical wave structure by merely presenting the information on the 'top' and 'bottom' levels, which to some extent are arbitrary. One cannot rule out the possibility that the lower level locations selected by Jonas 0981) and Douglas et al. (1972), being 0.11H and 0.12H, respectively, may be within the boundary layer mode. The notion that the maximum vertical tilt should correspond to the fastest growth rate is not generally consistent with either observational or numerical results. Two types of amplitude vacillation, with and without the occurrence of structural Fig. 9. Sameas Fig. 8, exceptfor the case of D = 1.72 s- I. The additionaltime coordinateindicatesthe times correspondingto the time sequenceof cross-sectionplots in Fig. 7.

502

H.-L Lu, T.L. Miller/Dynamics of Atmospheres and Oceans 27 (1997) 485-503

separation of the temperature waves have been found in this series of experiments. Structural separation is due to a stagnation of lower layer waves relative to the fast moving upper layer waves, causing a phase lag of one wave period when the wave amplitude nearly reaches its minimum in each vacillation cycle. This type of vacillation has a slower mean drift speed on the lower layer than on the upper layer, hence can be classified as the 'X'-type as defined in the experimental analyses (Buzyna et al., 1989). However, no such phase splitting occurs in either the pressure or vorticity waves. Dynamically, the three modes in the temperature wave interact indirectly through wave-mean flow interactions, i.e., as the zonal mean flow is modified by the wave, different regions of the fluid domain become most unstable (or least stable). The new cycle of vacillation begins with the growth of a slowly moving wave in the lower layers while the upper layer waves are still decaying. A transition from lower to upper mode takes place during the growing stage until the saturation amplitude with an equivalent barotropic configuration is reached. In the wave decaying stage, the upper layer waves generally have a weaker damping rate than the lower layer waves. These results, while having a degree of similarity to the view of Lindzen et al. (1982), present a different paradigm for amplitude vacillation. Rather than consisting of two near-neutral modes interfering constructively and destructively with each other, the present results indicate a structural change in the waves and mean state as they interact during the vacillation cycle. The mean state actually alternates between stability and instability during the cycle, while the structure of the most unstable wave alternates between 'top-heavy' and 'bottom-heavy'. Structural separation of the temperature wave does not apparently contribute any additional mechanisms to the amplitude vacillation; they are merely an indicator of the intensity and timing of the lower layer wave development. It appears that structural separation is caused by a significant difference between the existing wave speed and the corresponding linearly most unstable wave speed during the beginning of the growth stage. Notice that the existing wave during the time of separation is dominated by the upper mode traveling at a fast speed, while the linear wave is in the lower mode, traveling at a much slower speed. Fig. 8 and Fig. 9 indicate a trend in which the wave speed of the nonlinear solution decreases, while that of the linear solution slightly increases, as the rotation rate of the annulus increases. In the present numerical experiments, there is a cut-off g2 at approximately 1.45 s-l for the 'X'-type vacillation to occur at a slower rotation rate. However, the experimental measurements of Buzyna et al. (1989) indicated that the 'X'-type amplitude vacillation is pervasive throughout the amplitude vacillation wave regime.

Acknowledgements IGCRE is jointly operated by the University of Alabama in Huntsville and the Universities Space Research Association. This research was supported by NASA's Global Atmospheric Modeling and Analysis Program, Office of Mission to Planet Earth, administered by Dr. Ken Bergman. Dr. Lu's work in IGCRE was supported by Universities Space Research Association SUB93-216 and SUB94-218 under NASA Cooperative Agreement NCC8-22. Karen A. Butler provided programming assistance.

H.-I. Lu, T.L. Miller~Dynamics of Atmospheres and Oceans 27 (1997) 485-503

503

References Barcilon, A., Drazin, P.G., 1984. A weakly nonlinear theory of amplitude vacillation and baroclinic waves. J. Atmos. Sci. 41, 3314-3330. Buzyna, G., Pfeffer, R.L., Kung, R., 1989. Kinematic properties of wave amplitude vacillation in a thermally driven rotating fluid. J. Atmos Sci. 46, 2716-2729. Chou, S.-H., Loesch, A.Z., 1986. Supercritical dynamics of baroclinic disturbances in the presence of asymmetric Ekman dissipation. J. Atmos. Sci. 43, 1781-1795. Douglas, H.A., Hide, R., Mason, P.J., 1972. An investigation of the structure of baroclinic waves using three-level streak photography. Q. J. R. Met. Soc. 98, 247-263. Hignett, P., 1985. Characteristics of amplitude vacillation in a differentially heated rotating fluid annulus. Geophys. Astrophys. Fluid Dynamics 31,247-251. Hignett, P., White, A.A., Carter, R.D., Jackson, W.D.N., Small, R.M., 1985. A Comparison of laboratory measurements and numerical simulations of baroclinic wave flows in a rotating cylindrical annulus. Q. J. R. Met. Soc. 111, 131-154. James, I.N., Jonas, P.R., Farnell, L., 1981. A combined laboratory and numerical study of fully developed steady baroclinic waves in a cylindrical annulus. Q. J. R. Met. Soc, 107, 51-78. Jonas, P.R., 1981. Some effects of boundary conditions and fluid properties on vacillation in thermally driven rotating flow in an annulus. Geophys. Astrophys. Fluid Dynamics 18, 1-23. Kidson, J.W., 1988. Indices of the southern hemisphere zonal wind. J. Climate 1, 183-194. Lindzen, R.S., Farrel, B., Jacqmin, D., 1982. Vacillation due to wave interference: applications to the atmosphere and to annulus experiments. J. Atmos. Sci. 39, 14-23. Lu, H.-I., Miller, T.L., Butler, K.A., 1994. A numerical study of wavenumber selection in the baroclinic annulus flow system. Geophys. Astrophys. Fluid Dynamics 75, 1-19. McGuirk, J.P., Reiter, E., 1976. A vacillation in atmospheric energy parameters. J. Atmos. Sci. 33, 2079-2093. McGuirk, J.P., 1982. The climatology and physical mechanisms of atmospheric vacillation. Tech. Report., Dept. of Meteorology, Texas A&M University. Miller, T.L., Butler, K.A., 1991. Hysteresis and the transition between axisymmetric flow and wave flow in the baroclinic annulus. J. Atmos. Sci. 48, 811-823. Miller, T.L., Lu, H.-I., Butler, K.A., 1992. A fully nonlinear, mixed spectral and finite difference model for thermally driven, rotating flows. J. Comp. Physics 101,265-275. Pedlosky, J., 1970. Finite-amplitude baroclinic waves. J. Atmos. Sci. 27, 15-30. Pedlosky, J., 1979. Finite amplitude baroclinic waves in a continuous model of the atmosphere. J. Atmos. Sci. 36, 1908-1924. Pfeffer, R.L., Buzyna, G., Kung, R., 1974. Synoptic features and energetics of wave-amplitude vacillation in a rotating, differentially-heated fluid. J. Atmos. Sci. 31,622-645. Pfeffer, R.L., Buzyna, G., Kung, R., 1980a. Time-dependent modes of behaviors of thermally driven rotating fluids. J. Atmos. Sci. 37, 2129-2149. Pfeffer, R.L., Buzyna, G., Kung, R., 1980b. Relationships among eddy fluxes of heat, eddy temperature variances and basic-state temperature parameters in thermally driven rotating fluids. J. Atmos. Sci. 37, 2577-2599. Tamaki, K., Ukaji, K., 1985. Radial heat transport and azimuthally averaged temperature fields in a differentially heated rotating fluid annulus undergoing amplitude vacillation. J. Met. Soc. Japan. 63, 168-179. Ukaji, K., Tamaki, K., 1994. A numerical study of amplitude vacillation observed in a differentially heated rotating fluid annulus. J. Met. Soc. Japan 72, 1-9. Wallace, J.M., Hsu, H.-H., 1985. Another look at the index cycle. Tellus 37A, 478-486. Wang, B , Barcilon, A., 1986. The weakly nonlinear dynamics of a planetary green mode and atmospheric vacillation. J. Atmos. Sci. 31, 1275-1287. Webster, J., Keller, J.L., 1975. Atmospheric variations: vacillations and index cycles. J. Atmos. Sci. 32, 1283-1300.