Part 5
Advances in Tropical Cyclone Research
21
Introduction to Hurricane Dynamics: Tropical Cyclone Intensification
Michael T. Montgomery Department of Meteorology, Naval Postgraduate School Monterey, California, USA [email protected]
1. Introduction The broad aim of this set of two articles on tropical cyclone theory is to provide graduate students and post-doctoral researchers from India and other parts of the world with state-of-the-art knowledge and understanding of tropical cyclone dynamics. It is hoped that this knowledge base will be a useful tool to help these students and post docs understand and improve the forecasts of tropical cyclone genesis and intensification in various parts of the world affected by these storms. To me, tropical cyclones are magnificent and scientifically fascinating large-scale convective vortices that still hold many hidden secrets regarding their birth, intensification, mature evolution and decay. Observing the formation and intensification of these large-scale coherent structures fuels my scientific passion to obtain a deeper physical understanding of what many atmospheric scientists have called “the greatest storms on earth” (see examples in Fig. 1). Practical considerations, such as saving human life and property in the path of these tropical tempests are, of course, another important driving factor in the quest for knowledge about these storms. My interest in these storms spans all facets of these storms. Recent events surrounding Hurricane Sandy (2012) are a reminder that even tropical storms can wreak havoc on populated coastal communities, maritime assets and even inland populations (e.g., Lussier et al., 2015). As coastal communities continue to grow, there is an increasing demand throughout regions of the world for more accurate tropical cyclone forecasts. Typically, the response to such a demand relies heavily on financial resources devoted to develop and apply more comprehensive numerical models using 4 Michael T. Montgomery increased computer power. However, these efforts are no guarantee for success in improving tropical cyclone forecasts. It is the belief of the author that without a commensurate understanding of the dynamics and thermodynamics of these convective vortices, such efforts, even if grandiose, will not realize their full potential. It is only when the state-of-art computational methods and observations are combined with hypothesis testing and the development of conceptual models that a firm scientific foundation can be established to make measurable advances in hurricane intensity forecasts. In coordination with aircraft and satellite observations, knowledge of the structure of mature storms progressed rapidly from the earliest hurricane reconnaissance missions just after the Second World War. Understanding of the mature stage of these tropical tempests evolved largely in parallel with the collection and analysis of these observational data (e.g., Anthes, 1982; Emanuel, 1986; Willoughby, 1995; Montgomery et al., 2006; Houze et al., 2007; Smith et al., 2008; Marks et al., 2008; Bell et al., 2008) and also see Chapters 1 and 2 by Dr. Frank D. Marks Jr. and Dr. C.M. Kishtawal respectively. Today, our understanding of the birth, intensification and mature stages of these vortices is currently undergoing dramatic changes, not yet properly conveyed in current atmospheric science textbooks. One of the objectives of this article and associated references is to address this deficiency by summarizing recent scientific progress made on the intensification, mature and genesis stages of the tropical cyclone lifecycle. This article will begin by reviewing the current state of understanding of tropical cyclone intensification and will then summarize key elements of the dynamics and thermodynamics of tropical cyclone intensification in a new intensification paradigm.
2. Conceptual Models of Tropical Cyclone Intensification Over the past five decades substantial scientific efforts have been devoted to construct conceptual models of tropical cyclone (TC) intensification. A conceptual model (or paradigm) of TC intensification is an overarching framework of scientific ideas that may be used to direct and interpret observations, to design and develop numerical models, and to link together observations and numerical models in order to develop an improved tropical cyclone predictive system. Having a paradigm of TC intensification that correctly captures the essential dynamical and thermodynamical mechanisms occurring in nature is an essential ingredient to make forward progress in many aspects of TC research and operational activities. The four most prominent paradigms of TC intensification are the CISK model (Charney and Eliassen, 1964; Ooyama, 1964; Carrier, 1971), the cooperative intensification model (Ooyama, 1969, 1982; Willoughby, 1995, Introduction to Hurricane Dynamics: Tropical Cyclone Intensification 5
Fig. 1. Satellite imagery of (a) Typhoon Haiyan (November 07, 2013, western North Pacific), (b) Hurricane Isabel (September 13, 2003, Atlantic) and (c) Tropical Cyclone Bejisa (December 30, 2014, Northeast of Madagascar) in the southern Indian ocean. Image (a) is a colour enhanced infrared image at 13:45 UTC available at http:// rammb.cira.colostate.edu. Image (b) is a visible image at 1745 UTC from GOES super-rapid-scan operations. The bottom image (c) is an infrared image at 1213 UTC from the NOAA 18 satellite. The intensity of Haiyan (strongest tropical cyclone at landfall on record), the “pinwheel” pattern of mesovortices within Isabel’s eye, and the “pinhole-like” size of Bejisa´s eye are particularly striking and intriguing features. 6 Michael T. Montgomery
1998), the thermodynamic air-sea interaction model commonly known as Wind-Induced-Heat-Exchange (WISHE) (Emanuel, 1986, 1989; Rotunno and Emanuel, 1987; Emanuel et al., 1994; Emanuel 1995, 1997; Emanuel, 2012; Holton, 2004) and the new rotating convective model (Nguyen et al., 2008; Smith et al., 2009; Montgomery et al., 2009; Bui et al., 2009; Fang and Zhang, 2011; Persing et al., 2013; Montgomery and Smith, 2014). The first three models were presented originally assuming axisymmetric flow (no departures from axial symmetry about rotation axis, i.e., “no azimuthal eddies”), with a simple resting tropical environment without uniform flow or vertical shear. The simple resting environment has historically served as the canonical testing ground for obtaining a basic understanding of tropical cyclone intensification. Like the first three models, the rotating convective model is illustrated most simply in a quiescent tropical environment and includes air-sea transfer of momentum and heat. However, the rotating convective model is three- dimensional and contains an azimuthally-averaged (mean-field) dynamics that is forced by the azimuthally-averaged heat and momentum sources, including eddy momentum and eddy heat fluxes and their divergence, etc. The rotating convection model will be shown to be an extended cooperative intensification model and the explicit role of eddy processes as represented by the near-cloud resolving model is diagnosed in a newly published study by Persing et al. (2013). The four main intensification paradigms have been examined in Montgomery and Smith (2014) and the strengths and weaknesses of each of these paradigms are discussed there. The upshot is that only the rotating convective model properly captures the simulated behaviour of intensifying tropical cyclones in three dimensions and the dependence of storm intensification on key parameters pertinent to the convective-vortex phenomenology for realistic forecast time scales. In this lecture, only the most pertinent key points of the new paradigm and earlier models will be highlighted. To gain a broad picture of the physical elements of the new paradigm, it is first necessary to summarize the key ingredients of the first three intensification models.
2.1 The CISK, Cooperative Intensification and WISHE Paradigms The classical, or ‘conventional’ model of tropical cyclone spin-up, embodied in both the CISK and cooperative intensification models, is illustrated schematically in Fig. 2. The basic tenet is that, in an azimuthally-averaged sense, deep convection in the inner-core region generates inflow in the lower troposphere. Above the frictional boundary layer, the inflowing air materially conserves its absolute angular momentum (M) and spins faster. Here, M is defined as follows: Introduction to Hurricane Dynamics: Tropical Cyclone Intensification 7
M = r
Fig. 2. Schematic diagram illustrating the CISK and the cooperative intensification models (see text for details). Adapted from Montgomery and Smith (2014). 8 Michael T. Montgomery in a number of papers by Emanuel (1986), Raymond and Emanuel (1993), Emanuel et al. (1994), Craig and Gray (1996), Ooyama (1997) and Smith (1997). In particular, Raymond and Emanuel op cit. argued persuasively that the CISK closure fundamentally violates causality because convection is not caused by the macroscale water supply. In a landmark effort to overcome the problems with the CISK theory, a new model was developed emphasizing thermodynamic processes and focusing in particular on a postulated positive feedback loop involving the near-surface wind speed and the evaporation of water from the underlying ocean, with the evaporation rate being a function of wind speed and thermodynamic disequilibrium (Emanuel, 1989, 1995, 1997, 2003; see also Rotunno and Emanuel, 1987, p. 559; Holton, 2004). This evaporative-wind feedback mechanism is a form of air-sea interaction instability and is now commonly referred to as Wind-Induced-Heat-Exchange (WISHE) in modern textbooks. We have a number of issues with some of the popular articulations of the putative WISHE spin-up mechanism (e.g. that of Holton and Hakim, 2012) and present here our own interpretation of the mechanism summarizing that given in Montgomery et al. (2009). The feedback mechanism is illustrated schematically in Fig. 3. The pertinent variables shown in the figure are defined in the text of Montgomery and Smith (2014) and are otherwise standard. Here, thermal wind balance refers to the axisymmetric thermal wind equation in pressure coordinates that relates the radial gradient of azimuthally-averaged θ to the vertical shear of the azimuthally-averaged tangential velocity field. e
Fig. 3. Schematic diagram illustrating the WISHE feedback mechanism of tropical cyclone intensification as articulated in Montgomery et al. (2009) (see text for details). Introduction to Hurricane Dynamics: Tropical Cyclone Intensification 9
The argument assumes that for some reason the near-surface azimuthal mean mixing ratio is increased in the core region. This increase leads to an increase in the corresponding mean radial gradient and also an increase in the mean radial gradient of θ throughout the boundary layer. The secondary e circulation is imagined to loft this increase in θ into the vortex interior e thereby warming the core region of the vortex. By the thermal wind relation and its corollary, obtained upon applying a Maxwell thermodynamic relation (presuming reversible thermodynamics) and then integrating this equation along a surface of constant absolute angular momentum, this increased warmth is invoked to imply an increase in the mean tangential wind at the top of the vortex boundary layer. It is understood here that θe values at the boundary layer top correspond to saturated values. Turbulent mixing processes in the boundary layer are assumed to communicate this wind speed increase to the surface, and thereby increase the potential for increasing the sea-to-air water vapour flux F . The saturation mixing ratio must increase approximately qv in step with the putative increase in near-surface q so as to maintain a v thermodynamic disequilibrium. If this occurs then the surface mixing ratio will increase, thereby leading to a further increase in the mean tangential wind and so on.
2.2 Testing the WISHE Paradigm An in-depth test of the putative role of wind-speed dependent surface heat fluxes in the intensification process was carried out by Montgomery etal. (2009) using a non-hydrostatic mesoscale numerical model without cumulus parameterization1. Some of the principal results of this study are summarized in Fig. 4 (adapted from their Fig. 9). When the wind-speed dependence of the surface heat fluxes is capped at 10 ms-1, the maximum horizontal wind speed is a little less than in the uncapped experiments, but the characteristics of the vortex evolution are qualitatively similar, in both the pseudo-adiabatic experiments and when warm rain processes are included. In contrast, when the latent and sensible heat fluxes are suppressed altogether, the system- scale vortex does not intensify, even though there is some transient hot-tower convection and local wind-speed enhancement. Despite the non-zero CAPE present in the initial sounding, no system-scale amplification of the wind occurs. This finding rules out the possibility that vortex intensification with capped fluxes is an artifact of using an initially unstable moist atmosphere (cf. Rotunno and Emanuel, 1987). (These conclusions have been verified with further sensitivity experiments and supporting analyses using an initial sounding with nearly zero CAPE in the Appendix of Montgomery et al. (2009).) A minimum value of approximately 3 ms-1 was found necessary for the wind-speed cap at which the vortex can still intensify to hurricane strength in the pseudo-adiabatic configuration of the numerical model. 1 A recent study by Montgomery et al. (2014) has revisited this issue using a state-of- the-art cloud model and affirms the conclusions of Montgomery et al. (2009). 10 Michael T. Montgomery
No. Name Description 1 Warm rain Warm rain (control) expeirment on an f -plane. 2 Pseudo-adiabatic Explicit latent heat release with no water loading or evaporative cooling processes. 3 Capped warm rain Surface wind speed is capped at 10 ms–1 in formulae for latent and sensible heat fluxes. 4 Capped pseudo-adiabatic As Experiment 3, but using pseudo-adiabatic option.
Fig. 4. Time-series of maximum azimuthal-mean tangential wind component (Vmax) for experiments 1-4 (solid and dashed red/blue curves) from Montgomery et al. (2009).
Shown also is the time series of the local horizontal wind speed maximum VTmax (solid green curve) and Vmax (dashed green curve) of the related experiment when the surface latent- and sensible-heat fluxes are set to zero (Experiment S3; Table II of Nguyen et al. (2008)). Adapted from Montgomery et al. (2009).
When warm rain processes are included, the vortex still intensifies with a wind-speed cap of 5 ms-1 (not shown here, see Montgomery et al. (2009) for details). These experiments demonstrate that large latent heat fluxes in the core region are not necessary for cyclone intensification to hurricane strength and indicate that the evaporation-wind feedback process postulated in the WISHE intensification mechanism is not essential. The take away message from these experiments is that, although the wind- speed dependent heat fluxes do augment the intensification process somewhat, the basic spin up process operates without the postulated evaporative wind- speed feedback (WISHE) process. Therefore WISHE is not essential nor the dominant intensification mechanism for these idealized configurations without uniform flow or vertical shear, configurations that have been used to underpin the WISHE feedback hypothesis.
2.3 The Rotating Convection Paradigm The new intensification model is described in a series of papers by Nguyen (2008), Montgomery et al. (2009), Smith et al. (2009), Bui et al. (2009), and Persing et al. (2013) and highlights the presence of rotating convective Introduction to Hurricane Dynamics: Tropical Cyclone Intensification 11 structures2 and their vorticity remnants as well as their role in organizing the vorticity structure of the core of the storm. In essence, these clouds collectively drive the spin-up of the system-scale vortex by inducing radial inflow above the boundary layer, consistent with the previous three theories. As in the earlier hypotheses, radial convergence of M above the boundary layer in conjunction with its material conservation leads to spin-up of the bulk tangential winds and an increasing radial pressure gradient there. In contrast, this new theory stresses the fact that the spin-up of the maximum tangential winds actually occurs in the boundary layer below 1 km altitude. In the boundary layer, M is reduced by the frictional torque, but the radial inflow is much stronger than above this layer due to both friction and an increase in the radial pressure gradient at the top of the layer resulting from elevated radial inflow atop the boundary layer. Spin-up occurs in the boundary layer when the radial inflow is large enough to converge rings of air to small radii faster than the rate at which M can be removed by friction. The amplification of the tangential wind by this mechanism naturally leads to the development of supergradient flow and an accompanying outward agradient force. The agradient force acts to arrest the radial inflow at the base of the eyewall, whereupon air parcels turn upwards and carry their elevated tangential momentum into the eyewall. This process contributes also to the spin-up of the tangential winds in the bulk vortex (Smith et al., 2009; Bui et al., 2009; Smith and Thomsen, 2010; Smith and Montgomery, 2010; Persing et al., 2013). Further elaboration of the two system-scale mechanisms that comprise the extended cooperative intensification model is presented in Montgomery and Smith (2014, in section entitled ‘An axisymmetric view of the new paradigm’). The rest of this article will present an overview of the local dynamics of the new model.
3. The Prototype Experiment for Tropical Cyclone Intensification To fix ideas, our discussion here will be focused on understanding various aspects of the prototype problem for tropical cyclone intensification
2 The term “rotating convection” is used in lieu of vortical hot towers (VHTs), which were first described in Hendricks et al. (2004) and later studied in Montgomery et al. (2006a) and Nguyen et al. (2008), among others. To avoid any potential controversy that surrounds the definition of VHTs, the term “rotating convection” will be used throughout the rest of these lectures. Usage of the term VHT has given some the impression that only deep convection (>12 km cloud depth) has strong rotation. However, a recent study by Wissmeier and Smith (2011) showed that even moderate convection (6-12 km depth) in a background rotation rate typical of the undisturbed tropical atmosphere can have a comparable impact on the stretching of low-level relative vorticity in comparison to intense deep convection. Thus, a broad definition is required for studying the aggregate impact of these convective elements on tropical cyclone intensification. 12 Michael T. Montgomery
(Fig. 5), which examines the evolution of a prescribed, initially cloud free, axisymmetric, baroclinic vortex in a quiescent environment over a warm ocean on an f-plane. For simplicity, and for didactic reasons, the effects of an ambient flow, including those of ambient vertical shear, are not considered. Omitting the complicating effects of ambient and vertical shear flows permits a clear comparison between the new paradigm and the old paradigms (summarized above), which were historically formulated and interpreted for the case of a quiescent environment on an f-plane. The numerical experiments summarized here were carried out using a modified version of the Penn State/NCAR fifth-generation Mesoscale Model (MM5 version 3.6) (Dudhia,1993; Grell et al., 1995), which is suitable for the study of idealized problems. Details are provided in Nguyen et al. (2008). (The basic results and interpretations reported herein have been verified with more state-of-the-art three-dimensional mesoscale and cloud models, such as the Weather Research and Forecasting (WRF) model (see Fang and Zhang, 2011), the hurricane version of the WRF model (called HWRF) (see Bao et al., 2012; Gopalakrishnan et al., 2011) and George Bryan’s Cloud Model 1 (CM1; version 14) (see Persing et al., 2013). The MM5 model is configured with three domains: a coarse mesh of 45 km resolution, and two two-way-nested domains of 15 km and 5 km resolution. The domains are square, and are 5400 km, 1800 km and 600 km respectively on each side. For the experiments on the β-plane, the innermost domain moves to keep the vortex near the centre of the domain. There are 24 σ-levels in the vertical, seven of which are below
Fig. 5. The prototype experiment for understanding the dynamics and thermodynamics of tropical cyclone intensification following Nguyen et al. (2008). Introduction to Hurricane Dynamics: Tropical Cyclone Intensification 13
850 hPa. One experiment employing a fourth domain (300 km square) has been conducted, to permit increased spatial accuracy down to 1.67 km horizontal grid spacing on the innermost domain. Another experiment employing higher vertical resolution (45 σ-levels) has been conducted also. In order to keep the experiments as simple as possible in this presentation, the main physics options chosen are the bulk aerodynamic boundary-layer scheme (Braun and Tao, 2000) and the simplest explicit moisture scheme (the pseudo-adiabatic scheme, comprising instantaneous rainout after condensation). These schemes are applied in all domains. The sea-surface temperature is set to a constant 27 °C. The initial vortex is axisymmetric, with a maximum tangential wind speed of 15 ms-1 at the surface at a radius of 135 km. The strength of the swirling wind decreases sinusoidally with height, vanishing at the top model level (50 hPa). The temperature field is initialized to be in gradient wind balance with the wind field, using the method described by Smith (2006). The far-field temperature and humidity are based on Jordan’s Caribbean sounding during summertime conditions (Jordan, 1958), and are shown in a skew T-log p diagram in Fig. 6.
3.1 Illustrative Solutions The integration time for the main experiments described here is 96 h (four days). The intensification process is summarized in Fig. 6. The solid red curve in Fig. 6 shows the time series of the local horizontal wind speed in the pseudo-
Fig. 6. Far field temperature and humidity skew T-log P diagram, showing the temperature (solid line) and dew-point temperature (dashed line) of the initial sounding at a radius of 2,250 km from the domain centre (adapted from Nguyen et al., 2008). (b) Radius-height plot of initial tangential wind velocity. Contour intervals are shown every 2 ms-1. Adapted from Montgomery et al. (2009). Initial thermodynamical conditions are those in gradient wind balance with the wind field shown in part (b). The balanced fields are computed following Smith (2006), using the temperature and humidity shown in part (a) as the far field, reference sounding. 14 Michael T. Montgomery adiabatic control experiment (C0) from Nguyen et al. (2008). (This experiment is labelled as “Expt. 2” in the figure caption.) In essence, the evolution begins with a gestation period, during which the vortex slowly decays because of surface friction, but moistens in the boundary layer because of evaporation from the underlying sea surface. This period lasts approximately 9 h, during which time the minimum surface pressure rises slightly from its initial value of 1004 hPa to 1008 hPa (not shown), but the maximum tangential wind speed decreases only by a small amount (by less than 0.5 ms-1). The imposition of surface friction from the initial instant leads to an azimuthal-mean inflow in the boundary layer and a mean outflow above it, the outflow accounting for the decrease in tangential wind speed through the approximate material conservation of M. The inflow is moist, and as it rises out of the boundary layer and cools, condensation progressively occurs in some grid columns interior to the corresponding radius of maximum tangential wind speed. Latent heat release in these columns initiates deep convective updrafts with vertical velocities up to 5 ms-1 at 850 hPa. Figures 7 and 8 exemplify the local flow evolution, which shows the patterns of vertical velocity and the vertical component of relative vorticity at the 850 hPa and 500 hPa levels, respectively, at selected times for the control experiment of Nguyen et al. (2008). The updrafts form initially (Fig. 7, upper left panel) in an annular region inside the 135 km radius of the maximum initial tangential wind speed and their distribution shows a dominant azimuthal wavenumber-12 pattern around the circulation centre3. The updrafts rotate cyclonically around the vortex centre and have lifetimes on the order of an hour. As they develop, they tilt and stretch the local vorticity field and an approximate ring-like structure of intense, small-scale, vorticity dipoles emerges (lower left panel of Fig. 7). The dipoles are highly asymmetric in strength with strong cyclonic vorticity anomalies and much weaker anticyclonic vorticity anomalies. Early in the intensification phase, the strongest updrafts lie approximately in between the vorticity dipoles whereas later, they are often approximately collocated with the strong cyclonic anomalies. It is a useful reminder to point out here that these updrafts need not penetrate all the way to the tropopause to greatly enhance the local ambient rotation in the lower troposphere (z < 3 km) (Wissmeier and Smith, 2011). In the simulation, during the rapid intensification phase, the number of rotating updrafts decreases from twelve at 9 h to no more than three by the end of the period of rapid intensification, but their structure remains spatially irregular and during some periods, the upward motion occupies a contiguous region around the vortex centre. The cyclonic vorticity anomalies associated with the rotating updrafts grow horizontally in scale due to merger and axisymmetrization with neighbouring cyclonic vorticity anomalies. This
3 It turns out that the number of updrafts that form initially increases as the horizontal resolution increases, but the subsequent evolutionary picture is largely similar. Introduction to Hurricane Dynamics: Tropical Cyclone Intensification 15
Fig. 7. Vertical velocity (upper panels) and the vertical component of relative vorticity (lower panels) at 850 hPa at 9.75 hrs (left panels) and 24 hrs (right panels) in the control experiment C0 in Nguyen et al. (2008). Contour interval for vertical velocity: tick curves 2.0 ms-1 and thin solid curves 0.5 ms-1 with highest value 1.0 ms-1, thick dashed curves for negative values with interval 0.25 ms-1. Contour interval for relative vorticity in left panels is 5.0 × 10-3 s-1 (thick curves) and 1.0 × 10-3 s-1 (thin curves). For the right panels is 2.0 × 10-2 s-1 and 5.0 × 10-3 s-1, for thick and thin curves respectively. Adapted from Montgomery and Smith (2014). upscale growth occurs in tandem with the horizontal convergence of like- sign vorticity into the circulation of the rotating updrafts. They move slowly inwards also and become segregated from the anticyclonic vorticity anomalies, which move slowly outwards relative to them. As the anticyclonic anomalies move outwards they decrease in amplitude and undergo axisymmetrization by the parent vortex. Elements of the segregation process are discussed by Nguyen et al. in their section 3.1.5. The right panels of Fig. 7 show the situation at 24 hours. At this time there are five relatively strong updrafts (upper panel), each possessing greatly enhanced cyclonic vorticity (lower right panel), so that the initial monopole structure of the vortex is completely dwarfed by the local vorticity of the rotating updrafts. 16 Michael T. Montgomery
Comparing Figs 7 and 8 shows that the rotating updrafts move inwards over time. Spiral inertia-gravity waves propagating outwards occur throughout the vortex evolution and these are especially prominent in animations of the vertical velocity fields. The pattern of strong updraughts changes continuously with time, but is mainly monopolar, dipolar, or tripolar and always asymmetric. By the end of the rapid intensification period, no more than three rotating updrafts are active around the circulation centre. The simulated vortex, even with the simplest explicit moisture scheme (pseudo-adiabatic dynamics), exhibits many realistic features of a mature tropical cyclone (e.g., Willoughby, 1995; Kossin et al., 2002; Corbosiero et al., 2006; Marks et al., 2008) with spiral bands of convection surrounding an approximately symmetric eyewall and a central eye that is free of deep
Fig. 8. Vertical velocity (upper panels) and the vertical component of relative vorticity (lower panels) at 850 hPa at 48 h (left panels) and 96 h (right panels) in the control experiment C0 in Nguyen et al. (2008). Contour intervals for vertical velocity are 4.0 ms-1 (thick curves, with lowest absolute value 2.0 ms-1) and 0.5 ms-1 (thin curves, with highest absolute value 1.0 ms-1). Contour intervals for relative vorticity are 1.0 × 10-2 s-1 (thick curves) and 1.0 × 10-3 s-1 (thin curves). Positive values are solid curves in red and negative values are dashed curves in blue. The zero contour is not plotted. Adapted from Montgomery and Smith (2014). Introduction to Hurricane Dynamics: Tropical Cyclone Intensification 17 convection (Fig. 8 top row). In the mature phase, the evolution of the vortex core is characterized by an approximately axisymmetric circulation superimposed on which are: • small, but finite-amplitude vortex Rossby waves that propagate azimuthally, radially and vertically on the mean potential vorticity gradient of the system scale vortex (Shapiro and Montgomery, 1993; Guinn and Schubert, 1993; Montgomery and Kallenbach, 1997; Möller and Montgomery, 2000; Chen and Yau, 2001; Wang, 2002a, b; Chen et al., 2003; McWilliams et al., 2003; Martinez, 2008; Martinez et al., 2011)4; • barotropic-baroclinic shear instabilities (Schubert et al., 1999; Nolan and Montgomery, 2000); • eyewall mesovortices (Schubert et al., 1999; Montgomery et al., 2002; Braun et al., 2006); and • trochoidal “wobbling” motion of the inner-core region (Nolan and Montgomery, 2000; Nolan et al., 2001). Montgomery and Kallenbach (1997) presented rudimentary aspects of the vortex Rossby wave dynamics, the barotropic shear instability mechanism and related vorticity mixing processes that have been hypothesized to be important and operative in mature tropical cyclone vortices. However, we must be mindful that these processes as presented in that work, as well as the other processes cited in the foregoing list, are generally not adiabatic and inviscid because they are often coupled to the frictional boundary layer and convection (Schecter and Montgomery, 2007; Nguyen et al., 2011). A new study devoted in part to addressing these concerns was published by Persing et al. (2013). Those authors conducted a systematic diagnosis of the simulated eddy dynamics during the intensification process for the prototype intensification problem and found that eddies can contribute positively to the intensification process. Both sheared vortex Rossby waves and rotating convective structures were found to be important elements of the spin up dynamics. However, a full thermo-mechanical explanation of the eddy dynamics is still lacking.
3.2 Intermediate Summary From a three-dimensional fluid dynamics perspective, the intensification mechanism is via the generation of locally buoyant rotating updrafts and the corresponding convergence that the rotating updrafts induce both above and within the system-scale boundary layer. During the amplification process, Montgomery et al. (2009) found that the system-scale density temperature generally lagged behind the local density temperature within the rotating updraft cores. It is these cores that intrinsically drive the intensification
4 The restoring mechanism for vortex Rossby waves and shear instabilities related thereto is associated with the radial and vertical gradient of dry Ertel potential vorticity of the system-scale vortex. 18 Michael T. Montgomery process until the system-scale temperature along absolute angular momentum surfaces nearly coincides with the local temperature of the rotating updraft cores. The azimuthally-averaged aspects of this spin-up process have been presented in Smith et al. (2009) and Persing et al. (2013). We refer the readers to the above mentioned references, where we present a brief comparison of the azimuthally-averaged view with the prior intensification models.
3.3 Predictability Aspects Before closing this article we briefly discuss some of the implications of the foregoing findings on the predictability of tropical cyclone intensification. A very important practical issue is to investigate the predictability of the vortex intensity and asymmetric flow structure (e.g., Lorenz, 1969). In contrast to Lorenz (1969), the word “predictability” is used here in a qualitative sense to describe the degree of randomness of the ensuing vortex evolution. It is not unreasonable to expect that the asymmetric flow structures may be strongly influenced by deep convection, at least in the inner-core region, broadly defined as the region with radius less than twice the radius of maximum tangential wind speed. Since the pattern of deep convection is strongly influenced by the low-level moisture field, which is known to have significant variability on small space scales (e.g. Weckwerth, 2000), it is of interest to examine the variability of the inner-core asymmetries as a resultof random variations in the boundary-layer moisture distribution. The sensitivity of the flow asymmetries to low-level moisture in an intensifying model cyclone was demonstrated in Nguyen et al. (2008) in an ensemble of ten experiments in which a small (± 0.5 g/kg) random moisture perturbation was added at every horizontal grid point in the innermost domain of the model below 900 mb. As a simple and convenient measure of the local intensity, we show here a time series of the maximum horizontal wind speed
VTmax at 900 hPa, near the top of or just above the boundary layer. As shown in Fig. 9, the evolution in the local intensity as characterized by VTmax at 900 hPa is similar in all ensemble members, but there is a non-negligible spread, the maximum difference at any given time in the whole simulation being as high as 20 m/s. The pattern of evolution of the flow asymmetries is significantly different between ensemble members also. These differences are exemplified by the relative vorticity fields of the control experiment and three randomly-chosen realizations at 24 hours from the Nguyen et al. (2008) ensemble (Fig. 10). In general, inspection of the relative vorticity and vertical velocity fields (Fig. 11) of each ensemble member shows similar characteristics to those in the control experiment, the field being non-axisymmetric and still dominated by locally intense cyclonic updrafts. However, the detailed pattern of these updrafts is significantly different between the ensemble members. Introduction to Hurricane Dynamics: Tropical Cyclone Intensification 19
Fig. 9. Time-series of maximum total wind speed at 900 hPa in the control experiment in Nguyen et al. (2008) (blue) and in the 10-ensemble experiments therein (thin, red).
The foregoing results are supported by the calculations of Shin and Smith (2008), who performed similar ensemble calculations using the minimal three-layer hydrostatic model of Zhu and Smith (2003). Their calculations had twice the horizontal grid spacing used in Nguyen et al. (2008), and a much coarser vertical grid spacing (i.e., only three vertical levels). In fact, the calculations indicate a much larger variance between the ensembles than those in Nguyen et al., suggesting that the predictability limits may be resolution dependent also5. The problem of estimating the “true” intensity fluctuations for real-world forecasting of these severe storms warrants further systematic investigation. The implication of this work is as follows. Since the flow on the convective scales exhibits a degree of randomness, the convective scale asymmetries are intrinsically chaotic and unpredictable. Only the asymmetric features that survive in an ensemble average of many realizations should be regarded as robust.
5 Recent work of Fang and Zhang (2011) has broadly verified the foregoing conclusions, but show a more modest variability in wind-speed intensity due to small perturbations in the initial condition. Aside from numerical model differences and spatial resolution dependencies summarized here, the difference in magnitudes of the intensity fluctuations is believed to be due in part to the retention of liquid water in the more complex microphysics option used in their WRF simulation. Nguyen et al. (2008) attributed the lower intensity found in their own ‘warm-rain’ simulations compared to the pseudo-adiabatic limit to a reduction in the convective instability that results from downdrafts associated with the rain process and to the reduced buoyancy in clouds on account of water loading. Therefore it seems scientifically plausible that the associated water loading in the low-to-mid troposphere that tempers the convective updrafts during intensification would temper also the maximum wind fluctuations in these more complete cloud representations. 20 Michael T. Montgomery
Fig. 10. Relative-vorticity field at 850 hPa of (a) the control experiment C0 in Nguyen et al. (2008), and (b, c, d) three representative realizations from the f-plane ensemble at 24 h by perturbing the boundary layer moisture as described in the text. Positive values are indicated by red solid contours, with a contour interval of 0.5 × 10-3 s-1. Negative values are indicated by blue dashed contours, with a contour interval of 0.1 × 10-3 s-1. The zero contour is not plotted.
3.4 Summary of Rotating Convection Paradigm The foregoing simulations, both on an f-plane and on a β-plane (not shown here), show that the rotating updrafts dominate the intensification period at early times and that the progressive aggregation, merger and axisymmetrization of these cyclonic updrafts together with the low-level convergence they generate are prominent features of the tropical cyclone intensification process. As discussed in more detail by Montgomery and Smith (2014 and supporting refs.), the emerging flow asymmetries are not merely a manifestation of the numerical representation of the equations on a Cartesian coordinate grid mesh, but rather a reflection of a key physical process, namely moist convective instability. The results of Nguyen et al. (2008) support those of Montgomery et al. (2006) in an idealized study of the transformation of a relatively-weak, axisymmetric, middle-level, cold-cored vortex on an f-plane to a warm- cored vortex of tropical cyclone strength and also those of near cloud- resolving numerical simulations of tropical storm Diana (1984) (Davis and Introduction to Hurricane Dynamics: Tropical Cyclone Intensification 21
Fig. 11. Vertical-velocity field at 500 hPa of (a) the control experiment C0 in Nguyen et al. (2008), and (b, c, d) representative ensemble experiments at 24 h by perturbing boundary layer moisture as described in the text. Positive vertical velocity values are shown in red (solid contours); negative values are shown in blue (dashed contours). Contour interval is 0.25 ms−1 and the zero contour is not plotted.
Bosart, 2002; Hendricks et al., 2004), which drew attention to the important role of convectively-amplified vorticity in the spin-up process. In turn, the above results are supported by more recent calculations of the intensification process using a range of different cloud-representing models with a refined horizontal resolution (Fang and Zhang, 2011; Gopalakrishnan et al., 2011; Bao et al., 2012; Persing et al., 2013). All of these studies indicate that rotating convective updrafts are the basic coherent structures of the tropical cyclone intensification process.
Acknowledgements Much of the work presented in this article has been conducted and written in collaboration with my close colleague and friend, Professor Roger K. Smith. The foregoing material is a highly abridged version of the review by Montgomery and Smith (2014) and is based on work carried out with our student and research colleagues over the past several years. 22 Michael T. Montgomery
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