transport and mixing in idealized barotropic hurricane

QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETYQ. J. R. Meteorol. Soc. 135: 1456–1470 (2009)Published online 28 July 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/qj.467

Transport and mixing in idealized barotropichurricane-like vortices

E. A. Hendricks*,† and W. H. SchubertColorado State University, Fort Collins, CO, USA

ABSTRACT: The effective diffusivity diagnostic is used to obtain basic insight into the two-dimensional transport andmixing properties of idealized barotropic tropical-storm and hurricane-like vortices. Three flow configurations believed tobe relevant to hurricane dynamics are examined in a non-divergent barotropic model: (i) an elliptical vortex, (ii) a Rankinevortex in a turbulent background vorticity field, and (iii) unstable vorticity rings. During the evolution of these vorticalflows, effective diffusivity is used as a mixing diagnostic on a passive tracer field that also evolves in the non-divergentflow. The internal dynamical processes causing mixing, as well as the location and magnitude of both turbulent mixingand partial barrier regions, are identified in the evolving vortices. Breaking vortex Rossby waves (VRWs) are found tocreate turbulent mixing regions of finite radial extent. For monotonic vortices, which are analogous to tropical storms,the wave breaking and axisymmetrization creates a surf zone outside the radius of maximum wind, while the vortex coreremains a partial barrier or containment vessel. For unstable vorticity rings, which are analogous to intensifying hurricanes,two regimes of internal mixing are found. During barotropic instability of thick rings, the inner and outer breaking VRWscreate two local mixing regions, separated by a partial barrier region at the location of the tangential jet. For barotropicinstability of thin rings, the entire hurricane inner core becomes a turbulent mixing region, allowing passive tracers tobe radially mixed between the eye, eyewall and local environment. In either case, the horizontal mixing associated withthe inner, breaking VRW would support intensification, provided the passive tracer is equivalent potential temperaturewith a maximum in the eye. In addition to the insights obtained for internal mixing in hurricanes, effective diffusivity isshown to be a robust diagnostic for two-dimensional turbulence, complementing its previous use in large-scale atmosphericdynamics. Copyright c© 2009 Royal Meteorological Society

KEY WORDS hurricane; effective diffusivity; Rossby-wave breaking; mixing; barrier

Received 8 July 2008; Revised 31 March 2009; Accepted 27 May 2009

1. Introduction

Although the intensification and decay of tropicalcyclones is partially controlled by factors such aslarge-scale vertical wind shear and moist entropy fluxfrom the underlying ocean, internal dynamical processesare also important and not clearly understood (for asuccinct review of our current understanding, see Wangand Wu, 2004). Some important internal processes arewave-mean flow interaction due to convectively coupledvortex Rossby waves (Montgomery and Kallenbach,1997; Moller and Montgomery, 1999), potential vorticity(PV) mixing between the eyewall and eye (Schubertet al., 1999; Kossin and Schubert, 2001; Montgomeryet al., 2002), vortex Rossby-wave breaking and innerspiral rainbands (Guinn and Schubert, 1993; Chen andYau, 2001; Wang 2008), and eyewall replacement cycles(Willoughby et al., 1982; Houze et al., 2007). Accuratepredictions of hurricane intensity are currently limitedby the lack of a comprehensive understanding of these

∗Correspondence to: Dr Eric A. Hendricks, Naval Research Laboratory,7 Grace Hopper Ave., Monterey, CA 93943, USA.E-mail: [email protected]†Current affiliation: Naval Research Laboratory, Marine MeteorologyDivision, Monterey, CA, USA.

processes. In particular, these internal processes may beimportant factors governing the rapid intensification andweakening of hurricanes. As an example of the potentialimportance of mixing processes in the hurricane innercore, Figure 1 shows the persistent multiple mesovorticesthat occurred in Hurricane Isabel (2003). It is generallyagreed that such mesovortices can play an important rolein the radial transport of PV in the hurricane core. Atopic of recent research and debate (Persing and Mont-gomery, 2003; Aberson et al., 2006; Braun et al., 2006;Montgomery et al., 2006; Cram et al., 2007; Bryan andRotunno, 2009) is whether such multiple mesovorticescan transport high moist entropy from the eye to theeyewall, and thereby explain why both real and modelhurricanes often have an intensity that is significantlyhigher than that predicted by the axisymmetric MPItheory of Emanuel (1986, 1988, 1995).

Mixing is due to the combined effect of differentialadvection and turbulent (or, inevitably, molecular) dif-fusion. Differential advection (i.e. stirring) stretchesand deforms material lines, from which diffusionaccomplishes true irreversible mixing. The interplaybetween advection and diffusion makes mixing dif-ficult to quantify. Recent work has proposed theuse of a hybrid Eulerian–Lagrangian area coordinate

Copyright c© 2009 Royal Meteorological Society

TWO-DIMENSIONAL MIXING IN HURRICANES 1457

Figure 1. Visible satellite image of Hurricane Isabel at 1315 UTCon 12 September 2003 (reproduced from Kossin and Schubert, 2004).Eye mesovortices such as these can efficiently transport quasi-passive

tracers such as cloud water between the eyewall and the eye.

(Butchart and Remsberg, 1986; Nakamura, 1996; Wintersand D’Asaro, 1996; Shuckburgh and Haynes, 2003) thatcombines the irreversible effects of diffusion with thereversible effects of advection (which is absorbed into thecoordinate). When transforming the advection–diffusionequation into the area coordinate, an effective diffusion(i.e. diffusion-only) equation is obtained with a diag-nostic coefficient (Nakamura, 1996) that quantifies theequivalent length of a tracer contour. As this equivalentlength becomes large, there is more interface for diffu-sion to act and the ‘effective diffusivity’ is larger. Thus,the effective diffusivity encompasses aspects of bothdifferential advection and diffusion. Shuckburgh andHaynes (2003) further demonstrated effective diffusivityto be a quantitative measure of transport and mixing,in their study of chaotic time-periodic flows. In recentwork, the effective diffusivity diagnostic has been usedto quantify transport and mixing properties in the uppertropophere and stratosphere (Haynes and Shuckburgh,2000a, 2000b; Allen and Nakamura, 2001; Scott et al.,2003, and references therein). Such work complementsthe previous use of Lyapunov exponents (Lapeyre, 2002)in large-scale transport and mixing (Pierrehumbert andYang, 1993; Ngan and Shepherd, 1999a, 1999b).

It is well known that geophysical vortices act astransport barriers. However, in local regions of thevortices and their near environment, strong mixing canoccur. For example, it has been shown that planetaryRossby waves may break on wintertime stratosphericvortices, creating a nonlinear critical layer (or surf zone),by which filaments can be extruded from the edge of thevortex (McIntyre and Palmer, 1983, 1984). These longfilamentary structures can then mix chemical speciesfrom the vortex to the midlatitudes (Waugh et al., 1994).In hurricanes, horizontal mixing due to vortex Rossbywave activity is occurring at smaller scales, helping to

determine the spatial distributions of both quasi-passivetracers (e.g. moist entropy or total airborne moisture)and active tracers (e.g. vorticity or potential vorticity).

Insight into transport and mixing processes in a fullphysics numerical model simulation of a hurricaneundergoing vertical shear has been obtained by Cramet al. (2007). An important finding of that work wasthat high equivalent potential temperature (θe) air wastransported from the eye into the eyewall, therebyincreasing the efficiency of the hurricane heat engine. Inthe present work, we seek to obtain a more basic under-standing of transport and mixing processes in hurricanesby considering idealized vortices in a non-divergentbarotropic model framework. While the non-divergentbarotropic model is an oversimplification of the realatmosphere, it can capture low-frequency (Rossby wave)advective dynamics of tropical cyclone evolution in adynamically clean framework. Numerical solutions tothe non-divergent barotropic vorticity equation and theadvection–diffusion equation are obtained with suitableinitial conditions, and the effective diffusivity diagnosticis used to quantify the transport and mixing propertiesof the following tropical-storm and hurricane-like flows:(i) an elliptical vorticity field, (ii) a Rankine vortexembedded in a turbulent background vorticity field, and(iii) unstable vorticity rings. These configurations havebeen shown to exhibit some interesting internal dynamicsrelevant to tropical cyclone evolution, such as secondaryeyewall formation, vortex Rossby-wave-breaking surfzones and PV mixing between the eye and eyewall. Whilethe vorticity dynamics involved in these scenarios hasbeen studied before, here we use the effective diffusivitydiagnostic to quantify their mixing properties, leading toa better understanding of both passive tracer transport inhurricanes and the internal dynamics governing hurricaneintensity change. The location and magnitude of strongpartial barriers (where the time-scale for transport islarge), weak partial barriers (where the time-scale fortransport is small) and mixing regions (where trajectoriesare chaotic) are identified in these vortices. Implicationsfor the evolution of passive tracers and their relationshipto intensity change are discussed in light of the results.

The outline of this article is as follows. In section 2 thedynamical model and passive tracer equation used for thisstudy are described. In section 3 we review the derivationof the transformation of the advection–diffusion equationinto the area coordinate and the equivalent radius coor-dinate, yielding the effective diffusivity diagnostic in aform useful for hurricane studies. In section 4 we presentpseudospectral model results for several types of mixingscenarios believed to be relevant in hurricane dynamics.In section 5 we document the relative insensitivity ofthe effective diffusivity diagnostic to certain arbitrarychoices made in its calculation from solutions of thepassive tracer equation. Finally, the main conclusions ofthis study are presented in section 6.

Copyright c© 2009 Royal Meteorological Society Q. J. R. Meteorol. Soc. 135: 1456–1470 (2009)DOI: 10.1002/qj

1458 E. A. HENDRICKS AND W. H. SCHUBERT

2. Dynamical model and passive tracer equation

The dynamical model used here considers two-dimensional, non-divergent motions on a plane. Thegoverning vorticity equation is

∂ζ

∂t+ u · ∇ζ = ν∇2ζ, (1)

where u = k × ∇ψ is the horizontal, non-divergentvelocity, ζ = ∇2ψ is the relative vorticity and ν is theconstant viscosity. The solutions presented here wereobtained with a double Fourier pseudospectral codehaving 768 × 768 equally spaced points on a doublyperiodic 600 km × 600 km domain. Since the code wasrun with a de-aliased calculation of the nonlinear termin (1), there were 256 × 256 resolved Fourier modes.The wavelength of the highest Fourier mode is 2.3 km.A fourth-order Runge–Kutta scheme was used for time-differencing, with a 3.5 s time step. The value of viscositywas chosen to be ν = 50 m2 s−1, so the characteristicdamping time for modes having total wavenumber equalto 256 is 2.4 h, while the damping time for modes havingtotal wavenumber equal to 170 is 5.5 h.

As a way to understand the transport and mixing prop-erties of an evolving flow described by (1), it is usefulto also calculate the evolution of a passive tracer subjectto diffusion and advection by the non-divergent velocityu. The advection–diffusion equation for this passivetracer is

∂c

∂t+ u · ∇c = ∇ · (κ∇c), (2)

where c(x, y, t) is the concentration of the passive tracerand κ is the constant diffusivity. The numerical methodsused to solve (2) are identical to those used to solve(1). However, the results to be presented here have quitedifferent initial conditions on ζ and c. The passive tracerc is always initialized as an axisymmetric and monotonicfunction. We have chosen both linear and Gaussian func-tions with maxima at the vortex centre. In contrast, theinitial vorticity is not necessarily monotonic with radius(e.g. it may have the form of a barotropically unstablevorticity ring) and is not necessarily axisymmetric.

3. Area coordinate transformation and effectivediffusivity

Following Nakamura (1996) and Shuckburgh and Haynes(2003), here we present a brief derivation of the effec-tive diffusion equation and resulting effective diffusivitydiagnostic used in this study. To aid in the derivation, adiagram of the area coordinate is shown in Figure 2. Con-sider the transform from Cartesian (x, y) coordinates totracer (C, s) coordinates, where C is a particular contourof the c(x, y, t) field and s is the position along that con-tour. Let dC be the differential element of C and ds bethe differential element of s. Let A(C, t) denote the area

Figure 2. Diagram of the area coordinate. Two hypothetical contoursC of the tracer field c(x, y, t) are shown with corresponding area abovethe contours A(C, t). The other parameters used in the derivation are

illustrated as well.

of the region in which the tracer concentration satisfiesc(x, y, t) ≥ C, i.e.

A(C, t) =∫∫

c≥C

dx dy. (3)

Let γ (C, t) denote the boundary of this region. Note thatA(C, t) is a monotonically decreasing function of C andthat A(Cmax, t) = 0. This guarantees a unique inversefunction C(A, t) that is monotonic and decomposableinto isolines. Now define uC as the velocity of thecontour C, so that

∂c

∂t+ uC · ∇c = 0. (4)

Noting that ∇c/|∇c| is the unit vector normal to thecontour, we can use (3) and (4) to write

∂A(C, t)

∂t= ∂

∂t

∫∫c≥C

dx dy

= −∫

γ (C,t)

uC · ∇c

|∇c|ds

=∫

γ (C,t)

∂c

∂t

ds

|∇c| .

(5)

Using (2) in the last equality of (5), we obtain

∂A(C, t)

∂t=

∫γ (C,t)

∇ · (κ∇c)ds

|∇c| −∫

γ (C,t)

u · ∇cds

|∇c| .(6)

We now note that (since dx dy = ds dC′/|∇c|)∂

∂C

∫∫c≥C

( ) dx dy = ∂

∂C

∫∫c≥C

( )ds dC′

|∇c|= −

∫γ (C,t)

( )ds

|∇c| ,(7)



where the minus sign on the right-hand side comes fromour arbitrary definition of A(C). Using (7) in (6) whilenoting that u · ∇c = ∇ · (cu) because u is non-divergent,we obtain

∂A(C, t)

∂t= − ∂

∂C

∫∫c≥C

∇ · (κ∇c)ds dC′

|∇c|+ ∂

∂C

∫∫c≥C

∇ · (cu)ds dC′

|∇c|= − ∂

∂C

∫γ (C,t)

κ|∇c|ds

+ ∂

∂C

∫γ (C,t)

cu · ∇c

|∇c|ds.

(8)

The third and fourth lines of (8) are obtained usingthe divergence theorem. The fourth line of (8) vanishesbecause the factor c in the integrand can come outsidethe integral, leaving

∫γ (C,t)

u · (∇c/|∇c|)ds, whichvanishes because u is non-divergent.

Since A(C, t) is a monotonic function of C, there existsa unique inverse function C(A, t). We now transform (8)from a predictive equation for A(C, t) to a predictiveequation for C(A, t). This transformation is aided by

∂A(C, t)

∂t

∂C(A, t)

∂A= −∂C(A, t)

∂t, (9)

which, when used in (8), yields

∂C(A, t)

∂t= ∂C(A, t)

∂A

∂

∂C

∫γ (C,t)

κ|∇c| ds

= ∂

∂A

∫γ (C,t)

κ|∇c| ds.

(10)

Because of (7), the integral∫γ (C,t)

κ|∇c| ds on theright-hand side of (10) can be replaced by (∂/∂C)∫∫

c≥Cκ|∇c|2 dx dy. Then (10) can be written in the form

∂C(A, t)

∂t= ∂

∂A

(Keff(A, t)

∂C(A, t)

∂A

), (11)

where

Keff(A, t) =(

∂C

∂A

)−2∂

∂A

∫∫c≥C

κ|∇c|2dx dy. (12)

To summarize, the area coordinate has been used totransform the advection–diffusion equation (2) into thediffusion-only equation (11), in the process yieldingthe effective diffusivity Keff(A, t). Since Keff(A, t) canbe computed from (12), it can serve as a useful diagnostictool to help us understand the interplay of advection anddiffusion in (2). However, note that, because of the useof A as an independent variable, the effective diffusivityKeff(A, t) has the rather awkward units of m4 s−1. Thisis easily corrected by mapping the area coordinate into

the equivalent radius coordinate, re,† which is definedby πr2

e = A. Thus, transforming (11) to the equivalentradius using 2πre(∂/∂A) = (∂/∂re), we obtain

∂C(re, t)

∂t= ∂

re∂re

(reκeff(re, t)

∂C(re, t)

∂re

), (13)

where

κeff(re, t) = Keff(A, t)

4πA. (14)

Note that, with of the use of re as an independent vari-able, the effective diffusivity κeff(re, t) has the units ofm2 s−1. The effective diffusivity can also be interpretedin terms of the equivalent length of tracer contours(Nakamura, 1996), i.e.

Le(re, t) =(

κeff(re, t)

κ

)1/2

2πre. (15)

As shown by Nakamura (1996), Le reduces to the actualperimeter of the tracer contour when |∇c| is constantalong that contour, thereby reducing (15) to κ = κeff.During differential advection, Le can become quite largeas tracer contours are stretched and folded, exposingmore interface for diffusion to act.

The effective diffusivity diagnostics Keff(A, t),κeff(re, t) and Le(re, t) can be calculated at a given time t

from the output c(x, y, t) of the numerical solution of (2).The calculation of Keff(A, t) involves the following dis-crete approximation of the right-hand side of (12). First,the desired number of area coordinate points is chosen(nA = 200 for the results shown here). The tracer contourinterval is set using �C = [max(c) − min(c)]/nA. Next,|∇c|2 is calculated at each model grid point. Then, a dis-crete approximation of the function A(C, t) is determinedby adding up the area within each chosen C contour, i.e.by using a discrete approximation to (3). The discreteapproximation to A(C, t) is then converted to a discreteapproximation of its inverse, C(A, t). The denominatorof the effective diffusivity diagnostic, (dC/dA)2, is calcu-lated by taking second-order finite differences of C(A, t).The numerator of the right-hand side of (12) is then calcu-lated in the same manner, which completes the calculationof the effective diffusivity Keff(A, t). The remainingeffective diffusivity diagnostics κeff(re, t) and Le(re, t)

are then easily computed using (14)–(15). As will beshown, plots of these diagnostics reveal the locations ofpartial barrier and mixing regions within the vortex.

In the next section, two-dimensional plots of effectivediffusivity will be shown. This can be done becauseeffective diffusivity is constant along a tracer con-tour, and tracer contours meander in (x, y) space.

†It is important to note that re is a coordinate based on vorticityrearrangement, and therefore it does not necessarily correspond exactlyto the actual radius from the centre of the flow structure. However,it will be shown to be a reasonable approximation for our idealizedcases here, because the flow is mostly circumferential and has a naturalcentre.



From another point of view, κeff(re, t) can be mappedto κeff(x, y, t) because each horizontal grid point isassociated with an equivalent radius. Note, however, thatby using the area coordinate the dimensionality has beenreduced, so that these plots are effectively only showingone-dimensional information.

4. Pseudospectral model experiments and results

We now use the effective diffusivity diagnostic to under-stand the transport and mixing properties of a number offlows believed relevant to hurricane dynamics. The casesselected here are (i) an elliptical vortex, (ii) a Rankine-like vortex embedded in a random turbulent vorticity fieldand (iii) unstable vorticity rings. In the first two cases, theinitial vorticity is maximum at the vortex centre, whichis more characteristic of tropical storms rather than fullydeveloped hurricanes. In the third case, the initial vor-ticity is maximum in the ring, which is characteristic ofstrong or intensifying hurricanes. All of the experimentsare unforced and exhibit properties of two-dimensionalturbulence, in particular the preferential decay of enstro-phy over kinetic energy. In the following subsections, theinitial condition and parameters for each experiment areshown, and the results are presented and discussed.

4.1. Elliptical vortex

The initial elliptical vortex is constructed in a mannersimilar to Guinn (1992). In polar coordinates, the initialvorticity field is specified by

ζ(r, φ, 0) = ζ0

1, 0 ≤ r ≤ riα(φ),

1 − fλ(r′), riα(φ) ≤ r ≤ roα(φ),

0, roα(φ) ≤ r,

(16)

where α(φ) = [(1 − ε2)/{1 − ε2 cos2(φ)}]1/2 is an ellip-ticity augmentation factor for the ellipse (x/a)2 +

(y/b)2 = 1 (where a is the semi-major axis and b is thesemi-minor axis) with eccentricity ε = {1 − (b2/a2)}1/2.Here, ζ0 is the maximum vorticity at the centre, fλ(r

′) =exp[−(λ/r ′) exp{1/(r ′ − 1)}] is a monotonic shape func-tion with transition steepness parameter λ, r ′ = [r −riα(φ)]/[roα(φ) − riα(φ)] is a non-dimensional radius,and ri and ro are the radii where the vorticity beginsto decrease and where it vanishes, respectively. For thespecial case of α(φ) = 1 the field is axisymmetric. Forthe experiment conducted, ζ0 = 5.0 × 10−3 s−1, λ = 2.0,ε = 0.70, and ri and ro were set to 30 and 60 km,respectively. The initial tracer field was Gaussian: c(r) =c0 exp (−r/ra)

2, with c0 = 1000 and ra = 50 km.Plots of vorticity and effective diffusivity κeff at

t = 1.33 h during the evolution of the elliptical vortexare shown in Figure 3. At this time, two filaments ofhigh vorticity associated with breaking VRWs are clearlyvisible. Associated with these filaments are regions oflarge effective diffusivity. The effective diffusivity peaksjust upwind of the filaments and extends further upwind.The main vortex acts as a transport barrier during thefilamentation. In terms of an arbitrary passive tracer,these results indicate that the tracer will tend to be well-mixed horizontally in the wave-breaking surf zone, whiletracers initially in the vortex core will be trapped there.During its evolution, continued wave-breaking episodesoccur as the ellipse tries to axisymmetrize. However,axisymmetrization is not complete for t ≤ 48 h, andthe surf zone is a robust feature throughout the entiresimulation (not shown). The ability of an elliptical vortexto axisymmetrize (Melander et al., 1987) via invisciddynamics was shown to be determined by the sharpnessof its edge (Koumoutsakos, 1997; Dritschel, 1998). Ifthe vortex is more Rankine-like (i.e. possessing a sharpedge), it will tend to rotate and not generate filaments. If,on the other hand, the transition is not sharp, there willbe a tendency to generate filaments and axisymmetrize.

Figure 3. The relative vorticity ζ and the effective diffusivity κeff for the evolution of the elliptical vortex at t = 1.33 h.



Although it occurs on much smaller time- and length-scales, there is an analogy between this surf zone in tropi-cal cyclones and the planetary Rossby-wave-breaking surfzone associated with the wintertime stratospheric polarvortices (McIntyre and Palmer, 1983, 1984, 1985; Juckesand McIntyre, 1987; McIntyre, 1989; Bowman, 1993;Waugh et al., 1994). Planetary waves excited in the tropo-sphere may propagate vertically and cause wave breakingto occur on the edge of the stratospheric polar vortex,from which chemical constituents can be mixed into themidlatitudes. The wintertime stratospheric polar vorticesdisplay similar processes to our experiment: namely, thecore vortex is a transport barrier and the surf zone is achaotic mixing region. The existence of the main vortexbarrier was thought to be due to the strong PV gra-dient, a restoring mechanism for perturbations imposedupon it. Rossby-wave breaking has also been examinedin more idealized frameworks (see, for example, Polvaniand Plumb, 1992; Koh and Plumb, 2000).

In tropical cyclones, the deformation of an initiallycircular vortex core to an ellipse may happen due toexternal (e.g. vertical shear) or internal (e.g. PV gener-ation by asymmetric moist convection) processes. Therelaxation to axisymmetry will produce wave-breakingepisodes and, as we have shown here, moderate mixingregions in the associated surf zone.

4.2. Rankine vortex in a turbulent vorticity field

A Rankine vortex in a stirred vorticity field may berepresented mathematically by

ζ(x, y, 0) = ζm

1, 0 ≤ r ≤ r1,

S =(

r−r1r2−r1

), r1 ≤ r ≤ r2,

0, r2 ≤ r,

+ ζrand (x, y)

1, 0 ≤ r ≤ r3,

S(

r−r3r4−r3

), r3 ≤ r ≤ r4,

0, r4 ≤ r,

(17)

where ζm = 5 × 10−3 s−1 is the maximum vorticity ofthe Rankine vortex, S(x) = 1 − 3x2 + 2x3 is a cubicpolynomial shape function providing smooth transitionsfrom r1 = 20 km to r2 = 30 km and from r3 = 120 kmto r4 = 180 km, and ζrand(x, y) is a spatially randomturbulent vorticity field given by

ζrand(x, y) =kmax∑

k=−kmax

max∑ =− max

ζk, ei(2π/L)(kx+ y). (18)

Here, kmax and max are the spectral truncation limits in x

and y, L is the domain length, ζk, is random with a maxi-mum amplitude of 1.5 × 10−5 s−1 and the total wavenum-ber κ = (k2 + 2)1/2 is set for spatial scales primarilybetween 20 and 40 km. The initial tracer field was linear:c(r) = c0 + ar , where c0 = 1000 and a = −4.422 km−1.

As an analogy to real tropical cyclones, the Rankine-like vortex can be thought of as the tropical cyclone coreand the stirred vorticity field can be thought of as gener-ated by random convection. The initial condition for thisexperiment is shown in the top panel of Figure 4. As thesimulation evolves, the core vortex begins to axisym-metrize the random vorticity elements. At t = 9.5 h thecore vortex starts to become a partial barrier region.Also note that a thin partial barrier ring begins to format the radius of maximum winds just outside the centralvorticity core. Outside the vortex core, chaotic mixingis occurring as the random vorticity anomalies are beingaxisymmetrized by the shearing flow. By t = 40.0 h(bottom panels) the relative vorticity exhibits a centralmonopole, a low-vorticity moat and a secondary ring ofenhanced vorticity. Comparing the two bottom panels,the low-vorticity moat coincides with the ring of mod-erate effective diffusivity (100 ≤ κeff ≤ 500 m2 s−1). Inreal tropical cyclones, the moat region is a region of sup-pressed convective activity due to the combined effectsof subsidence (induced by return flow from the eyewall)and strain-dominated flow (Rozoff et al., 2006). Themoat here was identified as a region of enhanced mixing.The secondary ring of enhanced vorticity coincides withthe ring of low effective diffusivity (κeff ≤ 100 m2 s−1).The azimuthal mean wind (not shown) associated withthe bottom left panel of Figure 4 has two maxima. Thefirst is the primary azimuthal jet located at the edge ofthe central vorticity monopole, and the second is thesecondary azimuthal jet that occurs at the outer edge ofthe secondary ring of enhanced vorticity. In the effectivediffusivity plot, these jets are partial barriers (whiterings) with κeff ≤ 100 m2 s−1. Therefore, in this idealizedexperiment, azimuthal jets in hurricanes are shown to bepartial transport barriers, resistant to radial mixing.

4.3. Unstable vorticity rings

Five experiments were conducted for different dynami-cally unstable hurricane-like vortices. The initial vorticityfield consists of a vorticity ring (the eyewall) and arelatively low-vorticity centre (the eye). Observations(Kossin and Eastin, 2001; Mallen et al., 2005) indicatethat strong or intensifying hurricanes are often charac-terized by such vorticity fields. The average vorticityover the inner core was set to be ζav = 2.0 × 10−3 s−1,corresponding to a peak tangential wind of approximately40 m s−1 in each case.

The initial condition on the vorticity is given in polarcoordinates by ζ(r, φ) = ζ (r) + ζ ′(r, φ), where ζ (r) isan axisymmetric vorticity ring defined by

ζ (r, 0) =

ζ1, 0 ≤ r ≤ r1,

ζ1S(

r−r1r2−r1

)+ ζ2S

(r2−r

r2−r1

), r1 ≤ r ≤ r2,

ζ2, r2 ≤ r ≤ r3,

ζ2S(

r−r3r4−r3

)+ ζ3S

(r4−r

r4−r3

), r3 ≤ r ≤ r4,

ζ3, r4 ≤ r ≤ ∞,

(19)



(a)

(b) (c)

(d) (e)

Figure 4. The initial vorticity field (a) and side-by-side panels ((b)–(e); at 9.5 h and 40 h) of relative vorticity and effective diffusivity for theRankine-like vortex in a turbulent vorticity field. The model domain is 600 km by 600 km, but only the inner 200 km by 200 km is shown.



Table I. Unstable vorticity ring parameters: ζ values are in10−3 s−1 and r values are in km.

Exp. ζ1 ζ2 r1 r2 r3 r4 δ γ

A 0.8 2.7 22 26 38 42 0.60 0.40B 0.0 3.1 22 26 38 42 0.60 0.00C 0.0 4.6 28 32 38 42 0.75 0.00D 0.0 7.2 32 36 38 42 0.85 0.00E 0.2 6.7 32 36 38 42 0.85 0.10

where ζ1, ζ2, ζ3, r1, r2, r3 and r4 are constants, and S(x)

is the cubic polynomial interpolation function definedpreviously. The eyewall is defined as the region betweenr2 and r3. Schubert et al. (1999) defined two parametersto describe these hurricane-like vorticity rings: a ringthickness parameter δ = (r1 + r2)/(r3 + r4) and a ringhollowness parameter γ = ζ1/ζav. The parameters usedfor each of the five experiments are listed in Table I.Note that ζ3 is set to be slightly negative so that thedomain-averaged vorticity is equal to zero. Each ring isperturbed with a broadband impulse of the form

ζ ′(r, φ, 0) = ζamp

8∑m=1

cos(mφ + φm)

×

0, 0 ≤ r ≤ r1,

S(

r2−r

r2−r1

), r1 ≤ r ≤ r2,

1, r2 ≤ r ≤ r3,

S(

r−r3r4−r3

), r3 ≤ r ≤ r4,

0, r4 ≤ r ≤ ∞,

(20)

where ζamp = 1.0 × 10−5 s−1 is the amplitude and φm

the phase of azimuthal wavenumber m. For this setof experiments, the phase angles φm were chosen tobe random numbers in the range 0 ≤ φm ≤ 2π. In realhurricanes, such asymmetries are expected to developfrom a wide spectrum of background turbulent andconvective motions. The initial tracer field used was anaxisymmetric cone: c(r) = c0 + ar , where c0 = 1000and a = −4.422 km−1.

Two simulations from Table I are illustrated. The first(Exp. A) is a thick, filled ring, while the second (Exp. D)is a thin, hollow ring. According to Schubert et al. (1999),as the rings become thicker and filled disturbance growthrates become smaller and at lower wavenumber. As therings become very thin and hollow, they rapidly breakdown and sometimes evolve into persistent mesovortices(Kossin and Schubert, 2001). Experiment A is shown inFigure 5. At t = 13.0 h (middle left panel), the ring isbreaking down at azimuthal wavenumber m = 4, givingthe appearance of a polygonal eyewall with straight-linesegments. Note that both the inner and outer VRWs arebreaking at this time. The breaking of the inner VRWhas allowed vorticity to be pooled into four regions. Inthe effective diffusivity plot (middle right panel) thereare two distinct radial regions of mixing, separated by a

rather strong, thin barrier region. The inner mixing regionis at the location of the inner breaking VRW, while theouter mixing region exists at the location of the outerbreaking VRW. The waves are phase-locked and helpingeach other grow, resulting in radial air movement andmixing. During this time the passive tracer field becomesrelatively well mixed in the radial intervals of the VRWactivity. However, the initial gradient is maintained inthe barrier region in between (not shown). Progressing tot = 41.0 h, the magnitude of the mixing due to the waveactivity is smaller, but the barrier region remains intact.

The breakdown of the ring in Exp. D is shown inFigure 6. The disturbance growth rates are much largerin this case, allowing the ring to break down much faster.Multiple mesovortices initially form (middle left panel).During the formation stage, these mesovortices and asso-ciated filamentary structures are characterized as strongmixing regions by the diagnostic (middle right panelof Figure 6). This was a surprising result, since coher-ent vortices are known to be partial barriers. Howeverit is possible that the enhanced effective diffusivity isa result of mixing that is occurring at smaller scalesinside the mesovortices just after their formation (how-ever there is some uncertainty whether this is real oran artefact of the choice of the initial tracer profile–seethe discussion in section 5.2). The mesovortices persistfor a very long time, and at t = 20.0 h there are threemesovortices left after some mergers have occurred. Atthis time, the passive tracer has been homogenized locallyinside the mesovortices, causing low effective diffusivi-ties there. It is also important to note that the appear-ance that the vorticity field is more compact than theeffective diffusivity field in these plots is due to thearbitrary cut-off contour of the vorticity, 5.0 × 10−4 s−1.There is vorticity mixing occurring outside the mesovor-tices contributing to large effective diffusivities in, forexample, the bottom right panel of Figure 5. Basedon these results, in conjunction with the Rankine-likevortex in a turbulent vorticity field, we surmise thatbarotropic geophysical vortices of all horizontal scalestend to act as partial barrier regions when they are long-lived; however, strong mixing can occur in their forma-tive stages.

To illustrate further the two regimes of internal mix-ing, Hovmuller plots of κeff(re, t) are shown in Figure 7for Experiments A and D. For the A ring (left panel),there exist two distinct mixing regions at 20 km ≤ re ≤30 km and 40 km ≤ re ≤ 55 km. These mixing regionsare associated with the counter-propagating VRWs evi-dent in the middle panels of Figure 5. For the D ring,in which a rapid breakdown occurs, the entire hurricaneinner core (10 km ≤ re ≤ 60 km) is a chaotic mixingregion. These two types of mixing regimes are furtherclarified in Figure 8, which shows the time-averagedeffective diffusivity κeff for all five rings. For the ringswith slower growth rates (A and B) there exist two peaksin κeff(re) coincident with inner and outer VRW activ-ity. For the rings with faster growth rates (C, D andE), the entire inner core is a chaotic mixing region.



(a)

(b) (c)

(d) (e)

Figure 5. The initial vorticity field (a) and side-by-side panels ((b)–(e); at 13 h and 41 h) of relative vorticity and effective diffusivity κeff for aprototypical thick, filled unstable vorticity ring (Experiment A of Table I).



(a)

(b) (c)

(d) (e)

Figure 6. The initial vorticity field (a) and side-by-side panels ((b)–(e); at 6 h and 20 h) of relative vorticity and effective diffusivity κeff for aprototypical thin, hollow unstable vorticity ring (Experiment D of Table I).



Figure 7. Hovmuller plots depicting the temporal evolution of κeff(re, t) for the A ring (left) and the D ring (right).

Figure 8. Time-averaged (0–48 h) effective diffusivity as a function ofequivalent radius for all unstable rings. The radius of maximum windvaries during the evolution but generally lies in the region between

30 and 40 km.

During the evolution of each ring, the radius of maxi-mum wind varies, but it is generally confined to radiibetween 30 and 40 km. Thus for thick, filled rings thehurricane tangential jet acts as a partial barrier regionfor t ≤ 48 h, while for thin, hollow rings the hurricanetangential jet breaks down and turbulent mixing in theentire inner core ensues. The implication of this result forreal hurricanes is that if the eyewall is very thick, pas-sive tracers will not easily be mixed across the eyewallduring barotropic instability, but may be mixed betweenthe eye–eyewall and environment–eyewall by the innerand outer breaking VRWs, respectively. If, on the otherhand, the eyewall is thin, as in rapidly intensifying hur-ricanes (Kossin and Eastin, 2001), passive tracers can bemixed across the eye, eyewall and environment, and at amuch faster rate. Assuming hurricanes have a maximumof equivalent potential temperature (θe) at low levels inthe eye, our results indicate that the inner, breaking, VRWwill mix air parcels with high θe into the eyewall, sup-porting intensification and the hurricane superintensity

mechanism concept (Persing and Montgomery, 2003).This mixing will be more rapid for the breakdown ofthin rings.

The mixing regime in which the tangential jet acts asa partial barrier is analogous to the results of Bowmanand Chen (1994), who found that air poleward of abarotropically unstable stratospheric jet remained nearlyperfectly separated from midlatitude air. Our hurricaneresults are again analogous to planetary-scale mixing, andit appears that under certain conditions azimuthal jets inhurricanes can become asymmetric but still remain partial(but leaky) barriers to radial mixing.

One may wonder how these results generalize to theentire δ–γ plane, which covers all possible annularvorticity structures in two dimensions. From a vorticitydynamics perspective, Hendricks et al. (2009) showedthat the most vigorous mixing episodes occurred withthin, hollow rings, while only minor mixing occurredin thick and filled rings. These results would probablygeneralize in a similar manner, with complete mixingbetween the eye, eyewall and local environment occur-ring for thin, hollow rings, and partial barriers existingfor thicker and more filled rings. Interestingly, thatstudy documented large pressure drops for the barotropicinstability of thin, hollow rings. Considering that κeff isapproximately twice as large for breakdowns of thin ringsas for thick rings (4000 versus 2000 m2 s−1, respectively,Figure 8), the high θe air in the low-level eye may bemixed into the eyewall at a much faster rate for thinrings, thereby accentuating the intensification process.

5. Sensitivity tests

In order to assess the robustness of effective diffusivity asa diagnostic of the mixing properties of a flow, a numberof sensitivity tests were conducted: (i) tracer diffusioncoefficient, (ii) initial tracer distribution and (iii) the accu-racy of the discrete approximation to the diagnostic (12).



Figure 9. Effective diffusivity versus equivalent radius for varyingvalues of the tracer diffusivity κ (units of m2 s−1) for the C unstable

ring experiment at t = 6.3 h.

5.1. Tracer diffusion coefficient

In the area-based coordinate system, it is expected thatthe effective diffusivity will increase with increasingtracer diffusivity κ . As material lines are stretched andfolded there exists more interface for diffusion to produceirreversible mixing, and if the diffusion coefficient islarger then the level of mixing should be larger, as areacan diffuse faster between tracer contours. This is clearlyillustrated in Figure 9 for the unstable ring experiment C.This experiment is similar to Experiment D in that alarge radial segment becomes a chaotic mixing region(Figure 7, right panel). Four different values of the tracerdiffusivity are chosen: κ = 50, 25, 10 and 0.1 m2 s−1. Thelarger tracer diffusivities clearly have larger effectivediffusivities, and the radial character of the profiles isbroadly preserved for each case. For example, the κ =50, 25 and 10 m2 s−1 cases are able to capture the peakeffective diffusivity at re = 30 km. The κ = 0.1 m2 s−1

case is not seen on the figure because the peak effectivediffusivity associated with it is only κeff = 20 m2 s−1,too low to be visible with the plot scaling. The sameplot is shown in Figure 10 for the normalized effectivediffusivity κeff(re, t)/κ . Note that the normalized effective

Figure 10. Normalized effective diffusivity κeff(re, t)/κ versus equiva-lent radius for varying values of the tracer diffusivity κ (units of m2 s−1)

for the C unstable ring experiment at t = 6.3 h.

Figure 11. Sensitivity of effective diffusivity to the initial tracer field.Three curves are shown: a Gaussian profile with a maximum value of1000 and linearly decreasing profiles with maximum values of 1000

and 5000.

diffusivity is not very sensitive to varying κ , and thereforeit is the best measure of chaotic advection.

5.2. Initial tracer distribution

Since effective diffusivity maps out the mixing proper-ties of a flow, it is supposed to be mostly insensitive tothe initial tracer field, provided it is monotonic and wellbehaved. In order to illustrate this, plots of effective dif-fusivity versus equivalent radius are shown in Figure 11for three different initial axisymmetric tracer fields: aGaussian distribution with maximum value of 1000 (usedin the elliptical vorticity field and Rankine-like vortex ina turbulent vorticity field) and linearly decreasing dis-tributions with maximum values of 1000 (used in theunstable vorticity ring experiments) and 5000. Each ofthese curves has different dC/dre (or dC/dA), used inthe denominator of the effective diffusivity diagnostic.The κeff profiles are almost identical for the two linearcases, and only a slight variation is found for the Gaus-sian case. The Gaussian case departs from the linear casesslightly at small radii. The likely reason for this is thatthe slope of the tangent line (dC/dre) is very small there,causing the effective diffusivity diagnostic to be unre-alistically distorted. We feel that the linearly decreasinginitial tracer profile is the best to use because it guaranteesconstancy of the initial dC/dA in the domain. Overall,effective diffusivity is found to be mostly insensitive tothe initial tracer profile, provided it is monotonic.

The sensitivity of these results to non-monotonic tracerprofiles was also examined. Three additional experimentswere conducted for unstable rings A and D and theRankine vortex in which the initial tracer field wasexactly the same as the initial vorticity field. For exper-iment A, the partial barrier in the middle right panelof Figure 5 still existed, however there also existed athin polygonal region where κeff was large on the innerside of this barrier region. The four inner vorticity poolshad elevated effective diffusivity, consistent with thecontrol experiment. For experiment D, the mesovorticesat t = 6 h (middle panels of Figure 6) had anomalously



low (rather than high) effective diffusivities, consistentwith previous findings of small Lyapunov exponents incoherent vortices. Considering this finding, there existsuncertainty in whether these mesovortices are partialbarriers or mixing regions at this time. At the latertime t = 20 h in this experiment, the mesovortices wereidentified as partial barriers, consistent with the controlexperiment (bottom right panel of Figure 6). For theRankine vortex in a turbulent vorticity field, the coarse-grained results were largely unchanged, in that the vortexcore became a partial barrier in time while mixing wasoccurring outside. However, at t = 40 h this experimentproduced a stronger mixing region at the location of themoat and it also did not capture the partial barrier regionat the location of the secondary wind maximum. Basedon these results, there exists a rather strong sensitivity ofeffective diffusivity to the initial tracer field if it is notmonotonic and broadly distributed. However, in thesecases this method also caused dC/dA in (12) to be verysmall outside the vorticity cores. For the unstable rings,this created physically unrealistic large effective diffu-sivities there, which is clearly wrong since those tracercontours were not disturbed. These results are consistentwith Shuckburgh and Haynes (2003), who found a strongsensitivity to the initial tracer profile in regions wherethe spatial gradient was approximately zero (see theirsection III.B.2). The conclusion of these tests is that,while it may be desirable to make the initial tracer fieldexactly the same as the initial vorticity field, ultimatelyin certain instances (such as the vorticity configurationshere) the effective diffusivity diagnostic is not as reliablemacroscopically for the reasons stated above.

5.3. Number of area points

Sensitivity tests were performed using varying numbersof area points in the discrete approximation to the effec-tive diffusivity diagnostic (12). To illustrate the sensitivityto the discrete approximation, the effective diffusivityversus equivalent radius is shown in Figure 12 for theC unstable ring with nA = 50 and 200 (used in all theexperiments in this article). Both curves produce peaks

Figure 12. Sensitivity of effective diffusivity to the number of areapoints: 50 and 200. The case shown is the C unstable ring at t = 6.3 h.

near re = 30 km, but only the nA = 200 curve capturesthe peak value of κeff = 5000 m2 s−1 at re ≈ 35 km andthe dip at re = 40 km. The conclusion is that our choicenA = 200 is sufficient to resolve the mixing variabilityof the inner core for our chosen model resolution.

6. Conclusions

The two-dimensional mixing properties of non-divergentbarotropic flows resembling the idealized evolution ofboth tropical storms and hurricanes were quantified usingthe effective diffusivity diagnostic. The location and mag-nitude of both turbulent mixing and partial barrier regionswere identified, yielding insight into how passive trac-ers are asymmetrically mixed at low levels. The primaryfinding is that breaking vortex Rossby waves (VRWs),resulting from either axisymmetrization or dynamic insta-bility, are quite effective at mixing passive tracers overlarge horizontal distances in hurricanes, and that this mix-ing is likely an important internal mechanism of intensitychange.

For monotonic vortices that have profiles similar totropical storms, the wave breaking and mixing was gen-erally confined to outside of the radius of maximumwind. For the elliptical vortex, a 20–30 km wide surfzone existed which was characterized by turbulent mix-ing. For the Rankine vortex inside a convective vorticityfield, strong mixing occurred as the random vorticityelements were axisymmetrized. In both of these cases,the centre of the storm was a partial barrier, or contain-ment vessel, and air was not easily mixed with the localenvironment. For unstable rings, which are analogous tostrong or intensifying hurricanes, both the inner and outercounter-propagating VRWs break due to reinforcementfrom barotropic instability, causing two mixing regions:one between the eye and eyewall and one between theeyewall and local environment. In the case of thick rings,the disturbance growth rates are small and a long-livedasymmetric partial barrier region may exist between thetwo breaking waves, coincident with the tangential jet.In the case of thin rings that are very dynamically unsta-ble, the rapid breakdown created a strong chaotic mixingregion over the entire hurricane inner core (eye, eyewalland local environment). In this case, passive tracers maybe horizontally mixed over large radial distances (approx-imately 60–80 km in our experiments). Since observa-tions show a maximum of θe at low levels in the eye(Eastin et al. 2002a, 2002b), our results indicate that theinner, breaking VRW would be quite effective at mix-ing air parcels with high θe into the eyewall, causingintensification. For thick rings, the centre of the hurricaneeye remains a partial barrier during barotropic instabilitybecause turbulent mixing associated with the inner, break-ing VRW is confined to the outer eye. In these cases, itis possible that the highest θe air may never be mixedinto the eyewall, limiting the level of intensification viainternal mixing.

Both primary and secondary azimuthal wind maximawere identified as partial barriers in our simulations.



These jets act as mixing barriers because they are locatednear regions of strong radial PV gradients (cf. McIntyre,1989). A surprising result is that the primary jet barrierregion can be maintained for long times during barotropicinstability. Additionally, it can maintain itself as a partialbarrier when it is deformed asymmetrically to a polygonwith straight-line segments. Therefore, in this simpleframework the hurricane primary azimuthal jet appearsto be a robust transport barrier for both dynamicallystable and unstable vortices, provided the latter vorticesare characterized by thick annular vorticity structures.Whether or not this jet is such a robust barrier in realhurricanes, where moist processes and environmentalvertical shear are active, is an open question.

One way to extend the present work would be to useeffective diffusivity as a diagnostic for transport andmixing in axisymmetric hurricane models (Rotunno andEmanuel, 1987; Hausman et al., 2006). Of particularinterest would be examining the vertical structure ofmixing between the eye and the eyewall. The diagnosticcould also be used to understand aspects of the transportand mixing of water vapour in the frictional boundarylayer of hurricanes.

Acknowledgements

This research was supported by NSF Grants ATM-0530884 and ATM-0332197. We thank Gerhard Dan-gelmayr, Christopher Davis, Arthur Jamshidi, RichardJohnson, Michael Kirby, Brian McNoldy, Michael Mont-gomery, Kate Musgrave, John Persing, Roger PielkeSr, Christopher Rozoff, Blake Rutherford, Emily Shuck-burgh, Richard Taft and Jonathan Vigh for their commentsand assistance. This manuscript was improved by thehelpful comments of three anonymous reviewers.

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