Tropical cyclones are intense, cyclonically-rotating, low-pressure weather systems that form over the tropical oceans. Cyclonic means counterclockwise in the northern hemisphere and clockwise southern hemisphere while intense means that sustained wind speeds exceed 17 m s-1 (60 km h-1, 32 kn) near the surface. The convention for the definition of a "sustained wind speed" is a 10 min average value, except in the United States, which adopts a 1 min average. Severe tropical cyclones have near surface sustained wind speeds equal to or exceeding 33 m s-1 (120 km h-1, 64 kn): these are called hurricanes over the Atlantic Ocean, the East Pacific Ocean and the Caribbean Sea, and Typhoons over the Western North Pacific Ocean. Typically the strongest winds occur in a ring some tens of kilometres from the centre and there is a calm region near the centre, the eye, where winds are light. For moving storms, the wind distribution is asymmetric with the maximum winds in the forward right quadrant in the northern hemisphere and in the forward left quadrant in the southern hemisphere. The eye obtains its name because, in a mature storm, it is normally free of deep clouds, but is surrounded by a ring of deep convective clouds that slope outwards with height. This ring is called the eyewall cloud or simply the eyewall. At larger radii from the centre, storms usually show spiral bands of convective clouds. Figure 1 shows a satellite view of the eye and eyewall of a mature typhoon, together with photographs looking out at the eyewall cloud from the eye during aircraft reconnaissance flights.
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The mature tropical cyclone consists of a horizontal quasi-symmetric circulation on which is superposed a vertical, or transverse circulation. These are sometimes referred to as the primary and secondary circulations, respectively. When combined, these two component circulations result in a spiralling motion with inflow at low and middle levels and outflow at upper levels. The secondary circulation is mostly thermally-direct, which means that warm air rising, a process that releases potential energy. However subsidence occurs in the eye and the circulation there is thermally indirect, a process that requires energy to be supplied.
Figure 2 shows a schematic cross-section of prominent cloud features in a mature cyclone including the eyewall clouds that surround the largely cloud-free eye at the centre of the storm; the spiral bands of deep convective outside the eyewall; and the cirrus canopy in the upper troposphere. Other aspects of the storm structure are highlighted in Fig. 3. Air spirals into the storm at low levels, with much of the inflow confined to a shallow boundary layer, typically 500 m to 1 km deep, and it spirals out of the storm in the upper troposphere, where the circulation outside a radius of a few hundred kilometres is anticyclonic. The spiralling motions are often evident in cloud patterns seen in satellite imagery and in radar reflectivity displays.
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The majority of tropical cyclones develop from pre-existing deep convective cloud systems, essentialy clusters of thunderstorms, in the tropics, but a few originate from the transformation of subtropical weather systems. Only a small fraction of tropical cloud systems develop into tropical cyclones and, traditionally, the questions as to how tropical cyclones form (the genesis problem) and how they intensify (the intensification problem) have been treated separately. Nevertheless, there is accumulating evidence to suggest that the physical processes involved in tropical cyclogenesis and intensification are the same (Montgomery and Smith 2011). Even so, it is, perhaps useful, to consider the intensification problem first, if only because, historically, most previous paradigms for intensification were axisymmetric, an assumption that cannot be applied to the genesis problem.
As a pathway towards developing understanding the intensification of tropical cyclone, many studies have focussed on the so-called prototype problem for intensification. This problem asks: how does a prescribed, initially cloud free, axisymmetric vortex in a quiescent environment evolve when located over a warm ocean on an f-plane? The assumption of a quiescent environment has served historically as a simplification for isolating basic aspects of cyclone intensification that do not involve strong interactions with the storm environment. In a recent review paper (Montgomery and Smith 2014), we examined and compared four paradigms for tropical cyclone intensification relate to this question.
The four paradigms reviewed are:
the Conditional Instability of the Second Kind (or CISK-) paradigm;
the cooperative intensification paradigm;
a thermodynamic air-sea interaction instability paradigm (widely known as WISHE, an acronym for Wind Induced Surface Heat Exchange); and
a new rotating convection paradigm.
The first three paradigms assume the flow to be axisymmetric, including any explicitly-resolved deep convection. This assumption can be justified as a reasonable approximation to the inner core of well-developed mature storms with an approximately circular eyewall, but is likely to be poor under other circumstances such as in the outer part of such storms and in the developing stages of storms, where observations show that significant flow asymmetries are the norm. In fact, a recent investigation by Persing et al. (2013) suggests that previous studies using strictly axisymmetric models, and their attendant phenomenology of axisymmetric convective rings, have intrinsic limitations for understanding the intensification process.
Particular problems with the CISK paradigm are discussed in Montgomery and Smith (2014) and it would be a distraction to go into these here. Furthermore, calculations presented by Montgomery et al. (2009) and more recently by Montgomery et al. (2015) show that the WISHE paradigm, currently the most widely cited intensification paradigm, is not the dominant mode of intensification in the prototype intensification problem described above. For these reasons we will focus here on paradigms 2 and 4.
The rotating convection paradigm does not assume the flow to be axisymmetric, but has an azimuthally-averaged mean view that incorporates and extends the cooperative intensification paradigm. The paradigm recognizes the presence of localized, rotating deep convection that grows in the cyclonic rotation-rich environment of the incipient storm, a process that is believed to operate also during the genesis phase. The updrafts within these convective structures greatly amplify the vorticity locally by vortex-tube stretching and the patches of enhanced cyclonic vorticity subsequently aggregate to form a central monolith of cyclonic vorticity (Hendricks et al. 2004, Montgomery et al. 2006, Nguyen et al 2008).
The mean field dynamics of the rotating convection paradigm constitute an extended cooperative intensification paradigm in which eddy processes can contribute positively to amplifying the tangential winds of the vortex. In this azimuthally-averaged view, illustrated schematically in Fig. 3, there are two mechanisms for spin up besides the eddy processes.
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The first mechanism is a key element of the cooperative intensification paradigm in which the spin up of the winds above the boundary layer is accomplished by the convectively-induced inward radial advection of the surfaces of absolute angular momentum, M, where this quantity is approximately materially conserved.
The quantity M is defined in terms of the tangential wind speed v by the formula M = rv + ½fr2, where r is the radius and f is the Coriolis parameter. Alternatively, v = M/r - ½fr.
It is assumed that surface moisture fluxes are sufficient to maintain the required deep convective activity to maintain the convergence.
Supported by a scale analysis of the equations of motion, the winds above the boundary layer are widely held to be in approximate gradient wind balance (e.g. Willoughby 1979, 1995). The conventional spin up mechanism has its roots in studies by Ooyama (1969, 1982).
Perhaps counter intuitively, the spin up of the maximum tangential winds takes place within the frictional boundary layer, where M is not materially conserved and where the winds are no longer in approximate gradient wind balance. The breakdown of gradient wind balance by the frictional retardation of the tangential wind component leads to a net inward force in the boundary layer and, as it turns out, to a much stronger inflow than in the vortex above. The stronger the inflow, the shorter is the trajectory of air parcels as they spiral inwards and therefore the smaller is the loss of M caused by the frictional torque. Spin up of the maximum tangential winds in the boundary layer is possible if the fractional rate of reduction of M is less than the fractional rate of reduction of inward displacement for an air parcel. The possiblility of spin up occuring in the boundary layer was anticipated by Anthes (1971), who noted ... the paradox of the dual role of surface friction, with increased friction yielding more intense circulation ... and such spin up was shown to occur in a numerical simulation of Hurricane Andrew by Zhang et al. (2001). The generality of this mechanism was articulated by Smith et al. (2009).
The two mechanisms of spin up are coupled through boundary layer dynamics because the equations for motion in the boundary layer depend on the tangential wind speed at the top of the boundary layer and increase as the winds above the boundary layer increase. It follows that a spin up of the winds in the boundary layer requires a spin up of the winds above the boundary layer as well. The foregoing ideas provide an explanation for observations that the maximum storm-relative tangential winds occur in the boundary layer (Kepert 2006a,b, Montgomery et al. 2006, Schwendike and Kepert 2008, Sanger et al. 2014, Montgomery et al. 2014).
From an azimuthally-averaged perspective, in the absence of convective forcing, the frictionally-induced inflow within the boundary layer would be accompanied by a shallow layer of outflow above the boundary layer (shallow because the atmosphere is stably stratified) and, by the material conservation of M in this outflow, to a spin down of the vortex. This spin down would be accompanied, through approximate gradient wind balance, by a demise of the radial pressure gradient at the top of the boundary layer. This process of vortex spin down was articulated by Greenspan and Howard (1963) and was examined in the hurricane context by Eliassen (1971), Eliassen and Lystadt (1977), and Montgomery et al. (2001). Clearly, for a vortex to spin up, the convectively-induced inflow must be sufficient to outweigh the frictionally-induced outflow above the boundary layer. In other words, the convection itself must be strong enough to more than "ventilate" the mass converging in the boundary layer associated with friction: it must be strong enough to produce inflow above the boundary layer also.
Persing et al. (2013) demonstrated that, within the new intensification paradigm, eddy processes can contribute positively to amplifying the tangential winds of the vortex. This positive contribution to vortex spin up contrasts with previous assumptions and speculation of the downgradient action of asymmetric motions (referred to as "turbulence", but including vortical convection and vortex Rossby waves and their wave-mean-flow and wave-wave interactions), which would lead to spin down (Bryan et al. 2010). The findings of Persing et al. suggest that previous studies using strictly axisymmetric models and their attendant phenomenology of axisymmetric convective rings have intrinsic limitations for understanding the intensification process.
As is well known, approximations to one or more of the governing equations for tropical cyclone evolution may be invoked to simplify the problem. In fact, guided by a scale analysis of the azimuthally-averaged equations for the bulk flow about the storm centre expressed in cylindrical coordinates, it is frequently assumed that the system-scale vortex is approximately in hydrostatic and gradient wind balance (e.g. \citealt{Willoughby1979}). For an axisymmetric vortex, these assumptions constrain the primary (or tangential) circulation above the boundary layer to be in thermal wind balance at all times. This constraint determines an equation for the streamfunction of the secondary (or overturning) circulation, which is required to maintain balance in the presence of processes trying to drive the system away from balance. Such processes include radial and vertical gradients of diabatic heating associated with latent heat release in deep convection, or vertical gradients of any frictional force in the boundary layer. Because of the pioneering work of Eliassen (1951, 1952) and Sawyer (1956) for both circular vortex and frontal circulations, the equation for the overturning circulation is often referred to as the Sawyer-Eliassen equation. When combined with the remaining un-approximated component of the momentum equations, i.e. that for the tangential component, one can develop a prognostic system of equations governing the evolution of a balanced vortex when the forcing terms in the Sawyer-Eliassen equation are prescribed or parameterized.
As it turns out, there are technical issues that can arise in the solution of the Sawyer-Eliassen equation in localized regions where the equation ceases to be elliptic (Möller and Shapiro 2002, Bui et al. 2009).
The balance theory does not strictly apply to a steady state vortex because the derivation of the Sawyer-Eliassen equation is formally not possible in this case. However, the existence of a realistic globally-steady state for a tropical cyclone has been questioned recently (Smith et al. 2014). For one thing, such a state would require a steady supply of cyclonic relative angular momentum to replenish that lost to the system by friction.
One limitation of the balance theory described above is that it is neither accurate nor formally applicable in the boundary layer, where the
assumption of gradient wind balance breaks down. In principle one might think of applying the theory above the boundary layer and using a
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Explicit comparisons between two different full physics mesoscale models and the Sawyer-Eliassen model and corresponding tangential wind tendency have been carried out in Smith et al. (2009) and Abarca and Montgomery (2014). These studies have shown that during vortex spin up the radial inflow in the boundary layer region using the balance model was insufficient to offset the frictional spin down effect. In other words, the balance model cannot capture the spin of the tangential wind in the boundary layer as observed in the full physics models.
A related approach to the balance formulation just summarized is that based on the geopotential tendency equation (Shapiro and Montgomery 1993, McWilliams et al. 2003, Vigh and Schubert 2009, Persing et al. 2013). In particular, Vigh and Schubert (2009) demonstrated analytically that the surface pressure fall in the balance model is significantly larger when the imposed ring of diabatic heating lies inside the high vorticity region of the inner core. This finding reaffirmed the idea that heating within regions of high inertial stability is highly efficient for tropical cyclone spin up (\citealt{SchubertHack1982}). However, it should be pointed out that, irrespective of the efficiency argument, a ring of convection located at an inner radius has the potential to converge air parcels to a smaller radius than a ring located at an outer radius, Thus the inner ring would be able to draw M surfaces to a smaller radius than the outer one leading potentially to a more intense vortex both above and within the boundary layer.
The new intensification paradigm has already proved useful in understanding the latitudinal dependence of the intensification rate for the prototype problem (Smith et al. 2015) and we would argue that aspects of it provide a useful starting point for understanding intensification in more complex environments with a background flow. The foregoing Smith et al. study highlighted the fact that any interpretation of vortex evolution, even in the absence of an environmental flow, requires consistent consideration of all three components of Newton's equations of motion, constrained by a mass continuity equation, as well as a thermodynamic equation and possibly equations for species of water substance. Explanations of intensification that fail to consider any one or more of these equations must be viewed with suspicion.
Arguments based on the balance formulation provide a succinct means for providing an
Because of the tight coupling between the boundary layer and the vortex above it, the construction of cause and effect arguments to explain vortex behaviour is fraught with danger. For example, as noted earlier, the boundary layer dynamics and thermodynamics control the radial profiles of radial and tangential velocity components within the boundary layer as well as those of vertical velocity, horizontal momentum and equivalent potential temperature that exit its top into the eyewall. It is the radial profiles of vertical velocity and equivalent potential temperature that determine, in part, the radial gradient of diabatic heating rate in the eyewall. The diabatic heating rate, Qdot = Dθ/Dt, is approximately related to the vertical velocity, w and equivalent potential temperature θe by the formula Qdot = μw, where μ = -L(∂qv/∂z) θe = constant), where L is the latent heat of condensation and qv is the water vapor mixing ratio. In turn, it is the radial and vertical gradients of diabatic heating rate as well as the forcing from the vertical velocity at the top of the boundary layer that determine the balanced secondary circulation in the vortex above the boundary layer. It is the inward branch of this circulation that determines, inter alia, changes to the tangential wind profile at the top of the boundary layer which then feeds back to determine the flow in the boundary layer, itself. Finally, the thermodynamics of the boundary layer are controlled in the outer region by the subsidence of vortex air into the boundary layer and the surface enthalpy flux, which depends, in part, on the surface wind speed (see e.g. Smith and Vogl 2008).
If one doesn't invoke balance dynamics, any viable theory for intensification needs to consider
In some real-world cases of tropical-cyclone intensification, the ambient flow is not weak and the magnitude of the vertical shear impinging on a storm is one of the critical parameters sought by forecasters. Although the new paradigm should still provide a useful building block for understanding vortex spin up in these more complex circumstances, we would expect that important modifications to it would emerge in the form of coherent eddy processes associated with the interaction of the vortex with the impinging vertical shear (Reasor et al 2004, Reasor and Montgomery 2014 and refs.) and their coupling to the boundary layer and convection, as well as the projection of these eddy processes on the azimuthally-averaged vortex dynamics.
A recent study of tropical cyclogenesis in wind shear by Nolan and McGauley 2012 gives a review of five decades of empirical and numerical modeling research examining the effects of vertical and horizontal wind shear on tropical cyclogenesis. The paper discusses also a suite of new numerical experiments and diagnostic analyses for a sheared vortex undergoing genesis and intensification. Although the tropical cyclogenesis problem lies beyond the scope of the present paper, Nolan and McGauley give modelling results and insightful interpretations that would appear to apply to the intensification problem as well, after genesis occurred (There are supporting theoretical reasons to believe that a unified view of the genesis and intensification problems is meaningful (Montgomery and Smith 2011, Riemer and Montgomery 2011).). One of Nolan and McGauley's principal conclusions was that large vertical shear values "delayed or suppressed further development (intensification after genesis, our insertion), consistent with a substantial body of previous work regarding the effects of wind shear on developing and mature tropical cyclones (Frank and Ritchie 2001, Wong and Chan 2004, Riemer et al. 2010, DeMaria and Kaplan 1999, Tang and Emanuel 2010)".
Insofar as the role of vertical shear in the intensification problem, the emerging view from Nolan and McGauley (2012} and complementary theoretical work is that moderate or weak vertical shear excites new dynamic-thermodynamic pathways through which relatively dry air may be entrained into the moist envelope region of the vortex. These pathways act to generate mesoscale downdrafts that flush portions of the boundary layer with low-level moist equivalent potential temperature (θe) air that originate from above the boundary layer near the minimum of θe and outside of the moist envelope (Tang and Emanuel 2010, Riemer et al. 2010, Riemer et al. 2013, Riemer and Montgomery 2011). A boundary layer with reduced $\theta_e$ suppresses convective instability, and, unless the moisture fluxes can ameliorate the θe deficit for inward-spiraling air parcels, the vortex will begin to spin down until the boundary layer can recover to its pre-shear values and intensification can resume (Riemer et al. 2010, Riemer et al. 2013).
The foregoing is a broad-brush synthesis of what we know about the physical effects of vertical shear in tropical cyclone intensification. Without a doubt, the vertical shear intensification problem is an important scientific problem of societal relevance and further basic research on it is clearly warranted to better understand the dynamic-thermodynamic pathways that have been discovered in recent work. However, before embarking on a systematic programme, it is critical to have a solid understanding of the intensification problem in the prototype problem as defined above.
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Latest version: Munich 09 Feb 2015