Ma y 2012 Li et al. 1

Dependence of tropical intensification on the Coriolis parameter

Ti m Li a n d Xu y a n g Ge Department of Meteorology and IPRC, University of Hawaii, Honolulu, Hawaii

Me l i n d a Pe n g Naval Research Laboratory, Monterey, California

We i Wa n g Shanghai Minhang Meteorology Bureau, Shanghai, China

(Manuscript received , in final form )

ABSTRACT

The dependence of (TC) intensification on the Coriolis parameter was investigated in an idealized hurricane model. By specifying an initial balanced vortex on an f-plane, we observed faster TC de- velopment under lower planetary vorticity environment than under higher planetary vorticity environment. The diagnosis of the model outputs indicates that the distinctive evolution characteristics arise from the extent to which the boundary layer imbalance is formed and maintained in the presence of surface friction. Under lower planetary vorticity environment, stronger and deeper subgradient inflow develops due to Ekman pumping ef- fect, which leads to greater boundary layer moisture convergence and condensational heating. The strengthened heating further accelerates the inflow by lowing central pressure further. This positive feedback loop eventually leads to distinctive evolution characteristics. The outer size (represented by the radius of gale-force ) and the of the final TC state also depend on the Coriolis parameter. The TC tends to have larger (smaller) outer size and eye under higher (lower) planetary vorticity environment. Whereas the radius of maximum wind or the eye size in the current setting is primarily determined by inertial stability, the TC outer size is mainly controlled by environmental absolute angular momentum. Key words:

1. Introduction a hurricane is slower under a higher planetary vorticity en- The fact that tropical (TCs) typically form a few vironment. They attributed the slower intensification to the degrees away from the equator (Gray 1968) implies that a greater radius of inflow away from the vortex center. Using non-zero Coriolis parameter is necessary to spin up a vor- an axisymmetric model on an f-plane, Bister (2001) found tex. Beyond this low limit of planetary vorticity, however, that an initial weak vortex takes twice the time to develop it is not clear how dynamically TC development depends into hurricane intensity at 30°N than at 10°N. The study on the Coriolis parameter. From a pure dynamics point of indicated that the inner core remains weaker for view, the planetary vorticity may modulate the radius of a long time in a higher Coriolis parameter environment and Rossby deformation, the latter of which may determine the thus slows down the moistening of the inner region. geostrophic adjustment and thus affect the final state of bal- Wang and Zhou (2008) found that, climatologically, TCs anced flows in response to thermal forcing (Schubert and formed in November in western North Pacific have a great- Hack 1982; Nolan 2007). Using a simplified model, De- er chance of rapid intensification (RI) than those formed in Maria and Pickle (1988) found that the development into August, even though the TC frequency in August is greater than that in November. Given that the mean genesis loca- Corresponding author address: Tim Li, Department of Meteorology tion in August is nearly 8° latitude north of that in Novem- and IPRC, University of Hawaii, Honolulu, Hawaii, USA. Email: timli@ ber, it is likely that the planetary vorticity, in addition to hawaii.edu other environmental factors, may play a role in determining DOI: 2 Tropical Cyclone Research and Review Vo l u m e 2 the RI preference. gential wind field, the mass and thermodynamic fields are Motivated by the aforementioned observational and then obtained based on a nonlinear balance equation so that modeling studies, we intend to investigate specific dynamic the initial vortex satisfies both the hydrostatic and gradient processes through which TC development depends on the wind balance. Coriolis parameter. The specific science questions we at- To investigate the sensitivity of TC intensification to the tempt to address in this study are: How does the environ- Coriolis parameter, a series of experiments were designed mental planetary vorticity affect the TC intensification? by running the model on an f-plane centered at different lat- What are structural differences in inner core, boundary itudes of 5, 7.5, 10, 12.5, 15, 20, 25 and 30°N, respectively. layer thickness and outer size for TCs simulated with dif- The experiments are identified as F05, F075, and F30, etc. ferent planetary vorticity parameters? For each experiment, the model was run for 5 days in a The paper is organized as the follows: In section 2, the resting environment. model and experiments are described. In section 3 we analyze the cause of different TC development behaviors 3. Results from the boundary layer gradient wind imbalance perspec- The time evolutions of the center minimum sea level tive. A positive feedback among the radial inflow, moisture pressure (CMSLP) and maximum tangential (Vmax) convergence and surface evaporation, diabatic heating and in the eight experiments are shown Fig. 1. In all the experi- surface pressure drop is illustrated. Finally, the conclusion ments, the initial weak vortex develops into the typhoon and discussion are given in the last section. strength within 5 days. Note that, however, the timing of rapid intensification depends on the Coriolis parameter. 2. Model experiments The rapid intensification is referred to as the rapid drop of The numerical model used in this study is the triply nest- CMSLP or rapid increase of maximum tangential wind. A ed, movable mesh primitive equation model-TCM3 (Wang TC develops much earlier under lower planetary vorticity 2001). The model consists of 25 layers in the vertical from σ=0 to σ=1 (here σ is normalized pressure relevant to surface pressure) with higher resolutions in the planetary boundary layer (PBL). The model physics include a modi- fied Monin-Obukhov scheme for the surface flux calcula- tions (Fairall et al. 1996), an E-ε turbulence closure scheme for sub-grid scale vertical mixing above the surface layer (Langland and Liou 1996), an explicit treatment of mixed phase cloud microphysics (Lin et al. 1983), and a fourth- order horizontal diffusion with a deformation-dependent diffusion coefficient. The model prognostic variables con- sist of zonal and meridional winds, surface pressure, tem- perature, turbulence kinetic energy and its dissipation rate, and mixing ratios of water vapor, cloud water, water, cloud ice, snow and graupel. The model has been used for many studies on TC (e.g., Wang 2001, 2002; Li et al. 2003, 2006; Ge et al. 2007, 2008; Li 2012). The model is configured with three nested domains: a coarse mesh of 22.5 km resolution and two 2-way nested domains of 7.5 and 2.5 km resolution, re- spectively. The domains have a square shape, with a side of 5400, 1800, and 450 km respectively. A strong damping is specified within a sponge layer near the model lateral boundaries to minimize wave reflections. No cumulus pa- rameterization was used in the inner two domains. The sea surface temperature is set to be a constant of 29°C. The en- vironmental temperature and relative humidity profiles are the same as that in Holland (1997), representing the mean summertime conditions over the western Pacific. Fi g . 1. Time evolution of the center minimum sea level pres- The initial vortex is an axisymmetric vortex with a maxi- -1 sure (unit: hPa, top panel) and Vmax (unit: ms ; bottom panel) in F05 -1 mum tangential wind speed of 15 ms at a radius of 100 (dashed), F075 (red), F10 (green), F125 (solid curve with plus mark), km. The strength of the tangential wind decreases with the F15 (open circle), F20 (filled circle), F25 (open square) and F30 (filled height and vanishes at 100 hPa. Given the specified tan- square). Ma y 2012 Li et al. 3 environment than under planetary vorticity environment. inflow accelerates the development of local vorticity (and For instance, the CMSLP starts to drop rapidly shortly af- thus tangential wind) through a vorticity stretching effect. ter hour 24 in the F05 experiment, and it reaches a quasi- This effect can be clearly seen from maximum wind speed steady state around hour 60. As the Coriolis parameter difference between F05 and F30 at hour 12-24 (Fig. 1). For value increases, the timing of RI is delayed. For instance, example, at hour 24 the wind speed in F05 is approximate- the RI of TC in F30 starts after hour 48 and reaches a ma- ly twice as large as that in F30. The wind speed difference ture state after hour 84. may lead to a marked difference in the surface latent heat flux (or surface evaporation), the latter of which further a. Structure differences between high and low Coriolis impacts the moisture and condensational heating in the TC parameter cases core region. What causes the distinctive difference in the timing of TC To verify this wind speed effect, we examine the time- intensification among different Coriolis parameter values? radius cross section of the azimuthal mean surface heat flux To address this question we primarily examine the different difference between F05 and F30 (Fig. 3). It is clear that due development characteristics in two extreme cases (F05 and to the greater surface wind speed, the surface latent heat F30 cases). One of most striking differences is the distinc- flux in the TC core region is much greater in F05 than in tive evolutions of the boundary layer radial inflow. Figure 2 F30. As discussed by Emanuel (1988), the wind induced shows the time-radius cross section of the azimuthal mean surface heat exchange is crucial for rapid TC intensifica- radial flow at 500 m. Note that the radial inflow sets up tion. much earlier in F05 than in F30. For example, the inflow In addition to affecting the boundary layer moisture con- with a speed of 2 m/s has already set up at r=25 km prior to vergence and surface evaporation, the strengthened radial hour 24, whereas in F30 it does not establish until hour 48. inflow also enhance the secondary circulation and directly The establishment of the boundary layer radial flow affect vertical motion. The enhanced upward motion may is crucial for subsequent TC development. The increase further strengthen upward transport of moisture, leading to of boundary layer inflow leads to the strengthening of enhanced condensational latent heat release. moisture convergence into the TC core region (figure The increase of heating in the TC core region can further not shown), which can further impact the condensational enhance the radial inflow through the decrease of the cen- heating in the core region. The strengthening of the radial tral minimum pressure at the surface. Such a positive feed-

-1 Fi g . 2. Time-radius cross section of the radial flow (unit: ms ) at z=500m in F05 (left) and F30 (right). 4 Tropical Cyclone Research and Review Vo l u m e 2

momentum equation, i.e.

where p is the pressure, ρ is the air density and other quantities are as defined above. If F = 0, the tangential flow is in exact gradient wind balance; if F <0, this flow is sub- gradient, which means that there is a tendency to enhance the inflow toward the vortex center; and if F >0, it is super- gradient, which means that there is a tendency to enhance the outflow away from the vortex center. The radius–height cross-sections of the azimuthal mean net radial force field (F) in the F05 and F30 experiments at hours 45, 60 and 120 are shown in Fig. 5. At hours 45 and 60, the most prominent regions of inflow acceleration indeed occur in the atmospheric boundary layer. The su- pergradient flow emanates inside of and slightly above the Fi g . 3. Time-radius cross section of the surface heat flux difference subgradient flow (Fig. 5c). During the TC mature phase (at (unit: wm-2) between F05 and F30 (F05 minus F30). hour 120), the supergradient flow is co-located with strong ascending motion at low level (Fig. 5e-f). To illustrate distinctive evolution features between F05 and F30, we plotted the time-radius cross section of the back loop may be inferred from the distinctive structure azimuthal mean net radial force field at z=500 m (Fig. 6). evolutions of tangential wind and diabatic heating fields Note that in F05, the negative F (i.e., subgradient flow) between the F05 and F30 cases. Figure 4 illustrates the develops much earlier. It becomes evident around hour radial-height cross section of symmetric component of the 24, and continues to strengthen during the intensification tangential wind and condensational heating rate during the period. In contrast, the negative inflow tendency shows a intensification period. As the vortex intensifies, inner core significant delay (by about 24 hours) in F30. diabatic heating tends to concentrate towards the vortex Given the similar evolution characteristics between the center. It is noted that during the intensification period the radial flow and the net radial force (Figs. 2 and 6), it is like- heating source is located near the radius of maximum wind ly that difference in the radial wind between F05 and F30 (RMW) in both the cases, which may warm core develop- is primarily attributed to the extent by which the boundary ment (Vigh and Schubert 2009). In F05, the diabatic heat- layer imbalance is triggered. What causes the difference in ing near the RMW is stronger and develops earlier, com- the boundary layer imbalance between F05 and F30? We pared to that in F30. Thus, the evolutions of both the radial argue that the following factors may contribute. Firstly, a and tangential wind components and the heating patterns lower Coriolis parameter by definition corresponding to a under different planetary vorticity conditions are consistent smaller inertial stability may allow low-level inflow pen- with the distinctive CMSLP evolutions shown in Fig. 1. etrating closer to the vortex center. Secondly, the strength and depth of the boundary layer inflow associated with b. imbalances in the boundary layer flow the breakdown of the gradient wind balance depends on In the analysis above, we attribute the distinctive devel- the Coriolis parameter. It has been shown that the Ekman opment characteristics with different Coriolis parameter to pumping velocity (or boundary layer convergent flow) is the difference in the boundary layer inflow. A natural ques- reversely proportional to square root of f (Holton 1992). tion is what causes the radial wind difference? As we know, This means that for a specified cyclonic vorticity at top initially the radial flow is set to be zero for all the experi- of the boundary layer, friction induced inflow is stronger ments. The triggering of the radial flow lies in the break- when f is smaller. In addition, the Coriolis parameter also ing of a gradient wind balance due to surface friction. To controls the depth of the boundary layer. Figure 7 shows measure how differently the boundary layer gradient wind the structure difference of the azimuthal mean boundary imbalance is excited under different planetary vorticity layer inflow between F05 and F30 during TC intensification conditions, we examine a net radial force field, following and mature phases. Note that the depth of the inflow layer, -1 Smith and Vogl (2008). The net radial force field is defined as inferred from the Vr =-3 ms contour, are much greater as the residual term of the gradient wind balance, reflecting in F05 than in F30. For instance, at hour 60, the height of -1 a difference between the local radial pressure gradient and Vr =-3 ms contour in the inner-core region near the RMW the sum of the centrifugal and Coriolis forces in the radial reaches 2.5 km in F05, but it is only about 1 km in F30. Ma y 2012 Li et al. 5

Fi g . 4. Radial-height cross section of symmetric component of tangential wind (contour) and condensational heating rate (shaded) in F05 (left) and F30 (right).

Even in the mature phase at hour 120, the difference in the Coriolis parameter. The formula above indicates that a inflow depth is still clearly seen. lower Coriolis parameter leads to a deeper Ekman layer. Does the inflow depth difference appear during the initial A stronger, deeper inflow under lower planetary vorticity development stage? Figure 8 shows the time-height cross environment, in turn, brings about larger moisture conver- section of the radial wind at r = 20 km. It shows clearly that gence, leading to stronger condensational heating in the TC the difference of the inflow layer appears even in the initial core region. The stronger heating under lower planetary development period, say, prior to hour 24. The dependence vorticity environment is inferred from the time-radius cross of boundary layer depth on f can be readily interpreted in section of rainfall field (Fig. 9). Note that the rainfall rate the context of the Ekman pumping theory (Holton 1992). in the inner-core region is much greater in F05 than that in F30, particularly during initial development stage, whereas The Ekman layer depth may be define as , the rainfall rates in F10 and F15 are somehow in between. It is also noted that the radius of the maximum rainfall in where Az is an eddy viscosity or diffusivity, and f is the F05 is about 12.5 km, which is smaller than that in F30 6 Tropical Cyclone Research and Review Vo l u m e 2

-2 Fi g . 5. Radius-height cross section of the net radial force field (ms ) at hour 45 (top), 60 (middle) and 120 (bottom) from the F05 (left) and F30 (right) experiments. In (e) and (f), the red lines denote the vertical velocity (ms-1) at the same time.

(about 20 km). This confirms the inertial stability effect Under lower planetary vorticity environment, surface fric- hypothesis that a TC under lower planetary vorticity en- tion induced Ekman flow tends to be stronger and deeper, vironment may have a relatively small eye, due to weaker which results in greater moisture convergence and diabatic inertial stability. heating. The strengthened heating may further enhance the In summary, the dependence of timing of TC intensifica- boundary layer inflow. Through this positive feedback loop, tion on f is primarily attributed to the extent to which the a vortex develops faster. Besides, a lower Coriolis param- boundary layer imbalance is triggered and maintained. The eter allows greater contraction of the RMW due to weaker imbalance directly affects the strength of radial inflow. inertial stability. Ma y 2012 Li et al. 7

-2 Fi g . 6. Time-radius cross section of the net radial force field (ms ) in F05 (left) and F30 (right).

-1 Fi g . 7. Radial-height cross section of the radial flow (ms ) field in the lower 3 km in F05 (left) and F30 (right). 8 Tropical Cyclone Research and Review Vo l u m e 2

at different development stages. Generally, during the TC intensification period, the absolute angular momentum moves inward (outward) in the lower (upper) troposphere. The contours have a small angle to the horizontal at low- levels where the radial inflow is strong. The contours slope upwards and outwards in the region above the inflow layer where the TC flow is upward and outward, consistent with Smith and Montgomery (2008). In the absence of friction, it is materially conserved above the boundary layer. There- fore, the tangential wind speed will increase in the region where the mean absolute angular momentum isopleths move inwards. As a consequence, larger environmental mean absolute angular momentum results in a larger TC outer size. Compared to F05 (left panels), it is evident that the environmental absolute angular momentum is much greater in F30. To examine closely the inner core structure, we plotted the azimuthally mean tangential wind speed and vertical velocity at hour 120 within 50 km radius in Fig. 12. The eyewall tilts radially outward with height, especially above z=5 km. Meanwhile, the strong upward motion coincides with the radius of maximum tangential wind. It is noted that the radius of the eyewall is about 12.5 km in F05, whereas it is about 22.5 km in F30. In general, with other -1 Fi g . 8. Time-height cross section of the radial flow (unit: ms ) field environmental conditions fixed, the inner eyewall size de- at r=20 km. pends on the Coriolis parameter, with a smaller f leading to a smaller eye. The cause of the dependence is speculated to be related to inertial stability. A smaller f leads to a weaker inertial stability. As a result, the low-level inflow can pen- c. Difference in TC size etrate further into the vortex center region. In addition, TC Is the final TC size dependent on the Coriolis parameter? inner-core size may be sensitive to the radial distribution of To address this question, we examine the simulated TC the surface entropy flux (Xu and Wang 2010). The mecha- structures at the mature stage. Figure 10 shows the radius- nism responsible for the expansion in vortex outer size was height cross section of the azimuthal mean tangential wind discussed in previous studies (e.g., Ooyama 1969; Wil- speed at hour 120. As expected, in all the sensitivity experi- loughby 1988, 1995; Smith et al. 2010). However, the inner ments the maximum tangential wind speed appears at low core process may be independent of the spin-up of the outer levels. The TC size as defined by the radius of gale-force circulation, because the boundary-layer inflow is primarily wind (approximately 20 ms−1 contour line at the surface) affected by the gradient wind imbalance brought about by displays a greater dependence on the Coriolis parameter, the surface friction (Smith and Montgomery 2008). This that is, a TC formed under a higher Coriolis parameter has implies that different dynamic processes may determine a larger size than that under a lower Coriolis parameter. For the inner eyewall size and the outer size. The TC outer size example, the radius of the 20 ms−1 contour at the mature is largely determined by the outer environmental absolute stage is about 100km in F05, whereas it increases to 200km angular momentum above the boundary layer. The spin-up in F30. of the inner core, on the other hand, is associated with the To give a physical interpretation on the dependence, convergence of absolute angular momentum in the bound- we apply the absolute angular momentum conservation ary layer. The convergence may be produced by increasing concept. The absolute angular momentum is defined as system-scale radial buoyancy gradients associated with , where v is the symmetric component of deep, inner-core convection in the presence of enhanced surface moisture fluxes (Bui et al. 2009). tangential wind, r is the radius and f is the Coriolis parameter. Based on the definition, for a higher value of 4. Conclusion and discussion the Coriolis parameter, environmental absolute angular In this study, the dependence of tropical cyclone inten- momentum should be greater, and vice versa. sification on latitude at a constant f-plane is examined. By Figure 11 shows the radius-height cross section of the specifying different Coriolis parameters in a resting envi- symmetric component of the absolute angular momentum ronment, we found that vortex intensification under lower Ma y 2012 Li et al. 9

Fi g . 9. Time-radius cross section of 3-hourly accumulated rainfall (unit: mm) in F05, F10, F15 and F30 experiments.

-1 Fi g . 10. Radial-height cross section of symmetric component of the tangential wind field (unit: ms ) in F05, F10, F15 and F30 experiments. 10 Tropical Cyclone Research and Review Vo l u m e 2

5 2 -1 Fi g . 11. Radial-height cross section of symmetric component of absolute angular momentum (unit: 1×10 m s ) in F05 (left) and F30 (right) at hour 24, 48, 72 and 120.

planetary vorticity environment is faster than under higher TC has a larger (smaller) outer size under higher (lower) planetary vorticity environment. The distinctive develop- planetary vorticity environment, which is consistent with ment behaviors are attributed to the extent to which the previous model results (DeMaria and Pickle 1988) and ob- boundary layer gradient wind balance is broken. Given an servations (Merrill 1984). The dependence of the TC outer initial balanced vortex, surface friction destroys the gra- size on the Coriolis parameters is possibly attributed to the dient wind balance, leading to subgradient inflow in the effect of environmental absolute angular momentum. The boundary layer. The friction induced inflow is stronger and eye size or the RMW is also found to be f-dependent. It is deeper under lower planetary vorticity environment, which smaller (larger) under lower (higher) planetary vorticity en- brings about greater moisture convergence and leads to vironment. This dependence is possibly related the effect of greater condensational heating in the TC core region. The inertial stability, which regulates the extent to which low- strengthened heating lowers the central surface pressure, level inflow can penetrate into the core region. which further enhances the radial inflow. Through this posi- In this study, we mainly focus on the timing of TC inten- tive feedback loop, the vortex spins up at a faster rate under sification and associated structural change. The dependence lower planetary vorticity environment. of maximum intensity on the environmental planetary Besides the distinctive development behavior, the Corio- vorticity, on the other hand, is another important issue. Our lis parameter also controls TC size. It is found that the sensitivity numerical experiments show that at the final Ma y 2012 Li et al. 11

-1 Fi g . 12. Radial-height cross section of symmetric component of tangential velocity (contour; ms ) and vertical velocity (shading; ms-1) in (a) F05, (b) F10, (c) F15, and (d) F30.

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