Why must hurricanes have eyes?

Transcription

1 Robert Pearce Department of Meteorology, University of Reading Perhaps the most characteristic feature of hurricanes is their eye, an approximately circular region of relatively dry air about which the strongest winds circulate. An excellent example is shown in the satellite image (Fig. 1) in which the sloping eye wall is clearly marked by the tops of cumulonimbus bands diverging in the upper outflow. Such a clear image is not always obtained as the eye is sometimes covered by a thin cirrus shield. Our understanding of the structure of hurricanes and the reason for their occurrence has progressed considerably over the past 40 or so years. It is now generally accepted, for instance, that they can only form over an ocean with a surface temperature in excess of 26 C to enable inflowing air, taking up moisture (and some sensible heat) from the surface, to ascend, releasing its latent heat, all the way to the tropopause. It is also known that they can only form at latitudes greater than about 5 degrees north or south. This is because the earth s rotation component about the vertical must be present to provide a rotating environment enabling intensification to occur. Such a synoptic-scale environment must persist for at least a few days and vertical wind shear, which tends to inhibit concentration of a flow with axial symmetry, must also remain small over this period. Nevertheless, even when all of these conditions seem to apply, hurricanes do not always develop. Herbert Riehl, perhaps the most influential of hurricane researchers, used to say that the crucial question was not why hurricanes occur but why they do not occur more often. Observed temperature and flow structures At their mature stage hurricanes exhibit a strong degree of axial symmetry. Figure 2(a), redrawn with modification from the paper by Hawkins and Imbembo (1976), shows a vertical cross-section of wind speeds Fig. 1 Hurricane Floyd (1999) approaching North Carolina (from Elsberry (2002), reproduced by kind permission of the American Meteorological Society from the cover of its September 2002 Bulletin) obtained from aircraft traverses in Hurricane Inez on 28 September It shows clearly a wind speed maximum of about 60 m s 1 at a radius of about 15 km in the lower troposphere and, at all radii, a decrease of wind speed, essentially that of the tangential wind, with height. Figure 2(b) is a radial section of Inez at a height of 2 km, showing the wind speed and an approximate fit using a power law. Figure 3 shows vertical distributions of radial and tangential winds at a radius of 100 km for, respectively, composites of hurricanes and typhoons. Lower-level inflow and upper-level outflow, particularly near the tropopause, associated with the meridional circulation is clearly indicated (together with an inflow layer in the lower stratosphere). A vertical section of the observed temperature anomaly distribution in Inez (redrawn and modified from the paper by Hawkins and Imbembo (1976)) is shown in Fig. 4. Its main features are the temperature anomaly maxima in the eye and the large temperature gradient across the eye wall. The temperature everywhere decreases with the radius. A strong moisture gradient (not shown) exists across the eye wall between the eye itself, where the relative humidity is generally low, around 50%, and the saturated ascending air in the eye wall cloud. An eye in a simple two-fluid model A hurricane consists essentially of a ring of low-level air converging radially towards a central eye and then ascending in a saturated moist convective outflow. The inflowing tangential wind increases rapidly with decreasing radius as a result of spin up like a weight attached to one s finger on the end of a piece of string which, when swung round, wraps itself round the finger more 19

2 Fig. 2 (a) Vertical cross-section of wind speeds in Hurricane Inez on 28 September The dashed lines indicate the inferred position of the eye wall. (Redrawn from Hawkins and Imbembo 1976.) (b) Radial distribution of wind speeds at a height of 2 km extracted from (a). Also shown (dashed) is a profile fitted taking α = ½ and R = 70 km in the power law vr α = v R R α for the convective inflow. See text for explanation of symbols. and more rapidly. The air spins down in the outflow. In the eye, on the other hand, the wind drops to near zero on the axis. The eye wall separates these two flows. An important insight into the reason why such a flow configuration can exist is provided by constructing an imaginary, purely spun-up fluid, water say, embedded in a slightly less dense fluid, oil say, with a free surface (Fig. 5). Thus in the lower, slightly higher-density fluid the angular momentum is constant, i.e. the tangential velocity, v, is inversely proportional to the radius, r, and the upper fluid, with a free surface, is assumed to be at rest (Fig. 5). The condition which must be satisfied at the interface is that the pressure should be continuous across it. Thus, in Fig. 5, the pressure drop in the upper fluid along AB must be the same as that along the same section in the lower fluid. There must be a pressure increase along AC in the lower fluid balancing its centrifugal acceleration, but not in the upper fluid which is at rest. At the same time the hydrostatic pressure drop along CB in the lower, denser, fluid must be greater than that along AD in the upper fluid. The slope of the interface must adjust to a value which ensures that this pressure drop along CB just offsets the radial increase along AC, i.e. that the net difference in pressure between the two fluids along AB is zero. The mathematical derivation of the model interface is presented in Box 1. This analysis of the eye wall shape readily extends to the case of two compressible fluids and to the atmosphere, where the discontinuities in v and density, ρ, at the interface are replaced by sharp, but continuous, changes in these variables (see Pearce 2004). Also, crucially, the frictionless constraint used in the above model is removed, and eddy momentum and heat transfers are incorporated in an atmospheric model. The role of gravity in maintaining the hurricane circulation It is well known that bubbles of air warmer than, i.e. less dense than, their environment rise. Consider the flow in a vertical section of an idealised axisymmetric rising bubble (Fig. 6(a)). An element of air, A, on the axis simply rises. An element of air in the section T 1, on Fig. 3 Vertical distributions at 100 km radius of radial wind, u (taken from the vertical cross-section for a western Atlantic composite hurricane (Gray 1979)), and tangential wind, v, for a Pacific composite typhoon (Frank 1977) the other hand, which could be inside or on the edge of the bubble, may rise, but at the same time it rotates in a clockwise sense. Air in the section T 2 also rotates, but in an anticlockwise sense. Fig. 4 Vertical cross-section of the temperature anomaly for Hurricane Inez on 28 September The eye wall (dashed) is as in Fig. 2(a). (Redrawn from Hawkins and Imbembo 1976.) Fig. 5 Configuration of the interface and values of the pressure deficit (dashed, in arbitrary units) for an axisymmetric, spun-up, incompressible fluid of density ρ in a stagnant fluid of density ρ ρ. The box ACBD is referred to in the text. 20

3 Box 1 A highly idealised hurricane-type model If the density difference between the two fluids is ρ, the mass per unit area of the column CB of depth δz is ρδz greater than that of the column AD and the pressure drop along CB is ρgδz larger (where g is the acceleration due to gravity). The centrifugal acceleration in the lower fluid at A is v 2 /r and the pressure rise along AC approximately ρδrv 2 /r, whereas, with the upper fluid at rest, that along BD is zero. The pressure drop along AB in the upper fluid is thus equal to that in the lower fluid if ρgδz = ρv 2 (δr/r), i.e. if the slope of the interface at A, approximately δz/δr, is equal to (v 2 /r)/(g ρ/ρ). If we specify the constant angular momentum in the lower fluid at, say, 100 km as vr = v R R, where v R is the tangential velocity at radius R, then it follows that the slope of the interface must satisfy the equation dz/dr = 2C/r 3, where 2C = (v R 2 R 2 )/(g ρ/ρ). Thus the slope increases with decreasing radius, and if it tends to zero at some height, H, when r is large, it is seen to satisfy the relation z = H(1 (r e /r) 2 ) where the eye radius at the surface, r e, satisfies (r e /R) 2 = C/HR 2. The eye radius at the surface is thus proportional to v R but inversely proportional to ( ρ/ρ) 1/2. Consider now the meridional flow in the schematic cross-section of a hurricane depicted in Fig. 6(b). The maximum rate of ascent is along the boundary of the eye wall PQMSC so that fluid elements in the convection region rotate in a clockwise sense. In the eye wall itself they rotate in an anticlockwise sense. The essential difference between this flow and that in the bubble in Fig. 6(a) is that the warmer air at each level is not where the ascent rate has its maximum value, but on the central axis OD where the air is at rest. These flows are best analysed in terms of their vorticity component, η, about an axis perpendicular to the section; this is taken as positive in a clockwise sense and negative if anticlockwise. (The value of η at any point in the section* may be thought of as the circulation round a small, solid, rotating disc centred at the point and divided by its area. Thus if its radius is a and its angular velocity ω, the velocity on its rim is ωa, the length of its perimeter is 2πa and the circulation round it is 2πa 2 ω. The area of the disc is πa 2 so that η at the centre of the disc is 2ω.) The region ABCA in Fig. 6(b) may be thought of as comprising a very large number of these circular discs, each of which rotates with its own value of η. The circulation round the region, its meridional circulation, is then the sum of the circulations round the discs, i.e. is equal to the integral of η over the area. In a hurricane this is clockwise in this region, i.e. η>0. On the other hand, it is anticlockwise (η<0) in the region B'BCC'B'. Gravity is able to generate η-vorticity in a fluid if it contains horizontal density gradients such as those generated when latent heat is released in clouds. Consider the horizontal section of a fluid in Fig. 7(a) compare section T 1 in Fig. 6(a) in which the density increases, i.e. the temperature decreases, to the right. Since the mass of the right-hand half is greater than that of the *This is referred to as η-vorticity to distinguish this component from the components associated with (i) rotation about the vertical when there is tangential flow, v, about the axis and (ii) rotation about a horizontal radius when v varies with height. left-hand half, the effect of gravity is to accelerate its rotation in a clockwise sense, i.e. generate positive η. In the hurricane the radial decrease of temperature throughout its inner region (Fig. 4) implies a radial increase of density everywhere, i.e. for gravity to tend to increase η. Gravity also plays its part, however, in the eye where the air subsides during its generation. This is a statically stable region and vertical displacements of the air are subject to a gravitational restoring force in propagating waves so-called gravity waves (see Box 2 for a more detailed discussion). This effect is readily observed as waves on the surface of a pond when a stone is dropped into it. In the hurricane eye, however, the disturbing mechanism, located mainly in the eye wall, is more subtle and on a much larger scale. This is described in the next section. The role of vortex tilting in maintaining the hurricane circulation Generation of vorticity by gravity acting on a fluid with horizontal density gradients is sufficient to explain the essential dynamics of a transient rising bubble. However, for the hurricane, some other mechanism must be involved in maintaining its steady state. It is its tangential flow which provides this mechanism. This component of the flow implies rotation about a vertical axis and it is this rotation component which can be converted into rotation about the horizontal, i.e. can provide a source or sink of η-vorticity. Tilting of a horizontal tube of vorticity in the atmosphere into the vertical is discussed and illustrated in Holton (1972, p. 74). Here, however, we are concerned with vertical tubes of vorticity tilted towards the horizontal. Figure 7(b) shows such a vertical Why must hurricanes have eyes? (a) (b) Fig. 6 (a) The flow in a vertical cross-section of an idealised rising bubble of air. The dashed curve indicates the envelope of the rising bubble. The arrows indicate the speed and direction of the meridional motion of fluid elements at levels z 1, z 2, and, in the displaced ambient air, z 3. A denotes the axis; T 1 and T 2 denote typical fluid elements in a ring (torus) which may be inside or on the edge of the bubble (as shown). (For an element inside the bubble the air both rotates and ascends, the maximum rate of ascent being on the axis.) The warmest air at each level is on the axis. There is no tangential flow, i.e. rotation about the axis. (b) Schematic crosssection of a model hurricane. The various symbols are referred to in the text. The meridional flow only is depicted. There is, in addition, a dominating tangential flow. 21

4 22 Fig. 7 (a) A horizontal cylinder of fluid split into two sections, each of unit volume with upper and lower surfaces of area A. The left-hand section, with centre of gravity at G 1, is of mean density ρ ρ, which is also its mass. Thus the pressure difference between the upper and lower face times A needs to be equal to g(ρ ρ) to keep it in equilibrium. Similarly, the right-hand section, with centre of gravity at G 2, is of mass and mean density ρ+ ρ and needs a pressure difference times A equal to g(ρ+ ρ ) to keep it in equilibrium. The total mass of 2ρ is maintained in equilibrium under gravity by a mean pressure difference times 2A between the upper and lower faces equal to its weight, i.e. 2gρ. Thus, in the absence of a horizontal difference in pressure gradient between the two sections, the difference in density between the two halves results in a couple about the centre of mass G of strength g ρ times the distance G 1 G 2 tending to rotate the cylinder clockwise. (b) A cylinder of fluid, radius a, rotating about its vertical axis with angular velocity ω, moves to the right and is at the same time tilted in the tangential (s,z) plane by the tangential velocity, v, which decreases with height. In the vertical position the fluid on the boundary of the cylinder rotates anticlockwise (viewed from above) in a horizontal plane with speed ωa. In the tilted position it moves with the same speed, but in a sloping plane. Thus, in the absence of a resisting pressure field, it descends to the rear (looking towards the origin O) and ascends in the front. This corresponds to an anticlockwise meridional circulation, i.e. η<0. tube of vorticity in a fluid lying in the plane of the paper; this is conveniently regarded as a tangential plane at r in a hurricane circulation, i.e. with v positive to the right. The tube clearly moves to the right but, with v larger at the base than at the top, it is also tilted anticlockwise. Horizontal sections of the tube are thus also tilted and the circulation round them acquires a vertical component, with descent behind the plane of the paper and ascent in front. In a hurricane in the Northern Hemisphere rotation about the vertical relative to axes fixed in space is everywhere positive, i.e. in the sense shown in Fig. 7(b) so that, with v everywhere decreasing with height, vortex tilting tends to decrease η everywhere. By applying an argument similar to that used above for the area ACBD in Fig. 5 for the case in which v decreases continuously rather than discontinuously with height, it is seen that vortex tilting is simply the result of the atmosphere adjusting its mass distribution in order to enable the pressure field to accommodate the decrease of centrifugal force with height. In the relatively steady state reached at the mature stage of a hurricane the gravitational and vortex tilting effects must everywhere balance each other. But it is only in the convection region that heat is released, creating the temperature gradient generating positive vorticity and supporting the meridional circulation associated with the convection; here vortex tilting generates negative vorticity at a rate just sufficient to maintain a balance. In the eye, with virtually no heat source, it is the vortex tilting in the eye wall which controls the dynamics; balance can only be achieved by the air subsiding in the eye in response to this vortex tilting, thereby raising its temperature through adiabatic compression (see Box 2 for a more detailed discussion) and establishing the radial density gradient across the eye wall required to offset the tendency of the tilting there to increase its anticlockwise circulation. The contrasting dynamics of the convection region, and the eye with its eye wall The positive circulation in the convection region is driven by latent heat release in the clouds. In the steady mature state this is balanced by the rate of generation of an equal and opposite negative circulation by vortex tilting associated with the decrease of v with height. So why should this distribution of v occur here? How does the atmosphere adjust itself to adapt to the dynamic forcing of the heat input and achieve a steady state? It is now shown that this is essentially brought about as a result of surface drag extracting angular momentum from the inflowing air, whereas the air flowing out at upper levels is subject to only a relatively small extraction of its angular momentum Box 2 Gravity waves The energy content of an element of the atmosphere at rest is the sum of its potential energy, proportional to its height above the surface, and its internal energy, proportional to its absolute temperature. In a hydrostatic atmosphere the vertical pressure difference, δp, across depth, δz, of a column of unit area is gρδz. Thus, in vertical displacements, z, the change in potential energy per unit mass, g z, is balanced by δp/ρ; but, since the density decreases with height, air is compressed when it subsides, i.e. work is done by the pressure field. If there is no external source or sink of energy (i.e. the subsidence is adiabatic), this work simply increases the internal energy and its temperature rises. Vertical displacement of the air (such as, for example, that forced by vortex tilting) is associated with the propagation of what are referred to as internal gravity waves (see, for example, Holton 1972). These propagate both horizontally and vertically. For example, an internal wave with a vertical wavelength of, say, 20 km propagates horizontally at a phase speed of about 100 km per hour, and vertically with a phase speed of 5 10 km per hour. At a fixed level at which air is subsiding in a statically stable atmosphere the temperature increases. With gravity maintaining hydrostatic balance with pressure virtually constant at upper levels, the air expands laterally, resulting in a radial export of mass. This export propagates, essentially horizontally, at the speed of sound, approximately 1000 km per hour at the surface. This is usually referred to as an acoustic wave, but since gravity is intimately involved in the process it is also referred to as an external gravity wave. Its rapid rate of propagation implies that hydrostatic balance is rapidly established in the eye region of a hurricane as it exports mass and the pressure falls. and can be regarded as simply conserving it. The tangential velocity of the inflowing air (which contains large eddies) must spin up at a rate less than it would if it was not subjected to surface drag. Thus, in Fig. 6(b), v falls off less rapidly with r along the inflow streamline MQP than it does along the outflow streamline MSC. The magnitude of v at S is less than its magnitude at Q. This is demonstrated mathematically in Box 3. Observations (see, for example, Fig. 2) suggest that v follows a power law vr α = constant, where α lies between 0.5 and 0.6. Also, it may be shown by considering the heat and angular momentum balance of the system, maintained by the meridional circula-

5 Box 3 Vortex tilting Consider (Fig. 6(b)) air moving along the trajectory PQMSC along which v is a maximum at both inflow and outflow. If the inflow v is taken to satisfy a law vr α = constant = v R R α, say, where α is less than 1, then at Q, at radius r, v i = v R (R/r) α. At M, when its maximum value, v m, is reached, this is given by v m = v R (R/r m ) α. If, then, it conserves its angular momentum, its outflow velocity, v o, at S at r is v m (r m /r ). Thus v o /v i = (r m /r) 1 α and, with α less than 1 and r greater than r m, v o is less than v i. tion (Pearce 2004), that the value of α must be restricted to a narrow band embracing this range. The distribution of v in Inez with α = 0.5 is shown in Fig. 2(b). Thus, surface transfers of heat and angular momentum play a major role in determining the dynamics of the convection region, in particular in determining the strength of the meridional circulation (see Pearce (2004) for a detailed analysis). The situation in the eye and eye wall is quite different. Surface transfers of heat and angular momentum in the eye itself play a relatively minor role. Subsidence in the eye, which occurs during intensification of the system as a result of gravity-wave propagation, raises its temperature, setting up a temperature gradient across the eye wall. This generates sufficient (positive) η-vorticity there to offset its destruction by vortex tilting. There is virtually no meridional circulation in the eye at the mature stage. (The temperature gradients usually observed in the eye itself can only be associated with tangential flow within it, arising from horizontal convergence and spin up of the subsiding air or angular momentum transfer from the eye wall by eddies.) Thus, during intensification, it is vortex tilting in the eye wall, where v decreases with height, which forces subsidence in the eye. When the mature state is reached, subsidence virtually ceases. On the other hand, in the convection region, latent heat release forces the meridional circulation, vortex tilting providing the brake which ultimately determines its strength when the mature state is reached. How and why do hurricanes form? The question still remains as to how the above, dominantly axisymmetric, system sometimes develops and sometimes does not when there is little vertical wind shear over an appreciable area of a tropical ocean such as the Caribbean with a high sea surface temperature. Since this is a question which is still largely unresolved, attempts to address it here must be regarded as largely speculative. Dvorak (1975, 1984) has extensively analysed the satellite images obtained over periods of a few days preceding the development of a large number of hurricanes and classified them into six categories, two sequences of which, from his 1975 paper, are reproduced in Fig. 8. Each sequence shows the evolution of an initial, generally asymmetrical, distribution of distinct cloud masses developing into an almost symmetrical hurricane. These cloud masses are mainly associated with curved contracting bands of deep convection. However, it is only in the latter stages of intensification that these bands completely enclose a central eye. It can be shown using a simple model of a ring of deep convection of radius, say, 200 km, that because the air entering from the outside spins up and that from the inside spins down (Fig. 9) the system must contract. If the same argument is assumed to apply to a curved band of convection occupying only a sector of a ring, it is possible to envisage circumstances under which a sequence of these bands which develop and then decay eventually form, as a result of contraction, a completely closed loop which then further contracts to become the eye wall of a hurricane. With this concept in mind it is possible to understand how the eye forms. The outflow from the convection just below the tropopause must conserve most of its spunup tangential momentum, resulting in a decrease of v with height in this region if there is only a small tangential velocity in the lower stratosphere. Vortex tilting and gravity-wave propagation then induce subsidence in the inner region, resulting in eyewall formation as its outflow penetrates downwards and inwards (Fig. 9). However, if the convection band occupies only a partial sector of the region, the gravity waves disperse the subsidence over a wide area as, indeed, is generally the case with convective storms. It is only when the band of convection completely encircles a central region that these waves are trapped and the subsidence rate increases, resulting in a rapid intensification of the system. As a result of the subsidence the tropopause effectively lowers in the inner region as contraction of the convection ring proceeds, resulting in the eye-wall formation at its outflow penetrating downwards and inwards (Fig. 9). These ideas are consistent with Gray s (1993) conclusions from his analysis of observations. He suggests that hurricane development is a two-stage process. In Stage 1 a transient environmental wind surge establishes a mesoscale cloud cluster Why must hurricanes have eyes? T1 T2 T3 T4 T5 T6 Fig. 8 Two sequences of satellite image cloud patterns, each illustrating the development of a hurricane from an initial group of deep cumulus clusters (taken from Dvorak 1975) 23

6 Fig. 9 Schematic representation of a contracting ring of convection. The clockwise circulation (vorticity η 1 ) of the flow entering from the right and spinning up is stronger than that of the anticlockwise circulation (vorticity η 2 ) entering from the left and spinning down. The difference between the strengths of the two flows results in contraction of the ring. The arrows at the surface indicate the contrast between the surface heat input (and surface drag) between the two circulations. The arrows in the upper troposphere indicate subsidence induced by vortex tilting at the cumulus outflow level. The dashed line shows the lowering of the outflow level, which occurs as the ring contracts, generating an eye wall if the contraction proceeds to the hurricane stage. (MCC) from which low-level circulation centres (LLCCs) of order 100 km radius are generated, the latter being characterised by a high efficiency of warming and spin-up rate, and last for 1 3 days. (The scattered cloud patterns in the T1 to T4 images in Fig. 8 may be thought of as arising from Gray s wind surges.) Both MCCs and LLCCs are commonly observed. However, intensification to named hurricane status requires a further wind surge near an existing LLCC to produce a substantial central pressure drop (Stage 2) and this is a much less common event since the wind surges tend to occur randomly and over a much wider area. Concluding remarks Hurricanes (together with their companion typhoons in the Pacific and tropical cyclones in the Indian Ocean) are among the most destructive, yet scientifically fascinating, weather phenomena. Their structure, essentially an approximately axisymmetric organisation of deep, moist convection capable of surviving and moving to landfall as a steadystate system, is maintained against surface drag by heat supplied over a warm ocean. The generation of an eye, under the rarely occurring conditions enabling a closed ring of deep convection to form, is inevitable, indeed crucial. This is because the mass distribution associated with its structure is the only one capable of supporting this convection fed by air spinning up as it enters the circulation. The generation of the eye accompanies the redistribution of mass during intensification. The investigation of hurricane dynamics (Pearce 1998, 2004), on which much of this article is based, started by considering a simple model of a spun-up, frictionless, incompressible fluid lying beneath a stagnant fluid of slightly lower density and showing that it must possess a hurricanetype eye a kind of thought experiment. Having examined the conditions under which hurricanes form and are maintained, it is quite clear that the incompressible, frictionless, fluid model could never exist in nature or be constructed in the laboratory. Nevertheless, despite being unrealistic, it has led to an increase in our understanding of a fascinating atmospheric phenomenon. Indeed, perhaps it is only in the unique physical environment of the earth s atmosphere that systems with the structure of hurricanes can actually exist. Acknowledgements The author is most grateful to the referee whose thoughtful and helpful comments led him to expand and clarify parts of the paper, in particular the dynamics of the eye and eye wall; also to Derek Thomas for redrafting the figures. References Dvorak, V. F. (1975) Tropical cyclone intensity analysis and forecasting from satellite imagery. Mon. Wea. Rev., 103, pp (1984) Tropical cyclone intensity analysis using satellite data. NOAA Technical Report NESDIS 11 Elsberry, R. (2002) Predicting hurricane landfall precipitation: Optimistic and pessimistic views from the Symposium on Precipitation Extremes. Bull. Am. Meteorol. Soc., 83, pp Frank, W. M. (1977) The structure and energetics of the tropical cyclone, Paper 1: Storm structure. Mon. Wea. Rev., 105, pp Gray, W. M. (1979) Hurricanes: their formation, structure and likely role in the tropical circulation. In: Shaw, D. B. (Ed) Meteorology over the tropical oceans, Royal Meteorological Society, Bracknell, pp (1993) Tropical cyclone formation and intensity change. In: ICSU/WMO International Symposium on Tropical Cyclone Disasters, Peking University Press, Beijing, China, pp Hawkins, H. F. and Imbembo, S. M. (1976) The structure of a small, intense hurricane, Inez Mon. Wea. Rev., 104, pp Holton, J. R. (1972) An introduction to dynamic meteorology. Academic Press, New York and London Pearce, R. P. (1998) A study of hurricane dynamics using a two-fluid axisymmetric model. Meteorol. Atmos. Phys., 67, pp (2004) An axisymmetric model of a mature tropical cyclone incorporating azimuthal vorticity. Q. J. R. Meteorol. Soc., 130, pp Correspondence to: Prof. R. P. Pearce, 27 Copped Hall Way, Camberley, Surrey GU15 1PB. Royal Meteorological Society, doi: /wea