Self-Similarity of Surface Wave Developments Under Tropical Cyclones
Key Insight:
Despite fundamentally different methodological approaches and state-space definitions, both reduced-order wave-growth theory and observational manifold analysis independently indicate that tropical-cyclone wave fields evolve within constrained low-dimensional dynamical structures governed by storm-relative forcing geometry, resonance, and directional organization.
Primary Claims:
TC wave fields are not arbitrary responses to local wind forcing. They evolve along constrained dynamical pathways imposed by the geometry and the motion of the storm.
Trajectory based view: pattern emerges from resonance, residence time, directional turning, nonlinear spectral evolution and wave ray geometry. the internal organization is the result of some wave trains remaining trapped within forcing regions, some detach and escape, some converge, some diverge → different pathways through the storm produce different spectral outcomes → my work proves these trajectories are observable, validating the broader dynamical structure
The manifold exists because TC wave evolution itself is dynamically constrained
State space:
or (energy, peak wavelength, peak direction)
this system evolves through wind input, nonlinear downshift, dissipation and directional turning
framework: coherent storm relative asymmetry, preferred dynamical pathways, constrained evolution, resonance-controlled organization
My state space:
My state space: these quantities encode bulk wave scale, surface roughness, directional organization (both high and low frequency) → i am tracking the structure of the spectral state itself framework: coherent manifold trajectories, preferred occupancy regions, smooth nonlinear evolution, directional-state organization despite state space differences, the coherent resulting organizational structure suggests the manifold geometry is emergent from the underlaying wave dynamics, not from my analysis choices alone I suggest that the system possesses internal organizational modes beyond bulk wave development alone My manifold may encode different dynamical histories of wave development under storm-relative resonance conditions→ this could be responsible for producing distinct manifold occupancy, coherent lower sheet states, stronger HF tail organization How do i explore this? Coherence suggests that the system is not fragile to variable choice, it is intrinsic to the system → robust!!
Methods
- parametric dominant wave model, not a full spectral observational analysis
- limitations:
- high frequency tail slope
- MSS variability,
- multimodal spectral branch structure
- wave supported stress
- directional spread
Figures:
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setup: polar grid, wave seeding, establishment of storm wave geometry
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stationary TC ray evolution: non-moving TC scenario, rays grow rapidly near the storm, reach maximum energy near RMS, then transition into swell when wave age reaches a threshold. even without storm motion, wave development is a trajectory through energy, wavelength and directional space
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stationary TC ray superposition: when all rays are overlaid, you get an azimuthal symmetric wave field. swell propagates radially away
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Moving TC wave pattern: as translation speed increases, the wave ray field becomes asymmetric. trajectories become hook shaped, group velocity nearly matches storm translation, thus staying under forcing longer, grow larger and propagate through the front sector as swell (asymmetry depends on residence time and propagation geometry, not windspeed)
5-8. individual rays to coherent TC scale pattern: many individual ray histories combine into organized spatial structures (physical analog to individual storm trajectories occupying coherent paths in isomap space)
9- 12. multimodal and observed spectra: compare model ray distribution with observed directional spectra, find that front sectors tend to be dominated by more coherent single systems while back/right regions can contain multiple
13-18. self-similar collapse: across many idealized TC parameter combinations, the model outputs collapse when scaled by critical fetch. 18 is the universal 2D functions for normalized wavelength, energy and direction (all TCs share a common low dimensional organizing structure)
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comparison with observations: compare self-similar solutions to airborne observations of H_S, peak period and direction. agreement is good enough
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practical reconstructed fields: how universal functions can quickly reconstruct TC wave field from u_m, R_m, V.
Main conclusions:
- TC wave fields can be reduced to coherent, self-similar structures controlled by storm intensity, size, and translation speed. Slow and fast TCs separate into different wave-development regimes, with the largest possible waves occurring near the group velocity resonance condition
- Utility: you can predict constrained dynamical pathways from a reduced wave-ray model, I extend this low dimensional pathway to directional spread, HF tail energetics, MSS and coupling relevant structure