Many studies in the last decade have proposed mechanisms for SEF (e.g., Montgomery and Kallenbach 1997, Nong and Emanuel 2003, Kuo et al. 2004, 2008, Terwey and Montgomery 2008, Huang et al. 2012, Rozoff et al. 2012, Kepert 2013, Abarca and Montgomery 2013, Ge et al. 2016, Dai et al. 2017). Rozoff et al. (2012) and Sun et al. (2013) demonstrated the importance of the radial expansion of the tangential velocity field before SEF. Kepert (2013) argued the importance of the radial gradient of the vertical component of relative vorticity using a non-linear boundary-layer model. Kepert (2013) proposed a positive feedback between convection in the secondary eyewall and convergence in the boundary layer. However, this and other previous studies consider vorticity profiles after SEF begins, in which a secondary peak of tangential velocity is already present outside the RMW.
We propose that the initial process of SEF can be attributed to a linear instability via the interaction between Ekman pumping and flow in the free atmosphere. The Ekman pumping process can amplify the vertical velocity under some special conditions, which normally damps flow disturbances by diminishing vertical velocity. In particular, the Ekman velocity is proportional to relative vorticity in the classic Ekman theory leading to decay of the free-atmosphere vorticity. However, it will be negatively proportional to vorticity when the wind speed is significantly fast in the free atmosphere so that the basic-state motion is not negligible in the boundary layer. Once the usual relationship is reversed, flow disturbances can grow exponentially. A 1.5-layer linear model was developed and eigenvalue analyses of the linear model showed that the eigenmodes are consistent with the hypothesized flow interaction. The unstable condition for the linear instability that is theoretically obtained is tested in SEF events simulated in both axisymmetric and 3D full physics models.