## Abstract

This paper addresses the validity of the gradient wind balance approximation during the intensification phase of a tropical cyclone in Ooyama’s three-layer model. For this purpose, the sensitivity to various model modifications is examined, given by the inclusion of (i) unbalanced dynamics in the free atmosphere, (ii) unbalanced dynamics in the slab boundary layer, (iii) a height-parameterized boundary layer model, and (iv) a rigid lid. The most rapid intensification occurs when the model employs the unbalanced slab boundary layer, while the simulation with the balanced boundary layer reveals the slowest intensification. The simulation with the realistic height-parameterized boundary layer model exhibits an intensification rate that lies in between. Intensification is induced by a convective ring in all experiments, but a distinct contraction of the radius of maximum gradient wind only takes place with unbalanced boundary layer dynamics. In all experiments the rigid lid and the balance approximation for the free atmosphere have no crucial impact on intensification, and a linear stability analysis cannot explain the found sensitivity to intensification. Most likely the nonlinear momentum advection term plays an important role in the boundary layer. It is found on the basis of a diagnostic radial mass flux equation that the source term for latent heat provides the largest contribution to intensification and contraction. Furthermore, it turns out that the position of the convective ring inside or outside of the radius of maximum gradient wind (RMGW) is of vital importance for intensification and most likely explains the large impact of boundary layer imbalance.

## 1. Introduction

The mechanism for tropical cyclone intensification is still controversially debated [for a review, see Smith and Montgomery (2015)], although many numerical nonlinear models can capture this phenomenon properly. The numerical axisymmetric model developed by Ooyama (1969, hereafter O69) was one of the first to reveal tropical cyclone intensification. The model is based on the hydrostatic Boussinesq equations and has three layers of uniform density, where the lowest one is a slab boundary layer. This simple model conception enables a better understanding of the intensification process and has the advantage of a high numerical efficiency. However, the Ooyama model includes the balance approximation—that is, the tangential wind underlies the gradient wind balance—and it is questionable if this assumption is justified during the intensification stage, when the rate at which pressure falls becomes large in magnitude. In the present study we will analyze the impact of this assumption by relaxing the balance assumption within Ooyama’s model. Indeed, K.V. Ooyama (1968, unpublished manuscript)^{1} presented results of a modified three-layer model that includes an unbalanced boundary layer. He found more realistic solutions than with the original model (see also Smith and Montgomery 2008). An unbalanced nonaxisymmetric Ooyama model was also developed by Schecter and Dunkerton (2009) and compared to a cloud-resolving model by Schecter (2011). These studies investigated the sensitivity of tropical cyclone formation and maximum intensity with respect to various model parameters but did not address the impact of gradient wind imbalance.

Another aim of this study is to advance understanding of the intensification mechanism. The balance approximation has the advantage that the cause of tangential wind rise can be revealed by the Sawyer–Eliassen equation (Shapiro and Willoughby 1982; Bui et al. 2009). This diagnostic equation results from the time derivative of the thermal wind balance equation. In the Ooyama model a simpler diagnostic equation can be derived because of the three-layer formulation. We will see that time differentiation of the gradient wind balance equation yields a single ordinary differential equation for the radial mass flux when a rigid lid is assumed. Shapiro and Willoughby (1982) also analyzed tropical cyclone intensification, and they found that a single point source for heat can induce contraction of the wind maximum and intensification. We will see that this convective ring contraction scenario also does take place in the Ooyama model and that the findings by Shapiro and Willoughby (1982) are indeed relevant. However, less clear is the role of the boundary layer. The boundary layer supplies the necessary latent energy that is released in the contracting convective ring. Smith and Montgomery (2008) found that the gradient wind imbalance has a large impact on the wind profiles in a steady-state slab boundary layer model of a tropical cyclone. It is likely that this also has an effect on the strength and position of the convective ring fed by boundary layer air. Therefore, the balance assumption in the boundary layer could sensitively influence the intensification rate. On the other hand, Kepert (2010a,b) and Williams (2015) found by comparing the slab boundary layer model with a height-resolving boundary layer model that the former tends to overestimate supergradient winds and vertical velocities. Kepert (2010b) suggested using a height-parameterized boundary layer model instead, which can be coupled to the Ooyama model indeed, as we will demonstrate in this paper.

The paper is organized as follows. Section 2 contains the model description and the outline of the simulations. In section 3, results of the performed simulations are presented and the differences due to the balance approximation are identified. Section 4 clarifies the intensification mechanism by analyzing the linearized equations and evaluating the diagnostic equation for the induced secondary circulation. Finally, the conclusions are summarized in section 5.

## 2. Description of the model

Ooyama’s model includes three layers lying upon each other, which is sketched in Fig. 1. The interface between the middle and the upper layers is a free material surface, while the interface between the lower and middle layers is fixed but permeable. The lowermost layer forms the boundary layer where microturbulent exchange with the ocean surface is relevant. Convection can pervade all interfaces, and it gives rise to the vertical mass fluxes , , and . The density of the two lower layers is , while that of the upper layer takes the value with . In the following subsections the governing equations, the physical parameterization schemes, the rigid-lid modification, Kepert’s height-parameterized boundary layer model, and the performed experiments are outlined.

### a. Governing equations of the free-surface Ooyama model

The governing equations of the free-surface Ooyama model are as follows:

where *r* denotes the radius, *t* the time, *u* the radial wind, *υ* the tangential wind, *w* the vertical velocity at the top of the boundary layer, *h* the layer depth, *H* the mean layer depth, *P* the kinematic pressure anomaly, the equivalent potential temperature, *f* the Coriolis parameter, the absolute vorticity, the vertical mass flux from layer *j* into layer *k*^{2}, and *g* the gravity acceleration. The indices *b*, 1, and 2 denote the boundary layer, middle layer, and upper layer, respectively. The terms , , and describe the tendencies of quantity *X* as a result of vertical exchange between the layers, horizontal mixing, and surface fluxes, respectively. The switches , , and include additional terms that were absent in the original formulation by O69. With , the balance approximation for the free atmosphere (layers 1 and 2) is switched off. Switch includes an unbalanced slab boundary layer model. Setting in the case removes the local time derivatives of the boundary layer momentum equations so that the boundary layer model becomes diagnostic as in Smith and Montgomery (2008). With this switch, we can estimate the importance of the boundary layer adjustment time scale for intensification. For , the boundary layer model also includes the effect of local wind change in the momentum budget. For , the original Ooyama model is recovered, while for , the model formulation corresponds to that by Schecter and Dunkerton (2009).

### b. Parameterization of irreversible physical processes

The Ooyama model comprises parameterization schemes for updrafts, surface fluxes, and vertical and horizontal diffusion. The updraft parameterization yields the mass fluxes between the various layers. It is assumed that the upward mass fluxes are proportional to the upward boundary layer mass flux so that

where *η* denotes the so-called entrainment parameter. Mass of layer 1 will be entrained into the updraft for and transformed into mass of layer 2 having a lower density. Then deep convection takes place, while yields detrainment, which is characteristic for shallow convection. The entrainment parameter *η* is a function of the vertical thermal and moisture stratification, namely,

where is the saturation equivalent potential temperature. A constant value is assumed for , while results from the approximation

where the overbar denotes the ambient value and *a* is a thermodynamic constant.

The upward boundary layer mass flux must be compensated by a downward mass flux in order to conserve the mass of the boundary layer. This is ensured by setting

There is no downward mass flux from layer 2 to layer 1 in the original Ooyama model.

Note that this parameterization differs from conventional convective parameterization schemes since it does not include downdrafts. Indeed, Ooyama’s scheme is rather valid for a single convective cloud or convective ring than for an ensemble of many convective elements that usually include downdrafts. Therefore, a cloud-scale model resolution does not disagree with this scheme.

The tendencies due to surface fluxes are parameterized as follows:

where and denote the minimum surface transfer coefficients for momentum and enthalpy, respectively. Furthermore, is the wind speed in the boundary layer and the saturation equivalent potential temperature at the sea surface, which is a function of pressure and is given by

where *b* is another thermodynamic constant. Since we make use of a slab boundary layer model, the surface transfer is regulated by the depth-averaged wind instead of the conventional 10-m wind. This obvious deficiency is eliminated in Kepert’s height-parameterized boundary layer model, which will be described below.

For horizontal exchange, a simple diffusion scheme is applied: that is,

where denotes the horizontal diffusion coefficient.

Vertical exchange is related to vertical mass fluxes, causing the following tendencies:

O69 also includes shearing stress at the interface between the layers. However, we found a negligible impact of these stresses in the simulations performed, and, therefore, we left this process out here.

### c. Rigid-lid assumption

Some of the governing equations have to be modified when a rigid lid is assumed. This assumption has the consequence that

Therefore, we obtain for the pressure anomalies

where is the kinematic pressure anomaly at the rigid lid. These equations replace Eqs. (9) and (10). Because of the enforced volume conservation of the free atmosphere, the radial velocity becomes a function of , , and , namely,

With the rigid-lid assumption, mass conservation does not hold anymore when convection is included, since the expansion due to latent heat release cannot be consistent with the volume conservation enforced by the rigid lid. Then the model loses some mass in the course of time. However, the artificial mass loss is rather small, since only the fraction of the total upper layer mass gain is removed, and *ϵ* is usually close to 1.

The rigid-lid assumption has the advantage that the radial flow of the free-atmosphere layer can be determined by a single diagnostic equation in the balanced case . The time derivative of the gradient wind balance equation for the middle layer becomes the following by substituting Eqs. (2), (32), and (6):

where denotes the inward radial volume flux in the middle layer. The contribution of the lid pressure tendency can be evaluated by the gradient wind balance equation for the upper layer, and the result leads to the following diagnostic equation for the inward volume flux :

in which

is a factor measuring the inertial stability, and the various source terms are as follows:

The source term results from the mass fluxes between the middle layer and the boundary layer due to frictional convergence or divergence, while is associated with latent heat release due to deep convection. The source terms and refer to vertical flux of tangential momentum from the boundary layer into the middle layer and from the two lower layers into the upper layer, respectively. The source term arises as a result of horizontal diffusion, and is the effect of radial angular momentum advection in the upper layer due to frictional convergence. The diagnostic Eq. (36) forms the equivalent to the Sawyer–Eliassen equation that applies to a vertically continuous model. In the latter equation, however, vertical and horizontal momentum advection do not appear as source terms. These diagnostic equations can help in understanding tropical cyclone intensification, since they diagnose the inflow in the free atmosphere, which is required for the spinup of the vortex. The linear nature of Eq. (36) facilitates the assignment of the various source terms to associated tangential wind tendencies resulting from inward angular momentum advection.

In the rigid-lid case, the solution method for the free-atmosphere equations is as follows. First, the gradient wind balance equations are integrated inward from the outer boundary to the center to obtain the lid pressure anomaly and the middle-layer depth . The values and are assumed at the outer boundary for the integration. Then the diagnostic Eq. (36) is solved, and the resulting radial velocity is used for the time integration of the prognostic tangential wind equation [see Eq. (2)].

### d. Kepert’s height-parameterized boundary layer model

The slab boundary layer model used in the present study has some shortcomings, which were demonstrated by Kepert (2010a,b), and Williams (2015). They found by comparing the slab boundary model with a height-resolving model that the inflow has too large an amplitude and that the departure from gradient wind balance is overestimated as a result of excessive surface drag. Furthermore, the vertical and horizontal wind components can display unnaturally large oscillations for certain gradient wind profiles. Kepert (2010b) proposed a height-parameterized boundary layer model that resolves these issues. In this model, the vertical profiles of the boundary layer wind components are prescribed by the Ekman spiral as a function of height *z* and are given by the following:

where *d* denotes the height scale. Kepert (2010b) found that 2.5 × *d* approximately yields the boundary layer height . The vertical averages of these profiles becomes

Therefore, and are not the vertically averaged wind components, since . These components rather yield the scale of the boundary layer winds. The equations for the height-parameterized model become

where , , and respectively denote the radial wind, tangential wind, and wind speed at the surface (*z* = 0) deduced from Eqs. (44) and (45). Using these surface wind values instead of vertically averaged values yields a more realistic surface transfer parameterization. Most terms result by vertically averaging the terms of the height-dependent axisymmetric momentum and temperature equations for the given vertical wind profiles, but the vertically averaged vertical advection term has been simplified as in the original slab boundary layer model. The vertical velocity at the top of the boundary layer becomes approximately

Note that Kepert’s height-parameterized boundary layer model deviates from the slab boundary layer model in terms of (i) different expressions for the inertia forces, (ii) the use of surface wind for the surface transfer parameterization, and (iii) a 2.5-times-smaller height scale in the momentum equations. Kepert (2010b) also included a radial variation of boundary layer depth, which is omitted here. Furthermore, there is a small inconsistency in the boundary condition, since the direction of turbulent stresses becomes discontinuous immediately above the surface (Kepert 2010b). However, the height-parameterized boundary layer model still produces more realistic results compared to the slab boundary layer model.

### e. Numerical method and experimental design

For the numerical solution, a stretched staggered grid has been introduced, as described by Frisius (2015). The grid has an inner part and an outer part with gridpoint distances of 250 and 2500 m, respectively. Between these regions there is a smooth transition zone that is centered between the first and second quarter of all grid points. For time integration, a leapfrog scheme including a Robert–Asselin time filter is applied. The lateral boundary is located at *r* = 4647.08 km where the radial velocity and all horizontal diffusive fluxes vanish. These boundary conditions also apply at the center axis where the tangential wind becomes zero in addition. In the balanced case , the equations for the upper two layers are solved as described in O69. To solve the steady unbalanced boundary equations ( and ), the boundary layer model is integrated separately in time at each time step until the solution attains approximately a steady state.

The initial fields for all simulations are identical to those of case A in O69. The parameters were also taken from O69 (Table 1), except for the interfacial friction coefficient, which is set to zero in the present study. The density ratio appears too small for the troposphere. However, DeMaria and Pickle (1988) found that the Ooyama model can alternatively be interpreted in terms of compressible isentropic layers. Then the chosen *ϵ* value agrees indeed with a typical tropospheric stratification. Table 2 shows the designation and configuration of the various experiments. Experiment REF corresponds to the original model experiment performed by O69. Experiments BALBL and UNBALBL include unbalanced dynamics in the free atmosphere and in the boundary layer, respectively, while experiment UNBAL adopts unbalanced dynamics in both the boundary layer and free atmosphere. Experiment SBL has unbalanced dynamics in the boundary layer, but local time tendencies for momentum are switched off so that equations for a steady-state boundary layer are solved . Experiment KEPERTBL includes Kepert’s height-parameterized boundary layer model but is identical to UNBALBL otherwise. Experiments REF_RIGID, UNBALBL_RIGID, SBL_RIGID and KEPERTBL_RIGID employ a rigid lid and conform in other respects with REF, UNBALBL, SBL and KEPERTBL, respectively.

## 3. Results

Before discussing all experiments, the results of the reference experiment REF are compared with those of O69, which should be similar. However, some differences may arise because of the different numerical scheme, model resolution, and boundary conditions.

### a. Reference experiment

Figure 2 shows the maximum tangential wind of the middle layer and the radius where it is located. The former is taken as a measure for intensity while the latter constitutes the radius of maximum gradient wind (RMGW) in the runs based on the balanced free atmosphere . The present simulation reveals a similar result compared to that obtained by O69 (see his Fig. 4). Nevertheless, the maximal tangential wind peaks at 64.9 m s^{−1} instead of 58 m s^{−1}, as found by O69, and the final radius of maximum tangential wind becomes 158 km, which is much larger than in the simulation by O69. Figure 3 presents radial profiles of various model variables at time *t* = 81 h, when the intensification rate is close to its maximum. This figure should be compared to the middle panel of Fig. 5 in O69. The profiles for the displayed tangential and radial wind components are very similar in magnitude and shape to those found by O69. However, the peak vertical velocity of 1.7 m s^{−1} is about 0.5 m s^{−1} larger than in the original Ooyama model. This could possibly explain the higher peak intensity, as the vertical velocity *w* is proportional to the latent heat release in the developing eyewall. The shape of the profile for the entrainment parameter *η* resembles that displayed by O69, except near the center axis, where O69 found somewhat smaller values. This parameter is closely related to the convective available potential energy (CAPE). Therefore, CAPE is minimal close to the developing eyewall, and it increases toward the center as well as outwards. Three reasons can be responsible for the differences found between the simulations. First, the grid spacing is reduced by a factor of 20 in the inner part of the model compared to that in the original Ooyama model. Second, the model domain of O69 has only a radius of 1000 km, which is less than a quarter of the domain size used here. Furthermore, O69 treated the lateral wall differently by allowing fluxes across the boundary. Third, O69 did not make use of horizontal diffusion in the prognostic equation for the equivalent potential temperature in the boundary layer. We also made a simulation with uniform grid spacing of 5 km, no horizontal diffusion in Eq. (5), and a lateral wall at *r* = 1000 km. In this case, the tangential wind only reaches a maximum of 45m s^{−1}, and the RMGW does not expand in the decay phase. Furthermore, the profiles of vertical velocity *w* and entrainment parameter *η* exhibit gridpoint scale oscillations. However, the value of *η* at *t* = 81 h is now at the axis (*r* = 0) very similar to that found by O69. Using a lateral wall at *r* = 1750 km yields a time evolution of maximum and RMGW that is very similar to the result of O69. From these results, we can presume that both the different boundary conditions and grid spacing are responsible for differences in the evolution of intensity and the RMGW. On the other hand, horizontal diffusion in the thermodynamic equation causes a higher *η* value at the vortex axis. The reason why O69 did not detect gridpoint-scale oscillations remains unclear. It is likely that the different gridpoint discretization schemes are responsible for the deviations.^{3} We also performed a simulation at doubled gridpoint spacing and with a smaller model domain. In both cases, we found only very small differences in the results. Therefore, the positions of the lateral wall and the radial gridpoint resolution appear suitable for the present simulation. However, this was possibly not the case in the original simulation by O69.

### b. Sensitivity experiments

In this subsection we compare the outcome of all performed simulations. Figure 4a shows the time evolution of the maximum tangential wind of layer 1. Obviously, large differences arise between the various developments. The peak intensity varies between 58 and 80 m s^{−1}, and the time of maximum intensification rate ranges from 1.5 to 4 days. The intensification phase can be assigned to four experiment groups. The most rapid intensification takes place when the model includes the full unbalanced boundary layer model (experiments UNBAL, UNBALBL, and UNBALBL_RIGID), while a somewhat slower intensification is found for the model configurations neglecting the local momentum tendencies in the boundary layer model (experiments SBL and SBL_RIGID). The tangential wind in experiments KEPERTBL and KEPERTBL_RIGID intensifies in turn less rapidly compared to SBL and SBL_RIGID. The slowest growth emerges in the experiments based on the balanced boundary layer model (REF, BALB, and BALB_RIGID). In these simulations, intensification is slower, and a decay of vortex intensity takes place eventually in contrast to the other experiments. The cases UNBAL, UNBALBL, and UNBALBL_RIGID reveal a contraction of the radius of maximum , after which this radius increases again very slowly (see Fig. 4b). This behavior is similar to that typically observed in more sophisticated tropical cyclone models (e.g., Hausman et al. 2006; Hill and Lackmann 2009; Xu and Wang 2010; Persing et al. 2013; Frisius 2015; Smith et al. 2015; Stern et al. 2015). The developments in the experiments SBL, SBL_RIGID, KEPERTBL, and KEPERTBL_RIGID are qualitatively similar, but the contraction begins several hours later and is slower. In Stern et al. (2015), it is noted that the vortex contraction stops well before the intensity reaches its maximum. This is also fulfilled in the abovementioned experiments, although after contraction only little further intensification happens. The experiments REF, BALB, and BALB_RIGID reveal only a weak contraction and an untypically fast outward migration of the wind maximum afterward.

Figure 5 displays radius–time diagrams of tangential wind and vertical velocity *w* for selected experiments. The diagrams for UNBALBL and SBL_RIGID are very similar to UNBAL and SBL, respectively, and have, therefore, not been displayed in Fig. 5. Also in this figure the differences and commonalities suggest the subdivision into the four experiment groups. All unbalanced boundary layer experiments exhibit the single convective ring contraction scenario, as suggested by Shapiro and Willoughby (1982). The convective ring develops already at the beginning of the intensification phase just inside of the tangential wind maximum. Then the solution of the Sawyer–Eliassen equation reveals both contraction and intensification as a result of the latent heat release (Shapiro and Willoughby 1982). Frisius (2006) also detected intensification by a single convective ring in an axisymmetric nonhydrostatic cloud model. Such a scenario may even be seen in more realistic 3D models when the azimuthally averaged vertical velocity is analyzed (e.g., Braun et al. 2006; Persing et al. 2013). The single convective ring scenario could be disturbed by further convection outside the developing eyewall as a consequence of latent cooling in downdrafts (Wang 2002a; Frisius and Hasselbeck 2009). Furthermore, Wang (2002b) found that vortex Rossby waves can lead to outward-propagating spiral rainbands and eyewall breakdown. However, the effects of latent cooling and asymmetries like vortex Rossby waves are not considered in the Ooyama model, and, therefore, further convective cells do not develop in our experiments.

In the experiments UNBAL, UNBALBL_RIGID, and SBL the vertical velocity *w* takes values of up to 18 m s^{−1}, which appear unrealistically large. On the other hand, *w* is much smaller and more realistic in experiments KEPERTBL and KEPERTBL_RIGID. Hence, Kepert’s height-parameterized model obviously corrects an unwarranted feature of the slab boundary model in this context. The experiments based on a balanced boundary layer (REF, BALB, and BALB_RIGID) reveal a single convective ring too, but it does not contract significantly and is weaker. It is always located outside of the tangential wind maximum. This could be the reason why contraction and intensification are much weaker in these experiments. The results are consistent with Heng and Wang (2016), who found in a nonhydrostatic tropical cyclone model with a prescribed heating and the Sawyer–Eliassen equation that the balance approximation is fulfilled quite well above the boundary layer but that unbalanced boundary layer processes may enhance eyewall contraction and produce more realistic boundary layer structures. At last, we can draw the conclusion that the balance approximation above the boundary layer and the rigid-lid assumption may have some effect on the peak intensity but are not very crucial for the intensification phase. Therefore, we only consider the rigid-lid experiments in the remainder of this study.

Figure 6 shows radial profiles of the boundary layer wind components at the time of maximum intensification for the three experiments REF_RIGID, UNBALBL_RIGID, and KEPERTBL_RIGID. Note that the tangential boundary layer wind in REF_RIGID is identical to the gradient wind of the middle model layer. In this simulation, the maximum inflow is found at a radius of more than 50 km, while the RMGW is located at a significantly smaller radius. This has the consequence that the maximum vertical velocity also lies at a radius larger than the RMGW. The maximum inflow velocity of 36 m s^{−1} is very large, and the resulting inflow angle of about 40° appears unrealistically high (e.g., Frank 1977). In contrast, the simulations UNBALBL_RIGID and KEPERTBL_RIGID reveal a lower inflow angle, and the radius of maximum inflow is close to the RMGW. Therefore, the vertical wind peaks inside of this maximum. Furthermore, tangential winds become supergradient beneath the developing eyewall in UNBALBL_RIGID and also in KEPERTBL_RIGID but with a smaller amplitude in the latter. Experiment KEPERTBL_RIGID has smaller horizontal wind shear inside of the tangential wind maximum, and the peak vertical velocity is also much smaller in comparison with UNBALBL_RIGID. Boundary layer profiles similar to KEPERTBL_RIGID were also observed in more complex cloud-resolving models [e.g., Fig. 11a of Schecter (2011)]. We can also clearly see weaker tangential winds at the surface, with the consequence that the supergradient wind nearly vanishes there, which is consistent with results found in multilevel models [e.g., Fig. 10 of Bryan and Rotunno (2009)]. The radial surface wind becomes weaker too at most radii, but the reduction is smaller, and the radial surface wind minimum is located more inward. Ooyama found similar results in his unpublished work. In the modified experiment conforming to experiment SBL, he detected in comparison with the balanced model faster intensification, RMGW contraction, a shift of the updraft toward the center, and less expansion of the RMGW after the intensification phase. The smaller surface wind speed considered in KEPERTBL_RIGID can explain the slower intensification compared to UNBALBL_RIGID. Indeed, the surface wind intensification rate might be similar to that observed in the balanced boundary layer simulation REF_RIGID. Therefore, the balanced boundary layer model has, compared to a realistic model, two deficiencies: (i) it overestimates the radius of eyewall convection, and (ii) it overestimates the near-surface wind speed. The experiments show that these two effects have opposing impacts. Profiles of UNBALBL_RIGID and SBL_RIGID are almost identical at the time of maximum intensification (not shown). Consequently, the consideration of the local time tendency of boundary layer momentum only enhances the amplification rate but does not influence the radial wind structure. A plausible candidate for explaining the differences to REF_RIGID is the momentum advection term in Eq. (3). It leads to higher inertia and more inward intrusion of the inflow beyond the RMGW, where supergradient winds and an updraft arise [for more discussion, see Frisius et al. (2013)]. This updraft lying closer to the center provokes further contraction and intensification, as will be shown in the next section. However, the slab boundary model overestimates the overshoot by inertia (Kepert 2010b). The height-parameterized boundary layer model corrects this overestimation to a high degree, but the inertia of the inflow can still be important, since supergradient winds also appear in KEPERTBL_RIGID.

## 4. Analysis of the intensification processes

O69 found conditional instability of the second kind (CISK; Charney and Eliassen 1964) in his model by linearizing the equations with respect to a conditionally unstable atmosphere at rest. Growth due to CISK increases with decreasing perturbation size. This property is known as the “ultraviolet catastrophe” (Montgomery et al. 2006) and leads to the consequence that only horizontal diffusion can prevent shrinking of the perturbation to arbitrary small scales.^{4} However, the numerical results are not inconsistent with such a scenario indeed, since the width of the convective ring is close to the scale of the grid and only remains finite because of the inclusion of horizontal diffusion. The effect of horizontal diffusion on the updraft profile in the intensification phase for experiment UNBALBL_RIGID is displayed in Fig. 7. Obviously, the width and radius increases significantly with increasing horizontal diffusion, while the maximum updraft velocity decreases. For the low-diffusion case (*ν*_{h} = 200 m^{2} s^{−1}), a doubled grid resolution was actually necessary to resolve the updraft properly. This result is qualitatively consistent with CISK, except for the difference that the updraft does not develop at the vortex axis. On the other hand, Eliassen (1971) showed that the updraft forms at a finite radius when a quadratic drag law is used that is more similar to that employed in the Ooyama model than a linear drag law. Therefore, some aspects of intensification could possibly be understood in terms of the CISK theory. This possibility will be checked in the following.

### a. Linear stability analysis

The linear instability analysis has been performed in a similar way to O69. We premise a basic-state atmosphere that is at rest and conditionally unstable. Furthermore, the surface drag coefficient remains finite in the linear model by assuming a gustiness velocity *V*_{g} = 10 m s^{−1} so that

where the prime denotes the linear perturbation. The linearized model also includes latent cooling for besides latent heating for . Then the model may reveal very unrealistic states after a certain time period. However, linear instability models are only valid for a limited duration anyhow. With these assumptions, the linearized equations in the rigid-lid case read as follows:

where . By assuming exponentially growing perturbations of the form , one can derive a single equation in :

where *σ* denotes the growth rate, the horizontal Laplace operator, and the internal Rossby radius. For solving this equation, it is appropriate to assume a radial structure of the form

where is the zeroth-order Bessel function of the first kind, and *k* is the radial wavenumber. Inserting this solution yields the following cubic equation:

where has been introduced in O69 as a nondimensional measure for the horizontal scale of the disturbance. There is only one real root in the unstable case, since all factors in front of the exponential growth rates are positive for reasonable and . Figure 8 shows the resulting growth rate as a function of *ξ* for the three cases REF_RIGID , UNBALBL_RIGID , and SBL_RIGID . In all cases, the largest growth rate results at , and instability arises only for : that is, . The unstable modes have an updraft with the radius in the center, and the growth rate *σ* increases with decreasing updraft diameter. The largest growth rates occur for and (UNBALBL_RIGID). Therefore, the inclusion of the local time tendencies and in UNBALBL_RIGID enhances the instability as in the nonlinear experiments (see Fig. 4a). Equation (63) simplifies drastically in the case and has the following solution:

For , this formula is very similar to that obtained by O69 [see his Eq. (8.10)]. In his formula the additional summand shows up in the denominator of the second factor, but it has a small impact since . For , the growth rate becomes smaller than in the case . Therefore, the violation of the balance assumption in the steady boundary layer has a damping effect in the linearized system. Thus, only nonlinear terms in the boundary layer model can explain the larger intensification rate observed in the nonlinear experiments SBL_RIGID in comparison to REF_RIGID (see Fig. 4a). This outcome does not appear surprising, since the abovementioned radial momentum advection term is not part of the linear model, and it is, therefore, understandable that it cannot explain the unbalanced intensification phase properly. Furthermore, representation of a convective ring in terms of a Fourier–Bessel series does not exhibit the ring contraction when each coefficient grows with the corresponding growth rate (not shown). Consequently, the linear model cannot explain the contraction of the vortex either.

Further simulations have been performed in which the vertically averaged radial momentum advection term has been set to zero in the boundary layer to test if this term plays an important role for the intensification. The experiments called SBLMOD_RIGID and KEPERTBLMOD_RIGID are identical to SBL_RIGID and KEPERTBL_RIGID, respectively, except for this modification. Figure 9 shows the maximum tangential wind and the RMGW for these additional simulations as a function of time. The results for the unmodified runs are also displayed for comparison. Obviously, the development for SBLMOD_RIGID significantly differs from that for SBL_RIGID. Both intensity and intensification rate take much lower values than for SBL_RIGID. Furthermore, the contraction of the RMGW already stops at 37 km compared to the minimum RMGW of 12 km observed in SBL_RIGID. The final intensity is even lower than in REF_RIGID. The omission of the radial momentum advection term has a visibly smaller effect in Kepert’s height-parameterized boundary layer model, as can be seen in Fig. 9b. This brings again into mind that the slab boundary model overestimates the role of radial momentum advection. We also performed simulations in which both nonlinear advection terms and are omitted in the boundary layer and the time tendencies are retained. In the case of a steady slab boundary, these additional modifications have little effect, but neglecting reduces the intensification and contraction by another significant amount in Kepert’s height-parameterized boundary layer model. These results demonstrate the vital role of nonlinear momentum advection for intensification in the Ooyama model. Possibly, nonlinear momentum advection in the boundary layer could also explain some aspects of the finite-amplitude nature of tropical cyclogenesis.^{5}

### b. Evaluation of the middle-layer momentum budget

The source terms , , , , , and [see Eqs. (38)–(43)] cause a radial middle-layer flow, which in turn induces a tendency in tangential wind through angular momentum advection. Therefore, at least one of these source terms should be responsible for intensification, because the resulting radial flow should be inward to produce a local increase of angular momentum. However, vertical transport of boundary layer air could also contribute to a local increase of angular momentum, but it appears rather unlikely that this process solely accounts for intensification. Figure 10 displays the various contributions to the tendency of tangential wind as a function of radius and time for the experiment UNBALBL_RIGID. As expected, latent heat (source term ) release provides the major contribution to intensification and RMGW contraction. During the contraction phase, it has a distinct and narrow maximum inside the RMGW. The small horizontal extension of the response can be explained by the high inertial stability inside the RMGW where the maximum vertical mass flux occurs (see Fig. 5). At the end of the contraction phase, nonnegligible contributions due to frictional convergence (source term ) and upper-layer gradient wind change (source term ) become apparent. The former arises because of downward motion immediately inside of the eyewall leading to inward flow in the middle layer. This descent could be an artifact of the simple slab boundary layer model, since it emerges with a much smaller amplitude in KEPERTBL_RIGID (see Fig. 6). A corresponding upward flux from the boundary layer into the middle layer does not appear, since the entrainment parameter *η* is larger than 1 below the eyewall, and, therefore, the updraft at the top of the boundary layer transports air directly into the upper layer [see Eqs. (11)–(13)]. The upper-layer gradient wind change contributes to the tendency, since an alteration of the radial pressure gradient in the upper layer also modifies the radial pressure gradient in the middle layer, leading to additional inflow. A relevant process for upper-layer gradient wind intensification is given by the vertical momentum flux . However, a large portion of this intensification is compensated by outward advection of angular momentum so that finally a positive contribution remains, which is considerably smaller than that of latent heat release. The impact of upward momentum flux (source term ) and diffusion (source term ) on midlevel inflow is negligible. Nevertheless, both these processes also have a direct contribution to the tangential momentum budget. Diffusion mainly dampens intensification but also supports the contraction of the RMGW. The direct tendency due to upward momentum flux appears to be small compared to latent heat release. All the displayed tendencies add up to the total tendency (shown in Fig. 10h). The pattern of the total tendency resembles that due to latent heat release, but the radial extension of the maximum is somewhat larger, which is mainly a consequence of horizontal diffusion.

Figure 11 shows the middle-layer momentum budget for experiment KEPERTBL_RIGID. In this experiment, the entrainment parameter *η* remains above 1 during the complete simulation. Therefore, no shallow convection and no momentum transport from the boundary layer to the middle layer takes place. Consequently, the tendencies due to upward momentum flux vanish identically. The tendency due to frictional convergence attains much smaller values, presumably because of the smaller vertical velocity at the top of the boundary layer. The tendency due to horizontal diffusion induces, in contrast to UNBALBL_RIGID, a spinup of the tangential wind inside the RMGW but with a small amplitude and after the vortex has already terminated its contraction. The other tendency terms resemble qualitatively those of the experiment UNBALBL_RIGID. However, the tendencies have a significantly smaller amplitude, which explains the slower intensification in KEPERTBL_RIGID.

Figure 12 displays the contributions to the tendencies of for the experiment REF_RIGID. Note that upward momentum flux out of the boundary layer does not change the middle-layer tangential wind because in REF_RIGID it is assumed that . In contrast to UNBALBL_RIGID, the tendency due to latent heat release spreads over a larger radial range. This happens because the updraft has a larger width than in UNBALBL_RIGID and is located outside of the RMGW (see Fig. 5f), where a smaller inertial stability occurs. Furthermore, the maximum tendency due to heating is about a factor of 10 smaller compared to UNBALBL_RIGID. Although the total heating might be similar to UNBALBL_RIGID, the smaller peak value can explain the smaller intensification rate in REF_RIGID, since the radial gradient of latent heating and not the total heating induces the gradient wind increase [cf. Eq. (39)]. Frictional convergence only dampens intensification in REF_RIGID, and its magnitude is very small. The upper-layer gradient wind change contributes with a large amount to the midlevel inflow in the mature stage. The radial flow induced by diffusion also supports intensification in the mature stage, but the resulting tendencies are very small. The direct tendency due to diffusion is negative near the RMGW, and, therefore, it mainly dampens intensification. After maximum contraction, the RMGW migrates outward rapidly. This happens because latent heat release produces a positive tangential tendency outside of the RMGW, while frictional convergence decreases tangential wind inside of the RMGW.

Comparison of Figs. 10 and 11 with Fig. 12 manifests large qualitative and quantitative differences in the intensification dynamics. In UNBALBL_RIGID and KEPERTBL_RIGID the tendencies are mostly concentrated in a narrow ring inside of the RMGW, while in REF_RIGID the tendency profiles are smoother and sometimes are maximized outside the RMGW. The reason for the different behavior of REF_RIGID is the neglect of gradient wind imbalance in the boundary layer model. The consequence is that the maximum ascent out of the boundary layer occurs outside of the RMGW with a smooth radial profile (see Fig. 6a). This leads via latent heat release to a gradient wind intensification with little contraction and whose profile is also smooth.

To substantiate this conclusion further, steady-state solutions of the unbalanced slab boundary layer model and of Kepert’s height-parameterized boundary layer model have been calculated using the gradient wind field of REF_RIGID at *t* = 99 h. The results are shown in Fig. 13, which should be compared with Fig. 6a. As expected, the boundary layer wind profiles differ greatly to those of REF_RIGID. For the slab boundary layer model, the vertical wind maximum arises far inside of the RMGW, where strong supergradient winds are found. The radial inflow also maximizes inside of the RMGW and takes smaller values compared to the balanced boundary layer calculation in REF_RIGID. The steady-state solution of Kepert’s height-parameterized boundary layer model also reveals an inward shift of the vertical wind maximum and a decrease of the radial inflow velocity. However, the maximum vertical velocity and the supergradient wind appear to be much smaller than in the slab boundary layer model. Such differences have already been noted and are consistent with the findings of section 3. An inward shift of the frictional updraft location was also found by Kepert (2013) with a height-resolving boundary layer model when compared with a linearized one. Furthermore, he demonstrated that the linearized version yields vertical wind profiles that resemble those found with the balanced slab boundary layer model of O69. Running the model by using the gradient wind of REF_RIGID at *t* = 99 h as initial condition for would lead to immediate contraction and rapid intensification. These results support our conclusion on the importance of the nonlinear advection terms and show that it is of importance for intensification whether the convective heating occurs inside or outside of the RMGW.

## 5. Conclusions

In this study we have investigated the impact of the balance assumption for the intensification of a tropical cyclone in Ooyama’s three-layer model. Furthermore, the effects of a rigid lid, the neglect of local time derivatives in the boundary layer momentum equations, and the inclusion of Kepert’s height-parameterized boundary layer model have also been examined. We found that the balance approximation in the two upper layers and the rigid-lid assumption are of minor importance for the intensification phase, while the balance approximation in the boundary layer has a significant impact. With a balanced boundary layer (as given in experiment REF_RIGID), the tropical cyclone intensifies much more slowly than in the simulation employing an unbalanced boundary layer (as given in experiment UNBALBL_RIGID). Furthermore, the vortex contracts only slightly during intensification, and the RMGW increases dramatically after the intensity has maximized in REF_RIGID. In contrast, the simulation UNBALBL_RIGID reveals a substantial vortex contraction with little RMGW variation after the intensification stage. In both experiments intensification is accompanied by the occurrence of a convective ring in the vicinity of the RMGW. However, in REF_RIGID the convective ring is located outside and in UNBALBL_RIGID inside the RMGW. We suggest that this discrepancy is crucial for the different intensification rates. Because of the higher inertial stability, latent heating inside the RMGW leads to a larger and tighter tangential wind tendency than in the case with heating outside the RMGW. This can explain the faster growth and contraction in UNBALBL_RIGID compared to REF_RIGID. An enhancement of intensification with increasing inertial stability was already found by Schubert and Hack (1982) in an idealized analytical solution of the Sawyer–Eliassen equation. This result hints at the importance of nonlinearity in tropical cyclone intensification, and the mechanism was further elaborated by Hack and Schubert (1986) on the basis of nonlinear primitive equation and balanced models. However, they assumed a time-independent heat source, and, therefore, they excluded a possible feedback of inertial stability on heating. This could stem from a modification of the boundary layer inflow and the associated vertical mass fluxes. The results based on Ooyama’s three-layer model show that the increasing inflow in the unbalanced boundary layer supplies the eyewall with enough moist air so that heating is maintained or even increased at larger inertial stability. Eventually, the intensification stops because the frictional dissipation rate increases at a much faster rate than the energy input rate from the ocean (Wang 2012). The simulations with neglected time derivatives (as given in experiment SBL_RIGID) and with Kepert’s height-parameterized boundary layer model (as given in experiment KEPERTBL_RIGID) exhibit vortex contraction like in UNBALBL_RIGID, but the intensification rates are smaller. Experiment KEPERTBL_RIGID has more realistic wind profiles, and the smaller intensification rate in this experiment partially results because the surface flux parameterization includes surface winds instead of vertically averaged winds, as in the slab boundary layer model. Furthermore, the radial overshoot is also reduced in the height-parameterized boundary layer model (see Fig. 13) so that convection forms in a less inertially stable environment, where a diminished intensification rate results.

The linear instability analysis of the governing equations does not reveal an enhancement of the intensification rate by the relaxation of the balance assumption in the boundary layer, and it cannot explain the contraction of the intensifying vortex either. Therefore, nonlinear terms appear to enhance intensification in the Ooyama model as a result of gradient wind imbalance. The radial advection of radial momentum likely supports the intrusion of boundary layer air inside of the RMGW, where the eyewall develops. A simulation without this term in the slab boundary layer model exhibits a much slower development with little contraction. The reason for faster intensification and contraction in UNBALBL_RIGID compared to REF_RIGID has been found by analyzing the various terms in the tangential wind equation of the middle model layer. In both experiments latent heat release is the main driver for intensification. Gradient wind intensification by upward momentum flux from the middle layer to the upper layer also contributes to intensification in the middle layer in the later stages of the development. The frictionally induced downdraft inside the eyewall yields another contribution in UNBALBL_RIGID. However, the crucial difference between both experiments is the radial scale of the wind tendency. In UNBALBL_RIGID the radial extent is much smaller than in REF_RIGID because the eyewall emerges in the inertially stable region inside of the RMGW, while in REF_RIGID it arises outside of the RMGW, where weak inertial stability dominates. Although the intensification rate is smaller, experiment KEPERTBL_RIGID reveals an intensification mechanism resembling that of UNBALBL_RIGID. Therefore, a more realistic representation of boundary layer dynamics still supports the finding that gradient wind imbalance in the boundary layer is of importance for the intensification scenario and vortex contraction.

These results suggest the following intensification mechanism in Ooyama’s three-layer model. First, an incipient vortex generates inflow, and the inflow overshoots the RMGW as a result of its inertia. The overshoot causes an updraft inside the RMGW, which would occur outside the RMGW in the balanced and the linearized boundary layer model [as found by Kepert (2013)]. Then moist convective instability released by Ekman pumping generates a convective ring, and the entrainment of air above the boundary layer intensifies the tangential wind because of angular momentum import in the middle layer. Finally, the resulting increase of gradient wind enhances the inflow with more inward intrusion, which causes an inward migration of the eyewall and further intensification. Fluxes of latent heat from the sea surface are, of course, necessary in this feedback loop to maintain the convective instability near the eyewall radius. This picture has some similarity with the CISK theory, since there is also a cooperation of a large-scale vortex with convection, and the latent heating depends on frictional convergence in the boundary layer. However, the CISK theory refers to an ensemble of individual convection cells, but in the present model convection only appears in the form of a single convective ring. Obviously, convection is in the nonlinear Ooyama model only triggered at the position where maximum frictional convergence appears. Furthermore, the linear CISK model by Charney and Eliassen (1964) predicts, like the linearized Ooyama model, a maximum heating at the vortex center, while in the nonlinear Ooyama model it appears at a finite radius where the contracting convective ring is located. This difference is in agreement with the finding by Eliassen (1971) that a quadratic drag law yields a boundary layer updraft maximum at a finite radius. Ooyama (1982) already appreciated the role of nonlinearities in his cooperation intensification theory, but he did not consider gradient wind imbalance in the boundary layer, although he found previously in his unpublished study that it enhances intensification. The frictionally induced intensification mechanism is also consistent with the study of Wang and Xu (2010), who found in a cloud-resolving model that inward boundary layer enthalpy transport is important to the energy balance in the eyewall.

The present study suggests that gradient wind imbalance must be taken into account for a proper understanding of the intensification process. However, one may ask why some balanced models already provide a reasonable picture for intensification and vortex contraction. For example, Emanuel (1989) developed a simple balanced hurricane model that reveals a behavior as in the unbalanced case discussed here. On the other hand, Emanuel (1989) applied a convective parameterization scheme in which the convective mass flux depends directly on buoyancy and is independent of moisture convergence in the boundary layer. Therefore, the placement of the developing eyewall follows other rules in his model than in the Ooyama model. A possible drawback of the Ooyama model is the nonexistence of ordinary conditional instability like that found by Lilly (1960). Convective instability can take place without frictional convergence in the boundary layer only if *η* becomes unrealistically large. This is unlike the situation in the vertically continuous atmosphere, where both kinds of conditional instability occur simultaneously, if at all (Fraedrich and McBride 1995). Possibly, convective ring formation can result from ordinary conditional instability in a more realistic model. This could potentially explain why intensification and contraction arise without surface drag in one of the axisymmetric model experiments by Craig and Gray (1996). Such an outcome does not invalidate the importance of unbalanced dynamics, since ordinary convection cannot evolve in a balanced model. Nevertheless, to reveal the relevance of the results of the present study and the possible limitations of the Ooyama model, it is necessary to investigate further the intensification mechanism in more complex models.

## Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the following individual research grant: The role of convective available potential energy for tropical cyclone intensification FR 1678/2-1. The first author also gratefully acknowledges support by the Cluster of Excellence CliSAP (EXC177) funded by the DFG. We thank M. T. Montgomery for the hint that K. V. Ooyama already indicated the importance of unbalanced boundary layer dynamics for intensification in an unpublished manuscript that was kindly provided to us. Furthermore, we thank three anonymous reviewers for their valuable comments.

## REFERENCES

*27th Conf. on Hurricanes and Tropical Meteorology*, Monterey, CA, Amer. Meteor. Soc., 14B.7. [Available online at https://ams.confex.com/ams/27Hurricanes/webprogram/Paper108695.html.]

## Footnotes

^{1}

“Numerical Simulation of Tropical Cyclones with an Axi-symmetric Model.”

^{2}

Strictly speaking, represents mass fluxes divided by reference density .

^{3}

The discretization of the prognostic equation for boundary layer equivalent potential temperature was not described in O69.

^{4}

It has to be noted that Charney and Eliassen (1964) found in their CISK model rather uniform growth rates for disturbance sizes smaller than about 100 km. Therefore, arbitrarily small disturbances grow only slightly faster than those having a scale of about 100 km.

^{5}

Substantial intensification within 10 days occurs in UNBALBL_RIGID only when the initial tangential wind maximum is above 3 m s^{−1} (not shown).