^{a}

^{*}

^{a}

This study addresses the role of convective available potential energy (CAPE) in the intensification of simulated tropical cyclones. Additionally, it also examines the ‘wind-induced surface heat exchange’ (WISHE) theory in which CAPE is non-existent during intensification. We use a hierarchy of models with different complexity. A low-order tropical cyclone model forms the simplest model. It is found that the damping of CAPE by fast convective exchange as assumed in the WISHE theory inhibits substantial intensification in the model. This result can be explained by the dominance of the secondary circulation over surface heat transfer in the growth stage. It leads to entrainment of low entropy air into the eyewall resulting in the weakening of the cyclone. Other simulations reveal that the intensification rate increases with increasing initial CAPE and that the inner core CAPE is smaller than that of the ambient region. Investigations with the more complex Ooyama model yield qualitatively similar results. In this model, two types of convection are considered. The first one is based on frictional convergence in the boundary layer and the second one describes a convective adjustment including a precipitation efficiency. Only frictionally induced convection supports tropical cyclone intensification while the second one strongly acts to dampen the cyclone. Finally, the complex nonhydrostatic cloud model CM1 is used where the initial CAPE is varied. This model also exposes the existence of radially increasing CAPE during intensification. The experiments of this study indicate a positive relationship between the radial CAPE gradient and the intensification rate which disagrees with the basic assumption of WISHE models. The results emphasise the role of the secondary circulation for transporting high entropy air into the tropical cyclone inner core, and therefore should be considered in a proper intensification theory as has been done in the rotating convection paradigm by Montgomery and Smith.

A complete and comprehensive theory on tropical cyclone intensification is still a work in progress. Many studies and extensive research have been dedicated to understanding the behaviour of tropical cyclones but in more recent times most of the focus is more on intensification (Montgomery and Smith,

Ooyama (

Later a new approach to cyclogenesis was proposed where Emanuel (

Later studies by Emanuel (

Although Montgomery et al. (

Persing and Montgomery (

In our paper, we investigate the role of CAPE for tropical cyclone intensification with a hierarchy of tropical cyclone models. This has the advantage that on one hand we can understand the mechanism in the simpler models while the results of the sophisticated models give us more confidence in the relevance of the mechanism. The comparison of the models must of course be sufficiently rigorous to avoid that similar results appear for different reasons. The outline of the paper is as follows. In Section ^{1}

We describe a conceptual model which highlights the key processes of intensification. The model corresponds to the azimuthally averaged view of the rotating convection paradigm by Montgomery and Smith (

Boundary layer inflow and spin up of ambient air mass;

Evaporation over the sea;

Rising of air in the eyewall;

Latent heat release in the eyewall;

Gradient wind adjustment to the modified temperature field.

Conceptual model of intensification.

These five reinforce each other in the intensification pr ocess as suggested by the results of a previous study based on the Ooyama-model (Frisius and Lee,

This mechanism should hold regardless of if the cyclone is truly axisymmetric or not.^{2}^{3}

As noted by Frisius and Schönemann (

In the following, we explain the possible impact of CAPE on intensification. The tropical cyclone’s most intense winds are located beneath the eyewall. In the eyewall, air flows approximately along slantwise surfaces of constant angular momentum (Willoughby,

where ^{2}

where _{b} is the tangential wind at the top of the boundary layer, _{s} the sea surface temperature, _{out} the outflow temperature, _{b} the specific boundary layer entropy and _{SC} the slantwise CAPE (SCAPE). The first term on the right-hand side of Equation (_{b}. However, there is no clear mechanism why it should do so. The surface enthalpy fluxes increase _{b} at the radius of maximum winds. Then, the radial gradient of _{b} becomes positive in the inner core of the cyclone so that Equation (_{out} is not a constant but depends on the outflow stratification. With the assumption of a constant Richardson number in the outflow he was able to describe intensification by WISHE without a correction factor. However, Persing et al. (

Schönemann and Frisius (_{1} and _{2} are constant in time. Consequently, the physical radii _{b1} and _{b2} enclosing the eyewall boundary layer move in the course of the development. The eye responds passively to the development of the eyewall. Therefore, no equations for the eye are needed in this model. It is assumed that above the boundary layer angular momentum conservation and gradient wind balance holds. Therefore, the eyewall boundary represents a material surface and its inward movement results from gradient wind adjustment. The moist potential vorticity vanishes because there is a congruence of angular momentum and moist entropy surfaces. This has the consequence that the tangential wind adapts to the predicted temperature field. The slab-boundary layer of the model has diagnostic balanced dynamics, i.e. the inflow velocity becomes a function of the gradient wind at the top of the boundary layer. The shortcomings of the balanced slab-boundary layer has been discussed by Smith and Montgomery (

Sketch of the low-order tropical cyclone model.

In these equations the indices _{b} the boundary layer height, Ψ_{b2} the radial boundary layer mass flux at radius _{b2}, _{H} the surface transfer coefficient for enthalpy, _{o} the entropy at the sea surface, _{C} the time scale of convection and _{E} the time scale of Newtonian cooling that represents a simple parameterisation of radiation in the model. _{b1} and _{b2} are the gradient winds at the inner and outer sides of the eyewall, respectively. The quantities _{b1}, _{b2}, Ψ_{b2}, _{bi}, _{oi} and _{oa} are functions of the saturation entropy in the eyewall,

The model has been modified by adding convective exchange in the eyewall besides the vertical transport due to the frictional convergence in the boundary layer. Furthermore, the convective exchange in the ambient region only takes place in the unstable case, i.e. when _{bi} > _{C}. Schönemann and Frisius (_{C} → 0. Then, we have _{bi} and _{ba} =

This equation can explain some intensification by WISHE at the beginning since the surface fluxes heat the eyewall immediately. It describes the adaption of the eyewall to sea surface entropy _{oi} with the time scale

Later, however, the frictionally induced secondary circulation entrains low entropy air so that intensification is hampered. This process takes place with the time scale

and it leads to transport of low entropy air of the environment (_{b2}. Obviously, the WISHE time scale _{W} is only at low vortex intensity larger than that of the secondary circulation, _{S}. The time scale _{S} becomes much smaller than _{W} in the range where _{b2} takes reasonable values for a typical intensification phase. From these circumstances we can conclude that the entropy fluxes of the secondary circulation dominate the budget of the eyewall entropy. This is in accordance with the study of Wang and Xu (

Time scale of WISHE _{W} (red curve) and of the secondary circulation _{S} (green curve) as a function of tangential wind _{b2}.

To investigate the mechanism of intensification in this model we performed five numerical experiments. In these experiments, the relative humidity of the boundary layer and the free atmosphere are set to 80% and the sea surface temperature is set to 28 °C. These are conditions favourable for tropical cyclogenesis. For the time scale of convective exchange the value _{C} = 10 h is used unless otherwise stated. All other model parameters are listed in Table _{bi} have a finite value (0.6 J/kg/K) while _{a} and _{ba} are set to zero. In this way a weak, initial vortex in a neutrally stratified undersaturated atmosphere is prescribed.

_{ba}

_{E}

_{C}

_{H}

_{D}

_{b}

^{−5}s

^{−1}

_{1}

_{2}

_{t}

_{s}

_{a}

_{b}

In the second experiment, initial environmental CAPE is included (HIGHCAPE) by reducing the temperature of the ambient region, _{a} by 2.5 K. In the third experiment (STABLE), the temperature _{a} is increased by 2.5 K which stabilises the atmosphere. The fourth experiment WISHE integrates Equation (_{C} so that no convective exchange does take place in the ambient region.

Figure _{b2} for the various experiments. All experiments start with a tangential wind _{b2} of about 11 m/s. Experiment REF and HIGHCAPE reveal substantial intensification to a hurricane-type vortex within two days while in experiment WISHE only a marginal increase of wind speed takes place. Experiment STABLE exhibits a very fast decay of the vortex. The final tangential wind in experiment HIGHCAPE is with _{b2} = 65 m/s much higher than in experiment REF where _{b2} = 51 m/s is observed. Therefore, the steady-state tropical cyclone does indeed depend on environmental CAPE which disagrees with Persing and Montgomery (_{C} does have a damping effect on intensification and final intensity. The latter result was already found by Schönemann and Frisius (

Tangential wind _{b2} as a function of time for the experiments REF (red curve), HIGHCAPE (green curve), STABLE (blue curve), WISHE (magenta curve) and NOEXCHANGE (black curve).

Experiment WISHE shows no intensification since the frictionally induced secondary circulation transports low entropy air into the eyewall. This is understandable since for an initial tangential wind of 11 m/s the time scale for the secondary circulation greatly falls below the WISHE time scale (see Fig.

It appears that CAPE does play a role for intensification in the box model. We calculate the following expressions:

which can approximately be identified with SCAPE of the eyewall (_{SC,i}) and the ambient region (_{SC,a}), respectively (see Emanuel, _{SC,i} and _{SC,a} for the experiments REF, HIGHCAPE and NOEXCHANGE. All cases reveal an increase of SCAPE in the intensification phase. After intensification SCAPE attains a nonzero and constant level. Furthermore, we see that SCAPE of the ambient region exceeds that of the eyewall region increasingly during the intensification phase. This causes an enhancement of gradient wind and intensification rate due to the thermal wind balance equation (_{bi} exceeds _{ba} only by a small amount. Therefore, an important key element of the WISHE and potential intensity theories building on vanishing CAPE is not fulfilled in the low order model. The substantial amount of eyewall SCAPE in the steady state turns out to be a somewhat unrealistic result that is related to the large mass of the eyewall region in the low order tropical cyclone model. From the results we can conclude that CAPE does play a crucial role for intensification in the simple low-order model. In the next section we check if this is true in the more complex Ooyama model.

_{SC,i} (solid curves) and _{SC,a} (dashed curves) as a function of time for the experiments REF (red curves), HIGHCAPE (green curves) and NOEXCHANGE (black curves).

Figure _{b}, _{1} and _{2}. A free interface separates the middle and the upper layer while the interface between the boundary layer and middle layer is fixed but permeable. The densities of the two lower layers are identical while the upper layer has a density which is by a factor _{b,1}, _{b,2} and _{1,2}. First, there can be a convective updraught from the boundary layer into the upper layer associated with the mass flux _{b,2}. The flux _{b,1} describes detrainment of mass from this updraught in the middle-layer which is characteristic of shallow convection. Deep convection is on the other hand accompanied by an entrainment mass flux _{1,2} from the middle layer into the updraught. Mass conservation in the boundary layer requires the existence of a downdraught mass flux _{1,b} from the middle layer into the boundary layer. Hydrostatic Boussinesq equations govern the flow in the model. The Ooyama model attracted wide attention in the scientific community (e.g. DeMaria and Schubert,

Sketch of Ooyama’s three-layer model.

The Ooyama model has a simple parameterisation scheme for the mass fluxes between the various layers.^{4}

where _{b} = 1

In this formula _{e} is the equivalent potential temperature (EPT) and the indices _{e,1} while _{b} a downward massflux

must occur due to mass conservation.

The Ooyama model only includes convective exchange triggered by frictional convergence in the boundary layer. Frisius and Lee (_{C} to infinity in the low-order model presented in the previous section. In the present paper, we wish to include also convection that does not depend on frictional convergence. Therefore, we have modified formulas for the mass fluxes in the case

where

where _{1} denotes the radial wind of the middle layer and the condition _{D} = _{H} = 0_{b}_{b} denotes the wind speed in the boundary layer. In the Ooyama model we use a simple expression for CAPE, namely:

where _{p} denotes the specific heat at constant pressure and _{C,O} must be positive for the appearance of deep convection.

A relationship similar to Equation (

Here, _{1} and _{2} denote middle- and upper-layer tangential winds, respectively. The expression on the left-hand side of this equation measures the intensity of the thermal wind and it is likely related to the maximum wind speed of the vortex. Similar to Equation (

For the numerical experiments we use this simple axisymmetric three-layer model in an unbalanced configuration as described in Frisius and Lee (_{h} = 1000 m^{2}/s. More information on the numerical method and model parameters are given in Frisius and Lee (_{e,b} which is set identical to

where _{max,0} = 10 m/s is the maximum tangential wind and _{0} = 50 km the radius where the maximum occurs. No flow is initially present in the upper layer. Table _{C} → ∞. Furthermore, the _{–}OOYAMA except for switching on the additional convective exchange with a time scale of _{C} = 24 h. The experiments HIGHCAPE and STABLE conform to REF but they include different initial stratifications. In HIGHCAPE the initial EPTs _{e,1} and _{C} = 1 min. Then, the time scale is so short that CAPE vanishes approximately during the complete simulation. Experiment WISHE_DRY includes a radially decreasing precipitation efficiency. At the beginning it has the following profile:

_{b}

_{1}

_{2}

^{−5}s

^{−1}

_{e,1}

This profile yields a dry region beyond the radius _{0} where the tangential wind is maximised initially. Then, the conditions for intensification according to the aforementioned WISHE models are fulfilled since the precipitation efficiency decreases radially at the radius of maximum winds.

Figure

(a) Maximum middle layer tangential wind as a function of time for the experiments REF (red curve), REF_OOYAMA (black curve), HIGHCAPE (green curve), STABLE (blue curve), WISHE (magenta curve) and WISHE _DRY (light blue curve). (b) as in (a) but the radius of maximum _{1} is shown.

The results clearly show that the WISHE mechanism cannot cause intensification within the Ooyama model. Relaxing CAPE to neutrality by the convection parameterisation reduces the intensification rate substantially and allowing only little CAPE by using a short time scale for convective exchange (about 1 min) does not lead to a mentionable increase of wind speed. Therefore, a significant amount of CAPE is necessary for intensification to take place in the Ooyama model. We can draw the conclusion that either the Ooyama model or the WISHE models does not explain tropical cyclone intensification properly. In the WISHE models the neglect of entrainment of low entropy air into the boundary layer could be a serious deficiency. On the other hand, the assumption of a time invariant middle layer entropy in Ooyama’s model could also be unsuitable.

The additional convective scheme used in the present model has some similarity with the quasi-equilibrium scheme included by Zehnder (

Figure _{1} and CAPE _{C,O} at the time when the profile maximum of tangential wind _{1} exceeds 50% of the time maximum of _{1}. All simulations reveal nonzero CAPE at nearly all radii. Narrow radial ranges with zero CAPE only occur inside the RMW in all shown experiments except for STABLE. These are located inside of the eyewall where a downdraught transports low entropy air into the boundary layer. It is noticeable that CAPE in experiment REF_OOYAMA has the largest values inside of the eyewall while in the other experiments this is not the case. This happens because CAPE accumulate in the inner core when frictional convergence does not occur in the absence of additional convective exchange. CAPE increases with radius immediately outside the RMW in all experiments except for STABLE. The associated radial CAPE gradient seems to correlate with the intensification rate. This result suggests a positive impact of a radially increasing CAPE on intensification within the Ooyama-model which is in agreement with the thermal wind balance equation.

Radial profiles of tangential wind _{1} (black curve) and CAPE _{C,O} (red curve) in the Ooyama-model at the time when the intensity reaches 50% of its time maximum for the experiments (a) REF_OOYAMA, (b) HIGHCAPE, (c) REF and (d) STABLE.

We conduct further experiments with the nonhydrostatic cloud resolving model CM1, version 18. The model was developed by George Bryan at NCAR (Bryan and Fritsch, _{v}), cloud water (_{c}), rain (_{r}), cloud ice (_{i}), snow (_{s}) and graupel (_{g}).^{5}^{−5} s^{−1}. The boundary conditions are periodic in all directions and at the surface we assume an ocean having a fixed sea surface temperature of 28 °C. For a more detailed description of CM1 see Bryan and Fritsch (

We use the default configured namelist for tropical cyclone simulation except for some changes as described below.^{6}_{H} = 0_{D} results from a parameterisation that is based on findings of Fairall et al. (

where _{10 m} denotes the horizontal wind speed at 10 m height, _{0} = 5 m/s and _{1} = 25 m/s. The grid spacing is set to 2 km with 600 grid points in each horizontal direction and 40 model levels in the vertical having a 500 m grid spacing are adopted. The model is time-integrated by adopting the Klemp-Wilhelmson time-splitting, vertically implicit scheme with a large time step of 7s and 4 intermediate short time steps. To test the sensitivity of the tropical cyclones intensification to CAPE we need to analyse its radial distribution at the time of maximum intensification. First we define CAPE as:

where _{LFC} is the level of free convection in the boundary layer, _{LNB} the level of neutral buoyancy, _{v,p} the virtual temperature parcel of the moist adiabat and _{v,e} the virtual environmental temperature. For the calculation we used an open source FORTRAN code.^{7}

Three experiments are conducted for different vertical temperature profiles. The moist tropical (MT) sounding found by Dunion (

where _{t} the temperature anomaly at a height of 10,000 m. Note that _{t} = _{t} = 5 K. The three experiments LOWCAPE, REF and HIGHCAPE have different initial CAPE values, namely 28.8 J/kg, 952 J/kg and 2435 J/kg, respectively. Therefore, we can assume that the initial atmosphere in LOWCAPE is nearly neutrally stratified. Dunion (

Figure _{t}. The initial vortex is identical to that used by Rotunno and Emanuel (

Initial vertical temperature profiles for the CM1 experiments REF (red curve), HIGHCAPE (green curve) and LOWCAPE (blue curve). The black curve shows the moist adiabat resulting from the initial condition of experiment REF.

Figure

Maximum azimuthally averaged wind speed as a function of time for CM1 experiments REF (red curve), HIGHCAPE (green curve) and LOWCAPE (blue curve). The black circles indicate time points representative for the intensification phase. These were selected for the further analysis.

Figure ^{8}

CAPE (coloured shadings) and surface pressure (isolines) for (a) experiment LOWCAPE at

Radial profiles of azimuthally averaged tangential wind (black curve) at the level of maximum wind speed (

Figure _{e}. Very low EPTs occur in the lower part of the troposphere immediately above the boundary layer. Fast convective exchange like in the WISHE models would likely decrease the boundary layer entropy in such a way that the eyewall would soon dissolve leading to a cyclone decay. The figure also shows a weak radial gradient of boundary layer entropy under and outside of the eyewall which accords with the low order tropical cyclone model but disagrees with the WISHE theory that must build on a substantial negative gradient of this quantity.

Radial cross section of azimuthally averaged EPT (coloured shadings, K), radial wind (black isolines, contour interval 2 m/s) and vertical wind (thick white isolines, contour interval 1 m/s) for CM1 experiment REF at t = 87 h.

In this study, we investigated the role of CAPE for tropical cyclone intensification by using a hierarchy of models having different complexity. With the conceptual model we explained the possible mechanism of intensification enhancement due to CAPE. It is assumed that generated CAPE by surface fluxes does not decay immediately by convection. Then, a substantial amount of high entropy air reaches the eyewall where it releases the latent heat. There are two aspects for enhanced intensification. First the higher amount of latent heat increases the negative radial temperature gradient. Second, the absence of convection outside the eyewall avoids a decrease of the negative radial temperature gradient. Both effects causes an increase of gradient wind which accelerates the intensification due to a stronger frictional inflow and higher surface heat fluxes. The WISHE theory excludes this mechanism since it is assumed that convection removes CAPE immediately. We challenged this assumption by testing it in two tropical cyclone models. The first one is a simple low-order model in which the vortex is divided into five regions. We showed that in this model the tropical cyclone cannot attain hurricane strength by assuming immediate convection although the WISHE mechanism works for small amplitude vortices. The reason for the absence of substantial intensification is that low entropy air of the ambient region gets into the eyewall by the frictionally induced secondary circulation. This mechanism forms an effective brake for intensification in the low-order model when CAPE is not present. On the other hand, it was found that the tropical cyclone intensifies significantly when CAPE can be generated by surface fluxes. Then, the frictionally induced secondary circulation supports intensification because it transports high entropy air from larger radii into the eyewall which is consistent with findings by Wang and Xu (

There are two types of convection that can occur within a tropical cyclone. The first one is convection induced by frictional convergence and the second one occurs due to other processes like precipitation induced downdrafts. For the latter we may use the term natural convection. Only frictionally induced convection supports tropical cyclone intensification while natural convection constrains the development because it reduces the entropy of the boundary layer air that will eventually be entrained into the eyewall. Models may reveal intensification where only frictionally induced convection occur. Besides the original Ooyama model, such a scenario can occur in axisymmetric nonhydrostatic models in which latent cooling processes like evaporation, sublimation and melting are ignored (see Frisius and Hasselbeck, _{C} has a strong damping effect in the low-order model and the Ooyama model. The absence of intensification in the limit _{C} → 0 suggests that the simple WISHE models by Emanuel (_{H} = 0^{9} kg/s ^{3}. On the other hand, only a mass of ^{2} – (10,000 m)^{2}] × 0_{s}^{9} kg/s _{s} evaporates from the surface below the circular ring where _{s} denotes the specific humidity difference between 2 m height and the surface. Obviously, the secondary circulation dominates the moisture budget in the eyewall while the contribution from surface fluxes is quite unimportant. This scale analysis is based on the same arguments as for the calculation of the time scales for WISHE and the secondary circulation in the low-order model (see Fig.

The results of this study do not mean that a wind-induced feedback between surface fluxes and wind intensification cannot take place. Montgomery et al. (

This work was supported through the Cluster of Excellence ‘CliSAP’ (EXC177), Universität Hamburg, funded by the German Science Foundation (DFG). Marguerite Lee was supported in part by the DFG within the individual research grant FR 1678/2.

No potential conflict of interest was reported by the authors.

The high-resolution CM1 simulations were performed at the DKRZ with the high performance computer IBM POWER6 ‘blizzard’. We thank two anonymous reviewers for their constructive comments.

For simplicity we refer the three-layer model to as the Ooyama model.

Montgomery and Smith (

We note that CAPE is not a fluid property, i.e. it does not exclusively depend on the properties of the specific fluid parcel. Therefore, generation refers to the area integrated CAPE.

Strictly speaking the _{j}_{k} are mass fluxes divided by reference density.

See ‘The governing equations for CM1’ on the web site

The model CM1 version 18 with the preconfigured name list can be downloaded via the link

A copy of the code is provided on the website

We did not calculate SCAPE because its determination is elaborate in a three-dimensional flow field. However, we expect that CAPE and SCAPE look qualitatively similar. Furthermore, SCAPE is always larger than CAPE in a cyclone with outward slanted angular momentum surfaces. Therefore, the existence of CAPE proves that also SCAPE is nonzero.

The Ooyama model comprises the following governing equations:

Radial momentum equation of the free atmosphere layers

Tangential momentum equation of the free atmosphere layers

Radial momentum equation of the boundary layer

Tangential momentum equation of the boundary layer

Budget equation for boundary layer EPT

Continuity equation of the middle layer

Continuity equation of the upper layer

Continuity equation of the boundary layer

Hydrostatic equation of the middle layer

Hydrostatic equation of the upper layer

In these equations ^{−1}_{v,X}, _{h,X} and _{s,X} describe the tendencies of quantity