Spirals in Potential Vorticity. Part II: Stability - AMS Journals - American ...

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Spirals in Potential Vorticity. Part II: Stability JOHN METHVEN Department of Meteorology, University of Reading, Reading, United Kingdom (Manuscript received 4 June 1997, in final form 14 October 1997) ABSTRACT A model of the linear stability of spiral-shaped potential vorticity (PV) filaments is constructed by using the Kolmogorov capacity as a time-independent characterization of their structure, assuming that the dynamics is essentially barotropic. The angular velocity ‘‘induced’’ by the PV spiral has a radial profile that is approximately consistent with the advective formation of the spiral itself. The background shear in angular velocity, at a position along the filament, arising from the net effect of the remainder of the spiral, suppresses the growth rate of barotropic instability. However, it is shown here that all such spiral-shaped PV filaments are unstable in isolation and that disturbance growth rate varies only weakly with spiral shape. Contour dynamics calculations verify these predictions, as well as illustrating the strong influence of far-field strain on growth rates. The implication is that persistent vortices, associated with PV spirals and to some extent isolated from external strain, will mix the air contained within them at a rate significantly enhanced by filamentary instability. It is also concluded that the Kolmogorov capacity provides a useful geometrical characterization of atmospheric spirals.

1. Introduction In this paper, the connection between the instability of a spiral-shaped vorticity filament and its geometrical structure, as characterized by the Kolmogorov capacity D9K, is investigated. The Kolmogorov capacity, or boxcounting dimension, is a measure of the ‘‘scale invariance’’ of structures or ‘‘space fillingness’’ (see Part I for more details). In the particular case of a spiral, the Kolmogorov capacity is a measure of the ‘‘rate of accumulation’’ of the turns of a spiral onto its center. Motivation for the study arises from satellite imagery showing spiral features and particularly from a spiral in Ertel potential vorticity (PV) that evolves in a baroclinic wave life cycle. This experiment is the same as the LC2 experiment of Thorncroft et al. (1993) except that the resolution is greatly enhanced. The structure of this PV spiral was the focus of Part I (Methven and Hoskins 1998), where it was demonstrated that the Kolmogorov capacity of the spiral in PV is nearly steady. This observation forms the basis for a simple model of spiral dynamics. Once formulated, the model is used to explore the linear stability of a wide range of spiral shapes to lateral disturbances. Fully nonlinear contour dynamics calculations are also performed to verify the predictions for the growth rates. The signature of the spiral instability in the LC2 ex-

Corresponding author address: Dr. John Methven, Department of Meteorology, University of Reading, P.O. Box 243, Earley Gate, Reading RG6 6BB, United Kingdom. E-mail: [email protected]

q 1998 American Meteorological Society

periment is revealed by PV on the 300-K isentropic surface (see Fig. 1 of Part I). The second coil of the spiral begins to undulate perceptibly after day 8. These undulations rapidly grow and eventually become so large that they interfere with the neighboring filaments and break, destroying the spiral in the process. The wavelike disturbances originate on the northernmost filament of the spiral and move cyclonically with an angular velocity approximately equal to p rad day21 . The disturbances are found to be approximately stationary with respect to the background flow. It can also be seen that the distance between consecutive disturbances is almost constant and therefore the instability has a welldefined wavelength (ø600 km). Disturbance amplitude, in terms of the displacement of a PV contour, was measured directly from plots of PV every 0.25 days. The amplitude was found to increase exponentially from day 8 to day 9 with a growth rate of 2.2 day21 6 0.2 day21 . After this time the disturbance saturated nonlinearly and the rate of amplitude increase slowed. Since the wavelength was almost steady, the wave slope of the disturbance increased at the same rate as its amplitude. The mechanism for the instability was thought to be barotropic for several reasons: the undulations in the PV filament resembled the rollup of PV strips due to shear instability (e.g., Dritschel 1989b) and the PV anomalies were seen to be very deep in the vertical, as illustrated in Fig. 2 of Part I. Consequently, Ertel PV anomalies within the spiral are dominated by the contribution from isentropic vorticity anomalies rather than that from static stability anomalies. However, the 300-K surface slopes downward at

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FIG. 1. Azimuthally averaged cross sections of angular velocity throughout the evolution of the LC2 vortex (T341 control winds truncated to T85) on the 300-K surface. The coordinate system has been transformed so that the pole lies at the vortex center.

the outermost spiral turn and the associated PV structure tilts, constituting a tropopause fold. The filament behavior is not expected to be barotropic at this location. Bush and Peltier (1994) also found that tropopause troughs were not folded under within cutoff vortices and it is assumed here that this is a generic feature of all synoptic-scale PV spirals. Disturbances that bear a strong resemblance to these

filamentary instabilities have been observed in the atmosphere. Ralph (1996) observed a set of five 750-km wavelength cyclonic disturbances wrapped around a synoptic-scale cyclone (surface diameter ;2500 km) using both water vapor channel (6.7 mm) and infrared (11.2 mm) satellite imagery. The disturbances moved around the cyclone at a speed of 19 6 1 m s21 relative to the cyclone center and were almost stationary with

FIG. 2. Two spirals with vorticity distributions specified by the Kolmogorov capacity D9K and width parameter g. Both spirals have g 5 0.15 and b(r 5 1) 5 98 giving c(r 5 1) 5 0.3. (a) A spiral with a capacity of 0.4 (a 5 1.5). (b) A spiral with a capacity of 0.2 and thus a much greater accumulation rate ( a 5 4.0).

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respect to the advecting wind field. Ralph also noted that the oldest disturbance faded out, leaving a wavenumber-4 pattern around the main cyclone. The water vapor imagery clearly showed the spiral nature of the dry region, which is associated with a stratospheric intrusion and therefore high PV. The disturbances grow rapidly over the timescale of a day, and two of the disturbances have hammer-shaped cloud formations associated with them when the waves break. The fact that the dry intrusion is deep enough for the water vapor imagery to pick out the spiral and filamentary instability clearly, and that these regions are largely cloud free, suggests that the mechanism for growth of the observed mesocyclones is dry and quasi-barotropic. However, Ralph does point out that the hammer-head cloud formations are also characteristic of baroclinically growing synoptic-scale cyclones. Other investigators have observed such relatively axisymmetric cyclonic disturbances, although they may be fairly rare events. For instance, Forbes and Lottes (1985) only noted one such event (with a 500-km wavelength) during a 36-day search for high-latitude vortices over the European side of the North Atlantic. The scarcity of these events seems most likely to be due to the low probability of the appearance of large cyclonic vortices that persist for as long as in the LC2 paradigm and that can therefore wrap up so many turns in PV. The signature of such spirals may also be obscured sometimes in satellite imagery by higher-level features. Kinematic considerations are employed, in section 2, to find the profile in angular velocity that results in a spiral in a material line with a particular value of capacity. Since PV is a dynamic tracer one should also expect the structure of the spiral itself to determine the angular velocity profile. In section 3, it is shown that the circulation ‘‘induced’’ by the PV spiral does indeed give a profile in angular velocity that is approximately consistent with the kinematic formation of the spiral itself. The dynamics of filaments in the model are assumed to be barotropic in nature as discussed above. The self-consistent model also relates the spiral shape to the shear in angular velocity, at the position of a filament, arising from the combined effect of the remainder of the spiral. Dritschel (1989b) showed that such a background shear can suppress the growth rates of lateral disturbances to vorticity filaments that lie along the shear. However, when filaments lie at an angle to the shear they stretch and thin at an algebraic rate. The thinning is expected to reduce growth rates by squashing the perturbation; stretching will result in an increase in disturbance wavelength that will eventually far exceed the favored length for rapid barotropic growth. In section 4, a linear stability calculation is performed for filaments of vorticity lying at an angle to pure shear, following methods employed by Dritschel et al. (1991), who investigated the instability of vorticity filaments in a strain flow dominated by deformation. The shear and initial strip angle are determined from

spiral shape and are then used for the stability analysis, resulting in a prediction for finite time disturbance growth. Finally, in section 5, these predictions are compared with the growth of disturbances on a spiral in vorticity in a contour dynamics integration of the 2D Euler equations and with the spiral evolution in the baroclinic wave (LC2). 2. Kinematic considerations In this section the kinematic formation of spiral structures in an axisymmetric vortex is investigated, and the relationship between the Kolmogorov capacity of the spiral and the shear in angular velocity is highlighted. The capacity is used as the fundamental measure of spiral structure because it is readily measured for spirals with as little as two turns, as demonstrated in Part I. Furthermore, it is simply interpreted in the case of a spiral in which the relationship between the number of turns n and the radius r is described by the formula 2a

12

r n 5 L nL

,

(1)

where n L is the number of turns at radius r 5 L and positive a describes the ‘‘rate of accumulation’’ of the spiral onto the origin. The expression relating the capacity and the accumulation rate, which was derived in the appendix of Vassilicos and Hunt (1991), is D9K 5

1 . 11a

(2)

Motivation for the axisymmetric flow assumption is derived from observations of the vortex in the LC2 experiment. The azimuthal (y a ) and radial (y r ) velocity components of the wind (coarse grained to T85 resolution), on the 300-K isentropic surface, were calculated by defining the vortex center as the point at which the velocity is equal to the bulk velocity of the vortex. It was found that y a k y r throughout the vortex. By taking azimuthal averages of angular velocity (v 5 y a /r) a representative cross section of the flow as a function of vortex radius was found. Comparing the cross sections shown in Fig. 1, from different times in the life cycle, it is clear that the vortex is reasonably steady from day 7 to day 9.5. The formation of spirals in a material line will now be investigated assuming that the vortex responsible is axisymmetric and nondivergent. In this case, the azimuthal coordinate of advected particles is given by integrating df /dt 5 v. The shape of the v (r) cross section determines the spiral widths. Regions where the cross section is relatively flat experience solid-body rotation, while where the cross section is steep, the shear is large and the filaments thin rapidly. In order to make analytical progress assume a scale-invariant form of angular velocity,

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v (r, t) 5 A

2q

12 r L

e bt ,

(3)

where L, A, and q are constants. Angular velocity distributions of this form have also been considered in the context of hurricanes (e.g., Smith and Montgomery 1995). Integrating (3) one finds 2q

12

A r f (t) 5 f (0) 1 (e bt 2 1) b L

5 f (0) 1 f (t)

2q

12 r L

. (4)

Any measure of the scale invariance of the spiral (e.g., the Kolmogorov capacity) depends upon the spacing of consecutive points of intersection between the spiral and a line cutting through its center. Equation (4) can be used to relate the interval between points of intersection to the number of turns that the particles (at those points of intersection) have turned through by time t, n 5 [f (t) 2 f (0)]/2p. For a large number of turns (n k 1), 2p f

2(1/q)

is due to the increasing fraction of contour length involved in the spiral as it winds up. 3. Dynamical consistency arguments Thus far the formation of a spiral in a material line has been considered without reference to the underlying dynamics. When a spiral in PV has established itself, the flow around the vortex typically has little variation with height. To the extent that the dynamics are barotropic, the circulation at a given radius must be related to the horizontal PV distribution and therefore to the spiral shape. Here a model is presented where the profiles of angular velocity are inferred from spiral-shaped vorticity distributions that are described by a small set of parameters including the (readily measured) Kolmogorov capacity. The background shear in angular velocity is also obtained; this will be used in the instability calculations of section 4. a. The vorticity distribution

2(1/q)

1 2 2p ø1 2 f

r n21 2 r n 5

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((n 2 1)2(1/q) 2 n2(1/q) ) 1 2(1 n q

1 1/q)

,

(5)

which illustrates the scale invariance of the spiral (a power law dependence upon n). Comparison between the spiral formula (1) and the form for r n in (5) shows that the ‘‘rate of accumulation’’ of the spiral is given by a 5 1/q, which is in turn related to the Kolmogorov capacity of the set of points of intersection via (2). The exponent of n is unaffected by f (t), since f (t) is not a function of position, and therefore growth of the circulation will not influence the scale-invariant nature of the spiral. Of course the rate of ‘‘coil-up’’ of the spiral will obviously depend upon the angular velocity experienced and therefore the growth rate of the vortex. In summary we have shown that D9K should remain constant, provided that the shape of the profile of angular velocity is constant. Furthermore, if the angular velocity profile has the form (3), the capacity D9K 5 q/ (q 1 1). The same kinematic result relating the exponent q to the scale invariance of (r n21 2 r n ) was obtained by Gilbert (1988), although he did not consider the growth of the vortex. It is also obvious that uniform divergence will not alter the power law dependence upon n in (5) and therefore the capacity. Interestingly, in 2D kinematic simulations of turbulent flows, Vassilicos and Fung (1995) found that the Kolmogorov capacity of a material line (D K ) eventually assumed a nonspace-filling asymptotic value, even if the flow was unsteady, while the length of the line increased indefinitely. Indeed, the capacity of a complete PV contour in the LC2 experiment, including the lengths outside the spiral, increases with time from D K 5 1 to D K ø D9K 1 1 (the value expected for the capacity of the spiral in isolation). This

The vorticity distribution is assumed to be piecewise constant with high and low values z h and z l , in approximate accord with the extremely high PV gradients at the edges of the filaments in the LC2 experiment. The general spiral form (1) is assumed, where the rate of accumulation of the spiral onto the origin (a) is related directly to the Kolmogorov capacity by (2). This is the first parameter of the model. In addition, the angle between the spiral arm and a circle of radius r is given by tanb 5

a . 2p n

(6)

For given a, angle b decreases as the number of turns, n, increases and one approaches the spiral center. Therefore b describes the location along a particular spiral. Smaller angles would be expected to be more closely associated with axisymmetric velocity, which will be assumed to wind up the spiral itself. The width of the high vorticity filament will be described by D 5 gr, where g is the second parameter describing the whole vorticity distribution (it is not a function of position). This choice will be justified in section 3b. One immediate consequence is the necessity to define an inner radius, c, at which the filament width is equal to half the filament spacing [D 5 (r nc21 2 c)/ 2]. Outside this radius the high vorticity filament is always narrower than the neighboring low vorticity regions, and within this radius the vorticity is made uniform with value z c . Assuming n k 1, one finds (r n21 2 r n )/L ø a/n L (n/n L )2(11a) directly from (1). By setting rn 5 c in this expression and then taking D 5 gc, the number of turns at the inner radius is found to be nc 5

a . 2g

(7)

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The function c, which combines some of the above information in a way that will prove useful later, is defined to be 2(1/a)

12

n g r c5 5 5 nc p tanb c

.

(8)

Here c is clearly a function of position around a spiral with given width parameter g. Two spirals specified by the parameters a and g are shown in Fig. 2. Distances are in units such that b(r 5 1) 5 98, which fixes the inner radius c, using (8). The width parameter g is the same for both spirals but the Kolmogorov capacity is different. For spiral (b) the rate of accumulation is greater and the spiral is less space filling with the effect that the inner radius is much smaller. Note that the functions r/c, b(r/c), and c (r/c) can be used interchangeably to locate a position on the spiral. b. The associated flow The circulation relates the vorticity and the angular velocity through Stokes’s theorem,

EE

R

z u dS 5

(u 1 V 3 r) · dl,

(9)

]S

so that at radius r . c,

[

ez h 1 1 2 e 2

12 c r

]

2

zl 1

12

2

c z 5 2v 1 f , r c

(10)

where e is the fraction of the area pr 2 occupied by the high vorticity filament, f is the Coriolis parameter at the latitude of the vortex center, and v is the azimuthally averaged angular velocity at radius r. The vorticity in the inner patch (r , c) is now set to be z c 5 ½(z l 1 z h ) so that the shear in angular velocity, S(r) 5 2rd v / dr, is a continuous, positive function of r. The expression is further simplified if it is assumed that z l 5 f, giving

[

12

v 1 1 c 5 e1 j 2 2 r

2

]

,

(11)

where the vorticity jump is equal to the relative vorticity j 5 z h 2 f. The area fraction e is given by integrating in the along-filament direction s:

eø

1 pr 2

E

ø

22

E

n

sumed that (a/2pn) 2 K 1 (i.e., tan 2 b K 1). Finally, substitution for e in (11) and some rearrangement gives

v a 1 5 c2 c 2a . j 2(2a 2 1) 4(2a 2 1)

nc

[

1 2

gr(n9) r(n9) a 11 c c 2p n9

a (c 2 c 2a ), 2a 2 1

]

2 1/2

2p dn9 (12)

where (8) was used to substitute for r/c and it was as-

(13)

The azimuthally averaged angular velocity has now been determined by the vorticity distribution, which is in turn uniquely defined by four parameters a, g, c, and vorticity jump j. Here c is used to give a scale to the spiral; the functions r/c, b(r/c), and c (r/c) can all be used interchangeably to represent position on the spiral. Moreover, when rescaling time with j, the angular velocity at radius r depends only upon the Kolmogorov capacity, through parameter a, and the function c (r), which is formed from a combination of the width parameter g (t) and the angle b(r, t) such that the time dependence is removed and the circulation is steady. Importantly, the first term in (13) is proportional to c, implying a radial dependence in angular velocity of the form r21/a . This only arises when the filament width is proportional to radius (D 5 gr), so that the high vorticity area fraction e decreases with radius. As discussed in section 2, the r21/a radial dependence is consistent with the advective formation of a spiral with the accumulation rate a [q 5 1/a in (3)]. The second term is not consistent with this picture but ensures that the angular velocity tends toward solid-body rotation (with relative vorticity j/2) as the inner vorticity patch is approached (r → c). Fortunately the second term is dominated by the first, provided that a . 0.5 and c K 1. Values of parameter a for spirals in flows are expected to exceed 0.5 since a 5 0.5 is equal to the rate of accumulation of a spiral formed in a point vortex by passive advection. Higher values of a (or D9K , 2/3) indicate spirals that converge more rapidly onto the center and are consistent with less steep angular velocity profiles. Figure 3 shows the angular velocity in D9K–c parameter space; it is apparent that v is greater for larger capacity and c. This is because a larger fraction of the area within a given radius (r/c , c2a ) is occupied by ‘‘high’’ vorticity as the spiral becomes more space filling and the width parameter g increases. Two further fields are now derived from the model and are also shown in Fig. 3. The nondimensional shear in angular velocity is crucial to the stability of the spiral-shaped filament and is given by L52

D ds

[ ]

1 r(n) ø p c

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r dv 1 5 (c 2 c 2a ). j dr 2(2a 2 1)

(14)

A measure of inconsistency in the model is given by the ratio of the second to the first term in the above expression for shear (DL 5 c 2a21 ) since only the first term is responsible for maintaining a spiral with constant capacity. This measure, shown as a percentage, illustrates that model consistency becomes poor as both D9K and c approach one-half.

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FIG. 3. The results of the spiral model as functions of the shape parameters D9K and c. The top four panels show angular velocity, shear, the inconsistency measure based on shear, and the time taken for the number of spiral turns to increase by one (t1 ). The bottom two panels show effective growth rate and initial wavenumber for the fastest growing disturbance in time 2t1 ; refer to section 4 for details of the instability calculation. Note that, for these two diagrams, angle b(t 0 ) 5 68. The initial wavenumber k m D, for each position in parameter space, is found by repeating the calculation for wavenumbers between 0 and 2, at intervals of 0.01, and selecting the wavenumber that results in maximum amplification by time t f . The discretization of wavenumber results in the jagged contours.

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c. Model time dependence The aim of the model is to study the evolution of a spiral whose capacity is constant in time and that is associated approximately with a steady, axisymmetric circulation. Since the motion is axisymmetric, dn(r)/dt 5 v (r)/2p. Invoking steadiness and (13) one sees that parameter c(r) must also be a constant in time, which implies [using (8)] that c(r) 5 v (r)/v (c). Of course the strip-width parameter g and angle b both decrease with time as the spiral wraps up. For instance the evolution of b is given by tanb(r, t) 5

a . 2p n(r, t0 ) 1 v (r)(t 2 t0 )

(15)

A timescale that will prove useful for the instability calculations is found by considering the increase in the number of spiral turns in the region c , r , L, in time t: Dn 5 [n c (t) 2 n L (t)] 2 [n c (0) 2 n L (0)] 5

v (L) [c (L)21 2 1]t. 2p

(16)

The time taken to wrap up one spiral turn in the region c , r , L is therefore t1 5

2p [c (L)21 2 1]21 . v (L)

(17)

Here t1 j is shown in D9K–c parameter space in Fig. 3, where it is seen that spirals with a large capacity wrap up one turn in a shorter time. Low values of c correspond to positions farther from the spiral center [refer to (8)] and thus the time taken to wrap up one turn is obviously shorter. In summary, by requiring consistency between the profile of angular velocity associated with the circulation of the vorticity spiral and the profile required to maintain the constant capacity of the spiral by advection, a simple model of spiral dynamics has been formulated. The coarse graining of the vorticity spiral, through inversion, is assumed to give the azimuthally averaged angular velocity, which, at any position on the high vorticity filament, is approximately equal to the net effect of the remainder of the spiral. This approximation will hold for tan 2b K 1, and consistency is ensured if a . 0.5 and c K 1. Since tanb is less than one the condition c K 1 also implies that the width parameter g K 1 [using (8)]. The shear obtained here will be used for the linear stability analysis below.

(1989b) derived the dispersion relation for the instability of a vorticity strip lying along a linear shear [u 5 (Ljy, 0)] for 2D barotropic flow:

j s (kD, L) 5 6 {[1 2 kD(1 2 L)] 2 2 e22kD }1/2 , 2

where j is the vorticity jump at the strip edges. Maximizing the growth rate with respect to wavenumber, he obtained the maximum growth rate for given shear, sˆ m 5 Im(s)/j, with corresponding wavenumber k m . In the absence of background shear, the most unstable normal mode occurs for k mD 5 0.797 and has a growth rate sˆ m 5 0.201. Positive shear (L) is ‘‘adverse’’ in the sense that it opposes the shear induced by the vorticity filament and reduces the growth rate of instabilities. When strip widths are much smaller than the radius of curvature (D K r, i.e., g K 1), Dritschel (1989b) showed that the dispersion relation for disturbances on a strip encircling a vortex with a shear in angular velocity is essentially the same as for a straight strip in linear shear. Indeed for g 5 0.2 the fractional increase1 in growth rates when accounting for filament curvature is smaller than 0.25%. Dritschel also showed that the change in growth rates due to the interaction of parallel filaments is small if D/S K 1, where S is the filament separation. For spirals of the form described in the last section D/(r n21 2 r n ) 5 c/2 and thus filament interaction is most important when c 5 0.5. A rough estimate of the greatest reduction in growth rate, when accounting for filament interactions, is found by taking D/S 5 0.25 but assuming two straight filaments; the reduction is much less than 1% (referring to Fig. 14 of Dritschel 1989b). Therefore for the purposes of this model it is sufficient to study the instability of a straight, isolated strip in linear shear [provided that one substitutes the shear in angular velocity as defined by (14)]. However, an important consideration is the angle that the strip makes with the shear. Indeed, in the spiral model the constraint c , 0.5 implies that tanb . 2g/ p. When a straight strip is at an angle, b, to linear shear [u 5 (Ljy, 0)] it will rotate toward the along-shear axis. By defining t 5 0 to be the instant when the strip is perpendicular to the shear, the time dependence of the angle is given by tanb 5

1 1 5 . 21 L jt [tanb(t0 )] 1 Lj (t 2 t0 )

(19)

The time dependence here is the same as in the spiral model [see (15)], where tanb(r, t 0 ) 5 a/[2pn(r, t 0 )] and L(r)j [ v (r)/a (which is indeed the dominant term in (14)). The width of the straight strip goes as

4. The instability calculation Strips of vorticity in a resting background fluid are unstable to lateral perturbations with favorable alongstrip wavelengths (kD ; 1); this is the classic Rayleigh shear instability. However, the growth of such instabilities can be suppressed by a background strain. Dritschel

(18)

D(t) 5 D(t0 )

1

sinb(t) sinb(t0 )

(20)

The largest difference occurs in the absence of background shear.

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and therefore for long times as t21 . The growth of lateral disturbances on the strip at an angle to shear is now examined following the methods of Dritschel et al. (1991, hereafter DHJS), who considered the instability of a vorticity strip in a general strain field [u 5 (lx 2 Vy, Vx 2 ly)]. Importantly, the basic strip on which disturbances grow is now time dependent, unlike the singular case where it lies along the shear. Equations for disturbance amplitudes on the edges of the strip are obtained after a lengthy analysis following DHJS. The calculation involves rotating to a coordinate frame for which the x9 axis lies long the strip and adding disturbances on each strip edge of the form

5

6

1 ˆ ˆ y96 (x9, t) 5 D(t) 6 1 eRe[hˆ 6 (t)e ikD(t)x9 ] , 2

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difference lies with the time dependence of strip width. DHJS consider cases where deformation exceeds rotation (l . |V|). In this situation the strip thins at an ˆ (t) ; exponential rate, for long times, such that D exp[2(l 2 2 V 2 )1/2 t]. However, in the pure shear case the time dependence of the strip is given by (20), which ˆ (t) ; t21 . The dependence for long times becomes D 2 ˆ upon kD reflects both the increase in wavelength and the decrease in strip width with time. The coupled, governing equations (23) are integrated numerically using a fifth-order Runge–Kutta scheme with adaptive step size (Press et al. 1992) starting from two independent initial conditions at time t 0 [which is defined by choosing b(t 0 ) and L]:

(21)

A1 (t0 ) 5 1,

B1 (t0 ) 5 0,

where hˆ 6 (t) is the complex disturbance amplitude and e is a small parameter. The disturbance wavenumber decreases with time at the same rate as strip width; therefore the initial strip width is used to nondimenˆ (t) 5 D(t)/D(t 0 ) is used to sionalize distances, and D scale the initial wavenumber k. The velocity field associated with the strip in vorticity is determined completely by the positions of the vorticity jumps. Since j 5 ¹ 2 c, where c is streamfunction, this velocity can be calculated by integrating the Green function for the Laplacian operator along both boundaries of the strip (C ):

A2 (t0 ) 5 0,

B2 (t0 ) 5 1.

[u9(x9, y9), y 9(x9, y9)] s s 5

E

j ln[(x9 2 s) 2 1 (y9 2 n) 2 ](ds, dn), 4p C

(22)

where (ds, dn) is the vector tangent to the strip edges so that the strip interior lies to the left. Finally, the kinematic boundary condition, describing the advection of the strip edges by the full velocity, is employed and expressions for the linearized governing equations are obtained [retaining only O(e) terms]. Recombining the complex disturbance amplitudes in terms of the quantities A 5 hˆ 1 1 hˆ 2 and B 5 i(hˆ 1 2 hˆ 2 ) one obtains governing equations with real coefficients: dA j ˆ 2 (1 2 L cos2b) 1 e2kDˆ 2 ]B, 5 2 [1 2 kD dt 2 dB j ˆ 2 (1 2 L cos2b) 2 e2kDˆ 2 ]A. 5 [1 2 kD dt 2

(23)

These are almost identical to Eqs. (11a) and (11b) of DHJS but differ in several respects. The shear measured in a frame rotating with the strip is given here by L cos2b while in DHJS it is given by 22l sin2f, where f is the angle between the strip and the dilatation axis of deformation l. However, these are equivalent since the pure shear considered here can be decomposed into a rotation (V) and deformation (l ) of equal magnitudes (l 5 V 5 2L/2) in a coordinate system rotated at 458 to the shear (i.e., b 5 f 1 p/4). The fundamental

(24)

Any other solution [A(t), B(t)] can be expressed as a linear combination of these two solutions by selecting the appropriate amplitude ratio and phase difference. One can then find an expression for the maximum amplification of the rms wave slope of the disturbance, A(t), over interval t, selecting from all initial conditions. Note that the definition is such that A(t 0 ) 5 1 (see DHJS for details). In DHJS A(t) was calculated, for one wavenumber, until the first maximum was reached. This computation was repeated for many wavenumbers and then the largest amplification selected and denoted by A m (t m , k m ), where t m and k m were the time and wavenumber corresponding to this maximum. This procedure was then repeated for a variety of initial strip angles, deformation rates, and rotation rates. The crucial assumption underpinning the whole procedure is that the twin effects of layer thinning and stretching of disturbance wavelength ultimately result in decay and therefore a maximum in amplification must occur if A(t) . 1. This was found to be true for their study where the thinning and stretching occurred at an exponential rate. Here, however, the thinning and wavenumber decrease only proceed as t21 . Disturbances with large initial wavenumbers achieve very large maxima but only after very long times. This is because, as the strip orients itself along the shear axis, the change in the basic state becomes extremely slow: dD/dt ; 2D(t 0 )/ (sinb(t 0 )Ljt 2 ). Although the high-wavenumber disturbances may initially have been decaying they eventually stretch to a favorable wavelength for barotropic growth (kD ø 0.8) and thence grow at an almost exponential rate. Since the basic state changes so slowly at these late times the exponential growth continues over a longer period and very large maxima are attained. However, in physical situations one is most interested in the disturbances that grow most over the period for which the basic state exists. Spirals in potential vorticity are generally destroyed before an exceptional number of turns are attained. A timescale of interest is therefore given

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by the time taken to wrap up one extra turn between the spiral arm at radius r and the inner core where vorticity is homogeneous, that is, t1 from (17). Note that for the low values of shear (L ; 0.1) experienced within spirals the suppression of the instability is weak and time t f 5 t 0 1 t1 is usually reached long before the maximum amplification of linear analysis. Therefore for this paper A(t f ) was calculated and maximized over wavenumber, giving A m[t f , k m , L, b(t 0 )]. The maximum amplification for disturbances on a particular section of a spiral-shaped filament depends upon the local shear L and the initial angle between the filament and shear, b(t 0 ). The shear in turn is a function of D9K and c. First the variation in amplification for fixed initial angle is examined. The effective growth rate [s m 5 ln(A m )/(t f 2 t 0 )] and initial wavenumber of the fastest growing disturbance over interval (t f 2 t 0 ) are presented in Fig. 3, for the case b(t 0 ) 5 68 and (t f 2 t 0 ) 5 2t1 . The variations in c, for given initial angle, reflect variations in the width parameter [i.e., g (t 0 ) 5 0.33c], and thus in the shear, at a fixed radius from the spiral center. The Kolmogorov capacity also results in changes in shear. For high D9K and c the shear is stronger and amplification is suppressed. The disturbances achieving highest amplification also have higher wavenumber because the along filament stretching is faster for stronger shear. Remarkably, the growth rate varies by less than a factor of 2 over the whole range of spiral shapes. This can be attributed to the fact that the shear is low over the whole D9K–c domain and therefore that the suppression of growth rate is small. The effects of strip orientation are now explored. The upper panel of Fig. 4 plots growth rate as a function of capacity for c 5 0.4. The dotted line shows the variation in normal mode growth rate predicted by the maximization of the dispersion relation (18) over wavenumber. These normal modes are valid for a strip lying along the shear, in other words, for a steady basic state. The decrease in growth rate as D9K increases reflects only the increase in shear. The other three curves are obtained by the linear stability analysis for the time-dependent basic state detailed above. For the solid line b(t 0 ) 5 6.08, while for the dashed line b(t 0 ) 5 4.58. The shear is equal in both cases, since L 5 L(D9K, c); thus the difference in growth is purely due to strip orientation. A larger angle implies a stronger along strip deformation [½ sin(2b)Lj], reducing growth rates by thinning the strip and stretching the wavelength. More detail on the variation of growth rate with strip orientation is given in the bottom panel of Fig. 4. Growth rate is shown as a function of c, for D9K 5 0.4. The solid curve arises from fixing the angle b(t 0 ) to 68 and varying the width parameter g, while the dashed curve arises from fixing g 5 0.1 and varying angle b(t 0 ) (equivalent to varying the position around the spiral). For c , 0.30, the fixed g case has a lower growth rate than for the fixed b case because the strip angle is greater, even though the shear is the same. Conversely, for

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FIG. 4. The top panel shows disturbance growth rate as a function of capacity, for c 5 0.4, and the bottom panel shows it as a function of c, for D9K 5 0.4. In both panels the dotted curve marks the normal mode growth rate for a filament lying along the shear. The other three curves show the effective growth rates for a filament at an angle to the shear: the solid curve for b 5 68 and t f 2 t 0 5 t1 , the dashed curve for g 5 0.1 and t f 2 t 0 5 t1 , and the dash–dot curve for b 5 68 and t f 2 t 0 5 2t1 . The growth rates measured from the contour dynamics simulation are shown as squares, for initial amplitude p 5 0.05, and diamonds, for p 5 0.10, along with their standard deviations.

c . 0.30 the fixed g case has a higher growth rate than the fixed b case. Surprisingly, the effect of changing strip orientation is small, although changing orientation has greater impact when the capacity is larger. Since the growth of the waves is not exactly exponential, the effective growth rate will depend upon the value chosen for t f . The dash–dot curves in Fig. 4 are calculated by fixing b(t 0 ) to 68 but calculating the effective growth rate over interval 2t1 . Note that this case is the one illustrated in Fig. 3. The effective growth rate is always slightly lower than for (t f 2 t 0 ) 5 t1 because the growth is slower than exponential. Note that the short-term growth rate can exceed that of the normal mode. This is a result of nonmodal growth, as discussed by Parker (1998). In the case of an unstrained vorticity

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strip the solutions for arbitrary initial phase difference converge onto the normal mode growth rate after roughly two e-folding times. Initially the fastest growth is obtained for the longest waves but these grow more slowly as they converge onto the exponentially growing normal mode. For a strip in strain there is competition between small wavenumbers, which grow fastest initially, and disturbances with larger wavenumbers, which will be stretched to the favorable wavelength (kD ø 0.8) for exponential barotropic growth. When attention is focused on later times (larger t f 2 t 0 ), one expects maximum amplification to occur for higher wavenumber disturbances with lower effective growth rates. A number of assumptions have been made en route to obtaining the growth rates as functions of the spiral shape. In the next section these predictions are compared with the results from nonlinear calculations. 5. Model verification with nonlinear calculations In this section the fully nonlinear evolution of a spiralshaped vorticity jump is examined using the contour surgery technique (Dritschel 1989a), in which the velocity at nodes on a jump is obtained by numerically integrating (22) along all vorticity jumps (see Zabusky et al. 1979). This velocity is then used to advect the jumps to their new positions using a fourth-order Runge–Kutta scheme for the time stepping. In order to maintain accuracy the nodes are redistributed with densities determined by contour curvature and the influence of other jumps. The spiral is set up exactly as in the spiral model of section 3. Figure 5a shows the initial spiral for the case: D9K 5 0.4, g 5 0.1, and b(r 5 1) 5 68. The evolution of this smooth spiral is used as a control run. The spiral winds up due to the shear in angular velocity and also slowly distorts and moves under the influence of the outermost end of the spiral arm. This distortion affects the instability in the perturbed cases and is discussed later. A small perturbation is then added whose initial amplitude scales with strip width,

h6 5 pgr sin

1 g (f 1 w )2 , km D

6

(25)

where p is the nondimensional amplitude. The initial phase difference w m 5 w2 2 w1 and nondimensional wavenumber k m D are obtained from the instability calculations of the last section. The fastest growing disturbance near r 5 1 (or c 5 0.30 for this spiral), over interval t1 , has a phase difference w m 5 668 and wavenumber k m D 5 0.93. The perturbation amplitude is arbitrary in the linear analysis but two nonlinear simulations are performed with p 5 0.05 and p 5 0.10. These simulations are shown in Figs. 5b and 5c, respectively, at time tj 5 12 when the maximum wave slope in the large amplitude case has reached unity. It is found that the perturbations are advected around the vortex at a

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rate that is commensurate with the azimuthally averaged angular velocity, as assumed in the spiral model. The amplification of a perturbation is measured using the rms wave slope, integrated over one wavelength, at a set of locations around the spiral. Since D9k and g are not functions of position, these initial locations correspond to different values of c. At each location the nearest three intersections between the control and perturbed contours were located in order to determine the local wavelength. The wave-slope measure was then defined as

E   A5  E ds  

1/2

tan 2d ds









,

(26)

where d is the angle between the tangents to the control and perturbed contours, s denotes distance along the control contour, and the integral is taken over one wavelength (delimited by each diamond and triangle in Fig. 5c). Throughout the evolution the disturbances are tracked around the vortex so that amplification is measured at a point on the spiral where the shear in angular velocity is approximately constant. It was found that A did increase nearly exponentially after a period of slower growth or, in some cases, decay. At all positions around the spiral the growth was approximately exponential once A (t)/A (0) . 2. The wave-slope measure is also expected to fail once the instability waves break; consequently the growth was only measured until the maximum wave slope reached unity. This limit is also associated with the breakdown of the linear stability analysis. The growth rate in this period was determined from a straight-line fit to ln(A (t)). The results from the fully nonlinear calculations are plotted as symbols in Fig. 4, together with the standard deviations in the least squares fits. The general trend is in accord with the linear stability analysis, although the growth rates are lower. It is immediately apparent from Fig. 5d that the spiral has been distorted. This occurs because the spiral arm must be truncated at some radius so that the contour is finite and the contour integration technique is not prohibitively expensive. This outer arm results in asymmetry in the initial conditions and velocities that are far from axisymmetric at this outer edge.2 The rollup of this end of the spiral-shaped filament imposes a strain on the main body of the spiral, which gradually distorts it, although this occurs on a much slower timescale than the instability. It is hypothesized that this external strain results in an extra suppression of the instability growth rates. For instance, Fig. 4 shows that the disturbance at c 5 0.20 [centered

2 The whole structure also moves about a point that is not the spiral center but this motion has been removed for the plots in Fig. 5.

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FIG. 5. The spiral evolution from the contour dynamics simulations. (a) The initial unperturbed spiral for parameters D9K 5 0.4, g(t 0 ) 5 0.1, and b(t 0 , r 5 1) 5 68. The background vorticity is zero, the high (narrow) vorticity filament has j 5 1, and the central patch (r , c) has j 5 0.5. (b) The perturbed spiral for low initial amplitude p 5 0.05. (c) The p 5 0.10 case at the same time as in (b), but with the unperturbed control overlaid. The intervals over which amplification factor is measured are delimited by the diamonds and triangles. At this stage the maximum wave slope has reached unity. (d) The whole spiral for the p 5 0.10 case at a late stage. Note the changes in axes between plots.

at (21.0, 21.3) in Fig. 5c] grows far more slowly than predicted by the spiral model. At this location the alongfilament stretching due to the far-field strain is large. The disturbances closer to the spiral center are less influenced by this strain and consequently the growth rates are higher. Kevlahan and Farge (1997) have investigated the shielding effect of a vorticity filament encircling a Gaussian vortex in a weak strain field. They found that the presence of the filament reduced the distortion of the inner vortex by the strain when the ring lies within the vortex and its circulation is sufficiently strong compared to that of the vortex. Since the background cir-

culation at a filament is the net effect of the remainder of the spiral, these conditions are always satisfied for the spiral model. Therefore it is reasonable to suppose that the outermost spiral turn partially shelters the inner turns from the influence of far-field strain. Note that the growth rates for the smaller initial perturbation are expected to be lower, at the inner turns, because the spiral distortion is more extreme by the time the wave slope has reached unity. The predictions of the spiral model can also be compared to the growth of spiral instability in the LC2 baroclinic life cycle. The LC2 spiral is found to have D9K ø

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0.4 and g ø 0.15. The function c(r) could be obtained at the location where disturbance growth was measured by measuring the angle between the filament and background shear [b(r)] but this angle is small and therefore measurements are too inaccurate. However, at day 8 the number of turns between the edge and the center (Dn) is 3. Using (7) and (8) we find parameter c 5 1 2 2gDn/a ø 0.4. In order to check the agreement between the spiral model, with parameters D9K 5 0.4 and c 5 0.4, and the LC2 case we find the time taken to wrap up one extra spiral turn. The spiral model predicts t1 j ø 30 (see Fig. 3). The relative vorticity of the spiralshaped filament in LC2 is roughly 25 days21 giving t1 ø 1.2 days, which is close to that observed. The predicted growth rate from the spiral model is s ø 4.2 days21 (see Fig. 4), which is almost twice as high as that observed (2.2 days21 ). The slowness of the growth in the LC2 experiment is attributed to the far-field strain imposed by the upstream and downstream vortices. Indeed, the instability is almost entirely suppressed on the outermost filament. In this section we have seen that the nonlinear evolution of an unperturbed vorticity spiral, whose initial state is given by the spiral model, progresses as if advected in an axisymmetric ‘‘background’’ vortex, as assumed in the spiral model. Moreover, when perturbed, the growth rate of the wave-slope measure is in reasonable accord with the spiral model, although far-field strain can significantly slow growth. The growth rate of instability on the spiral in the baroclinic wave LC2 is overpredicted by the spiral model, using appropriate parameter values, but by less than a factor of 2. The discrepancy is thought to lie with the influence of far-field strain in the LC2 case, although the growth may also be slower because the dynamics of the tropopause troughs are not entirely barotropic. 6. Conclusions Spiral structures are generic features of geophysical flows and are associated with persistent eddies. Many cases have been observed in satellite imagery of the atmosphere. Some are involved with synoptic-scale weather systems (e.g., Ralph 1996; Appenzeller et al. 1996) while Bowman and Mangus (1993) have observed spirals in the stratosphere in total column ozone. Such spirals have also been created in baroclinic wave life cycle simulations of weather systems (Polavarapu and Peltier 1990; Bush and Peltier 1994) and of Gulf Stream eddies (Bush et al. 1996). Here the baroclinic life cycle experiment LC2 of Thorncroft et al. (1993) was repeated at much higher resolution, revealing a synoptic-scale spiral in potential vorticity on isentropic surfaces that wound up many turns. The spiral-shaped filament was found to be unstable with a growth rate of 2.2 6 0.2 day21 . This paper examines the stability of such spiralshaped PV filaments using the Kolmogorov capacity to specify their geometry.

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The Kolmogorov capacity is directly related to the ‘‘rate of accumulation’’ of the turns of a spiral onto its center. In Part I it was demonstrated that the Kolmogorov capacity is a robust measure of spiral structure in the LC2 experiment and that its value is roughly constant while the spiral winds up. Here the steadiness of the capacity was used as the basis for a model of PV spiral instability. The dynamics of such spirals were assumed to be barotropic; a reasonable assumption from evidence in life cycle experiments and observations in the atmosphere. Thus the evolution of vorticity in 2D flow was considered. The shape of the vorticity spirals was described by two main parameters: the Kolmogorov capacity D9K and the filament width parameter g. The vorticity was assumed to be piecewise constant and consist of a spiral arm with relative vorticity j and a central patch with relative vorticity j/2. Relative vorticity j was used to scale time and the radius of the inner vorticity patch, c, was used as a distance scaling. The radial profile of azimuthally averaged angular velocity, v (r), was then obtained from this vorticity distribution. Importantly, the profile shape was approximately consistent with the kinematic formation of the spiral itself, assuming that the angular velocity at any angle around the vortex is approximately equal to v (r). Indeed, the angular velocity is expected to be a smoothed version of the vorticity distribution due to the scale effect of inversion (Hoskins et al. 1985). A linear stability analysis was then performed for a vorticity strip at an angle to shear. The shear and angle were obtained from the spiral model. Since the shear can be decomposed into a rotation and deformation of equal magnitudes this analysis closely follows the one presented by Dritschel et al. (1991) for a vorticity strip in a general strain. The important difference here is that the thinning of the filament has an algebraic time dependence (t21 ) rather than exponential, which occurs when deformation dominates rotation. Disturbances with short wavelengths initially decay kinematically until they stretch to the favored scale for barotropic instability (kD ; 1) when exponential growth occurs. Eventually the wavelength becomes too great and kinematic decay dominates barotropic growth. Since the basic state changes very slowly as the strip orients itself along the shear, high wavenumber disturbances, which reach the growth stage later, grow exponentially for longer periods and therefore achieve greater amplification. However, this may occur after a long period of decay. It is argued that we are interested in disturbances that amplify the most in a finite time and that the timescale of interest here is defined to be the time taken to wind up an extra spiral turn. The finite time amplification of wave slope is used to define an effective growth rate that is found to be quite insensitive to the time interval chosen (e.g., doubling the interval reduces the effective growth rates by less than 10% for all spiral shapes). The main conclusion arising from the model is that all spiral-shaped PV filaments are barotropically unsta-

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ble in the absence of external influences. The self-induced shear in angular velocity is sufficiently small that, although linear growth rates are suppressed, the growth rate is only weakly dependent upon spiral shape. In fact, the small sensitivity to the angle between the filament and shear means that the normal mode growth rates obtained for a filament along the shear are a reasonably good approximation. The nonlinear evolution is expected to result eventually in unsheared rollup of the filament as categorized by Dritschel (1989b), since the shear L , 0.2. The model is verified by a nonlinear contour surgery simulation of the evolution of a vorticity spiral. Growth rates are in accord with the linear stability analysis, although far-field strain arising from asymmetry in the initial spiral is found to reduce the growth rate further. However, the inner turns of the spiral are to some extent shielded from the effects of far-field strain by the outermost filament of the spiral. These conclusions are corroborated by the growth of disturbances around the LC2 spiral. Cyclonic disturbances that tend to form around large-scale cyclones (e.g., Ralph 1996) also indicate the instability of underlying spirals in PV. However, in the moist atmosphere it appears that the quasi-barotropic vortex rollup of spiralshaped PV filaments is sometimes followed by baroclinic development of the resulting cyclonic disturbances into mesocyclones; investigation of this process is left for future study. Geometrically based measures, including the Kolmogorov capacity, yield useful information on tracer structure. As demonstrated by the spiral instability model, the scaling property of a PV structure can be used to deduce the dynamical behavior of small-scale features in the associated flow. Although tracer contours only stretch at a linear rate in the differential rotation of persistent eddies, instability that reaches finite amplitude will result in rapid exponential stretching, thus dramatically increasing the area of the interfacial zone between air masses with different origins. Since all vorticity spirals are found to be unstable when there are no external influences, it is likely that mixing within persistent eddies may be much more rapid than the coarsegrained circulation would suggest. Therefore, although the small-scale dynamics may not be of interest to the energetics of the baroclinic waves themselves, such knowledge is crucial for the determination of mixing rates between air masses and the trace chemicals in them and for the decay rates of PV anomalies. Acknowledgments. I would like to thank Prof. Brian Hoskins for his support and encouragement throughout the course of my Ph.D. and for many helpful comments during the preparation of this paper. The high-resolution life cycle experiments were performed by Dr. Jeff Cole, Dr. Lois Steenman-Clark, and Dr. Mike Blackburn on the CRAY T3D at the Edinburgh Parallel Computing Centre, for which I am very grateful. Use of the contour

surgery code was kindly permitted by Dr. David Dritschel. Finally, I also acknowledge the constructive comments from the anonymous referees. This research was funded by Natural Environment Research Council Studentship GT4/92/284/P and more recently by a grant from the U.K. Meteorological Office. REFERENCES Appenzeller, C., H. Davies, and W. Norton, 1996: Fragmentation of stratospheric intrusions. J. Geophys. Res., 101(D1), 1435–1456. Bowman, K., and N. Mangus, 1993: Observations of deformation and mixing of the total ozone field in the antarctic polar vortex. J. Atmos. Sci., 50, 2915–2921. Bush, A., and W. Peltier, 1994: Tropopause folds and synoptic-scale baroclinic wave life cycles. J. Atmos. Sci., 51, 1581–1604. , J. McWilliams, and W. Peltier, 1996: The formation of oceanic eddies in symmetric and asymmetric jets. Part II: Late time evolution and coherent vortex formation. J. Phys. Oceanogr., 26, 1825–1848. Dritschel, D., 1989a: Contour dynamics and contour surgery: Numerical algorithms for extended, high-resolution modelling of vortex dynamics in 2-D, inviscid, incompressible flows. Comput. Phys. Rep., 10, 78–146. , 1989b: On the stabilization of a 2-D vortex strip by adverse shear. J. Fluid Mech., 206, 193–221. , P. Haynes, M. Juckes, and T. Shepherd, 1991: The stability of a 2-D vorticity filament under uniform strain. J. Fluid Mech., 230, 647–665. Forbes, G., and W. Lottes, 1985: Classification of mesoscale vortices in polar airstreams and the influence of the large-scale environment on their evolutions. Tellus, 37A, 132–155. Gilbert, A., 1988: Spiral singularities and spectra in 2-D turbulence. J. Fluid Mech., 193, 475–497. Hoskins, B., M. McIntyre, and A. Robertson, 1985: On the use and significance of isentropic PV maps. Quart. J. Roy. Meteor. Soc., 111, 877–946. Kevlahan, N., and M. Farge, 1997: Vorticity filaments in two-dimensional turbulence: Creation, stability and effect. J. Fluid Mech., 346, 49–76. Methven, J., and B. Hoskins, 1998: Spirals in potential vorticity. Part I: Measures of structure. J. Atmos. Sci., 55, 2053–2066. Parker, D., 1998: Barotropic instability in frontolytic strain. Quart. J. Roy. Meteor. Soc., in press. Polavarapu, S., and W. Peltier, 1990: The structure and nonlinear evolution of synoptic-scale cyclones: Life-cycle simulations with a cloud-scale model. J. Atmos. Sci., 47, 2645–2672. Press, W., B. Flannery, S. Teukolsky, and W. Vetterling, 1992: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, 702 pp. Ralph, F., 1996: Observations of 250-km wavelength clear-air eddies and 750-km wavelength mesocyclones associated with a synoptic-scale midlatitude cyclone. Mon. Wea. Rev., 124, 1199–1210. Smith, G., and M. Montgomery, 1995: Vortex axisymmetrization: Dependence on azimuthal wave-number or asymmetric radial structure changes. Quart. J. Roy. Meteor. Soc., 121, 1615–1650. Thorncroft, C., B. Hoskins, and M. McIntyre, 1993: Two paradigms of baroclinic-wave life-cycle behaviour. Quart. J. Roy. Meteor. Soc., 119, 17–55. Vassilicos, J., and J. Hunt, 1991: Fractal dimensions and spectra of interfaces with application to turbulence. Proc. Roy. Soc. London A, 435, 505–534. , and J. Fung, 1995: The self-similar topology of passive interfaces advected by 2-D turbulent-like flows. Phys. Fluids, 7, 1970–1998. Zabusky, N., M. Hughes, and K. Roberts, 1979: Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys., 30, 96–106.