What are the Balanced and Unbalanced Dynamics of Tropical Cyclone?

Recently, Ji and Qiao took into account the unbalanced components and derived an extended Sawyer–Eliassen (SE) equation. This study developed a new derivation of this extended SE equation from the perspective of restoring forces, and gives a physical interpretation for the coefficients that appear in the SE equation. For an unbalanced vortex, we demonstrated that the thermodynamic fields are only determined by the distribution of gradient wind, and thus the gradient wind and thermodynamic fields always remain in balance as the unbalanced vortex evolves. Consequently, we attributed the gradient wind imbalance to the agradient wind rather than to the thermodynamic fields. Subsequently, we explored the effect of the agradient wind on the secondary circulation, and showed that the agradient wind strengthens the secondary circulation in its vicinity, which can be explained as a consequence of the restoring forces and mass continuity. Furthermore, we speculated that the SE equation, together with the radial velocity equation could reproduce the primary characteristic of the axisymmetric boundary layer dynamics by prescribing the parameterization of subgrid–scale turbulent mixing. Specifically, the noU BL and noVa BL experiments conducted by Fei et al. in an article published in 2021 were reinterpreted, and the oscillation wavelength of the agradient wind in the eyewall was approximated based on this framework. Additionally, a new numerical solution algorithm to overcome the hyperbolicity near the boundary layer was proposed. This study attempts to develop a complete dynamic theory for TC in both qualitative and quantitative perspectives.