Tropical cyclone boundary layer shocks beyond the formation time: post-shock theory, shock-strength criteria, translating-storm geometry, and merger dynamics

In an idealized line-symmetric slab model of the tropical cyclone frictional boundary layer, Schubert, Slocum and Taft (2017) showed that the transient adjustment toward the steady Ekman solution generically forms shocks — discontinuities in the horizontal wind with singular boundary layer pumping — and obtained exact characteristic solutions valid only up to the shock formation time $t_s$ , connecting triangular-wave and N-wave asymptotics to single and concentric eyewalls. Four linked problems were left open: (i) extending the solutions beyond $t_s$ through jump conditions for the coupled, non-conservative slab system or through a matched small-horizontal-diffusion limit; (ii) a criterion for when a shock is dynamically strong enough that the boundary layer controls the organization of deep moist convection; (iii) shock formation under a translating pressure field, with non-circular (crescent- or spiral-shaped) geometry; and (iv) a systematic theory of shock merger for different initial convective-ring configurations.

Background

A tropical cyclone (hurricane or typhoon) owes its strongest winds and heaviest rainfall to the eyewall, the ring of deep precipitating convection surrounding the calm eye; many intense storms also develop a concentric (secondary) eyewall separated from the inner one by a moat of weak convection, and the replacement of the inner eyewall by the contracting outer one — the eyewall replacement cycle — is a major source of intensity change. The frictional boundary layer, the roughly 1 km deep layer adjacent to the ocean surface where drag decelerates the flow, is believed to control where this deep convection is organized: convergence of the boundary layer flow produces Ekman pumping (upward motion at the top of the frictional layer, named after the classical theory of spiral frictionally driven winds), which lifts moist air toward the level of free convection.

Schubert, Slocum and Taft (2017) study this mechanism with a slab boundary layer model: the boundary layer is treated as a single well-mixed layer of fixed depth $h$ carrying depth-integrated radial velocity $u$ and tangential velocity $v$ , forced by a specified pressure field (equivalently a gradient-balanced wind, the wind in cyclostrophic–Coriolis balance with that pressure), decelerated by quadratic surface drag $(c_D U/h)(u,v)$ with $U=(u^2+v^2)^{1/2}$ , and, in its simplest version, closed without horizontal mixing. Numerical predecessors (Williams, Taft, McNoldy and Schubert 2013; Slocum, Williams, Taft and Schubert 2014) showed that this model produces shock-like structures: the radial inflow drops by 20–30 m/s over a few kilometers near the radius of maximum wind, with a corresponding spike in Ekman pumping, and when the outer gradient-wind forcing is enhanced a second, outer shock forms that can cut off the inflow feeding the inner one — a dry prototype of the eyewall replacement cycle.

The 2017 paper supplies the analytical theory. On an $f$ -plane (constant Coriolis parameter $f$ ), the line-symmetric, horizontally non-diffusive slab equations

$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}-fv+\frac{c_D U}{h}u=-\frac{1}{\rho}\frac{\partial p}{\partial x},\qquad\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+fu+\frac{c_D U}{h}v=0,$

form a hyperbolic system solvable by the method of characteristics — the same technique used for Burgers' equation $\partial u/\partial t+u\,\partial u/\partial x=0$ , the canonical equation whose smooth solutions steepen and break into shocks (propagating discontinuities) at a finite shock formation time $t_s$ when characteristics first intersect. The paper's exact solutions describe the transient adjustment from specified initial $u$ and $v$ fields to the steady Ekman solution (the time-independent frictionally balanced state that the classical Ekman theory predicts for this forcing), and connect single eyewalls to the asymptotic triangular waves and concentric eyewalls to N-waves (the two universal large-time wave shapes of Burgers-type equations). Shocks are classified as divergence-preferred or vorticity-preferred according to whether the jump in the divergent component $u$ exceeds that in the rotational component $v$ , and critical thresholds on initial convergence and vorticity for shock formation are derived.

These solutions, however, are valid only up to $t_s$ ; beyond it they become multivalued and the associated Ekman pumping and boundary layer vorticity are singular. The authors enumerate what remains open: supplementing the equations with jump conditions (Rankine–Hugoniot-type relations connecting states across a shock, as in gas dynamics) or regularizing with a small horizontal diffusivity $K$ — neither carried out analytically in this framework, although small- $K$ diffusion and shock-capturing numerics in the spirit of Kuo and Polvani (1997) are known to work in practice; determining which shocks are dynamically strong enough to control the organization of deep moist convection, since a mathematically valid shock may be physically insignificant; extending the analysis to a translating storm, whose boundary layer shocks would take non-circular, crescent- or spiral-shaped forms; and a systematic exploration of the merger of two inward-propagating shocks, the mechanism by which two convective rings merge into a single eyewall in the axisymmetric Burgers-type solution. The slab framework itself neglects the curvature terms of the full axisymmetric model, vertical advection, a variable drag coefficient, and the feedback of convection onto the pressure field, so the eventual test of the theory is its quantitative applicability to realistic three-dimensional hurricane boundary layers.

Problem Statement

The problem is to complete the mathematical theory of tropical cyclone boundary layer shocks within the idealized dry slab framework of Schubert, Slocum and Taft (2017). All four parts are jointly required for completion and serve one objective — turning shock formation from a signal that the exact solution breaks down at $t_s$ into a usable theory of how the boundary layer organizes deep moist convection:

Post-shock continuation. Extend the line-symmetric slab solutions beyond the shock formation time $t_s$ : either derive the jump conditions across the discontinuity for the coupled system (settling the correct weak formulation of the non-conservative quadratic drag term and the Coriolis coupling) or construct the matched limit of the horizontally diffusive solutions as the diffusivity $K$ tends to zero, and obtain the post-shock solutions for specified initial data, including the shock trajectory.
Shock strength. Determine the conditions — a criterion in terms of the initial $u$ and $v$ fields, the specified forcing, and the parameters — under which the resulting shock is strong enough that the boundary layer takes control of the organization of the deep moist convection, distinguishing dynamically significant shocks from weak ones that are mathematically valid but physically negligible.
Translating storms. Extend the shock-formation analysis to a translating pressure field, characterizing when and where shocks form and their non-circular (crescent- or spiral-shaped) geometry as a function of translation speed and storm structure.
Merger. Systematically analyze the merger of two inward-propagating shocks for different initial ring configurations and parameter regimes via the axisymmetric Burgers-type solutions, delineating which configurations merge into a single eyewall.

The question of quantitative applicability to realistic three-dimensional boundary layers (curvature terms, vertical advection, variable drag coefficient, convection feedback onto the pressure field) frames the significance of the answer but is not itself the completion boundary.

References

Schubert, Wayne H.; Slocum, Christopher J.; Taft, Richard K. (2017). Basic Concepts Involved in Tropical Cyclone Boundary Layer Shocks. arXiv:1709.00101, DOI 10.48550/arXiv.1709.00101. https://arxiv.org/abs/1709.00101
Slocum, Christopher J.; Williams, Gabriel J.; Taft, Richard K.; Schubert, Wayne H. (2014). Tropical cyclone boundary layer shocks. arXiv:1405.7939, DOI 10.48550/arXiv.1405.7939. https://arxiv.org/abs/1405.7939
Williams, Gabriel J.; Taft, Richard K.; McNoldy, Brian D.; Schubert, Wayne H. (2013). Shock-like structures in the tropical cyclone boundary layer. Journal of Advances in Modeling Earth Systems, DOI 10.1002/jame.20028. https://doi.org/10.1002/jame.20028
Kuo, Allen C.; Polvani, Lorenzo M. (1997). Time-Dependent Fully Nonlinear Geostrophic Adjustment. Journal of Physical Oceanography 27(8), 1614–1634, DOI 10.1175/1520-0485(1997)027<1614:TDFNGA>2.0.CO;2. https://doi.org/10.1175/1520-0485(1997)027%3C1614:TDFNGA%3E2.0.CO%3B2
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Smith, Roger K.; Montgomery, Michael T. (2026). The slab boundary layer model underpinning the boundary layer control of translating tropical cyclones: comparison with an idealized three-dimensional numerical simulation. Tropical Cyclone Research and Review, DOI 10.1016/j.tcrr.2026.07.005. https://doi.org/10.1016/j.tcrr.2026.07.005
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