Recent Developments in the Fluid Dynamics of Tropical …

FL49CH22-Montgomery ARI 21 November 2016 8:6

Recent Developmentsin the Fluid Dynamicsof Tropical CyclonesMichael T. Montgomery1 and Roger K. Smith2

1Department of Meteorology, Naval Postgraduate School, Monterey, California 93943;email: [email protected] Institute, Ludwig Maximilians University of Munich, 80333 Munich, Germany

Annu. Rev. Fluid Mech. 2017. 49:541–74

The Annual Review of Fluid Mechanics is online atfluid.annualreviews.org

This article’s doi:10.1146/annurev-fluid-010816-060022

Copyright c© 2017 by Annual Reviews.All rights reserved

Keywords

tropical cyclone intensification, rotating deep convection, boundary layercontrol, size growth, potential intensity, quasi-steady-state hurricanes

Abstract

This article reviews progress in understanding the fluid dynamics and moistthermodynamics of tropical cyclone vortices. The focus is on the dynamicsand moist thermodynamics of vortex intensification and structure. We dis-cuss previous ideas on many facets of the subject and articulate also someopen questions. The advances reviewed herein provide new insight and toolsfor interpreting complex vortex-convective phenomenology in simulated andobserved tropical cyclones.

541

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ANNUAL REVIEWS Further

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FL49CH22-Montgomery ARI 27 January 2017 19:2

1. INTRODUCTION

Tropical cyclones are fascinating large-scale, organized, convective vortices that continue to holdmany scientific secrets regarding their birth, intensification, mature evolution, and decay. Thesemoist convective vortices comprise arguably all facets of classical fluid dynamics, ranging fromthe microscale flow in and around small droplets, coalescence of smaller droplets into larger ones,and precipitation and evaporation processes to the larger scales of buoyant thermals in a rotatingenvironment, their aggregate effects on the vortex circulation, and the even larger scale of vortexwaves and eddies, such as inertia-buoyancy waves, vortex Rossby waves, eyewall mesovortices, andtheir interaction with the vortex circulation. The large Reynolds numbers of these flows implythat turbulence of the Kolmogorov kind will be an element at small scales, but the presence ofstrong, spatially variable, vertical rotation in these systems suggests that quasi–two dimensional(quasi-2D) fluid dynamics and its associated turbulence phenomenology should be importantelements also with modifications due to the presence of deep moist convection, which is intrinsi-cally 3D. The more intense manifestations of these vortices (maximum near-surface wind speed>32 m s−1) are called hurricanes in the Atlantic and eastern Pacific basins and typhoons in thewestern North Pacific region.

Although flow speeds (typically <100 m s−1), are well below the sound speed, nonconservativeeffects, principally associated with friction at the ocean surface and wind-forced transfer of moistureand heat from the warm sea, make tropical cyclones a particularly interesting and challengingscientific problem to understand. Practical considerations, such as saving human lives and propertyin the path of these storms, are another important driving factor in the quest for more knowledgeabout them. Atlantic Hurricane Sandy (2012) is a reminder that even tropical storms (maximumnear-surface wind speed <32 m s−1) can wreak havoc on populated coastal communities, maritimeassets, and even inland populations (e.g., Lussier et al. 2015). As coastal communities continueto grow in tropical cyclone–affected regions, there is an increasing demand for more accuratetropical cyclone forecasts.

There are two main aspects of the forecasting problem: forecasting the storm track and fore-casting its intensity, typically characterized by the maximum near-surface wind speed. Trackforecasts have improved significantly in the past 25 years, but progress in intensity forecasting hasshown comparatively little improvement (DeMaria et al. 2005, Rogers et al. 2006). Because thetrack depends mainly on the large-scale flow in which the vortex is embedded, the improvement intrack forecasting may be attributed largely to the improvement in the representation of large-scaleflow around the vortex by global forecast models. In contrast, the intensity appears to depend onprocesses of wide-ranging scales spanning many orders of magnitude, as noted above.

Because of the challenges of forecasting the intensity change of tropical cyclones, the problem ofunderstanding how intensity change occurs has been at the forefront of tropical cyclone researchin recent years, especially in the context of the rapid intensification or decay of storms. Thesechallenges are motivated by the recently instigated Hurricane Forecast Improvement Projectby the National Oceanic and Atmospheric Administration (NOAA) and other US governmentagencies to coordinate hurricane research necessary to accelerate improvements in hurricanetrack and intensity forecasts (Gall et al. 2013).

There have been significant advances in understanding tropical cyclone behavior since theearlier reviews by Emanuel (1991) and Chan (2005), and the field has broadened significantly.Because of the breadth of the field, we focus on the dynamics and thermodynamics of the vortexwhen viewed as a coherent structure with embedded substructures. To begin, for those readersworking in other fields, we review briefly in Section 2 the equations of motion and some other basicconcepts involving zero-order force balances, moist thermodynamics, and deep convective cloudsin a rotating environment. This material provides a reference for much of the later discussion.

542 Montgomery · Smith

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Section 3 surveys progress made in understanding tropical cyclone intensification and structurefrom the perspective of the prototype intensification problem, which considers for simplicitythe spin-up of an initially balanced, axisymmetric, cloud-free, conditionally unstable, baroclinicvortex of near–tropical storm strength in a quiescent tropical environment on an f plane. Here,paradigms for vortex intensification, including emerging ideas pointing to the importance ofboundary layer control in vortex evolution, are discussed. Section 4 examines more deeply therole of cloud-generated vorticity in supporting vortex spin-up. Progress in understanding maturevortex intensity is reviewed in Section 5, and the steady-state problem is reviewed in Section 6.Section 7 concludes.

Owing to space constraints, we are unable to review aspects of vortex motion, vortex Rossbywaves, and their contribution to vortex resilience, nor the early stages of storm formation. For thesame reason, we do not address topics such as the interaction of storms with ambient vertical shear,helicity, secondary eyewall formation, ocean feedback effects, the interaction with neighboringweather systems including fronts and upper troughs, the extratropical transition when stormsmove into the middle latitudes, cloud microphysics, boundary layer rolls, wind wave coupling,and details of the surface layer (or emulsion layer).

2. PRELIMINARIES

To provide a common framework for this review, we present the equations of motion pertinentto understanding tropical cyclone behavior.

2.1. The Equations in Cylindrical Polar Coordinates

Because an intensifying tropical cyclone exhibits some degree of circular organization (althoughnot axially symmetric), it is advantageous to express the equations of motion in cylindrical polarcoordinates, (r , λ, z), with r the radius, λ the azimuthal angle, and z the height above the surface.Earth’s rotation is incorporated by the addition of Coriolis and centrifugal forces in the usualmanner (Gill 1982, Holton 2004). Because of the relatively limited horizontal scale of the tropicalcyclone circulation, the rotation rate is assumed to be independent of latitude (i.e., a so-called fplane, where f is the Coriolis parameter given by f = 2� sin φ, � is Earth’s rotation rate, and φ

is latitude). The governing equations are

∂u∂t

+ u∂u∂r

+ v

r∂u∂λ

+ w∂u∂z

−v2

r− f v = − 1

ρ

∂p∂r

+ Fr , (1)

∂v

∂t+ u

∂v

∂r+ v

r∂v

∂λ+ w

∂v

∂z+ uv

r+ f u = − 1

ρr∂p∂λ

+ Fλ, (2)

∂w

∂t+ u

∂w

∂r+ v

r∂w

∂λ+ w

∂w

∂z= − 1

ρ

∂p∂z

− g + Fz, (3)

∂ρ

∂t+ 1

r∂ρru∂r

+ 1r

∂ρv

∂λ+ ∂ρw

∂z= 0, (4)

∂θ

∂t+ u

∂θ

∂r+ v

r∂θ

∂λ+ w

∂θ

∂z= θ + Fθ , (5)

ρ = p∗π1κ −1/(Rθ ), (6)

www.annualreviews.org • Fluid Dynamics of Tropical Cyclones 543

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where u, v, and w are the velocity components in the three coordinate directions; θ is the potentialtemperature; θ is the diabatic heating rate (1/c pπ )Dh/Dt, with Dh/Dt the heating rate per unitmass expressed as J kg−1 s−1; π = (p/p∗)κ is the Exner function; p is the pressure; ρ is the density;g is the effective gravitational force per unit mass; R is the specific gas constant for dry air; c p isthe specific heat at constant pressure; κ = R/cp ; and p∗ = 1,000 mb is a reference pressure. Thetemperature is given by T = πθ . The terms (Fr , Fλ, Fz) represent unresolved processes associatedwith turbulent momentum diffusion. In the case of numerical models, these terms specifically rep-resent a divergence of subgrid-scale eddy momentum flux associated with unresolved processessuch as convection for a model that cannot resolve clouds and/or frictional stress at the lowersurface and related mixing processes in the frictional boundary layer. Similarly, Fθ represents theeffects of turbulent heat transport (again possibly including those associated with convection for acoarse-resolution model). Equations 1–6 comprise the three components of the momentum equa-tion, continuity equation, thermodynamic equation, and ideal gas equation of state, respectively.In these equations, the traditional approximation is made of neglecting the horizontal componentof Earth’s rotation rate and other metric terms associated with Earth’s approximate sphericity thatare small on account of the limited horizontal scale of a typical hurricane vortex compared to themean radius of the Earth. The equations assume that the origin of coordinates is located at somesuitably defined vortex center.

In moist flows, the equations need to be supplemented by tendency equations for the watervapor mixing ratio, qv, and for various species of water substance, whereas θ in the equation ofstate must be replaced by the virtual potential temperature, θv.

2.2. Solution for a Freely Spinning Vortex

For adiabatic frictionless flow [θ = 0, (Fr , Fλ, Fz) = (0, 0, 0)], Equations 1–5 have a solution,v(r , z), for a steady, freely spinning vortex in which u and w are identically zero and v(r , z) isan arbitrary function of r and z. Such a vortex is in gradient wind balance [(1/ρ)(∂p/∂r) = C]and hydrostatic balance [(1/ρ)(∂p/∂z) = −g], where C = v2/r + f v is the sum of the specificcentrifugal and Coriolis forces. The ratio of centrifugal to Coriolis forces is a local vortex Rossbynumber, Ro = v/ f r . Near the radius of maximum tangential wind, Ro is of order unity for atropical depression–strength vortex and can be as large as several hundred for a mature hurricaneor typhoon.

Multiplying the gradient wind and hydrostatic balance equations by ρ and cross-differentiatingto eliminate the pressure leads to the thermal wind equation

∂ log ρ

∂r+ C

g∂ log ρ

∂z= − 1

g∂C∂z

. (7)

This first-order linear partial differential equation relates the logarithm of density log ρ(r , z) (orequivalently log θ ) to the vertical gradient of C and hence the vertical shear of the swirling wind.The characteristics of the equation satisfy dz/dr = C/g and are just the isobaric surfaces. Forfurther details, readers are referred to Smith (2006, 2007).

2.3. Zero-Order Force Balances and the Agradient Force

A scale analysis of the equations of motion for a tropical cyclone vortex having a characteristicheight-to-width aspect ratio squared (H/L)2 and a ratio of radial to tangential velocity squared(U/V )2 much less than unity shows that, except near the surface and in the upper troposphere,the primary (tangential) circulation of a mature hurricane is approximately axisymmetric and in


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gradient wind and hydrostatic balance (i.e., the underlined terms in Equations 1 and 3). Ac-cordingly, the freely spinning vortex solution of Section 2.2 represents a meaningful zero-orderapproximation for the bulk vortex. It is then useful to enquire about the imbalance of forces inthe transverse (r , z) plane, the so-called agradient force, Fa. Defining density and pressure per-turbations, ρ ′ and p ′, relative to the corresponding quantities in the dynamically balanced stateas defined above, ρR(r , z) and p R(r , z), one may write the inviscid form of Equations 1 and 3 invector form:

DuDt

= Fa, (8)

where

Fa = − 1ρ

∇ p ′ + gρ ′

ρ, (9)

u is the transverse velocity vector (u, w), D/Dt is the full material derivative, and g = (C , 0, −g)is the generalized gravitational vector. The vector quantity on the right-hand side of Equation 9is the agradient force, and the term gρ ′/ρ is the generalized buoyancy force. Equation 9 showsthat air parcels lighter than the local density associated with the balanced vortex (i.e., ρ ′ < 0)are positively buoyant in the vertical direction and have an inward component of generalizedbuoyancy (Smith et al. 2005). Although the radial component of the generalized gravitationalvector is small compared to the vertical component in tropical cyclone vortices, the effect of theradial component of the generalized buoyancy force by itself is to move buoyant plumes inward.However, the inner-core clouds tilt outward so that the radial component of Fa must be dominatedby the perturbation pressure gradient, which is generally directed outward.

2.4. Absolute Angular Momentum and Centrifugal Stability

Multiplying Equation 2 by r and a little manipulation leads to the equation

∂M∂t

+ u∂M∂r

+ v

r∂M∂λ

+ w∂M∂z

= − 1ρ

∂p∂λ

+ r Fλ, (10)

where M = rv + 12 f r2 is the absolute angular momentum per unit mass of an air parcel about

the rotation axis. For axisymmetric (∂/∂λ = 0) and frictionless (Fλ = 0) flow, the right-hand sideof Equation 10 is zero, and M is materially conserved as rings of air move radially and vertically.

The freely spinning vortex solution of Section 2.2 is stable to small, axisymmetric radial dis-placements if the local inertial (centrifugal) stability parameter, I 2 = (1/r3)∂M 2/∂r , is positive.The quantity I 2 is a measure of the inertial stiffness of the vortex and is analogous to the static sta-bility parameter, N 2 = (g/θ )(dθ/dz), which is a measure of the resistance to vertical displacementsin a stably stratified fluid (Holton 2004, p. 54). Tropical cyclones generally have M distributionsthat increase monotonically with radius in the bulk of the troposphere (e.g., Franklin et al. 1993)and are therefore centrifugally stable. Even if it is stable to radial and vertical displacements, thevortex may be unstable to displacements in other directions, a condition known as symmetricinstability (e.g., Shapiro & Montgomery 1993).

2.5. Moist Deep Convection

Significant weather over the tropical oceans is generally associated with thunderstorms or clustersof thunderstorms that may be part of larger-scale circulations. Tropical cyclones are the end stageof a few of these storm clusters. Thunderstorms are manifestations of deep moist convection,which from a fluid dynamics perspective has properties that differ in important ways from those


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of dry convection (Emanuel 1994). An understanding of the dynamics of tropical cyclones reststo a considerable extent on an understanding of deep convection, which we do not discuss here.For in-depth discussions of moist convection, readers are referred to Emanuel (1994) and Houze(2014).

2.6. Rotating Deep Convection

When buoyant convection occurs in an environment with nonzero vertical vorticity, the convec-tive updraughts amplify the vorticity by the process of vortex-tube stretching (e.g., Julien et al.1996, Hendricks et al. 2004, Wissmeier & Smith 2011, Kilroy & Smith 2013). Typically, thevorticity may be amplified by between one and two orders of magnitude on the scale of the cloudupdraughts. Although the updraught strengths are generally much weaker than in mid-latitudesupercell thunderstorms, as are the associated local tangential wind components (see, e.g., Klemp1987, Rotunno 2013), the presence of these vortical cores appears to be important in the genesisand intensification of tropical cyclones.

The role of rotating deep convective clouds and their aggregation in the amplification ofthe larger-scale vortex has been the subject of recent numerical and theoretical investigations(Hendricks et al. 2004, Montgomery et al. 2006b, Nguyen et al. 2008, Shin & Smith 2008, Braunet al. 2010, Fang & Zhang 2011, Gopalakrishnan et al. 2011, Schecter 2011), but questions remainabout the quantitative importance of the enhanced vorticity within the clouds themselves. Weexplore these issues further in Section 4.

2.7. Buoyancy in Rapidly Rotating Fluids

Aircraft reconnaissance measurements have shown that the eye of a mature tropical cyclone is thewarmest place in the storm, warmer indeed than the eyewall clouds (Hawkins & Rubsam 1968,Hawkins & Imbembo 1976). From a fluid dynamics perspective, the question then arises, Are theeyewall clouds buoyant? The balanced vortex itself has system buoyancy in the traditional sensewhen the reference density is set to that of the far-field environment. However, the eyewall cloudsare not buoyant in the vertical direction in the traditional sense because air parcels rising in theeyewall have temperatures less than those in the eye (one side of the cloud environment)! As shownby Smith et al. (2005), the issue is resolved when one defines local buoyancy relative to the densitydistribution of the axisymmetric balanced vortex, as in Section 2.3.

3. TROPICAL CYCLONE INTENSIFICATION AND STRUCTURE

Tropical cyclones are generally highly asymmetric during their intensification phase, and onlythe most intense storms exhibit a strong degree of axial symmetry and, even then, only in theirinner-core region. Observations show that rapidly developing storms are accompanied by burstsof deep moist convection, presumably driven by significant local buoyancy (e.g., Heymsfield et al.2001). The deep convection is maintained by appreciable moisture fluxes at the ocean-air interface,which sustain a conditionally unstable thermodynamic environment.

In a recent review paper, Montgomery & Smith (2014) examined and compared the four mainparadigms proposed to explain tropical cyclone intensification in the prototype problem for inten-sification (see Section 1): the CISK (conditional instability of the second kind) paradigm, the coop-erative intensification paradigm, a thermodynamic air-sea interaction instability paradigm [widelyknown as WISHE (wind-induced surface heat exchange)], and a rotating convection paradigm.The first three paradigms assume axisymmetric flow about the rotation axis and therefore no


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azimuthal eddy terms. This axisymmetric configuration with its attendant phenomenology ofaxisymmetric convective rings has certain intrinsic limitations for understanding the intensifi-cation process (Persing et al. 2013), which, as noted in Section 2.6, is quite asymmetric at thecloud scale. As discussed by Montgomery & Smith (2014), the CISK paradigm has a number ofwell-known issues, and it is not discussed here.

3.1. The Cooperative Intensification Paradigm

What might be regarded as the classical view of tropical cyclone intensification, the cooperativeintensification paradigm, emerged from a simple axisymmetric model for intensification formu-lated by Ooyama (1969). It assumes that the broad-scale aspects of a tropical cyclone may berepresented by an axisymmetric, balanced vortex in a stably stratified, moist atmosphere. Balancemeans that the primary circulation is governed approximately by the thermal wind equation ob-tained in Section 2.2, even in the presence of nonconservative forcing processes such as diabaticheating and friction, which tend to drive the flow away from balance. Under such circumstances,the stream function for the axisymmetric, secondary (overturning) circulation required to main-tain balance satisfies a second-order partial differential equation, the so-called Sawyer-Eliassenequation. The traditional vortex balance equations are obtained from Equations 1–6 by retainingthe axisymmetric limit of Equations 2 and 5 together with the simplified equations given by theunderlined terms in Equations 1, 3, and 4 (see Montgomery & Smith 2014, pp. 39–41, for furtherdetails).

The cooperative intensification paradigm was explained succinctly by Ooyama (1969, p. 18):

If a weak cyclonic vortex is initially given, there will be organised convective activity in the regionwhere the frictionally-induced inflow converges. The differential heating due to the organised con-vection introduces changes in the pressure field, which generate a slow transverse circulation in thefree atmosphere in order to re-establish the balance between the pressure and motion fields. If theequivalent potential temperature of the boundary layer is sufficiently high for the moist convection tobe unstable, the transverse circulation in the lower layer will bring in more absolute angular momentumthan is lost to the sea by surface friction. Then the resulting increase of cyclonic circulation in the lowerlayer and the corresponding reduction of the central pressure will cause the boundary layer inflow toincrease; thus, more intense convective activity will follow.

Montgomery & Smith (2014, pp. 45–46) assessed this paradigm, and we discuss an extensionthereof in Section 3.4. Presumably, the conclusion that “more intense convective activity willfollow” is related to the closure scheme adopted for his representation of deep convection, whichassumes that all the air that converges in the boundary layer is ventilated by the eyewall convection,given that some degree of instability is maintained. Alternative closures in a minimal tropicalcyclone model are discussed by Zhu et al. (2001).

3.2. The WISHE Paradigm

The WISHE paradigm for intensification is based on the idea of an air-sea interaction instabilitycomprising a postulated multistep feedback loop involving, in part, the near-surface wind speedand the evaporation of water from the underlying ocean, with the evaporation rate a function ofthe wind speed and thermodynamic disequilibrium (Emanuel et al. 1994; Emanuel 1997, 2003,2012). Montgomery & Smith (2014, figure 6) provided a schematic illustration of this feedbackmechanism. In some subcircles, however, the term WISHE mechanism is used more loosely as


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simply the bulk-aerodynamic transfer of moist enthalpy from the ocean to the atmosphere bythe local prevailing winds. When used in this way, it does not constitute a mechanism of vortexintensification. Although the WISHE mechanism viewed as a feedback loop is widely held tobe the explanation as to how tropical cyclones intensify, when the mechanism is suppressed inmodels by capping the wind-speed dependence of the heat fluxes, the vortices still intensify.Therefore, the mechanism is unnecessary to explain intensification in the prototype problem, butthe simulated vortices do become stronger when the wind-speed dependence of the heat fluxes isretained (Montgomery et al. 2009, 2015).

3.3. The Rotating Convection Paradigm

The rotating convection paradigm recognizes the presence of localized, rotating deep convectionthat grows in the cyclonic rotation-rich environment of the incipient storm, structures that areintrinsically 3D. The paradigm recognizes also the stochastic nature of deep convection, whichhas implications for the predictability of local asymmetric features of the developing vortex. Theconvective updraughts greatly amplify the vertical vorticity locally by vortex-tube stretching, andthe patches of enhanced cyclonic vorticity subsequently aggregate to form a central monolithof cyclonic vorticity. An azimuthally averaged view of this paradigm constitutes an extension ofthe cooperative intensification paradigm in which the boundary layer and eddy processes cancontribute positively to producing the maximum tangential winds of the vortex.

In the context of the rotating convection paradigm, an important question arises as to whetherthere are important differences between 3D tropical cyclones and their purely axisymmetric coun-terparts. A hint that there may be follows from a finding by Moeng et al. (2004) of an excessiveconvective entrainment rate in a 2D planetary boundary layer with vertical shear relative to a 3Dmodel. Their results suggest that in the tropical cyclone context, axisymmetric convection occur-ring in concentric rings may be likewise overly efficient in generating buoyancy fluxes comparedto 3D convection in isolated thermals, leading to excessive condensation heating and an overlyrapid spin-up. Persing et al. (2013) obtained support for this hypothesis in explicit comparisonsbetween 3D and axisymmetric simulations of tropical cyclones.

Before reviewing the salient features of the rotating convection paradigm and its azimuthallyaveraged view in detail, we review first the dynamics and thermodynamics of the frictional bound-ary layer.

3.4. Boundary Layer Dynamics

The frictional boundary layer is loosely defined as a surface-based layer in which the effects ofthe turbulent transfer of momentum to the surface are important. (Issues surrounding attemptsto define this layer more precisely are discussed in Smith & Montgomery 2010, Zhang et al.2011, Abarca et al. 2015, and Kepert et al. 2016.) It plays an important role in the dynamicsand thermodynamics of tropical cyclones and is an essential component of all the paradigms forintensification referred to above. A brief review of the essentials is necessary here to discuss theseparadigms further (see Sections 3.5 and 3.6).

Reinforced by the situation in many simple fluid flows, where the boundary layer is one inwhich the flow speed is reduced by friction to below the free-stream value (Schlichting 1968), ithas been widely (and reasonably) assumed that friction acts everywhere to reduce the wind speedin a tropical cyclone (see Montgomery & Smith 2014, p. 56). However, the maximum tangentialwind speed in a tropical cyclone is found to occur within the inner-core boundary layer (Zhanget al. 2001, Smith et al. 2009). The reasons for this surprising behavior were anticipated by Anthes(1974, p. 506) and further elucidated by Smith (2003), Smith & Vogl (2008), and Smith et al. (2009).


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As a preliminary for understanding the role of the boundary layer and the surprising behaviorreferred to above, it is insightful to consider the spin-down of the freely spinning vortex discussedin Section 2.2. A scale analysis of the momentum equations (Equations 1–3) for a boundarylayer in general indicates that the pressure gradient force is transmitted approximately unchangedthrough the boundary layer to the surface (see, e.g., Jones & Watson 1963). Because of this, theappropriate balanced pressure field, PR(r , z), in the definition of the agradient force (Section 2.3) inthe boundary layer is that at the top of the boundary layer. At this level, the radial pressure gradientper unit mass is just the sum of the centrifugal and Coriolis forces (C) (Section 2.2). Beyond someradius outside that of the maximum gradient wind, C is reduced in the boundary layer because ofthe frictional retardation of the tangential wind, v. Thus, the radial component of the agradientforce, Far, is negative. (Note that, as we have defined it, the agradient force is a measure of theeffective pressure gradient force, but an alternative definition might include frictional forces also.)Additionally, because v decreases toward the surface, Far has a maximum magnitude at the surface.At these outer radii, the boundary layer flow is subgradient, and the negative agradient force (theradial component of the first term on the right-hand side of Equation 9) generates inflow in theboundary layer, with the largest inflow near the surface.

As air parcels converge in the boundary layer, they lose absolute angular momentum, M , to thesurface. However, if the rate of loss of M is sufficiently small (i.e., less than the rate of decrease ofthe radius), the corresponding tangential velocity (given by v = M /r − 1

2 f r2) may increase so thatat some inner radii, the tangential wind speed in the boundary layer exceeds the local value abovethe boundary layer. We refer to this process as the boundary layer spin-up mechanism. At suchradii, one finds that Far > 0 and the boundary layer flow is supergradient. Then, all forces in theradial momentum equation are outward and the radial inflow rapidly decelerates, leading to upflowat the top of the boundary layer. For these reasons, the boundary layer exerts strong control on theradii at which the inflow turns up into the eyewall clouds (see Section 3.8). Observational supportfor the occurrence of the maximum tangential wind within the boundary layer is provided byKepert (2006a,b), Bell & Montgomery (2008), Montgomery et al. (2014), and Sanger et al. (2014).Support for the boundary layer spin-up mechanism is provided by numerical model studies inidealized axisymmetric (Nguyen et al. 2002, Schmidt & Smith 2016) and 3D (Smith et al. 2009,Abarca & Montgomery 2013, Persing et al. 2013, Zhang & Marks 2015) configurations and innumerical simulations of real cases (Zhang et al. 2001). The mechanism is not found in modelsthat use an overly diffusive boundary layer scheme (Smith & Thomsen 2010).

3.5. Spin-Down, Spin-Up, and an Extended CooperativeIntensification Paradigm

If friction were the only effect acting on a vortex, the boundary layer would induce radial out-flow in a layer above it, and the vortex would spin down as air parcels move to larger radii whileconserving their absolute angular momentum. This mechanism of vortex spin-down was articu-lated by Greenspan & Howard (1963). If the air in the vortex is stably stratified (as in a tropicalcyclone), the vertical extent of the outflow will be restricted by the static stability. Clearly, fora vortex to spin up, there must be some mechanism to produce strong enough inflow above theboundary layer to reverse the outflow that would be produced there by the boundary layer alone.The only physically conceivable process capable of producing such inflow in a tropical cycloneis the collective effect of buoyant deep convection in the inner region of the vortex, as envisagedin the cooperative intensification paradigm (Section 3.1). Typically, a region of deep convectionproduces an overturning circulation with inflow toward it in the lower half of the troposphereand outflow in the upper half. In other words, for the vortex to spin up, the convective mass flux


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must be more than strong enough to ventilate the mass converging in the boundary layer, therebyoverpowering the tendency of the boundary layer to induce outflow above it.

In the early stages of tropical cyclone intensification, when the primary circulation of the vortexis comparatively weak, the boundary layer–induced inflow and outflow are weak, and the secondarycirculation is dominated by the convectively induced inflow throughout the lower troposphere.Above the boundary layer, where to a first approximation M is materially conserved, the vortexspins up. In these stages, the flow in the boundary layer is largely subgradient. However, as theprimary circulation increases in strength, the boundary layer–induced convergence progressivelyincreases, ultimately leading to a spin-up of the maximum tangential winds within the boundarylayer, as described above. Moreover, the convectively induced inflow may become progressivelyunable to oppose the low-level outflow induced by the boundary layer; i.e, the convection willbecome less able to ventilate the mass converging in the boundary layer, thereby slowing downor reversing the rate of intensification of the vortex.

Apart from the boundary layer spin-up mechanism and its consequences, the foregoing pro-cesses broadly constitute the cooperative intensification paradigm as articulated by Ooyama (1969).Figure 1a illustrates the extended cooperative intensification paradigm, and Figure 1b highlightsthe low-level secondary flow feeding into the eyewall of the vortex. These illustrations are consis-tent with the azimuthal average of fully 3D solutions of the governing fluid dynamical equations(e.g., Smith et al. 2009) and with observations (e.g., Montgomery et al. 2014).

Figure 2 illustrates the primary and secondary circulation of a simulated tropical cyclonevortex undergoing intensification in a state-of-the-art cloud model (Persing et al. 2013). In thisazimuthally averaged flow, there is weak inflow through much of the lower troposphere, withstrong inflow in a shallow boundary layer and strong outflow just above it where the flow eruptsinto the eyewall. There is outflow in the upper troposphere also. The maximum tangential wind

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Figure 1(a) An axisymmetric view of tropical cyclone intensification in the new paradigm. Above the boundary layer, spin-up of the vortex occursas air parcels are drawn inward by the inner-core convection. Air parcels spiralling inward in the boundary layer may reach small radiiquickly (minimizing the loss of absolute angular momentum, M , during spiral circuits) and acquire a larger tangential wind speed v thanthat above the boundary layer. (b) The hurricane inner-core region during intensification in relation to the broader-scale overturningcirculation. Air subsides into the boundary layer at large and moderate radii and ascends out of the boundary layer at inner radii. Thefrictionally induced net inward force in the boundary layer produces a radially inward jet. The subsequent evolution of this jet dependson the bulk radial pressure gradient that can be sustained by the mass distribution at the top of the boundary layer. The jet eventuallygenerates supergradient tangential winds, whereafter the radial inflow rapidly decelerates. As it does so, the boundary layer separates,and the flow there turns upward and outward to enter the eyewall. As this air ascends in the eyewall, the system-scale tangential windand radial pressure gradient come into gradient wind balance. This adjustment region has the nature of an unsteady centrifugal wavewith a vertical scale of several kilometers, akin to the vortex breakdown phenomenon (Rotunno 2014, and references therein).


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Figure 2Radius-height cross sections of the azimuthally averaged velocity components in the simulation described by Persing et al. (2013), timeaveraged during an intensification phase (144–148 h) of the 3D calculation: (a) radial velocity (contour interval 2 m s−1), (b) tangentialvelocity (contour interval 5 m s−1), and (c) vertical velocity (contour interval 0.5 m s−1 for positive values and 0.1 m s−1 for negativevalues). Positive values are represented in red and by solid lines, and negative values are represented in blue and by dashed lines. Theblack dotted circles in each plot show the location of the maximum tangential wind speed at each height. Figure adapted withpermission from Persing et al. (2013).

is found within the layer of strong inflow. Above this height, the tangential wind decreases withheight, and the radius of the maximum tangential wind at a given height increases. The verticalvelocity field shows a region of strong ascent into the upper troposphere where the boundary layererupts into the interior vortex. This updraught region is essentially moist saturated (not shownin the figure), and the inner edge of this cloudy region is referred to as the eyewall of the storm.Inside and outside of the main updraught region, there is weak subsidence. Near the top of theboundary layer where the flow turns into the eyewall updraught, there is a secondary maximumof vertical velocity.

As the vortex intensifies, the boundary layer exerts ever-increasing control on the pattern ofconvection, as well as on the ability of the convection to ventilate the mass converging in theboundary layer (Ooyama 1982). Accordingly, there is a subtle interplay through boundary layerdynamics between the spin-up of the circulation above the boundary layer, which depends on thestrength and location of the convection, and the boundary layer response, which exerts control onthe radii at which air ascends. The fate of the ascending air depends in part on thermodynamicprocesses, which affect the ability of convection to evacuate the increasing mass flux within theboundary layer and continue to produce inflow above the boundary layer (Kilroy et al. 2016,Schmidt & Smith 2016).

3.6. Boundary Layer Thermodynamics

Ooyama’s articulation of the cooperative intensification paradigm assumed that the boundary layerθe would remain high enough to sustain deep convection as the vortex developed (Section 3.1). Inhis model, the required high θe values were sustained by wind-speed-dependent surface moisturefluxes. In the WISHE paradigm, spin-up depends crucially on a progressive increase of the surfacemoisture fluxes with wind speed (see, e.g., Montgomery & Smith 2014, figure 6).

As in the cooperative intensification paradigm, the rotating convection paradigm for spin-uprequires a modest elevation of low-level moisture and hence θe to sustain deep convection at


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radii where air is being lofted from the boundary layer into the eyewall. For reasons discussedabove, the maintenance of buoyant deep convection is a prerequisite for convection to ventilatethe increasing amount of air being lofted from the boundary layer as the vortex intensifies and theupper-level warm core aloft strengthens (Kilroy et al. 2016). The ability of deep convection toventilate the mass that is expelled by the boundary layer depends on the convective mass flux, andthe mass flux must depend, e.g., on the buoyancy of the cloud updraughts. However, it dependsalso on the area of the updraughts. Clearly, to gain further insight that is free from speculationabout its postulated behavior, one needs to calculate the changes in convective mass flux using anumerical model (see, e.g., Kilroy et al. 2016).

3.7. Outer-Core Size

From a forecasting perspective, the prediction of tropical cyclone size (i.e., the extent of gale forcewinds, speeds greater than 17 m s−1) is comparable in importance with the prediction of intensity.For example, Atlantic Hurricane Sandy (2012) was only a Category 3 storm, but was accompaniedby an enormous area of gales that led to extensive damage along the US East Coast. The practicalimportance of the size problem has motivated several theoretical and numerical studies examiningfactors that determine tropical cyclone size (e.g., Yamasaki 1968, Rotunno & Emanuel 1987,DeMaria & Pickle 1988, Xu & Wang 2010, Hakim 2011, Rappin et al. 2011, Smith et al. 2011,Li et al. 2012, Chan & Chan 2014, Chavas & Emanuel 2014, Frisius 2015, Kilroy et al. 2016). Anassessment of many of these studies has been given by Kilroy et al. (2016).

An underlying assumption of most of these studies is that there exists a global quasi-steadysolution for storms, which would require, e.g., that the storm environment be quasi-steady. Thisrequirement is unlikely to be satisfied (see Section 6). In fact, according to the conventionalparadigm for tropical cyclone intensification (Section 3.5), one would anticipate that the outercirculation will expand as long as the aggregate effect of deep convection [including the eyewalland convective rainbands (Fudeyasu & Wang 2011)] remains strong enough to maintain the inwardmigration of absolute angular momentum surfaces. As demonstrated by Kilroy et al. (2016), thebroadening circulation has consequences for the boundary layer dynamics, which play a role indetermining the radii at which air ascends into the eyewall and the maximum tangential windspeed, which occurs within the boundary layer (Section 3.4). The broadening circulation hasconsequences also for the boundary layer thermodynamics, which affects the spatial distributionof diabatic heating above the boundary layer (Section 3.6).

3.8. Boundary Layer Control on Inner-Core Size

Kilroy et al. (2016) examined the long-term behavior of tropical cyclones in the prototype problemfor cyclone intensification on an f plane using a nonhydrostatic, 3D numerical model. Afterreaching a mature intensity, the model storms progressively decay while the size of both the innercore (characterized by the radius of the eyewall) and the outer circulation (measured, e.g., bythe radius of gale-force winds) progressively increases. This behavior was explained in terms ofa boundary layer control mechanism in which the expansion of the swirling wind in the lowertroposphere leads, through boundary layer dynamics, to an increase in the radii of forced eyewallascent, as well as to a reduction in the maximum tangential wind speed in the layer. These changesare accompanied by ones in the radial and vertical distributions of diabatic heating, which influencethe inflow in the lower troposphere and thereby the expansion of the swirling wind there.

Kilroy et al. (2016) pointed out that the tight coupling between the flow above the bound-ary layer and that within the boundary layer generally makes it impossible to present simple


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cause-and-effect arguments to explain vortex behavior. The best one can do is to articulate theindividual elements of the coupling, which might be described as a set of coupled mechanisms.They employed a simple, steady, slab boundary layer model (detailed in Smith et al. 2015) as away to break into the chain of coupled mechanisms referred to above. The assumption is that,because the boundary layer is relatively shallow, it adjusts rapidly to the flow above it. Given thatthe partial differential equations from which the slab boundary layer model is derived are parabolicin the radially inward direction, the inflow, and hence the ascent (or descent) at the top of theboundary layer at a given radius R, knows only about the tangential wind profile at radii r > R (seeFigure 3). The inflow at radius R knows nothing directly about the vertical motion at the top of theboundary layer at radii r < R, including the pattern of ascent into the eyewall cloud associated withconvection under the eyewall. In contrast, the numerical simulation does not solve the boundarylayer equations separately, and it does not make any special boundary layer approximation. Thus,the ability of the slab boundary layer model to replicate a radial distribution of radial, tangential,and vertical motion close to those in the time-dependent numerical simulation provides a usefulmeasure of the degree of boundary layer control in the evolution of the vortex. Indeed, it is a wayto approximately separate the sucking effect of deep convection from the boundary layer–inducedinflow.

As an example, Figure 4 compares velocity fields from the slab boundary layer calculations withthe corresponding azimuthally averaged fields from a full numerical simulation at latitude 20◦N.The radial and tangential wind components from the numerical simulation are averaged over thelowest 1-km depth, corresponding to an average over the depth of the boundary layer, to providea fair comparison with the slab boundary layer fields. The slab boundary layer calculations areperformed every 12 h using the smoothed, azimuthally averaged tangential wind profile extractedfrom the numerical simulation in Kilroy et al. (2016, figure 5). Even though the integration of theslab boundary layer equations breaks down at some inner radius where the radial velocity tendsto zero and the vertical velocity becomes large, the calculations capture many important featuresof the corresponding depth-averaged boundary layer fields from the numerical simulations. Forexample, they capture the broadening of the vortex core with time, that is, the increase in the


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Figure 4(a–c) Hovmoller plots of tangential and radial velocities in the boundary layer and the vertical velocity at the top of the boundary layerfrom the slab boundary layer model, with a constant depth of 1,000 m. (d–f ) The corresponding azimuthally averaged and temporallysmoothed quantities from the MM5 output. Contour intervals are 5 m s−1 in panels a, b, d, and e and ±2 cm s−1 and 10 cm s−1 inpanels c and f. In panels a and d, the black curve extending to the large radius highlights the 17 m s−1 contour. In panels b and e, and thefirst dashed contour is −1 m s−1. In panels c and f, the first (red ) contour is 2 cm s−1. Darker shading is from 10 cm s−1 to 100 cm s−1.The darkest (red ) shading is enclosed by a 100 cm s−1 black contour. Regions of downward motion are shaded blue and enclosed by a−2 cm s−1 blue contour. The solid contours represent positive values, whereas the dashed contours represent negative ones. Figureadapted from Kilroy et al. (2016).

radii of the maximum tangential wind speed and eyewall location, the latter characterized bythe location of maximum vertical velocity. They demonstrate also the broadening of the outerradial and tangential wind field. However, they overestimate the radial extent of the subsidenceoutside the eyewall (see Figure 4e,f ). For reasons articulated above, these results provide strongsupport for the existence of dynamical control by the boundary layer on the evolution of thevortex. Kilroy et al. (2016) investigated also the thermodynamic control of the boundary layerand other aspects of the coupling discussed above. This study provides new insight on the factorscontrolling the evolution of the size and intensity of a tropical cyclone, as well as a plausible and


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simpler explanation for the expansion of the inner core of Hurricane Isabel (2003) and TyphoonMegi (2010) than given previously.

3.9. Eyewall Spin-Up

Schmidt & Smith (2016) developed an improved version of a minimal axisymmetric model for atropical cyclone and used it to revisit some fundamental aspects of vortex behavior in the proto-type problem for tropical cyclone intensification. The calculation highlights the pivotal role ofthe boundary layer in spinning up the tangential winds in the eyewall updraught. As discussedin Section 3.4, the spin-up in the boundary layer is associated with the development there ofsupergradient winds. The spin-up of the eyewall updraught occurs by the vertical advection of thehigh tangential momentum associated with the supergradient winds in the boundary layer. Theseboundary layer and eyewall spin-up mechanisms, although consistent with some recently reportedresults (e.g., Kilroy et al. 2016), are not part of the classical theory of tropical cyclone spin-up(see Section 3.1). In fact, in the eyewall updraught, the flow is outward (typifying the outwardslope of the eyewall) so that the radial advection of absolute angular momentum (or radial flux ofabsolute vorticity) makes a negative contribution to spin-up in this region. Even so, there is inflowin the middle layer at large radii where the classical mechanism operates to spin up the tangentialwinds. Based on the results of Kilroy et al. (2016), the spin-up at large radii, where the flow inthe boundary layer is subgradient, leads to feedback on the inner-core vertical motion throughboundary layer dynamics and to a change in the spatial distribution of diabatic heating, mostlyabove the boundary layer, through boundary layer thermodynamics.

3.10. Efficiency Arguments

Following pioneering studies of Schubert & Hack (1982), Hack & Schubert (1986), and Vigh &Schubert (2009) in the context of the inviscid, axisymmetric, balance equations forced by pre-scribed diabatic heating, there are widely held arguments that attribute the increasingly rapidintensification of tropical cyclones to the increasing efficiency of diabatic heating in the cyclone’sinner-core region associated with deep convection (e.g., Vigh & Schubert 2009, Rozoff et al.2012). The efficiency, in essence the amount of temperature warming compared to the amount oflatent heat released, is argued to increase as the vortex strengthens on account of the strengthen-ing inertial stability. Assuming that the diabatic heating rate does not change, the strengtheninginertial stability progressively weakens the secondary circulation, which in turn is argued to re-duce the rate of adiabatic cooling of rising air. Thus, more of the heating is available to raise thetemperature of the air parcel. Another aspect concerns the location of the heating in relation tothe radius of maximum tangential wind speed, with heating inside this radius seen to be moreefficient in developing a warm core thermal structure and presumably an increase in tangentialwind.

Recently, Smith & Montgomery (2016) provided a more direct interpretation of the increasedspin-up rate when the diabatic heating is located inside the radius of maximum tangential windspeed. Furthermore, they drew attention to the limitations of assuming a fixed diabatic heating rateas the vortex intensifies, and on these grounds alone, they offered reasons why it is questionableto apply the efficiency arguments to interpret the results of observations or numerical modelsimulations of tropical cyclones. Given that the spin-up of the maximum tangential winds ina tropical cyclone takes place in the boundary layer and that the spin-up of the eyewall is aresult of the vertical advection of high angular momentum from the boundary layer, Smith &Montgomery questioned whether deductions about efficiency in theories that neglect the boundary


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layer dynamics and thermodynamics are relevant to reality. Rogers et al. (2015) discussed (but didnot endorse) the efficiency idea in a multiscale analysis of the rapid intensification of HurricaneEarl (2010).

4. MORE ON THE ROTATING CONVECTION PARADIGM

As noted in Section 3.3, the findings of Persing et al. (2013) suggest that previous studies usingstrictly axisymmetric models and their attendant phenomenology of axisymmetric convective ringshave intrinsic limitations for understanding the intensification process. To further understand therole of eddy dynamics during tropical cyclone intensification, we return now to the rotatingconvection paradigm. There is accumulating observational evidence supporting the hypothesisthat convective bursts in predepression disturbances and tropical cyclones act to spin up localizedcyclonic vorticity anomalies in the lower troposphere (Reasor et al. 2005, Sippel et al. 2006, Bell& Montgomery 2010, Raymond & Carillo 2011, Sanger et al. 2014, Kilroy & Smith 2015). Thequestion then arises, What is the role of these cyclonic vorticity anomalies in the spin-up process,and in what way might they modify the azimuthally averaged view of the rotating convectionparadigm?

4.1. Role of Cloud-Generated Vorticity

To help motivate one of the issues involved in understanding the role of in-cloud vorticity in thedynamics of a developing tropical cyclone, we recall Stokes’ theorem, which equates the area-integrated vertical vorticity to the circulation defined by the line integral around a closed-circuitloop within the fluid on a horizontal height surface. At any given instant in time, the circulationis of course given by the area-integrated vorticity within the loop. Vortical convective processesoccurring within the loop, such as adiabatic vortex merger (Melander et al. 1988, Dritschel &Waugh 1992, Lansky et al. 1997), vortex axisymmetrization processes (Melander et al. 1988,Montgomery & Enagonio 1998), and diabatic modifications thereof (Hendricks et al. 2004, Toryet al. 2006), certainly contribute to the consolidation and upscale growth of cyclonic vorticitywithin the loop. However, they appear to be unimportant to the net circulation unless theseprocesses have an influence on the flow normal to the loop.

In general, the change in circulation (and vorticity) is governed by the divergence of a horizontalflux, and the flux comprises an advective and nonadvective contribution (Haynes & McIntyre 1987).For the purposes of this discussion, we adopt standard geometric coordinates, with z denoting theheight above the ocean surface. The equation for the local tendency of absolute vertical vorticityζa may be written as

∂ζa

∂t= −∇h · Fζa , (11)

where Fζa = Faf + Fnaf , Faf = uhζa, and Fnaf = −ζhw + k ∧ Ffri. Here uh is the horizontal velocityvector, ζh is the horizontal vorticity vector, w is the vertical velocity, Ffri is the horizontal force perunit mass due to molecular effects and subgrid-scale eddy momentum flux divergences, and k is aunit vector in the vertical. In this form of the vertical vorticity equation, the baroclinic term thatwould ordinarily appear as an additional term on the right-hand side of Equation 11 is neglectedbecause it is generally small in the tropics (Raymond et al. 2014). From these definitions, it followsthat the advective flux is given by Faf and the nonadvective flux by Fnaf . The nonadvective flux isassociated with vortex-tube-tilting processes, as well as friction associated with subgrid-scale eddymomentum transfer.


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The physics of the nonadvective fluxes is described elegantly by Raymond et al. (2014,figure 1). The foregoing formalism is mathematically equivalent to the material form of the equa-tion for the vertical vorticity (e.g., Batchelor 1967, Holton 2004). The material form proves usefulin understanding the local amplification of vorticity following fluid parcels by deep convectiveupdraughts, in contrast to the concentration of vorticity inferred from Equation 11 within a fixedclosed circuit (assuming of course that −∇h · Fζa > 0). It is only the concentration of vorticity thatleads to an increase of circulation about a fixed circuit. The local amplification of vorticity doesnot by itself increase the circulation because the increase of vorticity by stretching is accompaniedby a decrease in the area of the material vortex tube that is stretched with zero change of circula-tion about the corresponding material circuit. However, by mass continuity, stretching must beaccompanied by flow convergence across the fixed circuit, which, if −∇h · Fζa > 0, does lead to anincrease in circulation about the fixed circuit.

The material form of the vorticity equation does not explicitly convey the area-integratedconstraint contained by the flux form expressed by Equation 11. In contrast, calculating thedivergence of the advective flux does not distinguish between local changes in vorticity associatedwith pure advection and the generation of vorticity by stretching. One would need to calculatethe stretching effect separately (see, e.g., Raymond & Carillo 2011, p. 156, column 2).

For the simple thought experiment posed here (see Figure 5), the fixed loop is first imaginedto lie outside of the convecting region; thus, the nonadvective contribution to the net circulationtendency is negligible compared to the advective vorticity flux. In this situation, the key factorresponsible for changing the net circulation is the flux of absolute vorticity across the loop. In anazimuthally averaged viewpoint taken with respect to an approximate invariant center of circula-tion, there will be both axisymmetric (or mean) and nonaxisymmetric (or eddy) contributions tothe absolute vorticity flux across the loop. If, conversely, the loop resides within the convectiveregion, the nonadvective flux contribution around the loop may no longer be small in regionswhere there is mean ascent or localized ascent spatially correlated with horizontal vorticity. Re-cent findings summarized in Section 4.2 show that eddy processes in the active cumulus zone of adeveloping vortex contribute positively to both the advective and nonadvective fluxes of vorticityand therefore also to the amplification of system-scale circulation there.

∫V·ds = ∫∫ζdA AC

C

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Figure 5Schematic illustration of a region of deep rotating updrafts with two hypothetical circuits ( gray circles). ByStokes’ theorem, the circulation about either circle is equal to the areal integral of the vorticity enclosed bythat circuit. Wide dark yellow arrows denote the local rotational flow associated with the rotating convectiveupdraughts.


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4.2. Role of Cloud-Scale Eddies

Motivated by the foregoing discussion regarding in-cloud vertical vorticity and the associated fluxform of the vertical vorticity equation, we summarize recent findings using near cloud-resolving,3D simulations of an intensifying tropical cyclone. To this end, we integrate the vorticity equation(Equation 11) over a horizontal circle of radius r , with the radius defined relative to an instanta-neous center of circulation. Then, dividing by 2πr , one obtains the equation for the azimuthallyaveraged tangential velocity tendency (i.e., the azimuthal average of Equation 2):

∂ 〈v〉∂t

= − 〈u〉〈 f + ζ 〉︸︷︷︸V mζ

−〈w〉 ∂ 〈v〉∂z︸︷︷︸

V mv

− ⟨u′ζ ′⟩︸︷︷︸V eζ

−⟨w′ ∂v′

∂z

⟩︸︷︷︸

V ev

+〈Dv〉︸︷︷︸V d

. (12)

Here, and elsewhere, the prime denotes a departure from the azimuthal mean (or eddy). Theazimuthal average of some quantity Q, denoted by angled brackets, is defined by 〈Q〉(r , z, t) =

12π

∫ 2π

0 Q(r , λ, z, t)dλ, where λ is the azimuth (in radians).The terms on the right-hand side of Equation 12 are the azimuthally averaged advective and

nonadvective vorticity fluxes in the Haynes and McIntyre form of the vorticity substance equation(in geometric coordinates). The mean and eddy terms in Equation 12 are, respectively, the meanradial influx of absolute vertical vorticity (V mζ ), the mean vertical advection of mean tangentialmomentum (V mv), the eddy radial vorticity flux (V eζ ), the vertical advection of eddy tangentialmomentum (V ev), and the combined diffusive and planetary boundary layer tendency (V d). Theazimuthally averaged perturbation pressure gradient force in Equation 2 has been neglected onaccount of its relative smallness, as discussed above. This methodology represents the traditionalEulerian approach to eddy-mean partitioning in the tangential wind equation (e.g., Hendrickset al. 2004, Montgomery et al. 2006b, Yang et al. 2007). Although we do not depart from thisapproach here, we note that, in principle, highly localized asymmetric features can project uponwhat are termed here as mean terms. For example, a single, large-amplitude, positive anomalyin vertical motion imposed on an otherwise axisymmetric vortex will project onto both the eddyand mean terms. (The larger the strength of the anomaly and the smaller its azimuthal extent,the larger will be the contribution to the eddy terms.) This formalism is analogous to a Reynoldsaveraging of the fluid equations for turbulent flow.

The subgrid-scale diffusive tendency of the tangential wind component may be separated intoradial (V dr ) and vertical (V dz) contributions:

〈Dv〉 = 1r2

∂⟨r2τrλ

⟩∂r︸︷︷︸

V dr

+ 1ρ0

∂ 〈ρ0τλz〉∂z︸︷︷︸

V dz

, (13)

where the subgrid-scale momentum fluxes are related to the mean strain-rate tensor in cylindricalcoordinates by a simple K-theory closure taking the form of local eddy diffusion relations (writtenhere in cylindrical-polar coordinates):

〈τrλ〉 =⟨Km,h

(1r

∂u∂λ

+ r∂v/r∂r

)⟩, 〈τλz〉 =

⟨Km,v

(1r

∂w

∂λ+ ∂v

∂z

)⟩, (14)

with parameterization formulas for horizontal and vertical eddy diffusivities, Km,h and Km,v (notwritten here; see Persing et al. 2013 for details). The analogous specification for 〈τrz〉 is

〈τrz〉 =⟨Km,v

(∂u∂z

+ r∂w/r∂r

)⟩, (15)


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and ρ0(z) is the basic state density profile. As discussed above, in the flux form of the vorticityequation, the subgrid-scale turbulent momentum fluxes 〈τrλ〉 and 〈τλz〉 are regarded as part of thenonadvective vorticity flux.

Persing et al. (2013, section 6) found that, although the mean vorticity influx and verticaladvection terms comprise leading terms of the mean tangential wind tendency (thus supportingthe revised illustration of spin-up in Figure 1), the resolved and parameterized (subgrid) eddyprocesses contribute significantly also to the mean spin-up tendency around the eyewall andtangential wind maximum throughout the troposphere. The resolved-eddy momentum fluxes areassociated with the in-cloud vorticity structures. Persing et al. (2013) obtained insight into thephysical nature of the in-cloud vorticity structures by using a flux equivalent form of Equation 12,adopting a Boussinesq approximation for simplicity. Equation 12 may then be rewritten as follows:

∂〈v〉∂t

= − 1r2

∂(r2 〈u〉〈v〉)

∂r− ∂(〈w〉〈v〉)

∂z− f 〈u〉 − 1

r2

∂(r2 〈u′v′〉)

∂r− ∂(〈v′w′〉)

∂z+ 〈Dv〉. (16)

Again, Dv is the subgrid-scale tendency expressed as a radius-height divergence of the subgridmomentum fluxes τ :

〈Dv〉 = 1r2

∂r2 〈τrλ〉∂r

+ ∂ 〈τλz〉∂z

, (17)

where, for consistency with the Boussinesq type of approximation, the vertical variation of thebasic state density has been neglected in the vertical derivative term. Comparison of Equations 16and 17 shows the direct analogy of resolved −〈u′v′〉 and −〈v′w′〉 with subgrid 〈τrλ〉 and 〈τλz〉. Inaddition, in the mean radial and vertical momentum tendency equations (not written), the resolved−〈u′w′〉 is the analog of subgrid 〈τrz〉.

4.2.1. Horizontal eddy momentum fluxes. Figure 6 shows the horizontal (radial) eddy mo-mentum flux in a radius-height format and is focused on the inner-core region of the vortex wherethe deep convection is active. It shows also the corresponding subgrid-scale momentum fluxesparameterized by the turbulence closure scheme given by Equations 14 and 15 and the radius ofthe maximum azimuthally averaged tangential velocity, i.e., the radius of maximum wind (RMW),at each height. The data are time averaged over a 4-h interval (144–148 h) during an intensificationphase of the vortex, but other time intervals during intensification produce similar results.

Figure 6a shows that during spin-up, the resolved-eddy momentum flux, −〈u′v′〉, has a co-herent region of positive values around the RMW within and just above the boundary layer andextending upward and outward in the mean updraught to the middle troposphere. This impliesan inward eddy tangential (and angular) momentum transport that is directed in the same sense asthe gradient of mean angular velocity (see Equation 14), which has its maximum value at or nearthe center of circulation during the spin-up phase. In contrast, Figure 6d shows that the corre-sponding subgrid-scale momentum flux is predominantly negative and weaker in magnitude thanthe resolved-eddy flux near the RMW and within the mean updraught. Therefore, the resolvedflux in the lower troposphere acts in a direction opposite to the averaged local angular velocitygradient assumed by the subgrid-scale model (i.e., it is countergradient). [The azimthally averagedradial subgrid-scale momentum flux (Equation 14) is dominated by Km,hr∂/∂r(〈v〉/r). AlthoughKm,h does vary with radius and height (see Persing et al. 2013, figure 15), the radial derivative of〈v〉/r dominates.] Simply, the resolved horizontal eddy momentum flux does not act diffusivelyto weaken the mean vortex. Rather, it amplifies the low-level tangential winds inside the RMWand contributes to a contracting RMW with time.


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4– 4 20

Horizontal eddy flux, 〈–u'v' 〉(all panel units are in m2 s–2)

4– 4 20

Vertical eddy flux (radial), 〈–u'w' 〉

4– 4 20

Vertical eddy flux (tangential), 〈–v'w' 〉

2– 2 10

Subgrid momentum flux (panel a), 〈τrλ〉 Subgrid momentum flux (panel b), 〈τr z〉

0.20– 0.20 0.01– 0.01

Subgrid momentum flux (panel c), 〈τλz〉

15

10

5

0

Hei

ght,

z (k

m)

15

10

5

0

Hei

ght,

z (k

m)

0 80 1006020 40Radius, r (km)

0 80 1006020 40Radius, r (km)

0 80 1006020 40Radius, r (km)

0 80 1006020 40Radius, r (km)

0 80 1006020 40Radius, r (km)

0 80 1006020 40Radius, r (km)

dd ee ff

aa bb cc

0.20– 0.20 0.01– 0.01

0

00

0

0

0 0

0 0

0

0

0

00

0

0

0

0

0

0

0

00

–2

–2

–2

–2

0

0–0.1 0.1

0.1

0.05

–0.05

0.05

0.1

–0.1

1.0

0.1

Figure 6Radius-height contour plots of resolved and subgrid-scale eddy momentum fluxes and related quantities from the 3D3k simulation ofPersing et al. (2013, figure 15) averaged over the second intensification interval (144–148 h). (a) Resolved horizontal momentum flux〈−u′v′〉. (b) Resolved vertical eddy flux of radial momentum 〈−u′w′〉. (c) Resolved vertical eddy flux of tangential momentum 〈−v′w′〉.(d ) Subgrid momentum flux corresponding to panel a, 〈τrλ〉. (e) Subgrid momentum flux corresponding to panel b, 〈τrz〉. ( f ) Subgridmomentum flux corresponding to panel c, 〈τλz〉. Contour intervals are 2 m2 s−2 for panels a–c, 0.5 m2 s−2 for panel d, and 0.01 m2 s−2

between −0.1 and +0.1 (thin) and 0.2 m2 s−2 above 0.1 and below −0.1 (thick) for panels e and f. The dotted black circles in each plotshow the location of the maximum azimuthally averaged tangential wind speed at each height.

Persing et al. (2013, section 6.5) provided insight into the nature of the upgradient horizontaleddy momentum fluxes by using high-resolution model output to examine the temporal evo-lution of the convective vorticity structures in horizontal planes. The evolutionary behavior ofthese structures is dominated by deep convection episodes, cyclonic vorticity enhancement by


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vortex-tube stretching in convective updraughts near the RMW, and vortex wave-like dynamicsinvolving the progressive shearing of cyclonic vorticity anomalies.

4.2.2. Vertical eddy momentum fluxes. The patterns of the resolved vertical eddy fluxes (i.e.,−〈u′w′〉 and −〈v′w′〉 in Figure 6b,c) are tall, outward-sloping columns concentrated around theRMW and mean updraught. This location is where the vortical convective updraughts are most ac-tive, and they are presumably the primary agents of these flux columns. The tendency of these fluxcolumns contributes to extending the region of strong tangential wind higher in the troposphere.

Persing et al. (2013) found that the spin-up tendency from V ev (and the corresponding verticaldivergence of −〈w′v′〉) is roughly three times larger than that for V eζ . That is, in the regionof vortical convection, the contribution from the nonadvective eddy vorticity flux to spin-up iscomparable with, or greater than, the contribution from the advective eddy vorticity flux. Theyfound also that the resolved-eddy flux, −〈u′w′〉, does not act like eddy diffusion, but rather acts tolocally strengthen the mean overturning circulation.

The subgrid vertical fluxes (Figure 6e, f ) have the expected large extrema in the boundarylayer, but they show nothing in the tropospheric RMW-updraught region where the resolved-eddy fluxes are active. There is some pattern similarity in the negative −〈v′w′〉 and 〈τλz〉 in theupper-troposphere updraught region, but the latter is much smaller in magnitude. 〈τrz〉 has a weakvertical dipole pattern in the upper-tropospheric outflow region also. This pattern implies a weaktendency in 〈u〉 to decrease the outflow altitude.

4.2.3. Synthesis and illustration of revised spin-up. During the spin-up of the vortex, theresolved-eddy momentum fluxes associated with the cloud-scale eddies act to strengthen the meantangential and radial circulation in the developing eyewall region. The largest resolved-eddy fluxesoccur in the RMW-updraught region where vortical convection is most active. The resolved-eddyfluxes generally do not act like eddy diffusion, but represent a nonlocal flux associated with thevortical updraughts and downdraughts. The resolved-eddy component of the nonadvective vor-ticity flux is as important in the vorticity (and circulation) dynamics as the corresponding advectivevorticity flux. The foregoing results show that both radial and vertical resolved-eddy fluxes havequalitatively different patterns above the boundary layer compared to the subgrid-scale eddy-diffusive fluxes, and the resolved-eddy fluxes are generally larger in magnitude, especially thevertical fluxes. The disparity between the resolved and subgrid patterns belies a simple interpre-tation as local momentum mixing.

During intensification, the multiple vortical updraughts excite vortex Rossby and inertia-buoyancy waves (as discussed, e.g., in Chen et al. 2003, Reasor & Montgomery 2015), whichin turn contribute to the sign and structure of the eddy momentum fluxes. The intensificationprocess generally comprises a turbulent system of rotating, deep moist convection and vortexwaves. A more complete understanding of the complex eddy dynamics is certainly warranted. Asa first step in this direction, Kilroy & Smith (2016) investigated the effects of eddy momentumfluxes associated with a single updraught on the tangential-mean velocity tendency and provideda conceptual framework for the interpretation of these eddy fluxes.

Zhang & Marks (2015) recently confirmed the foregoing findings regarding the positive con-tribution of the eddy processes to vortex spin-up by using an idealized configuration of the NOAAoperational hurricane forecast model. For realistic settings in the subgrid-scale parameteriza-tions suggested by recent observations, Zhang & Marks (2015, p. 3392) found that “angularmomentum budget analyses during the intensification phase suggest that the eddy transportof angular momentum contributes substantially to the total tendency of angular momentum,


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15

10

5

00 50

Radius, r (km)H

eigh

t, z

(km

)

B O U N D A R Y L AY E R

E Y ESpin-up of circulation

beyond eyewall

100

ν = M/r–½ frM is materially conserved

M is reduced by friction,but strong convergence

→ small r → large ν

Eyewall updraft spun up by the vertical advection of Mfrom the boundary layer and by eddy momentum fluxes

Figure 7Schematic illustration of the revised view of system-scale spin-up in the rotating convection paradigm. Graygradient arrows denote the sense of transverse (secondary) circulation. Blue shading denotes cloudy areas.The size of the arrows is not proportional to the flow speed.

especially at low levels (<4 km) inside the radius of the maximum tangential wind speed when Lh

(the horizontal mixing length, our insertion) is small.”These findings, together with those discussed in Section 3.5, suggest a further revision of

the spin-up schematic illustration in Figure 1, which is shown in Figure 7. The conventionalmechanism of spin-up helps explain the spin-up of the circulation outside the eyewall updraught.The eyewall updraught itself is spun up by the mean vertical advection of high tangential mo-mentum from the boundary layer and by the resolved-eddy momentum fluxes discussed above.

5. POTENTIAL INTENSITY THEORY

Our understanding of hurricanes has been strongly influenced by the simple, axisymmetric,steady-state hurricane model described in a pioneering study by Emanuel (1986). This model hasunderpinned many ideas about how tropical cyclones function. It provided the foundation forthe so-called potential intensity (PI) theory of tropical cyclones (Emanuel 1988, 1995; Bister &Emanuel 1998), and its time-dependent extension led to the formulation of the WISHE paradigmfor intensification (Emanuel 1989, 1997, 2003, 2012) referred to in Section 3.

PI theory refers to a prediction of the maximum possible intensity that a storm could achieve ina particular environment, based on the maximum possible tangential wind component (specificallythe maximum gradient wind). [Emanuel’s (1988) formulation for the maximum intensity of hurri-canes characterized the intensity by the minimum surface pressure. Subsequent papers revised theformulation for the minimum surface pressure and shifted focus to the maximum gradient wind.]That such an upper bound on intensity should exist follows from global energy considerations.Under normal circumstances, the energy dissipation associated with surface friction scales as thecube of the tangential winds, whereas the energy input via moist entropy fluxes scales generallywith the first power of the wind. (There is a subtle caveat with this scaling argument because thelinear dependence of the energy input on wind speed may be suppressed if the degree of mois-ture disequilibrium at the sea surface is reduced as the wind speed increases.) It follows that the


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frictional dissipation will exceed the input of latent heat energy to the vortex from the underlyingocean at some point during the cyclone’s intensification.

It is presently thought that there is an exception to the foregoing argument when the sea surfacetemperature (SST) is sufficiently warm or the upper-tropospheric temperature is sufficiently cold,or some combination of the two prevails (Emanuel 1988). Under such extreme conditions, thevortex is believed capable of generating enough latent heat energy via surface moisture fluxesto more than offset the dissipation of energy, and a runaway hurricane, the so-called hypercaneregime, is predicted. However, these predictions have been formulated only in the context ofaxisymmetric theory and simulated using subsonic, axisymmetric flow codes. It remains an openquestion whether hypercanes are dynamically realizable in a realistic, 3D flow configuration.

Most storms never reach their PI (Merrill 1988, figure 1; DeMaria & Kaplan 1994, figure 1;Emanuel 1999). This is attributed to the deleterious effects of vertical shear, which tends totilt/deform a developing vortex and open pathways for dry air intrusion (Riemer et al. 2010, Tang& Emanuel 2010, Riemer & Montgomery 2011).

Despite uncertainties regarding PI theory, discussed below, it has been widely used to estimatethe impact of global climate change on tropical cyclone intensity and structure change. As anexample of its far-reaching influences, Emanuel’s (1986) theory is still used as a basis for derivingupdates to the a priori PI theory (Bryan & Rotunno 2009a, Garner 2015) as well as for estimatingthe impact of tropical cyclone intensity and structure change due to global warming scenarios(Emanuel 1988, Camargo et al. 2014).

5.1. Emanuel’s Steady-State Model

Figure 8 schematically illustrates Emanuel’s (1986) steady-state hurricane model. The energeticsof this model are often compared to that of a Carnot cycle in which the inflowing air near thesurface acquires heat (principally latent heat) while remaining approximately isothermal. Theascending air is assumed to be moist adiabatic, and the outflowing air at large radii is assumedto descend isothermally in the upper atmosphere. The final leg in the cycle is assumed to followa reversible moist adiabat. Bister et al. (2010) pointed out that this hypothetical dissipative heatengine does no useful work on its environment.

Emanuel’s (1986) model assumes hydrostatic balance and gradient wind balance above theboundary layer and uses a quasi-linear slab boundary model in which departures from gradientwind balance are negligibly small. The boundary layer is assumed to have constant depth h inwhich M and the pseudo-equivalent potential temperature θe are well mixed. This layer is dividedinto three regions (Figure 8): the eye (region I); the eyewall (region II); and outside the eyewall(region III), where spiral rainbands and shallow convection are assumed to form in the vortexabove. The quantities M and θe are assumed to be materially conserved after the parcel leavesthe boundary layer and ascends in the eyewall cloud. [Contrary to statements made by Emanuel(1986), the formulation assumes pseudo-adiabatic rather than reversible thermodynamics in whichall condensate instantaneously rains out (Bryan & Rotunno 2009a, p. 3044). It is not a true Carnotcycle, in part because of the irreversible nature of the precipitation process in the eyewall regionof the vortex.]

In the steady model, the parcel trajectories are streamlines of the secondary circulation alongwhich M and θe are materially conserved. The precise values of these quantities at a particularradius are determined by the frictional boundary layer. The model assumes that the radius ofmaximum tangential wind speed, rm, is located at the outer edge of the eyewall cloud, althoughrecent observations indicate it is closer to the inner edge (Marks et al. 2008). The gain of relativeangular moment (RAM; rv) is needed to replace the frictional loss of angular momentum at the


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Vor tex sheet

Vor tex sheetM,θ*

eM,θ*e

h

o

d

o'

a

rm ro r1Radius

Region I Region II Region III

Hei

ght

Figure 8Schematic diagram of Emanuel’s (1986) model for a steady-state mature hurricane. The arrows indicate thedirection of the overturning circulation. The middle dashed curve emanating from rm is the M surface alongwhich the vertical velocity is zero and demarcates the region of ascent in the eyewall from that of large-scaledescent outside the eyewall. The outer dashed curve indicates the location of the vortex sheet as described bySmith et al. (2014). The flow segment between o and o ′ in the upper right corner of the figure represents theassumed isothermal leg and is the location at which air parcels are assumed to steadily gain cyclonic relativeangular momentum (rv) from the environment.

surface where the flow is cyclonic. This source of RAM is required for a steady state to exist, andit is pertinent to enquire whether this source is physically plausible (see Section 6).

Aside from the assumption of axial symmetry and realism of the source of RAM, the modelsuffers a range of issues, as discussed by Smith et al. (2008), Bryan & Rotunno (2009b), Emanuel(2012), and Montgomery & Smith (2014) and summarized below.

The original formulation for the PI leads to an equation for V 2max (Emanuel 1986, equation 43):

V 2max = Cθ

CDεLq ∗

a (1 − RH as)1 − f 2r2

04β RTB

1 − 12

Cθ

CDε

Lq∗a (1−RHas)β RTs

, (18)

where V max is the maximum gradient wind, L is the latent heat of condensation of water vapor,T s is the SST, Cθ is the surface exchange coefficient of moist entropy (and enthalpy), CD is thedrag coefficient, ε = (TB − T0)/TB is the thermodynamic efficiency factor, TB is the averagedtemperature of the boundary layer (assumed constant with radius), and T0 is the average outflowtemperature weighted with the saturated moist entropy of the outflow angular momentumsurfaces (Emanuel 1986, equation 19). [T0 is the temperature at which air parcels are assumed todescend approximately isothermally in the upper atmosphere.] Additionally, q ∗

a is the saturationmixing ratio at the top of the surface layer in the environment, RH as is the ambient relativehumidity at the top of the surface layer, β = 1 − ε(1 + Lq ∗

a RH as/RT s), and r0 is the radial extentof the storm near sea level (nominally the radius at which V = 0). [The mathematical definitionfor r0 is given by Emanuel (1986, equation 20).]

From Equation 18, Emanuel constructed curves for V max as a function of outflow temperatureand SST. For example, for an SST of 28◦C and an outflow temperature of −60◦C, the formula pre-dicts a V max of approximately 58 m s−1 (see Figure 9). This calculation assumes that Cθ /CD = 1,but the latest field observations and laboratory measurements synthesized in Bell et al. (2012)suggest a mean value of approximately Cθ /CD = 0.5 in the high-wind speed range. AlthoughBell et al. (2012) acknowledged the scatter in the latest observational estimates of Cθ /CD in


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20202525

3030

3535

4040

4545

5050

5555

6060

6565

70707575

30

32

28

26

24

22

20

0 –80–60–40–20 –70–50–30–10

To (°C)

Ts (°C)

Figure 9Predicted V max from Equation 18 as a function of sea surface temperature (T s) and outflow temperature(To) from Emanuel’s (1986) model for a mature steady-state hurricane. The ratio of moist entropy tomomentum transfer coefficients Cθ /CD is assumed to be unity. Calculations assume an ambient surfacepressure of 1,015 mb, an ambient relative humidity of 80%, a Coriolis parameter f evaluated at 20◦ latitude,and an outer radius r0 equal to 500 km. Figure adapted from Emanuel (1986).

the high-wind speed regime, these data represent our best estimates. For this reduced ratio ofexchange coefficients, Equation 18 predicts a reduced V max of approximately 42 m s−1.

One puzzling feature of both the Emanuel (1986) derivation and its extensions discussed belowis that there seems to be no constraint that ∂v/∂r = 0 at the radius of maximum tangential wind.Moreover, all derivations within this formalism appear not to predict the radius of maximumtangential wind, at least in terms of a priori known quantities.

Some major hurricanes can significantly exceed the value predicted by Equation 18. For exam-ple, Hurricane Isabel (2003) exceeded its predicted PI for three consecutive days (Montgomeryet al. 2006a, Bell & Montgomery 2008). On September 13, 2003, Isabel had an observed maxi-mum tangential wind of approximately 76 m s−1, yet a best estimate based on Equation 18 using inretrospect an arguably liberal Cθ /CD = 1 gives only 56.6 m s−1. [This estimate takes into accountthe uncertainty of the exchange coefficients and the ocean cooling effect by turbulence-inducedupwelling of cooler water from below the thermocline by assuming a ratio of exchange coefficientsof unity and that dissipative heating offsets ocean cooling (see Montgomery et al. 2006a and Bell& Montgomery 2008 for further details and the discussion below regarding an update on thematter of dissipative heating).] In this case, the discrepancy, approximately 20 m s−1, betweentheory and observations spans at least two intensity categories on the Saffir-Simpson HurricaneWind Scale (Categories 3–5). That some major hurricanes in regions of the world oceans cansignificantly exceed this theoretical predicted intensity for SSTs of 28◦C (or higher) is presumablya consequence of certain assumptions made in formulating Emanuel’s (1986) model.

High-resolution axisymmetric numerical simulations have been conducted to test the originalPI theory in a controlled setting (Persing & Montgomery 2003, Hausman et al. 2006, Bryan& Rotunno 2009a). For horizontal mixing lengths consistent with recently observed estimatesfrom flight-level data in major hurricanes (Zhang & Montgomery 2012), the numerical studies ofPersing & Montgomery (2003) and Bryan & Rotunno (2009a) confirm the tendency for solutions


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to exceed significantly the theoretically predicted PI given by Equation 18 and its modificationssummarized below.

5.2. Unbalanced Effects

Important limitations of Emanuel’s (1986) theory include its neglect of unbalanced dynamicsin the frictional boundary layer (Smith et al. 2008) and above the boundary layer (Bryan &Rotunno 2009b). Bryan & Rotunno (2009a) derived a modified formula for V max that accountsfor unbalanced processes above the boundary layer: V 2

max = EPI2 + γ , where EPI is V max as givenby Equation 18, γ = αrmaxηbwb, and these latter terms are evaluated at the top of the boundarylayer at the location of maximum tangential velocity (which, as noted above, is not predicted bythe theory itself ). Here, η = ∂u/∂z − ∂w/∂r is the azimuthal component of vorticity, rmax is theradius of V max, wb is the vertical velocity at the top of the boundary layer at this same radius andα = T s/To, where Ts and To are as defined above. The α term is associated with the inclusionof dissipative heating (the increase in internal energy associated with the dissipation of kineticenergy) in the PI formulation of Bister & Emanuel (1998).

In a recent theoretical study, Kieu suggested an inconsistency of the Bister & Emanuel (1998)formulation and related assumption that all dissipative heating in the atmospheric surface layer canreturn to the atmosphere as an additional heat source that acts to augment the maximum gradientwind of the vortex. Kieu (2015, p. 2504) recommended the use of the original PI formationof Emanuel (1986), as “it can be reinterpreted as a rational estimation of the [tropical cyclonemaximum potential intensity] even in the presence of the internal dissipative heating.” Drawing onthe observational analyses of Zhang (2010), Kieu suggested that the dissipative heating formulationused by Bister & Emanuel (1998) and Bryan & Rotunno (2009a,b) overestimates the true dissipativeheating, in part, because of an inaccurate estimate of the viscous work term in the boundary layerand because of the radiation of energy out of the hurricane by wind-induced surface gravity wavesat the air-sea interface. This interesting topic merits further study.

In the context of the modified formula for V max introduced above, Bryan & Rotunno (2009a)showed that

ηbwb ≈(

v2

r− v2

g

r

), (19)

where v is the total tangential velocity and vg is the tangential velocity in gradient wind balancewith the radial pressure gradient force per unit mass, both of which are evaluated at the top of theboundary layer. In a supergradient flow, the right-hand side of this equation is positive, and γ onthe right-hand side of the modified formula for V max is a positive contribution. In the limiting caseof small horizontal mixing length, Bryan & Rotunno (2009a, p. 3055) found that γ ≈ EPI so that“the effects of unbalanced flow contribute as much to maximum intensity as balanced flow for thiscase.” In the case of Hurricane Isabel, the new formula was a significant improvement, and despiteuncertainties in the observations, Bryan & Rotunno (2009a, p. 3056) concluded that “unbalancedflow effects are not negligible in some tropical cyclones and that they contribute significantly tomaximum intensity.”

Bryan & Rotunno’s (2009a) extended analytical theory is not an a priori form of PI in the senseof using only environmental conditions as input: It requires also knowledge of ηb, rmax, and wb.Thus, one cannot graph V max as functions of SST and outflow temperature, similar to Figure 9.The theory continues to use the same boundary layer formulation as Emanuel (1986) (see theirsection 2b). In essence, the flow in the boundary layer is assumed to be in approximate gradient


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wind balance. For this reason, the new theory does not address the concerns raised about theoriginal theory by Smith et al. (2008).

Clearly, a more complete boundary layer formulation that includes a radial momentum equa-tion would be required to determine ηb, rmax, and wb, and thereby V max. Indeed, these three quan-tities (ηb, rmax, and wb) characterize the effect of boundary layer control on the eyewall dynamicsdiscussed by Kilroy et al. (2016) and Schmidt & Smith (2016) (see Section 3.8). In this sense, Bryan& Rotunno’s (2009a) analysis provides further evidence that unbalanced effects in the boundarylayer are responsible for those in the eyewall. When ηb, rmax, and wb are diagnosed from a numeri-cal simulation (with an unbalanced boundary layer), the new formula for V max provides an accurateestimate for the maximum intensity of numerically simulated vortices (Bryan & Rotunno 2009b)for a range of values of the horizontal mixing length (see Bryan & Rotunno 2009a, figure 12).

Notwithstanding the good agreement, there appear to be two issues with the comparison of PItheory with their numerical calculation, both acknowledged in Bryan & Rotunno (2009a). First,the PI is calculated at the radius of V max in the model rather than at the radius of maximumgradient wind of either the theory or model. Second, the calculation of the gradient wind fromthe pressure field in the full model output incorporates substantial unbalanced effects, as well asthe balanced effects contained in Emanuel’s (1986) model.

Bryan & Rotunno (2009a), and subsequently Rotunno & Bryan (2012) and Bryan (2013),have emphasized the strong dependence of the simulated intensity in axisymmetric models to thehorizontal mixing length (and related diffusivity) used to parameterize asymmetric mixing andsmall-scale turbulence.

5.3. A Revised Theory

Emanuel & Rotunno (2011) and Emanuel (2012) questioned the assumption of the original modelthat the air parcels rising in the eyewall exit in the lower stratosphere in a region of approximatelyconstant absolute temperature. Emanuel (2012, p. 989) stated that Emanuel & Rotunno (2011)“demonstrated that in numerically simulated tropical cyclones, the assumption of constant outflowtemperature is poor and that, in the simulations, the outflow temperature increases rapidly with an-gular momentum.” A revised theory was proposed in which the absolute temperature stratificationof the outflow is determined by small-scale turbulence that limits the gradient Richardson numberto a critical value. Ordinarily, the Richardson number criterion demarcates the boundary betweenstratified shear stability and instability/turbulence. Here it seems that small-scale turbulence inthe outflow layer is presumed to operate and limit the Richardson number to a critical value.

The new theory represents a major shift in the way a storm is influenced by its environment.In the previous version, it was assumed that the thermal structure of the lower stratospheredetermined the (constant) outflow temperature. In the revised theory, the vertical structure ofthe outflow temperature is set internally within the vortex and, in principal, no longer matchesthe temperature structure of the environment. This shift in the formulation appears to haveramifications for the theory advanced by Nong & Emanuel (2003) as to how upper troughsinteract with the vortex and excite the process of inner-core wind amplification.

For Cθ /CD near unity, the revised theory predicts a reduced intensity by a factor of 1/√

2compared with the original formula given above (see Emanuel & Rotunno 2011, pp. 2246–47).However, it is difficult to assess the precise change in V max between the two theories for the caseof Hurricane Isabel or other observed storms because the new theory excludes the effects of dis-sipative heating and the increase of the saturation specific humidity with decreasing pressure. Inaxisymmetric numerical model simulations used to test the new theory, Emanuel (2012, p. 994)used also large values of vertical mixing length to “prevent the boundary flow from becoming


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appreciably supergradient,” thus keeping the tests consistent with a key assumption of the the-ory. Because the axisymmetric numerical model includes the foregoing effects and provides goodagreement with the theory, they argued that these effects must approximately cancel. This can-cellation would imply that the revised theory is not an improvement to explain the discrepancybetween the theory and observations in Hurricane Isabel, as discussed above.

5.4. 3D Effects

There are significant differences in behavior between tropical cyclone simulations in axisymmetricand 3D models (Yang et al. 2007, Bryan et al. 2009, Persing et al. 2013). 3D models predict asignificantly reduced intensity (15–20%) compared to their axisymmetric counterparts. In par-ticular, in 3D model simulations with parameter settings consistent with recent observations ofturbulence in hurricanes, Persing et al. (2013) showed little support for the upper-tropospheremixing hypothesis of Emanuel & Rotunno (2011). Persing et al. found that values of the gradientRichardson number were generally far from criticality with correspondingly little turbulent mix-ing in the upper-level outflow region within approximately 100 km from the storm center: Onlymarginal criticality was suggested during the mature stage. Based on these findings, 3D effectsshould be accounted for properly in a consistent formulation of the maximum intensity problem.

Bryan & Rotunno (2009b, p. 1771) pointed out that some of their own reported results “mightbe specific to axisymmetric models and should someday be re-evaluated using three-dimensionalsimulations.” This remark would seem to apply not only to their results.

6. REVISITING STEADY-STATE TROPICAL CYCLONES

Smith et al. (2014) questioned the existence of a realistic globally steady-state tropical cyclone.By global steady state, we mean that the macroscale flow does not vary systematically with time.Echoing Anthes (1974, p. 511), Smith et al. showed that if such a state existed, then a source ofcyclonic RAM would be necessary to maintain the vortex against the frictional loss of angularmomentum at the sea surface. Although a supply of cyclonic RAM is a necessary condition fora globally steady state cyclone, it is not sufficient. The vanishing of the spin-up function abovethe boundary layer and outside regions where turbulent diffusion is significant would also benecessary. A logical consequence would be that, without a steady source of cyclonic RAM, tropicalcyclones must be globally transient, a deduction that agrees with observations. The limitation ofthe steady-state assumption for all but the inner area of a mature tropical cyclone was recognizedby Ooyama (1982, section 2), but this issue seems to have been subsequently overlooked.

In simulated tropical cyclones that attain a quasi-steady state, a natural question arises as to thesource of RAM needed to support this state. Chavas & Emanuel (2014) presented solutions forlong-time sustained hurricanes using the Bryan cloud model in an axisymmetric configuration.Although they noted the limitations of their findings because of the long time required to achievethe equilibrium regime and the unlikelihood of observing quasi-steady hurricanes in reality overa several-day period, they did consider the primary source of RAM in one of their solutions.Using a surface torque balance for their control experiment, they reported that their sustainedhurricane simulation managed to maintain itself indefinitely via the replenishment of RAM byvertical diffusion at the surface in the anticyclonic portion of the outer vortex, as predicted bySmith et al. (2014).

Solutions for one or more sustained hurricanes lasting for 20 or more days have been foundto exist in 3D doubly periodic rectangular domains (Khairoutdinov & Emanuel 2013, Zhuoet al. 2014). Although these solutions appear to be plausible sustained hurricanes, the theoretical


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considerations of Smith et al. (2014) should still provide a means of identifying where the sourceof cyclonic RAM must originate for each individual hurricane to maintain itself. However, theneeded source of cyclonic RAM was not investigated by Khairoutdinov & Emanuel and Zhuoet al., and the precise source of RAM in these solutions remains unknown.

7. CONCLUSIONS

Above we review progress over the past 25 years in understanding of the fluid dynamics and moistthermodynamics of tropical cyclone vortices, focusing on vortex intensification and structure andthe role of coherent eddy structures in the evolution of these vortices. Important topics, such as theformation of these vortices, their movement, their interaction with larger-scale weather systems,and their extratropical transition, as well as the formation of vortex substructures such as vortexRossby waves, eyewall mesovortices, or secondary (outer) eyewalls, have not been covered.

The main emphasis is on a new rotating convection paradigm in which cloud buoyancy andvortex-tube-stretching are key elements. The frictional boundary layer plays a crucial role in theconvective organization and system-scale dynamics of the spin-up process. This boundary layerexerts strong control also on the location of the eyewall updraught, and the thermal properties ofthe updraught. A newly recognized feature of the spin-up process is that a sloping eyewall is spunup by the vertical advection of high tangential momentum from the boundary layer and not by theconventional mechanism of spin-up (i.e., through the radial import of absolute vorticity). Anothernewly recognized feature of the spin-up process is through the radial and vertical divergence ofeddy momentum fluxes, the latter of which is identified with the azimuthally averaged nonadvectivevorticity flux. During spin-up, these fluxes are typically countergradient and have no counterpartin strictly axisymmetric descriptions of these vortices.

Pioneering work describing mature hurricanes as axisymmetric, dissipative heat engines led toa theory for the maximum potential intensity of these storms. Several limitations to the theoryexist, resulting in recent modifications of the original theory. Despite efforts to revise the theory,limitations of the revised theory are noted also. Finally, other recent work calls into question theexistence of a globally quasi-steady tropical cyclone.

DISCLOSURE STATEMENT

The authors are not aware of any biases that might be perceived as affecting the objectivity of thisreview.

ACKNOWLEDGMENTS

M.T.M. acknowledges the support of NSF AGS-1313948, NOAA HFIP grantN0017315WR00048, NASA grant NNG11PK021, and the US Naval Postgraduate School.R.K.S. acknowledges financial support from the German Research Council (grant SM30-23) andthe Office of Naval Research Global (grant N62909-15-1-N021). The authors thank J. Persingand G. Kilroy for their valuable comments on a near-final draft of the manuscript. The viewsexpressed herein are those of the authors and do not represent sponsoring agencies or institutions.

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Annual Review ofFluid Mechanics

Volume 49, 2017Contents

An Appreciation of the Life and Work of William C. Reynolds(1933–2004)Parviz Moin and G.M. Homsy � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Inflow Turbulence Generation MethodsXiaohua Wu � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �23

Space-Time Correlations and Dynamic Coupling in Turbulent FlowsGuowei He, Guodong Jin, and Yue Yang � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �51

Motion of Deformable Drops Through Porous MediaAlexander Z. Zinchenko and Robert H. Davis � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �71

Recent Advances in Understanding of Thermal Expansion Effects inPremixed Turbulent FlamesVladimir A. Sabelnikov and Andrei N. Lipatnikov � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �91

Incompressible Rayleigh–Taylor TurbulenceGuido Boffetta and Andrea Mazzino � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 119

Cloud-Top Entrainment in Stratocumulus CloudsJuan Pedro Mellado � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 145

Simulation Methods for Particulate Flows and Concentrated SuspensionsMartin Maxey � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 171

From Topographic Internal Gravity Waves to TurbulenceS. Sarkar and A. Scotti � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 195

Vapor BubblesAndrea Prosperetti � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 221

Anisotropic Particles in TurbulenceGreg A. Voth and Alfredo Soldati � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 249

Combustion and Engine-Core NoiseMatthias Ihme � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 277

Flow Structure and Turbulence in Wind FarmsRichard J.A.M. Stevens and Charles Meneveau � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 311

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Particle Migration due to Viscoelasticity of the Suspending Liquid,and Its Relevance in Microfluidic DevicesGaetano D’Avino, Francesco Greco, and Pier Luca Maffettone � � � � � � � � � � � � � � � � � � � � � � � � � � 341

Uncertainty Quantification in AeroelasticityPhilip Beran, Bret Stanford, and Christopher Schrock � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 361

Model Reduction for Flow Analysis and ControlClarence W. Rowley and Scott T.M. Dawson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 387

Physics and Measurement of Aero-Optical Effects: Past and PresentEric J. Jumper and Stanislav Gordeyev � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 419

Blood Flow in the MicrocirculationTimothy W. Secomb � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 443

Impact on Granular BedsDevaraj van der Meer � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 463

The Clustering Instability in Rapid Granular and Gas-Solid FlowsWilliam D. Fullmer and Christine M. Hrenya � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 485

Phoretic Self-PropulsionJeffrey L. Moran and Jonathan D. Posner � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 511

Recent Developments in the Fluid Dynamics of Tropical CyclonesMichael T. Montgomery and Roger K. Smith � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 541

Saph and Schoder and the Friction Law of BlasiusPaul Steen and Wilfried Brutsaert � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 575

Indexes

Cumulative Index of Contributing Authors, Volumes 1–49 � � � � � � � � � � � � � � � � � � � � � � � � � � � � 583

Cumulative Index of Article Titles, Volumes 1–49 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 593

Errata

An online log of corrections to Annual Review of Fluid Mechanics articles may be foundat http://www.annualreviews.org/errata/fluid

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Documents

Recent Developments in the Fluid Dynamics of Tropical …