## 1. Introduction

Strong tropical cyclones (TCs) often experience secondary eyewall formation (SEF; Willoughby et al. 1982), in which a new eyewall forms outside the original eyewall. It is usually followed by an eyewall replacement cycle, and hence, predicting SEF is important for accurate TC forecasts. However, there is no widely accepted theory for SEF despite the significant effort by many researches, especially in the last decade. We here present a new theory focusing on flow interaction between the boundary layer and the free atmosphere.

SEF often occurs at a radius of 2–4 times the radius of maximum wind (RMW) in recently intensified TCs (Willoughby et al. 1982; Black and Willoughby 1992; Dodge et al. 1999; Houze et al. 2007; Kossin and Sitkowski 2009; Kuo et al. 2009; Wu et al. 2016). The key parameters and processes for SEF proposed by previous studies are inertial instability (Willoughby et al. 1982), terrain effect (Hawkins 1983), ice microphysics (Willoughby et al. 1984), inflow surge (Molinari and Skubis 1985; Molinari and Vollaro 1989), vortex Rossby waves (Montgomery and Kallenbach 1997; Terwey and Montgomery 2003), potential-vorticity intrusion from upper layer (Nong and Emanuel 2003), axisymmetrization of vorticity disturbance (Kuo et al. 2004, 2008), the rectification of eddies in the *β* skirt outside the primary eyewall (Terwey and Montgomery 2008), supergradient wind (Huang et al. 2012; Wu et al. 2012), wind field expansion (Rozoff et al. 2012; Sun et al. 2013), radial vorticity gradient outside the RMW where relative vorticity is low (Kepert 2013), top-down process associated with interaction with upper-level trough (Leroux et al. 2013), rainband heating (Zhu and Zhu 2014; Zhang et al. 2016), and upper-level outflow interactions (Dai et al. 2017).

Kepert (2013) showed the importance of the radial gradient of relative vorticity in a low-vorticity environment using both linear and nonlinear boundary layer models. According to the Ekman theory for a rotating fluid, Ekman pumping velocity is proportional to the radial vorticity gradient and inversely proportional to the square of the absolute vorticity itself. Kepert (2013) and Kepert and Nolan (2014) proposed a positive feedback that can lead to SEF: when the radial gradient of vertical relative vorticity is steep, Ekman pumping velocity is large, the upward velocity is enhanced by convection and causes further convergence, and the vorticity gradient steepens. However, the previous studies consider vorticity profiles *after* SEF and secondary wind maximum, in which a secondary peak of tangential velocity is already present outside the RMW. From ensembles of numerical simulations, Zhang et al. (2016) showed that the diabatic heating outside the RMW in the upper layer proceeds downward, changing the radial vorticity profile, and the secondary eyewall develops through the interaction between convection and boundary layer processes. Although the interaction might work once the vorticity gradient is steep, the proposed mechanism is incomplete because it offers no explanation of conditions under which outer rainbands produce such a strong vorticity gradient (otherwise, rainbands would always cause SEF). Therefore, the underlying mechanism of SEF, especially for the initial process (i.e., when and where SEF occurs), is unclear.

This study proposes a linear instability via an interaction between Ekman pumping and the flow in the free atmosphere, which is not associated with moist convection, unlike the previous theories, and hypothesizes that very early processes of SEF are attributed to this instability. Once the initial processes of SEF proposed in this study create a local maximum of vertical velocity or tangential velocity outside the primary eyewall, the heating–convergence feedback might work to develop the secondary eyewall, followed by an eyewall replacement cycle (Kepert 2013; Kepert and Nolan 2014).

The rest of the paper is organized as follows. We develop a linear theory for a 1.5-layer shallow-water system in Cartesian coordinates and eigenvalue analyses and numerical integrations for a nonlinear equation set are conducted in section 2. The theory is extended to cylindrical coordinates and a condition for the instability to occur is derived and linear stability analyses of an axisymmetric 1.5-layer shallow-water system are performed in section 3. The unstable condition is tested by applying it to simulation results by two nonhydrostatic full-physics models in section 4. The results are discussed by comparing them with previous studies in section 5. This study is concluded in section 6.

## 2. Extension of the Ekman theory in Cartesian coordinates to high–wind speed conditions

### a. Classic Ekman theory

^{1}This can be explained by considering a one-dimensional 1.5-layer shallow-water system on an

*f*plane with a semislip lower-boundary condition: a free atmosphere plus a boundary layer (cf. Fig. 1, but velocity in the free atmosphere is weak and its horizontal variations are slow). We can obtain an analytical solution for the evolution of perturbations of relative vorticity, and the solution indicates that the vorticity perturbation decays exponentially. A key relationship in the classic Ekman theory is that perturbation vertical velocity at the top of boundary layer

*y*-velocity perturbation

The Ekman damping process for velocity perturbations can be interpreted as follows: once a positive *x*-velocity perturbation *x* and negative *x*, which decelerates and accelerates, respectively, *y*-velocity perturbation *y* velocity

The classic Ekman theory is widely accepted and has provided many insights into geophysical fluid dynamics. For example, the effects of Ekman pumping on Rossby waves are modeled by adding vertical velocity at the top of the boundary layer that is proportional to vertical relative vorticity (e.g., James 1995; Vallis 2017). Although the theory can capture fundamental processes for various phenomena, the assumptions may limit its application. In particular, it is assumed that the background flow is not fast and its spatial gradient (relative vorticity) is not large. Hence, the theory may not be applicable to some circumstances such as jets or tropical cyclones in which the horizontal velocity is fast in the lower free atmosphere.

### b. Extension of Ekman theory to fast velocity conditions

We here hypothesize that the relationship in (1) can possibly be reversed when we include the advection in the boundary layer. If this were to occur, *positive*

*f*plane [(A7)–(A11)] are linearized. The basic state satisfies the geostrophic balance in the free atmosphere. In addition, we here assume that the basic-state

*y*velocity

*x*velocity

*y*direction [(A11)] as

*y*-velocity [

*f*, which results in

A series of numerical calculations for the equations for the free atmosphere [(3)–(5)] and vertical velocity [(8)] have been performed to test the hypothesized instability by artificially setting the coefficients

### c. The condition for instability

*x*than the peak velocity) where the velocity gradient is negative and the second-order derivative is also negative up to the inflection point. The unstable condition (12) can be rewritten as

### d. Eigenvalue analysis

*x*and time

*t*is decomposed as

*σ*is the growth rate. When Re

The eigenvalue equation [(16)] was discretized on a staggered grid system and eigenvalues and eigenvectors were computed with MATLAB. As in the test calculations in appendix C, the spatial derivatives in (15) were discretized by the fourth-order central difference scheme, and periodic boundary conditions were used. We have tested the second-order difference scheme and found that the results are insensitive to the discretization scheme. The nondimensional domain width was 8 with 201 grid points and uniform grid spacing of 0.04. The initial velocity field was a jetlike profile *L* is a length scale. The velocity was computed from *x* = 0 to 4, and then the value at *x* = 0 to 4 with the same peak magnitude and width but opposite sign was added from *x* = to 8; that is,

Figures 3a and 3b show spatial profiles of *V* = 20 m s^{−1}, *x* = 1.52 and 2.48. *x* up to the maximum at *x* = 1.88 and 2.36. As discussed in the previous section, the condition is more likely satisfied to the right of the peak of the jet where both the first-order and second-order derivatives of the basic-state velocity are negative. The range satisfying the unstable condition is highlighted in Fig. 3.

The eigenmodes with the maximum eigenvalue are shown in Figs. 3c and 3d. In this case, the real part of the maximum eigenvalue (i.e., growth rate) is 18.73 and corresponds to an *e*-folding time of 1.33 × 10^{4} s (≈3.7 h). The vertical velocity

We conducted a series of eigenvalue analyses for wide ranges of Ro and the spatial parameter

List of parameters in the eigenvalue analysis for Cartesian coordinates.

Figure 4 displays a phase diagram in space of two parameters: Ro and the domain maximum of

### e. Numerical integrations of nonlinear equation set

*x*-direction in the free atmosphere (A8) to allow ageostrophic components. The dimensionless form of

*u*equation used here is

*y*-momentum equations in the boundary layer [(A10) and (A11)] were combined into an equation for

*w*.

List of parameters that are constant in this study.

The equations were solved numerically using a finite difference method. As in the eigenvalue analysis, the quantities were allocated on a staggered grid system. The time and space derivatives in the equations were discretized by the third-order Runge–Kutta method and by the fourth-order central difference scheme, respectively. At the first time step, *w* was estimated from *υ*, and then the equations for the free atmosphere were stepped forward. The integration was conducted up to *υ* used in the eigenvalue analysis (Fig. 3a) was also applied to the initial field, which was balanced with the fluid depth. The experimental parameters and their range were also identical to the eigenvalue analysis.

Figure 5 shows spatial distributions of velocities at *t* = 0.02 in the control case, in which Ro = 0.80 and *x* = 2.40, which is the same location as the eigenvalue analysis (cf. Fig. 3c) and a negative peak near the positive one. There is a sharp peak in the *u* distribution around the positive peak of *w*, whereas *u* is negative at the both sides of the peak. The negative *u* transports absolute vertical vorticity around the peak of *υ* where the vorticity is not weak compared with the negative *u* region outside of the peak. As a result, *υ* is amplified around the peak of initial distribution. The velocity distributions in the nonlinear integration are qualitatively consistent with the eigenvalue analysis and the present hypothesis.

Figure 6a displays a space–time cross section of *υ* in the control case and time series of domain-maximum *w*. The maximum *w* amplifies with time. The maximum *w* keeps growing until the end of the simulation, and the locations of velocity peaks seen in Fig. 5 are nearly fixed in space. The spatial profiles of velocity during the integration (Fig. 5) strongly suggest that the proposed feedback works to amplify *w*. Since no explicit diffusion is incorporated into the discretized equation set, the linearly unstable modes continue to grow.

Figure 7 depicts a phase diagram in Ro and the domain-maximum *w* amplifies and exceeds 5 m s^{−1} are observed in the unstable regime derived from the linear theory, Ro*υ* decays after the initiation of integration (× marks in the figure). The theoretically derived unstable regime also includes a regime in which the domain-minimum absolute vertical vorticity is negative at the initial time. These features are quite similar to those obtained in the eigenvalue analyses (cf. Fig. 4). It should be noted that unlike the eigenvalue analyses, *w* does not amplify in several cases close to the neutral curve. In these cases, the maximum *w* decays rather than amplifies. The failure of unstable modes to grow when the parameters are close to the neutral curve may be due to nonlinear terms.

## 3. Theory in cylindrical coordinates

*υ*does not vary in space,

### Eigenvalue analysis in a cylindrical coordinate

*σ*is the growth rate, into (35) yields the following eigenvalue equation:

^{−5}s

^{−1}. A radial profile of dimensional tangential velocity

The radial dimension was discretized on a staggered grid system with uniform grid spacing of 0.04. The domain size was 80.0 (i.e., 80 times the RMW) with 2001 grid points. The radial derivatives were discretized by a fourth-order central difference scheme. At the boundaries (*r* = 0 and 80),

Figures 8a and 8b show radial profiles of basic-state *r* = 4.0. However, this jump is more than 10 times smaller than that in Kepert (2013). Hence, unlike most of the cases in Kepert (2013), *r* = 4.0. The ratio of angular velocity to absolute vorticity *r* = 4.5. *r* = 4.0, where the radial gradient of

The eigenmodes of *r* > 1.5 × RMW. The *e*-folding time is *r* = 4.0 where *r* = 4.0, which is opposite to

We examine the sensitivity of the solution to Ro and vortex parameters *υ*_{m} = 25.0 and 109.4 m s^{−1}, provided that *r*_{m} = 25 km and *α* is changed from 0.1 to 1.0.

List of parameters in the eigenvalue analysis for cylindrical coordinates.

Figure 9 shows the growth rate as a function of experimental parameters. The growth rate increases with Ro, *r* = 2.5 and 7.0: the disturbance cannot grow if it is too close to the RMW or too far from the RMW. The growth rate is large with large

*σ*, the figure indicates

## 4. SEF simulated in full-physics models

### a. Axisymmetric model

A long-term simulation was performed using an axisymmetric version of Cloud Model 1 (CM1; version 17; Bryan and Fritsch 2002), which solves fully compressible, nonhydrostatic equations. The experimental setting was basically the same as Hakim (2013) in which a number of eyewall replacement cycle events were simulated during his long-term simulation for 500 days. The differences from Hakim’s (2013) simulation are listed below. The initial mass, temperature, and water vapor mixing ratio was obtained from Jordan’s (1958) tropical mean sounding for hurricane season. The Coriolis parameter was 5 × 10^{−5} s^{−1}, and the sea surface temperature was 301.15 K, both of which were fixed during integration. The radiative effects were incorporated by the NASA Goddard scheme, which was called every 5 min. The integration period was 60 days (=1440 h) with an output interval of 3 h for prognostic quantities. The grid spacing for the radial direction was 2 km, whereas the spacing for the vertical direction was stretched from the bottom to top. The minimum grid spacing for the vertical direction was 0.25 km at the lowest model layer, stretching to 1.0 km near the top.

Figurea 11a and 11b show time series of TC intensity defined as the maximum tangential velocity at *z* = 2 km and its radius. The TC achieves maximum intensity around *t* = 340 h after an intensification phase, decreases to about 60 m s^{−1}, and then is approximately steady afterward, which is qualitatively consistent with the long-term simulations in previous studies (Hakim 2011, 2013). We shall call the period after *t* = 500 h a quasi-steady state. During the quasi-steady state, rapid changes in intensity and RMW are observed several times. As pointed out by Hakim (2011), the intensity oscillation appears to be due to the eyewall replacement cycle associated with SEF.

Figure 11c depicts a radius–height cross section of tangential and radial velocities and mixing ratio of hydrometeors averaged during the quasi-steady state. Both the velocity and cloud fields are consistent with the observational studies (e.g., Hawkins and Imbembo 1976): the tangential velocity has a peak at the top of the boundary layer (~1.5 km) and decays in both the radial and vertical directions, the inflow and outflow regions appear at the surface and around the 15-km level, and the hydrometeor mixing ratio is large around the RMW from the 1.5- to 15-km altitudes.

Figure 11d depicts a radius–time cross section of tangential velocity at *z* = 2 km. The RMW is about 60 km during the quasi-steady state as seen in Fig. 11b. Interestingly, a strong velocity region periodically forms about every 80 h, and the region proceeds inward after formation. The mixing ratio of hydrometeors and vorticity fields shows similar behavior to the tangential velocity (not shown). It is suggested that this cyclic process is the eyewall replacement cycle and formation of strong tangential velocity outside the RMW is SEF.

The SEF events were extracted from the simulation by applying the following conditions: the RMW is greater than that at the previous and next output time, the increase from the previous step is greater than 10 km, and the new RMW is more than 20 km beyond its average value during the quasi-steady state (~60 km in this case). Applying the conditions, six SEF events were detected in the quasi-steady period, and the six events were combined to make composites.

Figure 12a displays a radius–time cross section of tangential velocity at *z* = 2 km subtracted from the temporal average from *z* = 2 km at each output time after taking 1–2–1 average for *r* = 163 km, which is about 100 km outside the RMW, 20 h before the time of SEF. The peak amplifies as it propagates inward. The

Figure 12b shows the time series of *r* = 135 to 160 km, where the secondary peak of tangential velocity forms;

### b. Three-dimensional model

We conducted a numerical simulation using a three-dimensional full-physics numerical model, Weather Research and Forecasting (WRF) Model, version 3.4.1 (Skamarock et al. 2008). The experimental setting is described in appendix E.

Figures 13a and 13b show time series of TC intensity and RMW of the simulated TC. The TC experiences two different intensification phases and becomes quasi steady after *t* = 45 h. The intensity weakens from *t* = 65 to 76 h and reintensifies afterward. The RMW gradually decreases, jumps outward at *t* = 76 h, and then decreases again. Structural evolution of the simulated TC (shown below) indicates that the sudden changes in TC intensity and RMW are associated with SEF and the eyewall replacement process. We define *t* = 76 h as the time of SEF.

Figure 13c depicts a radius–height cross section of the azimuthally averaged hydrometeor mixing ratio, tangential velocity, and radial velocity, which are temporally averaged from *t* = 48 to 54 h. Figure 13d depicts a radius–time cross section of azimuthally averaged tangential velocity at *z* = 2 km and diabatic heating rate that is averaged vertically from *z* = 2 to 10 km. The tangential velocity intensifies with decreasing RMW to *t* = 45 h, and then the RMW is approximately constant. A notable point is that the tangential velocity field expands with time after the TC achieves the maximum intensity. The temporal changes in structure and intensify during the SEF event are consistent with observations.

Figure 14a displays a radius–time cross section of the deviation of tangential velocity from the average between *t* = 48 and 96 h, diabatic heating averaged vertically from *z* = 1 to 10 km, and *r* = 125 km from *t* = −17 h (*t* = 59 h of simulation time). The secondary peak intensifies while proceeding inward, which is accompanied with strong diabatic heating. The rapid inward motion of the secondary peak is consistent with Kepert (2017). It suggests that the convection–convergence feedback plays a role in intensifying the secondary peak of tangential velocity. The time of secondary-peak formation is approximately the same as the increase in diabatic heating in the outer region. The point is that

Figure 14b shows a times series of *r* = 115 and 135 km where the secondary peak forms;

## 5. Discussion

The theory indicates that a small gradient of vorticity can produce large

The initial amplification of the radial gradient of vorticity to satisfy the unstable condition might be caused by some processes such as diabatic heating in rainbands (Rozoff et al. 2012; Zhang et al. 2016). The present 3D simulation shows little convective organization before the secondary peak of tangential velocity forms (Fig. 14a). Rather, many isolated convective cells are present, but the accumulated effect of these can produce the needed vorticity gradient. Once a secondary peak of tangential velocity forms, a symmetric convective region is organized around that radius. Subsequently, the secondary peak and organized convection may amplify together through the feedback.

Figure 15 illustrates how the proposed feedback mechanism works, with a comparison to the classic Ekman feedback (and damping). Starting from perturbation of vertical velocity with a positive peak, the radial and tangential velocities in the free atmosphere will be produced (cf. Fig. 2). This change will result in faster velocity in the *x*-direction in the boundary layer to the left of the peak of the initial perturbation of vertical velocity and slower velocity to the right. Thus, the responding flow in the boundary layer tends to cause divergence and decrease *w*. On the other hand, when the unstable condition is satisfied, the change in flow fields in the free atmosphere rather produces slower velocity to the left and faster to the right, which increases in *w*. This would happen if the decrease in

## 6. Conclusions

We demonstrate an instability via interaction between the free atmosphere and the Ekman layer based on the classic Ekman pumping theory and hypothesize that the very early process of SEF in TCs can be attributed to this instability. The classic Ekman theory indicates that vertical motion at the top of the boundary layer is proportional to the spatial gradient of velocity (relative vorticity), and this results in a decay of velocity perturbations in the free atmosphere. Here, it was hypothesized that once the Ekman pumping velocity is *negatively* proportional to the velocity gradient under some special conditions, disturbances can grow exponentially.

The instability happens when velocity is fast and changes rapidly in the horizontal direction. Using a 1.5-layer shallow-water model on the *f* plane [(3)–(7)], we analytically derived an unstable condition (13), in which the proportionality coefficient of relative vorticity perturbation in the

Eigenvalue analyses were performed for the linearized equation set. A positive eigenvalue (growth rate) was obtained, and the eigenvector for the maximum eigenvalue showed a velocity field consistent with the hypothesis. In the parameter space of Ro and

We extended the theory to cylindrical coordinates. By linearizing the nondimensionalized equations, it was shown that the instability occurs where Ro

The hypothesis was tested by computing the dimensional values of

Another SEF was simulated using WRF in an idealized environment with nonzero shear. A secondary peak of tangential velocity formed around 5.0–5.5 times the RMW and intensified while proceeding inward, while accompanied by strong convective heating. The local value of

The two modeling results suggest that the initial process of SEF, the formation of a secondary peak of a tangential velocity, may be attributed to the instability, and thus

## Acknowledgments

Y. Miyamoto was supported by JSPS Scientific Research 26–358 for the JSPS fellowship program for overseas researchers. D. S. Nolan was supported by the NSF through Grant AGS-1654831. This work was supported by Keio University Academic Development Funds for Joint Research. The authors thank Dr. Jeff Kepert and two anonymous reviewers for their careful reviews and Drs. Hiroshi Taniguchi and Shigenori Otsuka for fruitful discussions.

## APPENDIX A

### Equations for the One-Dimensional 1.5-Layer Shallow-Water System in Cartesian Coordinates

*f*plane (cf. Fig. 1) considered in the main text. For the

*x*direction that is orthogonal to the direction with fast flow speed, fluid depth and velocity are in geostrophic wind balance in both the free atmosphere and the boundary layer. A bulk drag formula is used to represent the surface friction. No internal diffusion is included in either layers. The mass and momentum conservation equations for the free atmosphere are

*x*and

*y*directions,

*b*represents the boundary layer. We have assumed that the

*y*-velocity and horizontal gradient of fluid depth in the boundary layer are equal to those in the free atmosphere.

*L*,

*H*,

*T*, and

*V*are the representative scales of length, depth, time, and velocity, respectively. The dimensionless form of the equations is

*w*′:

## APPENDIX B

### Condition for the Instability in Cartesian Coordinates

## APPENDIX C

### Test Calculations for the Instability

To verify the hypothesized sensitivity of solutions to the coefficient

The initial perturbation horizontal velocities were zero everywhere at the initial time. White noise perturbations with amplitude of 1.0 were added to the vertical velocity. Given

The present hypothesis indicates that an instability occurs when

## APPENDIX D

### Equations for the Rapidly Rotating, Axisymmetric, 1.5-Layer Shallow-Water System

*b*represents the boundary layer, and the asterisk indicates dimensional quantities. Equation (D5) indicates the balance between frictional destruction of absolute angular momentum and radial advection. This assumption has been applied by previous studies (Ooyama 1969). The equations are normalized as

*T*are the scales for radius, the depth of free atmosphere, the boundary layer depth, and time, respectively.

## APPENDIX E

### Details of the WRF Simulation

The three-dimensional, full-physics simulation used in section 4b was produced using the WRF Model, version 3.4.1. The simulation depicts the early development and then rapid intensification of a tropical cyclone in a large, zonally periodic channel using the idealized modeling framework of Nolan (2011), with wind shear balanced by a meridional temperature gradient. However, in many other aspects, the simulation is modeled after the hurricane nature run simulations described by Nolan et al. (2013) and Nolan and Mattocks (2014). The simulation was originally intended to be one of a larger set of idealized simulations used by Koltz and Nolan (2018, manuscript submitted to *Wea. Forecasting*) for observing system tests, but its robust SEF and ERC made it useful for this paper.

The outer domain has 240 × 180 grid points in the zonal and meridional directions with 27-km grid spacing. Three nested, vortex-following grids are used with 9-, 3-, and 1-km grid spacings, with 180 × 180, 360 × 360, and 480 × 480 grid points, respectively. There are 60 vertical levels between the surface and 20-km altitude. The grid spacings in the WRF vertical coordinate are the same as used in the nature runs (see Fig. 2 of Nolan et al. 2013), and the physical parameterizations are also all identical, including the use of the one-dimensional mixed-layer cooling model of Pollard et al. (1972) available in WRF, version 3.4.1.

The initial vortex and background flow are very similar to the simulations described in Nolan (2011) and Nolan and McGauley (2012). The outer domain is initialized with mean easterly flow of 5 m s^{−1} at the surface, which smoothly increases to 5 m s^{−1} of westerly flow between 850 and 20 hPa. An axisymmetric vortex, balanced by temperature and pressure anomalies, is introduced into the eastern end of the domain. The symmetric initial vortex has a maximum tangential wind of 15 m s^{−1} at *r* = 135 km and *z* = 1.5 km. The radial profile of the tangential wind varies as a modified Rankine vortex with decay parameter *a* = 1/3 and decreases with height using the same analytical function as in (4.2) of Stern and Nolan (2011).

The Coriolis parameter is constant across the channel domain with value

Finally, as the simulation proceeds, the wind, temperature, and moisture fields on the outer domain are relaxed back to their initial values with a 24-h relaxation time scale. This relaxation does not apply on the nested grids. The relaxation keeps the environmental sounding and wind shear profile around the tropical cyclone roughly constant as the storm develops.

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^{1}

We here consider the atmosphere in which the interior layer is located above the boundary layer. The basic concept of the present theory can be applied to the ocean and other flows.