This study proposes that secondary eyewall formation (SEF) of tropical cyclones (TCs) can be attributed to an instability of flow in the free atmosphere coupled with Ekman pumping. Unstable solutions of a 1.5-layer shallow-water system are obtained under fast-wind speed conditions in the free atmosphere. The instability condition derived in the linear model indicates the importance of the ratio of angular velocity to vorticity, and the condition is more likely to be satisfied when the ratio is large and its radial gradient is positive. Thus, fast angular velocity, low absolute vertical vorticity, small negative radial gradient of angular velocity, and large negative gradient of vertical vorticity are favorable. Eigenvalue analyses are performed over a wide range of parameters using a vorticity profile with an infinitesimal secondary maximum. The growth rate increases with vorticity outside the radius of maximum wind (RMW), the radius of the secondary vorticity maximum, its magnitude, and the Rossby number defined by maximum tangential velocity, the RMW, and the Coriolis parameter. Furthermore, the growth rate is positive only between 2 and 7 times the RMW, and it is negative close to or far outside the RMW. These features are consistent with previous studies on SEF. A dimensionless quantity γ obtained from the unstable condition in the linear theory is applied to SEF events simulated by two different full-physics numerical models;γ increases several hours before a secondary peak of tangential velocity forms, suggesting that the initial process of SEF can be attributed to the proposed mechanism.
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