Dynamics of Atmospheres and Oceans, 17 (1993) 225-241
225
Elsevier Science Publishers B.V., A m s t e r d a m
A parcel theory approach to/3-plane inertial stability D.J. Parker Department of Meteorology, University of Reading, 2 Earley Gate, Whiteknights,ReadingRG6 2AU, UK (Received 8 November 1991; revised 8 June 1992; accepted 9 October 1992)
ABSTRACT Parker, D.J., 1993. A parcel theory approach to fl-plane inertial stability. Dyn. Atmos. Oceans, 17: 225-241. The problem of the orbit of a parcel from its initial location at which there might be a momentum anomaly is here considered for a/3-plane. The solutions represent a generalisation of the recent work of Wan and Yang (1990, Adv. Atmos. Sci., 7: 409-422). A physical interpretation of the orbits, in which they are related to flow stability, is given for both extratropics and tropics. The possible effect of dissipation on these results is also discussed.
1. I N T R O D U C T I O N
Parcel models have been found useful in the general understanding of symmetric and baroclinic instability (Thorpe et al., 1989) and may be thought of as providing a counterpart to linearisation methods. In both cases incomplete physical systems are studied. In a linearised analysis the broad assumption of small nonlinear terms, including those involving gradients of small terms, is invoked. With a parcel model it is assumed that the given fluid element does not influence its surroundings. In some sense this may be interpreted as a limiting case of an infinitesimal localised perturbation to a basic flow. However, it should be understood that the links between the two forms of analysis may be tenuous. For example, the present nonlinear study concurs with the linearised results of Dunkerton (1981) for flow at the equator but in a linear form it would fail to do so. In this regime the nonlinear terms seem to act in a similar way to the pressure (and three-dimensional) terms of linearised analyses. The parcel model also has the attraction that the supposed disturbance need not be assumed Correspondence to: D.J. Parker, D e p a r t m e n t of Meteorology, University of Reading, 2 Earley Gate, Whiteknights, R e a d i n g R G 6 2AU, UK. 0377-0265/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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D.J.PARKER
to be symmetric. The consequences of a localised anomaly to the flow are investigated and the large-scale interpretation need not exhibit axisymmetry. Wan and Yang (1990) (WY) have investigated a parcel model for two-dimensional motion in a linear zonal shear ~(y) on a /3-plane. By factorising the meridional m o m e n t u m equation, their eqn. (7), it is possible to obtain general solutions describing the behaviour of parcels including initial perturbations of position and zonal velocity. This may be further extended to a quadratic zonal flow. In this problem, neglect of the perturbation pressure gradient terms may be tentatively justified if one considers Stevens' (1983) analysis of the dynamic balances of unstable equatorial modes in which inertial forces can be about three times greater than pressure gradient forces, depending on the equivalent depth. In Section 2 an outline of the analysis of the mathematical system is given, in Section 3 the physical interpretation of these results is discussed, in Sections 4 and 5 specific applications to the extratropics and tropics are considered, in Section 6 outline of the extension to quadratic zonal flow is given and in the final section the possible effects of dissipation on the results are briefly discussed. 2. PARCEL MODEL ON/3-PLANE WY reduce the parcel model du --
=fv
dt dv d---t-=f(u-u)
(WY, eqn. (3)) (WY, eqn. (4))
for two-dimensional flow in the presence of a linear background zonal wind u = u o + (O~/Oy)(y -Y0), to the system d~7 -- =v dt dv = f ° u l + (/3Ul -f0~)~7 - (~ + f 0 / 2 ) / 3 ~ 2 - ~ P 1 ,-'~2~7 3 -dt -
(WY, eqn. (6)) (WY, eqn. (7))
Here n = y -Y0 is the meridional displacement from an initial position, v is the meridional component of velocity, u I = U 0 - - u ( Y 0 ) is the zonal velocity discrepancy of the parcel at y = Y0 (ul = a in the WY notation), = f 0 - (0~/ay) is the background absolute vorticity at Yo, and f0 and /3 are the usual Coriolis and Rossby parameters. It is important to remember that u 1 is a fixed initial condition for each parcel. The zonal velocity, u, has
227
13-PLANE INERTIAL STABILITY
been decoupled by integrating (WY, eqn. (3)) to obtain an explicit form for
u( r/) u=u(Yo) +for/+ ~r/2
(1)
which is a fl-plane form of the conservation of 'absolute angular momentum' in two dimensions. First, it should be noted that the pair of coupled equations (WY, eqns. (6) and (7)) represents a Hamiltonian system: defining H(r/, v ) = gV ' 2 - - f o U l r l -- ~(~U 1 1,-~2r/ 4 1 --fo~)r/2 + 1(so +fo/2)/3r/3 + gp
(2)
yields dr/
OH
dv
OH
dt
Ov' dt
Or/
A parcel specified by u I and Y0 will have its trajectories in phase space given by contours of the Hamiltonian, H, and critical points of (WY eqns. (6) and (7)) are given by stationary points of H. The polynomial H given above will yield, in general, closed, bounded contours. As the problem is Hamiltonian there can be no attractors. Critical points can be saddles or centres (Guckenheimer and Holmes, .1983). The period of closed parcel trajectories is given by the integral T=¢;r/
(3)
around the orbit, where v(r/) is found by setting H(r/, v) constant for the orbit. T will be infinite for trajectories which intersect saddle points (these trajectories are separatrices) and large for orbits close to these. Near centres of the system the limiting frequency of orbits is given by the square root of - d e t where det is the determinant of the linearisation matrix of (WY, eqns. (6) and (7)). It may also be shown that for large orbits T behaves like n - 1 / 4 as H ~ oo; that is, it tends to zero. Taking the two solutions of H = H 1 with v = 0 to be r/l, r/2 = +-(8H1/fl 2)1/4 as H 1 ---, % the integral for the period may be written
T = 2 f nln27d n "" (2)1/2 fn72
=
dr/ [ H 1 -- ( ~ 2 r / 4 / 2 ) ]
1/2
dr
nl~ 2 I
J0 (1 -- r4) 1/2 (4)
yielding the stated result, as the final integral is a finite constant.
228
D.J. PARKER
By returning to (WY, eqn. (4)), it becomes evident that (WY, eqn. (7)) will factorise simply, one root being
f=fo+
n
=0
The two other roots correspond to latitudes at which K - u = 0, that is, the latitudes at which the parcel's zonal velocity equals that of the background flow. At this stage, it is helpful to non-dimensionalise (WY, eqns. (6) and (7)) using f o 1 as a time-scale and fo/fl as a length-scale. The equatorial f0 = 0 results will be obtained dimensionally in Section 5. Using a tilde to denote dimensionless variables (e.g. ~: =f0sc), the equations become
(5)
dt d5
--dt = 51
"[-(/~1-()~- (("[-1)~'12- ~T]1"3
(6)
The other two critical points of (WY, eqns. (6) and (7)) may now be written ~=0, = _ (_+ ((2 + 2fi 1)1/2 for (2 + 251 > 0
(7)
T h e r e must be a bifurcation at ( 2 + 2fi I = 0, where the two roots for become imaginary. Consideration of the linear behaviour near critical points shows that the bifurcations lie on /~1 + ( = 1
(8)
(2 -'b 2t~ 1 = 0
(9)
AS the above are codimension-1 bifurcations (i.e. 'one-dimensional in two-dimensional p a r a m e t e r space'), except at the point ( = 1, f i l 21, only four canonical forms may occur. The 'Hopf' bifurcation is the case in which the stability of a fixed point is reversed by the development of a closed orbit around it. This case is excluded, as there are no attractors in the system. The 'pitchfork' bifurcation is that in which the stability of a fixed point is reversed by the creation of a symmetric pair of fixed points. The canonical system for this bifurcation is the case 2 = x ( / z - x 2) with fixed points x = 0 (stable) for tz < 0 and x = 0 (unstable) with x = +/z 1/2 (stable) f o r / z > 0. The pitchfork bifurcation requires a symmetry, as in the above example with x ~ - x , and so must be excluded from the current consideration. The bifurcation at (8) may be seen to be 'transcritical', in which two fixed points m e e t and exchange stabilities. The case 2 = x(/~ - x ) illustrates this: fixed points are x = 0 and x = t z which are, respectively, stable and unstable for /~ < 0 and unstable and stable for p. > 0. In the
/3-PLANE INERTIAL STABILITY
229
present study, the two fixed points will be a centre and a saddle point. The bifurcation at (9) is of the remaining possible form, the 'saddle-node'. In this case, a saddle point and another fixed point (here, a centre) are created as a pair as the parameters are varied. To illustrate this, let us consider the case k = ( ~ - x 2 ) . Fixed points do not exist for tz < 0; for /x > 0 they are x = +/z (unstable) and x = - ~ (stable). Locating (8) and (9) in the (ill, ( ) plane (Fig. 1) illustrates that there exists a codimension-2 bifurcation at the point ( - ½, 1). It is important to note that in region A of the (ill, ~) plane the system admits only one critical point, with f = 0; elsewhere, three critical points exist. Stability diagrams for critical points when sc varies (Fig. 2) illuminate the two significant forms of bifurcation. In Fig. 2(a), as ~ is varied close to the transcritical bifurcation, instability is transferred b e t w e e n the two fixed points. In Fig. 2(c) near the saddle-node bifurcations, as ( is varied the
REGION A
~. SADDLE-NODE BIFURCATION
I I I I-It
~ = 1 2
TRANSCRITICAL~ BIFURCATION ~
~2 + 2ux= 0
Fig. 1. The loci of bifurcations in parameter space. The lines of bifurcations in the plane of non-dimensional parameters ~ (absolute vorticity) and fii (initial zonal velocity perturbation) separate regions of essentially different flow structures. The three forms of flow in phase space are illustrated as insets in their respective domains. On these phase-space diagrams the parcel trajectories are exactly the contours of the Hamiltonian, H (2), and = 1 corresponds to the equator on the non-dimensionalised /3-plane.
230
D.J. PARr~R
(a)
%.
(b)
TRANSCRITICAL
-1
Fig. 2. Stability diagrams as ( ( n o n - d i m e n s i o n a l basic state vorticity) varies for fixed /~1" 17 is non-dimensional latitude on the /3-plane and -~ = - 1 corresponds to the equator. The curves represent the meridional positions of the fixed points of the system at each (. Dotted lines represent saddles; continuous lines represent centres about which parcels will tend to orbit in closed trajectories (see insets of Fig. 1). Points at which lines meet and exchange stability are the bifurcations of the system. Three cases are given, to illustrate essentially different, constant ill, sections of Fig. 1. (a) fil > 0; (b) fil = 0; ( c ) - 0.5 < t~1 < 0.
231
/3-PLANE INERTIAL STABILITY
(c)
]
J
SADDLE-NODE
SADDLE-NODEC
Fig. 2. continued. neutral and unstable fixed points may be seen to merge and annihilate each other. At this stage it should be noted that the result for fia = 0 is not generic. The system near ( = 0 is structurally unstable to small changes in fil and this has implications for the stability of flows with general zonal velocity variations. Similar stability diagrams for variable fil at fixed ~ may also be constructed, a particular case of which was discussed by Wan and Yang. The determinant of the linearisation matrix of eqns. (2) and (3) is det = fil +~7_ ~1 , f o r ~ - - - 1
(1o)
and det = (~2 ÷ 2fil)1/2 • I(~2 ÷ 2fil)1/2-T-(~ - 1)]
(11)
for the other two roots. The frequencies of orbits close to a centre are given by ( - det) 1/2 for that centre. The periods of orbits may range from being of the order of 2~-/( - det) 1/2 for those close to centres to infinite for those intersecting a saddle (separatrices), and fall to zero as the horizontal extent of orbits becomes very large. It is possible for T to be much larger than the inertial period, 2 ~ r / f o.
232
D.J. PARKER
The actual parcel trajectories, illustrated as inserts in Fig. 1, show that the time evolution of ~ can exhibit oscillations with single or double peaks, as in the weakly nonlinear solutions of Zhao and Ghil (1991). The periods and magnitudes of these oscillations will be determined by the initial conditions and the local values of f0 and /3, to be considered in the following sections. 3. PHYSICAL INTERPRETATION The physical interpretation of this p r o b l e m - - a parcel model with a velocity discrepancy at the 'home' position y = y0--requires consideration of the nature of perturbations both of position and zonal velocity. A parcel with given u 1 will have a particular stability to virtual displacements (those which preserve u I and ~ for the parcel). In other words, a parcel undergoing a virtual displacement may be caused to occupy an orbit which takes it far from its 'home' state ( r / = 0, v = 0). Alternatively, a small non-zero value of u 1, with no virtual displacement, may cause a parcel to travel far from its home position, as in the case ~ = 0 when u I is slightly negative. Here the home state of the parcel lies on an orbit which will carry it significantly far from that state. It will be informative to consider two different types of disturbance. One will be an isolated perturbation for which the motion of the perturbed air will be considered, and the basic flow is assumed to remain undisturbed otherwise. The other case will be that of small, general, random fluctuation of velocity and position in the basic shear flow. If these small perturbations are found to grow with time then the flow itself could be said to be unstable. When u 1 =/3 = 0, in the ~ > 0 regime a virtually displaced parcel will execute oscillations about r / = 0, the home state; this is said to be a stable state (as in WY). When /3 is non-zero, trajectories are closed contours for any values of ~. However, these contours may be sufficiently large that they can be said to represent an instability. Whether large orbits actually represent an instability of the basic flow is not well illuminated by the parcel model, as interaction between parcels may not be considered. However, the periods of orbits may be just as pertinent to the rate of onset of instability as to the period of regular oscillations. The physical relevance of the variable parameters depends on the local length and velocity scales
L =fo/#
18-PLANE INERTIAL STABILITY
233
For the present, only the n o r t h e r n hemisphere will be considered, so that L and V are both positive. On a perfect /3-plane, L is the distance of Y0 from the e q u a t o r - - i n middle and high latitudes L will be much larger than the scale of the idealised zonal shear flow assumed in this problem and the scale over which the /3-plane approximation is valid. At 30°N, V = 270 m s -1 and increases as sin ~b tan 4) (where 4' is the latitude) towards the pole; V decreases towards the equator as 0.28~b 2, and at 10°N it is roughly 30 m s -1. The results also d e p e n d significantly on the given scale for u 1, U. This will d e p e n d on the form of disturbance to be considered and will be discussed below. The linear phase of unstable growth will be governed by the magnitude of /~u I -fo~, from (WY, eqn. (7)). For the /3-effect to be significant in the early stage of flow instability, it is required that
t~ul >--f0~ or
--<--
Ul
fo-V
(12)
It is to be expected that the 3-effect will alter the f-plane stability results only near sc = 0 or in the tropics, w h e r e Ul/V may be suitably large.
3.1. ~<0 Taking u I = 0 shows (Fig. 2(b)), as in the W Y treatment, that shear flows with ~: < 0 are unstable to virtual perturbations of ('0, v) in accordance with the f-plane result. By considering Fig. 2 for ~ < 0 it may be seen that any parcel with negative u 1 and ~: will be in an orbit crossing the equator, however close u 1 is to zero. Clearly, for extratropical flows, the model will break down before the equator is reached, but in the tropics it seems possible that cross-equatorial flow may occur as a result of ~ < 0 inertial instability. 3.Z ~ - ~ 0 In all cases, ~ = 0 means that any parcel with u 1 < 0 will tend to move equatorwards, as r / = -fo/fl is the only critical point. In this sense, any negative u 1 perturbation of the sc = 0 zonal flow will cause an instability. This effect will also occur for non-zero ~ if u I is sufficiently negative, that is, if the parcel is within region A of Fig. 1. This equatorward instability occupies a band a r o u n d ~ = 0 given by ~:2< e2, where ~ is given by e 2 = 2U/3. To what extent this instability is relevant for geophysical situations at non-zero sc depends on the width of the band, e, relative to typical
234
D.J. PARKER
values of £. If it is assumed that £ = f0 (away from the equator itself), then defining a nondimensional g -- e l f o gives
(U )-'-'1/2 '2 =
(COS 4)) 1/2 at s i n ~b
1 -- - - a s 4 , ~ 0 4'
which evidently can b e c o m e large close to the equator, regardless of the (non-zero) choice of U. T h a t is to say that the range of p a r a m e t e r values, e, for which this cross-equatorial instability may occur becomes large compared with the local planetary vorticity as ~b ~ 0.
3.3. ~>0 W h e n £ > e, parcel trajectories from the h o m e state will orbit the critical point 1 r/c= ~ [ _ s c + (~:2 +
2Ulfl)l/q
(13)
In the special case £ = f 0 (8~/bY -= 0) a n d / 3 = 0, the solution given here reproduces the well-known 'inertial oscillation' (Gill, 1982, pp. 252-254) which likewise relies on there being a velocity perturbation u 1 to the basic state and in which the stationary point tic is given by r/c = ul/fo. In the full problem, every trajectory is an inertial oscillation but the scale may be altered. As £ > e, for large U1 r/c-- s¢: (14) as in the special f plane case above, but for smaller ~: this will not be so and the horizontal scale of the orbit can be extremely large. For instance, with u 1 = 10 m s -1 and s¢ = 10 -5 s -i, r/c ~- 1000 km. This could well be large c o m p a r e d with the scale of the overall shear flow and can be interpreted as instability rather than a neutral oscillation. H o w this interpretation is m a d e must d e p e n d on the assumed m a g n i t u d e and extent of the initial perturbation. For infinitesimal values of u~, ~/c is close to zero so that oscillations are small and may not be interpreted as having a great influence on the basic state. Likewise, a localised perturbation of significant u 1 may follow an orbit which takes it beyond the horizontal regime considered and be itself 'unstable' but not greatly influence the basic state. Stability of flows at ~: > 0 seems to d e p e n d on w h e t h e r perturbations are sufficiently large and widespread to modify the bulk shear substantially. T h e balance with dissipative effects is not clear, although it is also likely to be related to the nature and scale of flow fluctuations (for instance, the likelihood of gravity wave radiation from a disturbance).
0-PLANE INERTIAL STABILITY
235
By setting /3 = 0 in (WY, eqn. (7)) it is apparent that the above asymptote for large ~, (14), is the same as the exact critical point for the f-plane problem, so the observations on the choice of u I apply to the parcel theory of inertial instability as a whole. 4. EXTRATROPICS
Consideration of a dimensionalisation of Fig. 1 shows that the equatorward instability described in Section 3.2 occupies only a narrow band a r o u n d ~ = 0, given by I ~: I < e. A perturbation velocity of 10 m s- 1, taking 13 ~ 10 -11 m -1 s -1, yields • ~ 10 -5 S - 1 and g = 10 - 1 . Consequently, this form of instability seems to have little significance away from the equator. It should be recalled from (12) that in these latitudes Ul/V is likely to be m u c h smaller than ~/fo so that the linear (parcel theory) behaviour is expected to concur with that for/3 = 0. In mid-latitudes the/3-plane inertial stability results reduce essentially to the f-plane results, notwithstanding the c o m m e n t s of the previous section. 5. TROPICS
In these latitudes, negative values of Ul, for sc close to zero, may cause a parcel to move towards and even across the equator. This p h e n o m e n o n would appear to be possible for values of ~: up to O(fo), i.e. near-zero zonal wind shear, provided the scale V is sufficiently small (that is, if perturbations u 1 are of order V/2, from a dimensionalisation of Fig. 1). For f0 an order of m a g n i t u d e smaller than in mid-latitudes the ratio g may be large; possibly of order unity. T h e implication is that close to the equator a zonal shear flow must have ~ greater than a critical value which may be O(fo), for stability. For any ~ less than this critical value, crossequator flow seems possible as a form of instability (or wave) if the idealised shear flow has large e n o u g h meridional extent, as the only critical point is at ~ = - 1 . Now, considering the southern h e m i s p h e r e results for which f0 < 0 and imposing here ~: < f0 for stability, it seems that, matching at 4~ = 0, sc must be zero on the equator for a possible stable flow. This would agree with the observations referenced in Stevens (1983). (In fact, consideration of stability of potential vorticity (PV) anomalies would immediately suggest this result, as the P V is expected to change sign in the vicinity of ~b = 0.) T h e p h e n o m e n o n may be also of relevance to the trajectories of localised anomalies~ such as the outflow from a cumulus cluster. A further form of 'instability' is possible if a parcel of given u I is virtually displaced far e n o u g h to give it a cross-equatorial orbit. In this
236
D.J. PARKER
case, the virtual displacement must be relatively large, and consequently it is only of obvious interest in imagining the cross-equatorial trajectory of an isolated disturbed parcel. Studying the degenerate f0 = 0 case for a parcel originating precisely on the equator shows the limit of the above discussions as ~b ~ 0. Figure 3 shows the dimensional stability diagrams for this case. The roots corresponding to (~ - u) -- 0 are unchanged by setting f0 -- 0; now ~ = - ~ / O y . Because, in the u 1 > 0 case (Fig. 3(a)), a small perturbation of position may cause a parcel to be carried around the larger of the two possible orbits, then, bearing in mind the discussion of scales given in Section 3, this flow is unstable for all ~ (in accordance with Dunkerton (1981), whose analysis included pressure gradient terms). The 'least unstable' flow is s¢ = 0, where the two possible orbits are of equal magnitude, with centres at r / =
++_(2Ul//3) 1/2. Given the explicit, quadratic form for u(r/), it is apparent that d u / d y = f, so that a parcel's physical trajectory will tend to appear anticyclonic in either hemisphere, as in the f-plane inertial oscillation (Gill, 1982, pp. 252-254). Beyond this observation, the parcel model (WY, eqns. (3) and (4)) gives no indications of the behaviour of vorticity and potential vorticity for fluid which is carried across the equator. 6. Q U A D R A T I C B A S I C Z O N A L F L O W
A similar analysis will hold for a quadratic ~(y), defining u(Y) = u 0 + ~--y-yr / + ~,/2
(15)
where • is the curvature of the basic flow and O~/Oy is the shear at y = Y0 (as in the study of Stevens (1983)). In this case -t5 dt d~ dt = ( 1 + ~ ) ( t ~ l - ~ -
(16) 1 " ~2 2fll'O )
(17)
where/31 = fill~3 and/31 =/3 - & The bifurcations of this system (6)-(7) are found at t~1 + ~ = / 3 1 / 2 (transcritical)
(18)
and ~ + 2t~1/31 = 0 (saddle-node)
(19)
237
0-PLANE INERTIAL STABILITY
(a)
(b)
SADDLE-NODE
f SADDLE-NODE
Fig. 3. Stability diagrams of ~ with ~:, at f0 = 0. (a) fil > 0; (b) Ul < 0 (in this case, ~ = ¢ is a centre). Notation as in Fig. 2.
238
D.J. PARKER
Small variations o f ~1 from unity will not alter the structure of the system of critical points, although the interpretation may be altered because of changes in physical scales. It is interesting to look at the/31 = 0 case, as, in the conventional study of barotropic instability, it is required that 131 be zero somewhere in the flow (Gill, 1982, pp. 565-568) for growing modes. Now the critical points are given by tS=0 =
- 1, a a / (
and the bifurcations are at fil + ( = 0 (transcritical) ( = 0 (saddle-node) as in Fig. 4. Inspection of Figs. 4(a) and 4(b) indicates that the parcel will be 'unstable' unless sc > I fil l, that is, ~/fo >-- Ifil/V I. As mentioned above, ~/fo < I f i J V I is likely only in the tropics because of the variation of V with latitude. It seems that the curvature of the basic zonal flow does not alter the extratropical stability results. In the subtropics, however, there may be a possibility of stability to disturbances if ~/fo is sufficiently large. When/31 is negative, solutions are easily inferred by setting ~: ~ - ~ and fil ~ - i l l in (6) and (7), noting that, as the sign of d S / d t has changed, saddles become centres and centres become saddles. The qualitative form of the results is the same for/31 negative as for/3~ positive. 7. E F F E C T S O F D I S S I P A T I O N
If, for mathematical convenience and with no attempt at physical justification, a Rayleigh drag term is added to the right-hand side of (WY, eqn. (4)), but not (WY, eqn. (3)), these equations become d~7 dt dv
=U
=foa + (/3a_fo~)rl _ [so+ (f0/2)]/3r/z - s P1--2 n 3 - A v
(20)
where A is a suitably scaled constant. The critical points of this system are the same as those of the inertial system. Now a form of Lyapunov function, L(r/, v), may be constructed by adding a function of u 1 and ~: to H, so that L is positive semidefinite with value zero only at a critical point. Contours of H are contours of L but the new system has the property that d L / d t = - h v 2 ~ 0. The centres of the inertial problem now become stable
239
0-PLANE INERTIAL STABILITY
(a)
TRANSCRITICAL'~
I I I
I I I I
(b)
I
/
/
/
I
I
/ I ! I I ! ! ! ! i
/
/
. f
I
Fig. 4. Stability diagrams for @ with ~ at ~ = 0. (a) fil > 0; (b) fii < 0. Notation as in Figs. 2 and 3.
240
D.J. P A R K E R
fixed points and any given parcel will ultimately arrive at one of these states, as they represent minima of L. In this case, the maximum of v on the initial contour will be an upper bound for v throughout the parcel trajectory. Other, physically plausible forms of dissipation, e.g. adding Rayleigh drag to both (WY, eqn. (3)) and (WY, eqn. (4)), will tend to destroy the simple solutions for critical points obtained herein. In this sense, the solutions obtained are not structurally stable. However, outside the regimes in which dissipation is significant, and as an indication of shear flow inertial stability, they are expected to be useful. Although physical parcels will not follow the orbits illustrated here, these orbits may be helpful in the understanding of the dynamics of inertial instability. For a more complete discussion of the role of dissipation in this problem, readers are referred to Dunkerton (1981). 8. SUMMARY AND CONCLUSIONS By means of a parcel model it has been possible to obtain results for nonlinear inertial instability on a /3-plane. The simplifications inherent in the problem mean that it cannot yield the details of the whole flow behaviour: extension to an interpretation of flow stability must involve some reasoned heuristic arguments. However, results obtained by this parcel method concur surprisingly well with linear and weakly nonlinear solutions for symmetric flows and may give extra insight into the fully nonlinear and asymmetric regime. Given a choice of parcel latitude and initial conditions, the subsequent parcel orbit may be obtained by an evaluation of the Hamiltonian, H, given by (2). In terms of basic structure these contours may be more readily inferred by reference to the bifurcation diagram (Fig. 1). The zonal flow structure follows from eqn. (1). Using physical arguments based on the scales of orbits in the various parameter regimes, it has been shown that these solutions give, away from the tropics, inertial stability for flows with positive absolute vorticity, ~, and instability for those with negative ~ (in agreement with f-plane inertial stability results), provided the initial velocity perturbation is small. It has also been argued that in the tropics inertial stability requires ~ to be greater than a positive value (northern hemisphere). On the equator, this has the corollary that all shear flows are unstable, with the 'least unstable' being at ~ = 0 (agreeing with Dunkerton (1981)). Beyond the linear analyses it has been possible to exhibit, in an asymmetric model, B-plane inertial oscillations of similar temporal structure to
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t h o s e o f Z h a o a n d G h i l (1991), w h o cited inertial instability as a possible trigger for the 3 0 - 6 0 day oscillation. Finally, by its v e r y n a t u r e , the p a r c e l m e t h o d m a y have s o m e use in the tracing o f localised flow a n o m a l i e s , w h e t h e r in physical a p p l i c a t i o n s or in ' t h o u g h t e x p e r i m e n t s ' . T h e H a m i l t o n i a n s o f (2) will rapidly yield simple flow t r a j e c t o r i e s for t h e m o t i o n o f the a n o m a l o u s air. ACKNOWLEDGEMENTS D u r i n g the p r o g r e s s o f this w o r k I have b e e n f u n d e d by the N a t u r a l E n v i r o n m e n t R e s e a r c h Council. I w o u l d also like to t h a n k P r o f e s s o r A.J. T h o r p e for his h e l p f u l c o m m e n t s o n the physical aspects o f this study. REFERENCES Dunkerton, T.J., 1981. On the inertial stability of the equatorial middle atmosphere. J. Atmos. Sci., 38: 2354-2364. Gill, A.E., 1982. Atmosphere-Ocean Dynamics, Academic Press, New York, pp. 252-254 and pp. 565-568. Guckenheimer, and Holmes, 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, Heidelberg, pp. 46-47. Stevens, D.E., 1983. On symmetric stability and instability of zonal mean flows near the equator. J. Atmos. Sci., 40: 882-893. Thorpe, A.J., Hoskins, B.J. and Innocentini, V., 1989. The parcel method in a baroclinic atmosphere. J. Atmos. $ci., 46: 1274-1284. Wan, J. and Yang, F., 1990. The phenomena of bifurcation and catastrophe of large-scale horizontal motion of the atmosphere under the effect of Rossby parameter. Adv. Atmos. Sci., 7: 409-422. Zhao, J.-X. and Ghil, M., 1991. Nonlinear symmetric instability and intraseasonal oscillations in the tropical atmosphere. J. Atmos. Sci., 48: 2552-2568.