The modular approach to tropical-cyclone structure, introduced by Carrier and coworkers in 1971 and revived by Fendell and Mokhasi (2013), divides the steady axisymmetric vortex at large Reynolds number into matched subdomains — inviscid bulk vortex, frictional ocean-contiguous boundary layer, saturated convecting core, and, for the most intense storms, an eye — and solves each at its dominant asymptotic balance. The 2013 work executed the boundary-layer module and derived the coupling conditions to the overlying vortex, but left the remainder of the program unexecuted. The surviving problem is to formulate and solve the quasi-linear second-order elliptic streamfunction problem for the saturated inviscid core, in which the saturation constraint Ys(T,p)Y_s(T,p) couples the elliptic equation to nonlinear algebraic relations for temperature, pressure, entropy and angular momentum along streamlines; extend the matched solution to intense, eyed vortices, including the geometry and thermodynamics of an eye descending to sea level and the question of whether eye insertion coincides with an intensity threshold such as a critical peak swirl; and replace the ad hoc inviscid corner-flow treatment by a resolved viscous–diffusive model of the region where boundary-layer inflow becomes eyewall updraft.

Background

A tropical cyclone (hurricane or typhoon) is a large, rapidly rotating storm powered by heat and moisture taken up from the sea. Its intensity is difficult to predict in part because the key physics is concentrated in the ocean-contiguous boundary layer — a frictional layer of order 1 km thickness within which the wind decelerates toward the surface, air spirals inward, and enthalpy (the heat content carried by temperature and water vapor) is drawn from the ocean — and in the inner core, where that air rises in the ring of intense convection called the eyewall surrounding the calm, subsiding eye. Operational models integrate the full three-dimensional equations on grids too coarse to resolve these regions, which motivates reduced descriptions.

In the early 1970s, G. F. Carrier and coworkers proposed a modular analysis of the mature, axisymmetric tropical cyclone in the limit of large Reynolds number (the ratio of inertial to viscous forces): the vortex is divided into subdomains — an inviscid bulk vortex (module I), a frictional boundary layer (module II), a strongly convective saturated inviscid core (module III) and, for the most intense storms, an eye (module IV) — each solved at its locally dominant force balance and matched across interfaces. Companion analyses of boundary layers induced by potential vortices (Burggraf, Stewartson & Belcher 1971; Belcher, Burggraf & Stewartson 1972) exposed the singular structure of the near-axis boundary layer and its corner, and the modular program then lay largely dormant for four decades.

Fendell & Mokhasi (2013) revived the program with modern computing. For module II they integrated the steady axisymmetric boundary-layer equations — a parabolic system for the radial mass-flux function, the absolute angular momentum per unit mass (the swirling motion plus the planetary-rotation contribution, conserved along inviscid axisymmetric streamlines) and the vertical velocity — in stretched coordinates x=r′2/ro′2x=r'^2/r_o'^2 and ζ=z′/(ν′/(2Ω′))1/2\zeta=z'/(\nu'/(2\Omega'))^{1/2} (with ν′\nu' an eddy viscosity and Ω′\Omega' the planetary rotation rate), marching in xx under a prescribed inviscid top-edge circulation Ψ(x)=1+ε−x\Psi(x)=1+\varepsilon-x, where ε=vo′/(Ω′ro′)≪1\varepsilon=v_o'/(\Omega' r_o')\ll 1 is the swirl parameter. They computed the entrainment across the boundary-layer top (nominally ∣wedge′∣∼0.7|w'_{edge}|\sim 0.7 cm s−1^{-1}), closed the core-side interface state (moist enthalpy Ecore′E'_{core} and entropy Score′S'_{core}) by a four-equation algebraic system using invariants transported from the periphery, and derived the elliptic partial differential equation for the bulk-vortex streamfunction η′(r′,z′)\eta'(r',z') (in their appendix, under a vanishing-azimuthal-vorticity approximation) that makes the inviscid module solvable with Dirichlet data from the boundary layer. Two features of the inner region were, however, treated only approximately or not at all: the corner flow — the short region where the boundary-layer inflow turns upward into the eyewall updraft — was handled by an inviscid, ad hoc approximation; and vortices intense enough to possess an eye were outside the solution obtained.

In the core module the air is saturated (its water-vapor mixing ratio equals the saturation value Ys(T,p)Y_s(T,p), a thermodynamic function of temperature TT and pressure pp), so condensation releases latent heat on the streamlines passing through the eyewall. The core problem is a quasi-linear second-order elliptic equation for the streamfunction — elliptic in the sense that, given vorticity and mass sources, the overturning circulation is determined by a Poisson-type balance — coupled through the saturation constraint to nonlinear algebraic relations for TT, pp, entropy and angular momentum along each streamline; its formulation and numerical solution were left unexecuted in that work, as were the eyed-vortex module and the question of whether eye insertion coincides with an intensity threshold. The same discussion flagged input uncertainties: the high-wind behavior of air–sea enthalpy and momentum exchange (addressed observationally by Zhang, Black, French & Drennan 2008; Haus, Jeong, Donelan, Zhang & Savelyev 2010; Bell, Montgomery & Emanuel 2012) and the eddy diffusivities of the hurricane boundary layer (Zhang, Marks, Montgomery & Lorsolo 2011; Zhang & Drennan 2012), which remain only loosely constrained.

Neighboring literature treats parts of this physics by other means. Height-resolving and slab boundary-layer models of the tropical-cyclone core (Kepert 2001; Kepert & Wang 2001; Smith & Montgomery 2010) exhibit supergradient winds — winds exceeding the gradient-wind balance among pressure-gradient, centrifugal and Coriolis forces — near the eyewall, and address the boundary-layer–eyewall transition by direct nonlinear integration (see also Fei, Wang & Li 2021; Snaiki & Wu 2020). Eye formation has been studied in dry rotating-convection models, where a vorticity-stripping mechanism forms an eye (Oruba, Davidson & Dormy 2017, 2018), and in a dry hurricane-like vortex with bulk surface fluxes (Dormy, Oruba & Emanuel 2024), with the moist case explicitly left open; Bryan & Rotunno (2009) showed by cloud-resolving simulation how near-surface high-entropy air in the eye limits maximum intensity; and Chen & Wu (2022) documented two phenomenological types of eye formation in observed storms. None of these executes the moist modular matched problem that Fendell & Mokhasi left open.

Problem Statement

Within the steady, axisymmetric, modular matched-asymptotic framework of Fendell & Mokhasi (2013) — in which the boundary-layer module has been solved and matched to an inviscid bulk vortex, and the vortex is treated at large Reynolds number as a set of subdomains joined across interfaces — carry the modular analysis of the tropical cyclone to completion: (i) formulate and numerically solve the quasi-linear second-order elliptic problem for the streamfunction of the saturated inviscid core (module III), in which the saturation constraint Ys(T,p)Y_s(T,p) couples the elliptic equation to nonlinear algebraic relations for temperature, pressure, entropy and absolute angular momentum along streamlines, with boundary data supplied by the boundary-layer module and the ambient periphery; (ii) extend the matched solution to more intense, eyed vortices (module IV): determine the geometry and thermodynamics of the eye–eyewall interface for an eye that descends to sea level, and establish whether the insertion of an eye coincides with an intensity threshold, for example a particular value of the peak swirl; and (iii) replace the inviscid, ad hoc treatment of the corner flow — the localized region in which boundary-layer inflow is converted into core updraft — by a resolved viscous–diffusive model, including mixing across streamlines during condensation, so that the exchange of momentum, entropy and moisture across the boundary-layer–core interface is quantified. The deliverable is a mutually consistent set of module solutions and interface conditions yielding a self-consistent steady axisymmetric tropical cyclone, with and without an eye; recalibrating high-wind air–sea flux and eddy-viscosity parameterizations from improved observations, and extending to unsteady or non-axisymmetric evolution, are surrounding refinements rather than components of this completion. All three parts (i)–(iii) are mandatory for this completion and none resolves the task alone — in particular, formulating and solving the saturated-core problem (i) without the eyed-vortex extension (ii) and the resolved corner-flow model (iii) does not complete the programme.

References

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