On the Applicability of Linear, Axisymmetric Dynamics in Intensifying and Mature Tropical Cyclones
The applicability of linearized axisymmetric dynamics to the intensification and structure change of tropical cyclones is investigated. The study is motivated by recent work that presented axisymmetric solutions to the linearized, non-hydrostatic, vortex-anelastic equations of motion (the so-called 3DVPAS model). The work called into question the importance of a recently proposed nonlinear, system-scale boundary-layer spinup mechanism both in intensifying storms and in mature storms undergoing
... storms undergoing secondary eyewall formation. The issue is examined using a three-dimensional mesoscale simulation of an intensifying tropical cyclone, alongside the linear 3DVPAS model. Solutions to the linear model, for imposed eddy forcing terms derived from the mesoscale simulation, are shown to be valid only for short times (t < 1 h) in the inner-core region of the vortex. At later times, the neglected nonlinear terms become significant and the linear results invalid. It follows that the linear results cannot be used to describe all aspects of the tropical cyclone dynamics at later times. In particular, they cannot be used (a) to dismiss the importance of the nonlinear boundary-layer spinup mechanism, nor (b) to isolate the separate effects of diabatic heating from those of friction, within the nonlinear boundary layer at least. Such separation depends on the linear superposition principle, which fails whenever nonlinearity is important. Similar caveats apply to the use of another linear model, the traditional Sawyer-Eliassen balance model. Its applicability is limited not only by linearity, but also by its assumption of strictly balanced motion. Both are incompatible with nonlinear spinup. Fluids 2017, 2, 69 2 of 15 the axisymmetric "convective ring model" of tropical cyclone intensification originally proposed in [3, 4] , but can be seen also as a generalization thereof because the original ring model assumed axisymmetric balance dynamics. Ref.  used the 3DVPAS model to diagnose the relative roles of the heat and tangential momentum forcing in the formation of a simulated secondary eyewall as predicted by the three-dimensional Weather Research and Forecasting (WRF) model. The solutions for the transverse circulation using the 3DVPAS model were found to be qualitatively similar to the transverse circulation diagnosed from the WRF model. However, the theoretical justification for the neglect of the nonlinear terms was not provided. At a minimum, one needs to verify that all the nonlinear terms, when evaluated using the linear solutions, are small in comparison to the retained linear terms. In a more recent study, Ref.  used the same methodology as employed in  to diagnose the relative roles of the heat and tangential momentum forcing distributions in the intensification of a tropical cyclone using again the WRF model, but for another intensifying storm. Because of the linearization built into the 3DVPAS model, this use of it cannot address the situation within the frictional boundary layer in the inner-core region of the storm where the flow is intrinsically nonlinear    . As discussed in earlier publications (e.g., [7-10]), there is an intrinsically nonlinear boundary layer spinup process, which, as discussed in , involves strong, non-Ekman-like nonlinearities in an essential way, with dramatic consequences such as the generation of strong supergradient winds within the boundary layer. These supergradient winds include the strongest and most destructive winds found anywhere within a mature tropical cyclone. The inherently linear nature of the 3DVPAS model means that its use in Refs. [1, 2] produces results that are misleading in two ways. The first is simply that important nonlinear terms are neglected, and the second is that, in using the results to examine the relative importance of heat and tangential momentum forcing, the linear superposition principle is tacitly assumed. That is, it is assumed that the heat and momentum forcing processes can be considered separately, with effects that are additive. The superposition principle fails as soon as nonlinearity is significant, and fails drastically when nonlinearity is strong. In particular, its use in Ref.  to cast doubt on the nonlinear boundary layer spinup mechanism is, in our view, unjustified. The objectives of this paper are twofold. The first is to examine the validity of the 3DVPAS model, summarized in Section 2, for a class of axisymmetric vortex flows forced by azimuthally-averaged diabatic and frictional terms diagnosed from a state-of-the art three-dimensional mesoscale numerical model (Section 3). The specific goal in this context is to quantify the errors introduced by the neglect of the nonlinear terms in the governing equations in the inner-core region of the storm (Section 4). We show that the inner-core nonlinear terms are already significant even at an early stage of tropical cyclone intensification, much earlier than expected from the recent results of  who examined a later stage of tropical cyclone intensification and secondary eyewall formation. Our findings, together with those of  , suggest that the emergence of strong nonlinearities outside the scope of 3DVPAS invalidates the use of 3DVPAS for diagnosing and interpreting complex numerical model solutions for maturing and mature tropical cyclones that possess such strong nonlinearities. A second, and subsidiary, objective (Section 5) is to examine the use of the traditional Sawyer-Eliassen (hereafter SE) model of transverse circulations, in this same context of maturing and mature tropical cyclones. The SE model was used in  to cast further doubt on the existence of nonlinear spinup. However, the SE model is linear, by assumption, in the same sense as the 3DVPAS model, and is subject to additional limitations through its assumption of strictly balanced motion. Neither assumption is compatible with the nonlinear boundary-layer dynamics. For instance, when the boundary-layer inflow separates and turns upward into the eyewall, not only are the non-Ekman-like nonlinearities strong, but departures from balance can become significant in the form of axisymmetric inertia-gravity waves launched into the eyewall upflow  . Section 5 discusses the extent to which these limitations might be overcome in future work. Section 6 offers concluding remarks.