Nonaxisymmetric Thermally Driven Circulations and Upper ...

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HSU AND PLUMB

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Nonaxisymmetric Thermally Driven Circulations and Upper-Tropospheric Monsoon Dynamics C. JUNO HSU

AND

R. ALAN PLUMB

Program in Atmosphere, Oceans and Climate, Massachusetts Institute of Technology, Cambridge, Massachusetts (Manuscript received 22 October 1998, in final form 21 June 1999) ABSTRACT The authors investigate the nonlinear dynamics of almost inviscid, thermally forced, divergent circulations in situations that are not axisymmetric. In shallow-water numerical calculations, asymmetry is imposed on a locally forced anticyclone by imposition of a mean wind, or a planetary vorticity gradient. Behavior is similar in the two cases. With weak asymmetry, the forced anticyclone is distorted but remains intact and is qualitatively unchanged from the symmetric response. For sufficiently large asymmetry, however, the elongated anticyclone becomes unstable and periodically sheds eddies. This behavior shows how the circulation constraint can be satisfied, even when the time-mean absolute vorticity remains finite in the divergent region, and provides a continuous evolution between the nonlinear (symmetric) and linear (highly asymmetric) limits. Westward shedding of anticyclones from the Tibetan anticyclone is indeed evident in NCEP reanalysis data. These eddies are trapped near the tropopause. Cutoff potential vorticity features are confined to within about 20 K of the tropopause; in geopotential, they extend somewhat further, but not below about 400 hPa.

1. Introduction The various theoretical paradigms used for understanding the response of the tropical atmosphere to differential heating are not as unified as the quasigeostrophic dynamical framework used for the midlatitudes. The widely accepted paradigm for the zonally averaged tropical circulation is built on the framework of axisymmetric, inviscid, and nonlinear flow (Schneider 1977; Held and Hou 1980; Lindzen and Hou 1988). In contrast, theories of the zonally asymmetric component rely on linear theory, which has been used, for example, to explain the equatorial Walker circulation (Gill 1980). Hoskins and Rodwell (1995) found that their (nonlinear) model results for the response to summertime diabatic heating were reasonably well reproduced by a linear simulation. Linear calculations produce satisfactory results provided they include sufficient dissipation. Neelin (1988) and Lindzen and Nigam (1987) showed that in the lower troposphere, this dissipation can be associated with that of the boundary layer. In the upper troposphere, the flow is remote from boundary layer dissipation. However, Sardeshmukh and Hoskins (1985) found that vorticity transport by upper-tropospheric transients was such as

Corresponding author address: Dr. C. Juno Hsu, Canadian Centre for Climate Modelling and Analysis, University of Victoria, P.O. Box 1700, Victoria, BC V8W 2Y2, Canada. E-mail: [email protected]

q 2000 American Meteorological Society

to damp tropical vorticity anomalies. Provided these transients can be regarded as external to the dynamics of interest, they appear to provide justification for uppertropospheric damping in linear calculations. There is, however, evidence for the nonlinearity of the upper-tropospheric tropical flow. Observations show that relative vorticity is comparable to the Coriolis parameter in monsoon regions, and it appears that there is no a priori justification for assuming linear dynamics. Sardeshmukh and Held (1984) found nonlinearity to be important in the vorticity budget of the Tibetan anticyclone in a general circulation model, and Sardeshmukh and Hoskins (1985) found absolute vorticity z a to be close to zero over large regions of the Tropics during the northern winter of 1982/83. In the following, we shall review theoretical results concerning constraints on steady, inviscid, divergent flows; in such flows there can be no net flow across nonzero absolute vorticity contours. In cases where large-scale divergent flow exists, monsoon circulations being perhaps the most dramatic example, it must be because either nonlinearities are sufficiently strong or the steady flow is not effectively inviscid. Most of our understanding of steady, inviscid, divergent circulations is based on the assumption of zonal symmetry, as in the theories of Schneider (1977) and Held and Hou (1980), with subsequent refinements by Lindzen and Hou (1988). Under this assumption, flow cannot cross contours of angular momentum, so a divergent flow can exist only in regions of uniform ab-

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solute angular momentum, that is, of zero absolute vorticity. Schneider (1987) showed that this constraint carries over to the nonsymmetric problem. Assuming that the vertical velocity and the vortex twisting terms are small in the tropical outflow layer, the circulation tendency for the time mean flow, u, around any closed contour C is

I C

]u · dl 1 ]t

52

EE

EE

= · (z au ) d A

A

= · (z9u9) d A 1

I F · d l,

(1)

c

A

where A is the area enclosed by that contour, and the terms on the right describe the contributions of transient eddies and friction, respectively. In an inviscid steady state (with no transient eddies), then, if we choose C to be a closed contour of absolute vorticity (z a 5 constant),

za

EE

= · u d A 5 0,

(2)

A

whence there can be no net divergence within any such contour if z a is nonzero. Sobel and Plumb (1999) have shown that this result holds true for inviscid flow even in the presence of transients, provided the area enclosed by the contour does not change with time. How is this constraint met (say, on a monthly mean basis) in the upper-tropospheric monsoon anticyclone? Since z a is not zero over any finite region [cf. Fig. 7.7 of James (1994), and section 6, below] and the anticyclone contains nonzero net divergence [e.g., the diabatic heating distribution shown in Hoskins and Rodwell (1995), and see Fig. 14 below], it appears that (2) is violated there, implying that the flow must be either unsteady or affected by viscosity. (Since molecular viscosity is utterly negligible, anything that might loosely be regarded as viscosity must occur through eddy effects and thus in reality also point to unsteadiness.) From a theoretical point of view, we frame the question a little differently and ask what will be the inviscid response to a localized divergence in situations that are not axially or zonally symmetric: will the response be characterized by a region of zero absolute vorticity (as in the symmetric case), or will the flow change character and become unsteady? To this end, we investigate the characteristics of nonlinear, nonsymmetric, divergent circulations in a shallow-water model. In order to maintain a link with the symmetric solutions, we start with a symmetric problem and then investigate how the characteristics of the forced divergent flow change as the symmetry is progressively destroyed. We find it expedient to take as a starting point an axisymmetric solution forced locally on a basic state at rest on an f plane, rather than a zonally symmetric case. The axisymmetric problem has a steady solution that is directly analogous

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to the zonally symmetric Hadley circulation problem studied by Held and Hou (1980),1 with a divergent flow confined to a finite, circular region of zero absolute vorticity. One difficulty that immediately presents itself is that the steady, inviscid, symmetric solution is unstable to nonaxisymmetric perturbations. This presents no difficulty in a purely axisymmetric model, but the instability inevitably manifests itself in nonaxisymmetric calculations. We have, in fact, found it relatively straightforward to suppress the instability through the agency of a weak viscosity, which has the effect of smoothing the vorticity spike at the edge of the divergent region and, while not completely eliminating the instability, controlling it to the extent that the perturbation remains weak. Moreover, we find that the direct effects of viscosity on the vorticity budget are weak (e.g., see Fig. 10), and so the flow remains close to the inviscid limit we wish to investigate. We break the symmetry of the problem in two ways. In one set of experiments, we add a uniform flow. When this flow is weak (subcritical, in a sense to be defined below) compared with the forced divergent flow, its effects are qualitatively modest: the divergent region is displaced and stretched downstream, but the divergent flow is still confined to a well-defined region close to the forcing within which absolute vorticity is close to zero. For sufficiently strong imposed flow, however, the elongated forced vortex becomes unstable and periodically rolls up at its tip, shedding anticyclones into the background flow. The character of the time-averaged flow is quite different in this case; the region of nonzero convergence extends much farther downstream than in the subcritical case, and there is no longer a well-defined region within which absolute vorticity is close to zero. In a second set of experiments, a planetary vorticity gradient b is added. Qualitatively, its effects are similar to those of advection. With sufficiently weak b, the divergent region of almost zero absolute vorticity is simply displaced westward; with b supercritical (in a sense to be defined), anticyclonic eddies are periodically shed away to the west, again leaving a time mean state that is very different from the steady, axisymmetric case. The periodic shedding of anticyclonic eddies from an unstable elongated anticyclone created by the localized forcing is dynamically important, as it provides a mechanism for signals to travel to the far field and thus to establish a time mean global-scale circulation without invoking a large mechanical damping—a scenario perhaps more likely to apply to the inviscid upper-level atmosphere. We have found that shedding of anticyclonic air from the Tibetan monsoon anticyclone is indeed evident in

1 The axisymmetric solution has recently been discussed independently by Wirth (1998).

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upper-tropospheric potential vorticity analyses. We illustrate this behavior by showing two episodes of uppertropospheric eddy shedding in the northern summer monsoon season of 1990; this observed behavior, which is qualitatively very similar to that found in the model calculations, is confined to a relatively thin region near the tropopause. The layout of the paper is as follows. In section 2, we discuss the axisymmetric system and its linear stability properties, and the dependence of both on viscosity. In section 3, the 2D shallow-water model for the nonaxisymmetric simulations is introduced, and a scale analysis to derive the control parameter for the nonaxisymmetric problem is discussed. Results of the nonaxisymmetric simulations are presented and analyzed in section 4. In section 6, we present a case study of eddy shedding in the upper troposphere in July 1990. Section 7 provides concluding remarks.

a. Analytical solutions We can derive the equivalent of Held and Hou’s (1980) solution in an f -plane shallow-water model of mean depth D 0 when a ‘‘radiative–convective equilibrium’’ axisymmetric forcing is imposed through a mass relaxation. The governing equations, nondimensionalized with a timescale given by the inverse Coriolis parameter, f 21 0 , with a length scale given by the deformation radius, L R [ Ï F 0 / f 0 , and with a reference geopotential, F 0 5 gD 0 , can be written as follows: ]u ]u y 1u 2 2y ]t ]r r

]

1 2

]y uy ]y 1u 1 1u ]t ]r r

[

1 2

E

r

0

F e (r) 2 f (r) r dr t

for r # R and

y (r) 5

1

1 2r 1 2

f (r) 5 F e (r), u(r) 5 0

!

12

r2 2 8

2

2

r F e (r) , b

for r . R.

E1 R

0

2

F e (r) 2 f (r) r dr 5 0. t

This allows us to express the amplitude of the forcing, ˆ e , in terms of R and b: F

]

1 ] ]y y r 2 2 , r ]r ]r r

ˆe 5 1 F 16

]f ]f (1 1 f ) ] 1u 1 (ru) ]t ]r r ]r 5

1 r(1 1 f )

u(r) 5

]f 1 ] ]u u 1n r 2 2 , ]r r ]r ]r r

52

5n

1 y (r) 5 2 r, 2

The radius R is determined by requiring that there is no net mass loss across r 5 R, which leads to (cf. Held and Hou 1980)

2

[

ficient, where n* is the dimensional viscosity. The imposed equilibrium geopotential (4) has a localized peak ˆ e and length scale b (which will in these of amplitude F calculations be small compared with other important length scales). Geopotential is relaxed toward this equilibrium with timescale t . ˆ e exceeds the threshold value (1/8)b 2 , a Provided F divergent circulation must exist within a nonlinear region of uniform angular momentum in the vicinity of the forcing [r # R; see Plumb and Hou (1992) for the criterion for deriving the threshold]. By assuming a balanced azimuthal wind (i.e., dropping the small quadratic term due to the divergent flow, u]u/]r) and that the geopotential is continuous at r 5 R, we can write down the inviscid steady-state solutions as

1 f (r) 5 (R 2 2 r 2 ) 1 F e (R), 8

2. One-dimensional axisymmetric model

with

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[ 1 2] [ 1 2]

F e (r) 2 f 1 ] ]f 1n r , t r ]r ]r

ˆ e exp 2 r F e (r) 5 F b

[1 2 ]

R b 2 2 b 2 exp2 b

2

[1 2 ]

R 2 R 2 exp2 b

2

.

(5)

Note that if R k b, which is valid for the cases we will ˆ 1/4 discuss, R . 2b1/2 F e . (3) b. Time-dependent simulation and linear instability analysis

2

.

R4

(4)

The equations are in polar coordinates and depend only on the radius r measured from the origin. The radial and azimuthal velocity are denoted by (u, y ); geopotential is 1 1 f ; and n [ n*/ f 0 L R2 is the viscosity coef-

Evidently the axisymmetric and inviscid analytical solution is unstable to any azimuthal perturbation since the absolute vorticity is infinite at R. To understand the solution’s stability properties, we run a time-dependent model and analyze the linear normal modes of the flow as it evolves, with various choices of viscosity coefficient.

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FIG. 1. (a), (b), (c), (d) Plots of absolute vorticity in units of f 0 from the output of the 1D time-dependent axisymmetric model with the viscosity coefficient, (a) n 5 0, (b) n 5 4 3 1025 , (c) n 5 4 3 1024 , and (d) n 5 4 3 1023 . The 20 curves in each figure show the maximum progressing from left to right as time increases. The ‘‘X’’ in panel (a) marks the edge of the cross circulation to exist for the steady state at R 5 1. (e), (f ), (g), (h) Plots of growth rate in units of f 0 as a function of the azimuthal wavenumber, m, for the instantaneous basic states shown in (a)–(d). The growth rates increase as time increases (i.e., the curve with the largest growth rate corresponds to the largest time).

First, we construct a steady-state solution by choosing ˆe5 t 5 8.64, b 5 0.2, and R 5 1 such that, by (5), F 1.5625. Next, we run the time-dependent model from rest, defined by (3), with viscosity values n 5 0, 4 3 1025 , 4 3 1024 , and 4 3 1023 , respectively. The total domain for the calculation is three times larger than R; the geopotential perturbation and the velocities are set to zero at the outer boundary. Each case is run for a time 864 f 21 0 , and the absolute vorticity z a is plotted at time intervals of 43.2 f 21 in Figs. 1a–d. As shown in 0 the figure, for the first three cases, the region of zero z a advances farther away from the origin with time, the peak vorticity increases, and the slope of the vorticity becomes steeper. For the case with n 5 0, the region of zero absolute vorticity asymptotically approaches r 5 R (51) as time progresses. For the cases with n # 4 3 1024 , the zero absolute vorticity patch covers progressively larger regions but does not reach the r 5 R (51) radius. For the case with n 5 4 3 1023 , the nature of inviscid flow conserving angular momentum (zero absolute vorticity) is completely lost. We also calculate the normal modes for wave dis-

turbances on each flow. Figures 1e–h show the scaled growth rate of the normal modes with respect to the corresponding basic state shown in Figs. 1a–d. Wavenumber 2 becomes unstable first, and higher modes become unstable as the width of the jet decreases. For n # 4 3 1025 , the growth rate ranges from 0.1 f 0 to f 0 , which would allow waves to grow rapidly in the 2D simulation over a timescale of several times f 21 0 (as will be demonstrated in section 4b). Clearly, the basic state with a viscosity coefficient of 4 3 1023 is completely stabilized, but its inviscid characteristics are lost. Therefore, for the almost inviscid calculations to follow, we choose n 5 4 3 1024 , since this case is only weakly unstable and (as will be shown in section 4b) the perturbations saturate at weak amplitudes and thus do not seriously alter the axisymmetric basic state. 3. Nonaxisymmetric flow a. The model In our two-dimensional calculations, the forcing has the same profile as in the axisymmetric model but is

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placed in the center of the domain. Asymmetries are introduced in one of two ways: through a uniform planetary vorticity gradient b*, or by moving the forcing [the F e (r) distribution] with constant velocity 2U*m . In Cartesian coordinates, the governing equations, also scaled by f 0 , L R , and F 0 and expressed in a frame moving with the forcing, become

]u 5 y (1 1 by 1 z ) 2 ]t

[

]

1 ] f 1 (u 2 1 y 2 ) 2

]y 5 2u(1 1 by 1 z ) 2 ]t

]x

[

1 n¹ 2 u,

]

1 ] f 1 (u 2 1 y 2 ) 2 ]y

1 Um

1 n¹ 2y , ]f F (r) 2 f 5 2V · =f 2 (1 1 f )= · V 1 e ]t t 1 n= 2f.

(6)

In general, the notation is the same as in the previous section. The horizontal velocity is V 5 (u, y ), with u and y being the eastward and northward components, respectively; z is the relative vorticity; b 5 b*L R / f 0 ; and U m 5 U*m / f 0 L R .

4) We anticipate that nonlinearity is important, so

z ; f 0. 5) Nonlinear balance requires

f ; L2. We take L as the characteristic length scale, and the timescale for a particle to cross this region (T 5 D21 con ) as the characteristic timescale. By manipulating the above four approximations, these can be written as follows: 1/2 ˆ 1/4 ˆ 21/2b21 t . L;F , T [ D21 e b con ; F e For the above balances to hold, there are three requirements to be met. First, a strong localized mass source is required to generate a broad mass-sink area, that is, Ddiv k Dcon . Second, a small Froude number L 2 , which is the ratio between the radiative timescale and the characteristic timescale, that is, L 2 5 t /T K 1, is required to ensure that the ‘‘diabatic’’ cooling term is more important than the advection terms in the geopotential equation [thus preserving condition (2), above]. Third, for nonlinear balance to hold, the rotational energy must contribute most of the kinetic energy, that is, u x [ L/T K V c 5 L. These requirements lead to three inequalities as follows: ˆ e k b2 Ddiv k Dcon ⇒ F ˆ e K b22 Tkt ⇒F ˆ e K b22t 2. Tk1⇒F

b. Scale analysis We are interested in exploring the parameter range in which a steady, nearly inviscid, and nonaxisymmetric circulation can be established by perturbing the known axisymmetric circulation. Assuming that the qualitative dynamic properties remain the same as those of the symmetric case, we can use these balances to derive scales and thus to identify the key control parameters for the general case. To a first-order approximation, the following dynamical balances hold. 1) For the cases we investigate, for which the forcing area is small, the local divergence in this region must balance the forcing: ˆ F Ddiv ; e . t 2) Outside the mass source area, where F e (r) ; 0, the convergence (Dcon ) balances the dissipation: Dcon

f ;2 . t

3) The mass flux out of the forcing region (of area ; b 2 ) is balanced by the mass sink within an area ; L 2 , where L is the size of the convergent region. Then mass conservation requires Ddiv b 2 ; Dcon L 2 .

Together, these imply the constraint ˆ e K b22 min[1, t 2 ]. b2 K F In all the numerical simulations, we will use external parameters that satisfy the above inequalities. In the vorticity equation, we therefore adopt u x 5 L/T as the characteristic scale for the divergent flow, and u c 5 L for the rotational wind. We thus obtain dimensionless vorticity budgets as follows. 1) UNIFORM

FLOW EXPERIMENTS

For b 5 0, we obtain ]z a ]z 1 J(c, z a ) 1 Vx · =z a 1 m m a ]t ]x 5 2gz a= · Vx 1 k¹ 2z a ,

(7)

where

mm [

Um ˆ e )23/4 U mt , 5 (b 2F ux  Ddiv

 D 5 Fˆ 1/e 2 b21 , g 5  con   1,

ˆ e b 2 )21tn. k 5 (F

r#b r . b,

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Here z a is the absolute vorticity, V x the divergent wind, c the streamfunction of rotational wind, and r the radial distance from the origin. In the inviscid limit, k → 0, there are two control variables, m m and g. However, increasing g increases the strength of the divergence only within the small source area (r # b K L), where z a is small in equilibrium. Thus, while the stronger divergence shortens the evolutionary timescale, the term involving g contributes little to the steady budget. We therefore expect the parameter m m , which expresses the ratio of the translation velocity of the forcing to the magnitude of the divergent flow, to be, in practice, the dominant control parameter affecting the dynamics, which has been confirmed by numerical experiments (Hsu 1998). 2) b-PLANE

EXPERIMENTS

The nondimensional vorticity equation for the case U m 5 0 is ]z 1 J(c, z ) 1 Vx · =z 1 mby c 1 qy x ]t 5 2g (z 1 1 1 qy)= · Vx 1 k¹ 2z,

(8)

where

mb [

bL 2 ˆ e b 2 )21/4bt , 5 (F ux  Ddiv

 D 5 Fˆ 1/e 2 b21 , g 5  con   1,

q[

r#b r . b,

bL ˆ e b 2 )1/4b. 5 (F f0

The parameter m b expresses (as will be discussed further, below) the ratio of the ‘‘b-drift’’ speed of an eddy of radius L to the magnitude of the divergent flow and is thus directly analogous to m m in the previous case. The b effect appears in three places: advection of planetary vorticity by the rotational wind (as m b ) and by the divergent wind (as q), and in the vorticity generation term (as q). The ratio of the rotational to divergent advection terms is large provided T k 1, which, as noted earlier, is already satisfied if nonlinear balance holds. The parameter q expresses the fractional variation of f over the length scale L. The conventional midlatitude b plane corresponds to the limit q K 1, in which case the only remaining parameter is m b . We will concentrate on this regime in the first set of b experiments. When, on the other hand, q * 1, f changes sign within the dynamically active region, and equatorial b plane dynamics becomes involved. In section 5, we will describe some experiments in this regime.

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4. 2D numerical experiments a. Experiment outline We first create a nearly axisymmetric state with the same parameter settings as in the 1D simulation and use it as a starting point for the nonaxisymmetric study. Most external parameters are fixed throughout the series ˆ e 5 1.5625, b 5 0.2, t 5 8.64, n 5 of experiments (F 4 3 1024 , g 5 6.25, k 5 0.055). Asymmetry is then introduced progressively in each series of experiments, increasing either the imposed uniform flow (up to m m 5 2.48) or b (up to m b 5 4.32, for the midlatitude b-plane cases; in all of which q did not exceed 0.13). Each run takes as its initial condition the final state from the previous run in the sequence and is run for a period of time long enough to establish a steady or periodic equilibrium. We find that the flow takes the form of a steady, localized circulation or a more global, unsteady circulation, depending on whether or not the parameter m (m m or m b ) exceeds some critical value m c (noted as m mc or m bc later, in sections 4c and 4d). Each parameter (m m or m b ) can be thought as the ratio of the relative strength of the asymmetric element to the strength of the imposed axisymmetric forcing. We will refer to the cases with m , m c as ‘‘subcritical’’ and to cases m . m c ‘‘supercritical.’’ For the external flow experiments with b 5 0, the domain has doubly periodic boundary conditions, and 200 3 200 grid points with a grid point distance of 0.03L R . For the experiments with nonzero b, U m 5 0, and the northern and southern boundary conditions are replaced by two rigid walls, each with a sponge layer. The sponge layer has a 0.432 f 21 0 damping timescale and a three-gridpoint e-folding scale from the wall. The domain has 151 3 151 grid points with a gridpoint distance of 0.04L R . By changing the resolutions of a few runs, we confirmed that the model domain in either series of experiments is sufficiently large that the boundary conditions do not affect the response of interest. b. Instability problem for the f-plane system The two-dimensional model was first run with b 5 U m 5 0 to test the linear instability calculation of section 2b. Without viscosity, the vortex patch with zero absolute vorticity extends outward from the forcing center, shielded by a narrow high-vorticity band. An asymmetry with azimuthal wavenumber m 5 2 grows, and the vortex soon breaks into two dipolar structures that move away from the forced area. The same formation–breaking–leaving cycle repeats indefinitely. Figure 2 illustrates the second breaking cycle in our simulation. The forcing creates a shielded vortex with a narrow cyclonic band surrounding a broad anticyclonic patch, a basic state similar to that investigated by Flierl (1988), showing the m 5 2 instability of a piecewise-constant shielded vorticity patch (see Fig. 14 in that paper). As noted earlier, we avoid the complicating effects

FIG. 2. The second breaking cycle under axisymmetric forcing in the 2D model with t 5 ˆ e 5 1.5625, and n 5 0. The contours are absolute vorticity in units of f 0 8.64, b 5 0.2, F with contour values ranging from value 0 to 1 with a contour interval 0.25. The clean spaces next to the low-vorticity blobs are regions of high-vorticity not contoured for contrast. Each short tick mark indicates a grid point of the model as in all subsequent figures with tick marks.

FIG. 3. A wobbling cycle of a tripole under the axisymmetric forcing and with t 5 8.64, ˆ e 5 1.5625, and diffusivity coefficient n 5 4 3 1024 . The heavy lines denote b 5 0.2, F absolute vorticity in units of f 0 with an interval of 0.2 from the innermost line 0.2 to the satellites 1.2. The solid (dashed) thin lines denote the divergence (convergence). The solid thin lines have a contour 50 times larger than dashed lines.

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FIG. 4. Contour plots of the time mean absolute vorticity in units of f 0 and with the wind field indicated by arrows for the uniform flow experiments with the control parameters m m 5 0, m m 5 0.55, m m 5 1.38, and m m 5 2.48. The heavy line in each panel indicates the 0.5 f 0 contour. The contour interval is 0.1 with the minimum near the center of the forcing. The ‘‘X’’ denotes the center of the forcing.

of instability by employing a stabilizing diffusivity of n 5 4 3 1024 in both the momentum and height equations. From the linear calculation in section 2b, a viscosity of such magnitude causes the basic state to be weakly unstable to the azimuthal wavenumber m 5 2 mode but stable to all higher modes. In the two-dimensional simulation, the elliptical mode (m 5 2) indeed grows but remains weak: instead of breaking up, the vortex collapses into a tripole. Figure 3 shows the formation of a tripole from the shielded monopole with the associated divergence and convergence fields. This behavior is similar to that found by Carton et al. (1989) in a nondivergent barotropic system where they found that in a certain range of steepness of vorticity gradient, a shielded monopole collapses into a tripole. We take this tripole state to be the starting point for investigating the influence of asymmetrical elements on the otherwise symmetric thermally driven circulation. c. External uniform flow experiments The impact of the imposed external flow is evident in Fig. 4, which shows the time-mean absolute vorticity

and the time-mean total wind field for m m 5 0, 0.55, 1.38, and 2.48. As m m increases, the distribution of the vorticity evolves farther away from its original nearly circular pattern. The circular vortex evolves to a doughnut shape (the vortex center having a higher vorticity than the surrounding ring, e.g., m m 5 0.55), then to a comma shape (e.g., m m $ 1.38). For m m 5 0.55 and 1.38, the absolute vorticity contours intersect the total wind vector with a greater angle near the inner patch of the vortex, an indication of a net nonzero advection to be balanced by the generation term. Near the outer rim or tail where there is strong absolute vorticity gradient, the total wind field aligns well with the absolute vorticity contours, indicating a near-zero net advection resulting from the cancellation of the external wind and the rotational wind field. For m m 5 2.48 in Fig. 4, the time mean wind crosses the absolute vorticity contours even in the tail region at an angle, an indication (as we shall see below) of net nonzero advection balanced by transients. When the tongue is long enough in the zonal direction, the vorticity strip becomes unstable and rolls up

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FIG. 5. The phases of one wobbling cycle (t 5 1) at t 5 0, t 5 0.25, t 5 0.4, and t 5 0.6 for m m 5 0.55 in the 2D model domain. The color contours are the absolute vorticity with an interval of 0.025 f 0 . The inner circle marks the size of the imposed axisymmetric mass source, b, and the outer circle marks the limit of the corresponding axisymmetric divergent circulation, R. (Hereafter for similar plots, both circles are marked as in this figure.)

into a small vortex that is shed downstream. Figures 5 and 6 show a wobbling cycle for a subcritical case with m m 5 0.55 and a shedding cycle for a supercritical case with m m 5 1.93. From this series of the experiments, we found that the critical value, m mc , of m m separating the subcritical and supercritical cases lies in the range 1.38 , m mc , 1.65. The inference from the results is that, for a steady nonaxisymmetric circulation to exist under an external uniform flow, the magnitude of the external flow cannot be much larger than that of the forced divergent flow. Otherwise (when m m . m mc ), the potential vorticity (PV) distribution becomes zonally oriented, and the induced return flow is not strong enough to prevent the anticyclonic vorticity from being advected downstream by the external flow. d. Midlatitude b-effect experiments The immediate effect of adding a beta effect is to shift the vortex center southwest of the forcing center, and this is qualitatively similar to that of an imposed flow; in particular, a transition from steady to unsteady

behavior occurs when m b 5 m bc , where 0.86 , m bc , 2.16. Figure 7 shows a panel with six cases run, each showing the time-mean absolute vorticity and convergence. Notice that when m b . m bc , the convergence zone is greatly elongated zonally as a result of the consecutive passing of anticyclonic eddies. The resulting scale is much larger than the size of the corresponding axisymmetric case, thus ‘‘global-scale’’ will be used later to refer its size. A detailed vorticity balance will be analyzed in section 4e to illustrate the role of transient eddies in the time mean balance. The time evolution of a supercritical case is illustrated in Fig. 8, which shows one shedding cycle for m b 5 4.3. In these cases, the vortex pattern shifts farther southwestward and the convergence is enhanced. As a consequence, a thin anticyclonic strip is drawn out of the forcing area while a cyclonic strip is advected around the eastern flank of the anticyclone. Both strips are unstable and roll up at their tips into a pair of vortices that is soon shed westward and disappear gradually. As noted earlier, m b is the ratio between a speed C 5 bL 2 and the magnitude of the divergent flow u x ,

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FIG. 6. The phases of a shedding cycle (t 5 1) at t 5 0.25, t 5 0.5, t 5 0.75, and t 5 1.0 for a supercritical case with m m 5 1.93. Contouring is same as in Fig. 5.

where L is the characteristic length from the corresponding axisymmetric case. This ratio is physically unclear without an explanation for the speed C. One intuitive speculation is that C is the free-drifting speed of an unforced vortex. In such a case, the linear dispersion of Rossby waves could cause the phase to move predominately westward and a little northward or southward, depending on its shape and orientation (Flierl 1977). With finite amplitudes, nonlinear advection of the vortex center becomes important (e.g., Fiorino and Elsberry 1989; Sutyrin and Flierl 1994). Note that the ratio of advection to wave dispersion is large in our experiments, and so we cannot ascribe C 5 bL 2 only to linear dispersion. However, results (not shown here; see Hsu 1998) from a series of free-vortex experiments in which the forcing is shut off confirms that C 5 bL 2 is indeed a measure of the drift speed of the free vortex. e. Transients and the scale of the divergent circulation The experiments described above show that the existence of transient eddies shed by the forced anticyclone coincides with a change of character, and extent,

of the forced circulation. In this section, we shall quantify contributions to the vorticity budget from one of the supercritical cases. Figure 9a shows the time mean distribution of wind, absolute vorticity, and geopotential for the case m b 5 4.32, computed from the last 100 days of the model run. The geopotential center of the anticyclone is located to the southwest of the forcing center with a westerly jet to the north and an easterly jet to the south. There is a patch of low vorticity near the forcing center, and the vorticity minimum is approximately 20.5 f 0 . Each of the terms in the dimensional vorticity equation, ]z 5 2V · =z 2 = · (V9z9) 2 b y 2 (z 1 f )= · V ]t 1 n¹ 2z , is contoured in Fig. 10. The transient eddies are nearly nondivergent, so we write the transient term in its divergent form and ignore the transient stretching term. From the calculations, the time mean advection term dominates even after it is partly canceled in some areas by the term due to the transients. The stretching term is mostly balanced by the time mean advection term from the forcing area extending westward to the vicinity

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h5

EE

S

z = · (V9z9) dA

EE

,

z dA 2

S

FIG. 7. Contour plot of the time mean absolute vorticity (heavy lines) distribution in units of f 0 (heavy line) overlaid on the divergence and convergence distribution (solid and dashed thin overlaid lines) for the midlatitude b-plane experiments. The contour interval for the absolute vorticity is 0.25; the outermost line is the 1.0 contour. Contouring for divergence and convergence is same as in Fig. 3.

of the maximum convergence. To the east of the forcing, the planetary vorticity advection is balanced by the time mean advection term. To the west of the convergence maximum, Sverdrup balance nearly holds as the nonlinear terms tend to cancel each other. Sardeshmukh and Held (1984) calculated the vorticity budget of the Tibetan anticyclone at 200 mb from the output of the Geophysical Fluid Dynamics Laboratory GCM. They concluded that the balance is nonlinear and inviscid, that the time mean advection and the stretching term near the vicinity of the Tibetan anticyclone cancel each other, and that the total of the terms due to transients is not negligible in some regions. Those conclusions are consistent with our results. On the other hand, their distribution of transients has a less ordered pattern compared with our results, and their calculated damping coefficients are more sensitive to the choice of domain. From the figure, we see that the term representing divergence of the transient vorticity fluxes is not negligible. Since (as was discussed in the introduction) tropical upper-tropospheric circulations are often simulated reasonably well by linear, damped models, we investigated whether the net effect of the transients in our experiments can be represented in that way. If one replaces the transient vorticity flux divergence by a linear damping term, 2hz , one can estimate the rate coefficient h as

where S is either the area of the whole model domain or the area of a band where the transients have passed. The latter choice yields a somewhat larger value of h. We obtain a positive number, which indicates that the overall transients act to dissipate the vorticity. However, taking S to be the whole domain, the magnitude of h is about three times weaker than that required for 2hz to be comparable with the transient term, 2= · V9z9, in the region of large amplitudes. Figure 10h shows the distribution of time mean relative vorticity multiplied by h 0 5 3h. The three high- and low-vorticity strips in the figure crudely resemble the distributions of the transient term except for a phase shift. We then linearized the model by eliminating all nonlinear terms in the model and ran it with the same parameter settings as the nonlinear run with m b 5 4.32. Figure 9b shows the final steady state. By comparing with its fully nonlinear counterpart in Fig. 9a, both the shape and magnitude of the vorticity distributions are quite different: the absolute vorticity extremum in the linear model is nearly 22.5 f 0 , much stronger than the extremum 0.4 f 0 in the nonlinear run. The linear model generates a stronger circulation because the vortex stretching term can drive the absolute vorticity indefinitely until it is balanced by the beta term. In contrast, the vortex stretching term is shut down once the absolute vorticity becomes zero in the nonlinear model. However, in terms of the spatial scale of the geopotential distribution, the linear response to the forcing is quite similar to the nonlinear response. For the linear, ‘‘beta-plume’’ response (Rhines 1983), dissipation allows the flow to cross-latitudinal circles. In our case the dissipation is via thermal damping, which allows convergence and thus vortex stretching, which is then balanced by the advection of planetary vorticity. The linear response is characterized by an exponentially decaying geopotential with a length scale of bL R2 t . For the nonlinear run, the length scale of the geopotential anomaly is determined by the passage of the eddies that have been shed from the forcing region. The eddies, traveling with a speed of bL 2 , dissipate with the radiative timescale, t . Since L is comparable to the radius of deformation, the resulting length scale of the nonlinear geopotential response is similar to that of the linear case, although the dynamics are very different. Figure 9c shows the steady linear response with the inclusion of linear mechanical damping with a coefficient of magnitude h 0 mentioned above. The mechanical damping term has reduced the magnitude of the circulation to bring it closer to that of the nonlinear response. Meanwhile, the high-vorticity distribution to the

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FIG. 8. The phases of a shedding cycle (t 5 1) at t 5 0, t 5 0.25, t 5 0.5, and t 5 0.75 for a supercritical case of the midlatitude b-plane experiments with m b 5 4.32. The color contours are the absolute vorticity in units of f 0 . The line contours are geopotential perturbation plotted with a contour of 10 m 2 s22 . Arrows denote the wind field.

east of the forcing starts to resemble the one in the nonlinear run, although the negative vorticity still exists near the center. In terms of overall length scale, however, agreement with the full nonlinear run deteriorates. 5. Equatorial beta-plane experiments—Finite q If q is finite, the perturbation can reach the ‘‘equator’’ where the Coriolis parameter changes sign, and thereby trapped equatorial motions can be generated. Thus, the nature of the behavior is different from the midlatitude b-plane case. A sequence of three experiments with progressively increasing q is shown in Fig. 11. The first case is similar to the midlatitude b-plane case of Fig. 8. With q 5 0.25 (Fig. 11b), the shedding behavior is again similar, though the forced anticyclone is more elongated toward the west, and there is some weak cross-equatorial flow extending to the far field. With the exception of this latter component of the flow, there appears to be little interaction with the equator. Equatorial effects become more marked at q 5 0.52, as Fig. 11c shows. The instability that leads to the shedding appears to weaken,

and the anticyclone elongates farther westward. In fact, the instability now appears to be limited to cyclonic roll-up and westward movement of eddies on the cyclonic strip entrained by the anticyclonic circulation immediately to the east of the forcing. For the most part, the response is confined by the equator, with very little disturbance in the unforced hemisphere. However, there is a component of the flow trapped about the equator, extending eastward from the forcing region. In both structure and location, this feature is indicative of Kelvin waves propagating away from the forcing region and is reminiscent of the structure of linear solutions [e.g., see Fig. 7.5 of James (1994)]. Note, however, that the direction of the trapped flow is toward the forcing center, not away from it, as linear theory predicts in response to this sign of subtropical forcing. It seems that the Kelvin waves are generated by the near-equatorial convergence associated with the high vorticity advected from the subtropics, rather than directly by the divergence in the forcing region [cf. remote forcing of midlatitude Rossby waves as an indirect response to equatorial forcing, as described by Sardeshmukh and Hoskins (1985)].

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FIG. 9. The time mean distributions for (a) nonlinear run; (b) linear run; (c) linear run with a linear mechanical damping with the same parameter, m b 5 4.32. (left) Plots the absolute vorticity in units of f 0 with 0.5 f 0 emphasized by a thick line and the wind field. The dashed line denotes negative values. (right) Plots the geopotential distribution and wind field. The geopotential contour interval is 20 m 2 s22 .

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FIG. 10. (a)–(g) The time mean vorticity balance and (h) a calculated linear damping term for m b 5 4.32. (a) The time mean advection, (b) the transients, (c) the planetary advection, (d) the time mean stretching, (e) the diffusion, (f ) the time tendency, (g) the residual from the net of the above terms, (h) the relative vorticity multiplied by 3 3 h. The contour interval is the same for each figure.

In the final two experiments, we investigate the changing character of the solution as the linear limit (m b → `) is approached. For the case with m b 5 17.28 shown in Fig. 12, the circulation is similar to the supercritical cases of midlatitude beta-plane experiments, except that the eddies are shed more frequently and travel faster. As a consequence, there exist two or three eddies at any given instant. Each eddy travels mostly westward and slightly southward with the maximum of its geopotential located to the north of the minimum of its absolute vorticity. Note that in this figure the

shedding eddies have established a westerly jet to the north and an easterly jet to the south, patterns that extend around the periodic domain. When m b is increased to 172.8, the flow response takes the form of the linear solutions as shown in Fig. 13. Figure 13a shows the initial response—equatorial Rossby waves propagating westward and the Kelvin wave propagating eastward. Figure 13b shows the final steady state— Sverdrup balance with a weak anticyclone to the west of the heating and two zonal westerly jets off the equator. In this case, we reach the regime where the re-

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FIG. 11. A snapshot of the model output with (a) m b 5 2.16 and q 5 0.06, (b) m b 5 2.16 and q 5 0.25, and (c) m b 5 2.16 and q 5 0.52. The color contours show the absolute vorticity in units of f 0 . The geopotential contour interval is (a) 20 m 2 s22 , (b) and (c) 500 m22 s22 . Note that the unperturbed purple line approximately marks the position of the equator.

sponse is dominated by equatorial waves as in the theory of Gill (1980). 6. Observational evidence of eddy shedding from the Tibetan monsoon anticyclone Motivated by the numerical results, and by unpublished geopotential analyses of G. Postel and M. Hitchman (1997, personal communication), showing westward propagating upper-tropospheric anticyclone in the

northern summer of 1995, we sought evidence of similar occurrences of eddy shedding from the upper-tropospheric summer monsoon anticyclone. The data used were from the 4 times daily National Centers for Environmental Prediction reanalysis data, on 17 isobaric levels and 2.58 by 2.58 in longitude and latitude for the period of July 1990. Figure 14 shows the distributions of monthly mean streamfunction and precipitation for July 1990. The upper-level Tibetan anticyclone extends westward from the region of maximum precipitation

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FIG. 12. A snapshot of the model output with m b 5 17.28 and q 5 0.5. The plotted fields and contouring are same as in Fig. 8.

(and implied upper-level divergence) over Southeast Asia. While the western part of the anticyclone appears largely divergence-free, there is a nonzero net divergence within the eastern half (not shown here but implied by the precipitation pattern). Thus, the issues noted in the introduction, in connection with the constraint (2), are applicable here. Potential vorticity maps on upper-tropospheric isentropic surfaces were constructed every 6 h. Two episodes of shedding away from the main low PV body over the region of the Asian summer monsoon were found in July, 0000 UTC 11 July–1200 UTC 13 July and 0000 UTC 19 July–1800 UTC 20 July. During these episodes, the shedding events are clearly evident on the 360, 370, and 380 K isentropic surfaces but are less evident at and below 350 K. A corresponding splitting of the geopotential high is observed from 150 to 400 hPa. In Fig. 15 the first event is shown in the PV field on the 370 K surface and in the geopotential field at 200 hPa. By comparing Fig. 15 with shedding events from the model in Figs. 8 and 12, the PV field shows a similar distribution with a long low PV strip extending westward from the Asian summer monsoon region. The

monsoonal low PV is well separated from the low PV of the Tropics and surrounded by the high PV of the stratosphere (i.e., the tropopause is here bulging upward over the Asian monsoon region; the instability is apparently associated with this local PV minimum, as one would expect for parallel flow). On 0000 UTC 11 July, a low PV filament extends westward to 408E from the main PV body situated over 908E. By 1200 UTC 12 July, an anticyclonic vortex enclosed by the 1.0 PVU (1 PVU 5 1026 K m 2 kg21 s21 ) contour has detached from the strip and propagates westward. On 0600 UTC 13 July, the westward edge of the PV blob reaches about 208E. Above the monsoon region, the tropopause—and hence the local PV minimum—extends upward to about 380K, and so the eddy shedding is still visible there. This is evident in Fig. 16, although the shed eddy is smaller than at 360 and 370 K; 380 K is close to the upper level of this behavior. To ensure that the visualized shedding event from the observation is not an artifact of the PV analysis, contour advection calculations (Waugh and Plumb 1994) were used to reproduce the episode, advecting PV contours

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tion necessary to generate transient eddies in both the theoretical model and the observational shedding event on the 360, 370, and 380 K surfaces, thus demonstrating the consistency of the model and the observations. In the observations, the low PV at 350 K near the monsoon region is not separated from the low PV in the Tropics. Thus, the source region extends no more than about 20 K below the tropopause, and cutoff eddies in the PV field are not evident below this level. Consistent with the nature of PV inversion (e.g., Hoskins et al. 1985), the geopotential signal is visible down to somewhat lower altitudes (to about 400 hPa, in fact). This vertical scale is consistent with the horizontal scale of the eddies, which is rather less than the Rossby deformation radius based on local f and tropopause height. 7. Concluding remarks

FIG. 13. A snapshot of the model output with m b 5 172.8 and q 5 5.0 at (a) an early stage and (b) the steady state. The background contours are absolute vorticity in units of f 0 with a contour interval 4. On top of the absolute vorticity, the geopotential is plotted with a contour interval (a) 5 m 2 s22 and (b) 20 m 2 s22 . The contour of zero absolute vorticity is emphasized with a thick line. The equator is denoted by ‘‘E,’’ and the forcing center is denoted by ‘‘X.’’

from the initial analysis with the analyzed winds. Figures 17 and 18 show the time evolution of the 0.5, 1.0, and 1.5 PVU contours of PV on 370 and 380 K, which do indeed capture the shedding event. The westward propagation of the eddy stops after 12 UTC 13 July, at which time it is advected northward to eventually recirculate eastward and back toward the main anticyclone. The second episode is similar to that of the first; however, its recirculation occurs even sooner, and hence it is not as clear as the first. To the extent that instability of the zonally elongated anticyclone can be thought of as occurring on a locally parallel flow, a low PV strip embedded in a high PV background provides the dynamically unstable condi-

From a theoretical point of view, the results of the shallow-water calculations have provided an answer to the questions posed in the introduction, namely, how can a divergent anticyclone with inviscid dynamics satisfy the circulation constraint (2) when its time mean absolute vorticity is nonzero? In the experiments described here, PV transport by eddies, shed from the forced anticyclone, is the agency by which the constraint is relaxed. In the upper troposphere, while externally forced eddies also contribute to the PV transport (Sardeshmukh and Held 1984; Sardeshmukh and Hoskins 1985), we have found that the Tibetan monsoon anticyclone does indeed shed anticyclones in a way that is qualitatively similar to the behavior seen in the shallowwater experiments. One further theoretical implication is the link between the two extreme views of tropical dynamics: the fundamentally nonlinear, inviscid dynamics of axisymmetric theories in which absolute vorticity must vanish in divergent regions, and the linear, dissipative theories that have proved successful in describing the asymmetric part of the circulation. While we have not here attempted to produce a continuous documentation of the evolution from one extreme to the other, the progressive introduction of asymmetries, with the consequent weakening and expansion of the divergent circulation, does indeed provide an illustration of evolution from m 5 0 (the symmetric, nonlinear limit) to large m (m → ` being the linear limit). We have seen, in one example, that linear theory in fact captures the essence (though not the magnitude) of the solution rather well, even when nonlinearities are important in a budgetary sense. One of the unsatisfactory aspects of the modeling results is that it was necessary, for pragmatic reasons, to include some explicit viscosity in order to suppress instability of the forced vortex. One is led to ask the question as to whether this undermines our attempts to reveal the behavior of an inviscid forced vortex. In fact, there are two instabilities at work. The first is that (shown in Fig. 2) associated with the cyclonic PV ring

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FIG. 14. Surface precipitation rate shaded with a contour interval 4 mm day 21 and the streamfunction at 200 hPa plotted with a contour interval 5 3 10 6 m 2 s21 for July 1990.

FIG. 15. Time sequence of PV field at 370 K (color shading) and the geopotential height at 200 hPa (line contours) over the Asian summer monsoon at successive 18-h time integrals from 0000 UTC 11 Jul to 0600 UTC 13 Jul 1990. The color contour interval is 0.05 PVU. The black contour is 25 m.

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FIG. 16. Time sequence of PV field at 380 K (color shading) and the geopotential height at 150 hPa (line contours) over the Asian summer monsoon at successive 18-h time integrals from 0000 UTC 11 Jul to 0600 UTC 13 Jul 1990. Contouring is the same as in Fig. 15.

around the outer edge of the anticyclone; in the cases with axisymmetric external conditions, this is the only instability. In an axisymmetric model, this instability is suppressed simply as a result of being excluded by the geometry. In our nonaxisymmetric model, it was necessary to control this instability, through viscosity, in order to study the (perhaps hypothetical) characteristics of the forced vortex. The second instability is that of the elongated anticyclone (in the asymmetric case) that manifests itself as the anticyclonic eddy shedding that is the main focus of this paper. As it turns out, once eddy shedding starts, the cyclonic PV ring develops either weakly or not at all: as a result, the first instability does not materialize, and thus viscosity is not required to suppress this instability. This is illustrated in Fig. 19, which shows two snapshots of an experiment with the

same external conditions as that shown in Fig. 8, but with zero viscosity. The shedding behavior is essentially unchanged, and therefore, at least in this respect, the model experiments run with imposed viscosity do seem to capture the inviscid behavior. Just one example of observed eddy shedding2 was present here to link the dynamics of the upper-tropospheric monsoon anticyclone with the shallow-water calculations. However, there is one immediate problem in applying the theoretical result to the data: that of choosing the appropriate parameter range to put the nu-

2 Popovic (1999) has found two or three eddy shedding events in each of the years 1987–90.

FIG. 18. The PV evolution at 380 K over the Asian summer monsoon from CAS technique, plotted at four successive 18-h intervals. The initial field is 0.5, 0.75, and 1.0 PVU potential vorticity contours on 380 K at 1800 UTC 10 Jul 1990. Compare this figure with Fig. 16.

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FIG. 17. The PV evolution at 370 K over the Asian summer monsoon from Contour Advection with Surgery (CAS) technique, plotted at four successive 18-h intervals. The initial field is 0.5, 0.75, and 1.0 PVU potential vorticity contours on 370 K at 1800 UTC 10 Jul 1990. Compare this figure with Fig. 15.

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troposphere exchange associated with the filamentation that occurs (suggested in Figs. 15 and 16) as the shedding takes place and (not shown here) later as the eddies merge back into the main anticyclone. Whether the upper-tropospheric behavior has any impact on the monsoon circulation as a whole is unclear. Acknowledgments. This work forms part of the first author’s Ph.D. research. We would like to thank Prof. Glenn R. Flierl for guidance during the course of this work, Dr. Isaac M. Held for his helpful comments, Dr. Matt Hitchman for providing the authors with unpublished results showing westward propagating subtropical anticyclones in upper-tropospheric geopotential data, and Ms. Popovic for drafting Fig. 14. We also thank the three reviewers, whose comments led to an improved presentation. The research was supported by the National Science Foundation, through Grant ATM9528471. REFERENCES

21 FIG. 19. Two snapshots at t 5 52 f 21 0 and t 5 78 f 0 from the model output with the same external variables as in Fig. 8, but without viscosity, i.e., n 5 0.

merical experiments into the context of the real atmosphere. For one thing, the ‘‘radiative–convective equilibrium’’ heating profiles are difficult to estimate for the real atmosphere. However, if one accepts the theoretical assumptions and then estimates the values of the control parameters, m b and q applicable to the upper troposphere over the Asian summer monsoon, one obtains m b ; 10 and q ; 0.2, when f 0 and b are taken at 308N, m x ; 1 m s21 , and L* 5 R/2 ; 750 km. These estimates suggest that the upper-level circulation of the Asian summer monsoon is supercritical (and, in fact, somewhat similar to the case shown in Fig. 11b) and should therefore exhibit shedding, as observed. Other than its potential impact on the vorticity budget, we have not investigated what might be the impact of the eddy shedding. The eddies themselves are, for straightforward dynamical reasons, shallow and confined to the upper troposphere and thus are unlikely to interact directly with the surface and with moist convection. They may well have a significant impact at the tropopause, however, not least through the stratosphere–

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the tropical upper troposphere of a general circulation model. J. Atmos. Sci., 41, 768–778. , and B. J. Hoskins, 1985: Vorticity balances in the tropics during the 1982–83 El Nino–Southern Oscillation event. Quart. J. Roy. Meteor. Soc., 111, 261–278. Schneider, E. K., 1977: Axially symmetric steady-state models of the basic state for instability and climate studies. Part II: Nonlinear calculations. J. Atmos. Sci., 34, 280–297. , 1987: A simplified model of the modified Hadley circulation. J. Atmos. Sci., 44, 3311–3328. Sobel, A. H., and R. A. Plumb, 1999: Quantitative diagnostics of

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mixing in a shallow water model of the stratosphere. J. Atmos. Sci., 56, 2811–2829. Sutyrin, G. G., and G. R. Flierl, 1994: Intense vortex motion on the beta plane: Development of the beta gyres. J. Atmos. Sci., 51, 773–790. Waugh, D. W., and R. A. Plumb, 1994: Contour advection with surgery: A technique for investigating finescale structure in tracer transport. J. Atmos. Sci., 51, 530–540. Wirth, V., 1998: Thermally forced stationary axisymmetric flow on the f plane in a nearly frictionless atmosphere. J. Atmos. Sci., 55, 3024–3041.