Does Extended Sawyer–Eliassen Equation Effectively Capture the Secondary Circulation of a Simulated Tropical Cyclone?

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Abstract The validity of the gradient wind balance in a tropical cyclone (TC) remains controversial, especially for the boundary layer and the upper outflow layer, even though this balance is assumed in the derivation of the Sawyer–Eliassen (SE) equation. This study derives an extended SE equation with the relaxation of the gradient wind and hydrostatic balance in cylindrical r–z coordinates, and then we diagnose the secondary circulation using this unbalanced SE equation and the azimuthally averaged tangential wind and thermodynamical fields from a three-dimensional numerical simulation of an intensifying TC. The gradient wind and hydrostatic imbalance produce two additional time-dependent forcing terms on the right-hand side (rhs) of SE equation, which are proved to be negligible for diagnosing the secondary circulation, even as the storm evolves rapidly. The use of the unbalanced basic state deforms the fields of coefficients that appear in the SE equation, and thus the forced secondary flows. The results indicate that the unbalanced solution captures the boundary layer inflow better than the balanced solution described by Bui et al. and the pseudobalanced solution described by Heng et al. The unbalanced solution is closer to the simulation because more unbalanced components are included. Many previous studies always employ the thermal wind balance relation to simplify the SE equation, which is invalid in an unbalanced vortex and result in an overestimation of the boundary layer inflow. These unbalanced dynamics could provide a reliable diagnosis of the secondary flow near the boundary layer.