The next phase of the analysis examined whether the manifold organizes observations dynamically through preferred pathways and transition regions.
Methods: Dynamical Methods
Isomap Axis Interpretation
The two dominant physical evolution axes of the manifold separate cleanly into:
- a spectral scale evolution axis
- a directional organization evolution axis
Correlations between local manifold motion and physical evolution rates show:
- strongly tracks peak-frequency evolution,
- while primarily tracks directional-spread evolution.
Figure: Manifold Dynamic Diagnostics (streamlines, divergence, curl)
This indicates that spectral shifting and directional restructuring are related but partially independent modes of evolution.
The streamline structure indicates that the manifold core behaves as a transition nexus, while the outer branches support more coherent directional evolution. The upper-left branch exhibits slower, more rotational evolution, while the lower-right branch exhibits stronger coherent downstream flow and tighter directional organization.
This interpretation is consistent with the sheet diagnostics:
- the lower sheet exhibits stronger high-frequency organization and tighter coupling,
- while the upper sheet behaves more diffusively.
The divergence field identifies:
- convergent attractor-like behavior along the lower-right branch,
- and localized spreading near the manifold center where transitions occur.
The curl field reveals strong rotational structure near the central manifold junction, indicating hysteresis-like evolution and path dependence rather than simple linear progression through state space.
Curvature and Acceleration Diagnostics
The highest curvature occurs near the manifold center where the sheets overlap and trajectories rapidly reorganize. This indicates that transitions between directional states occur through localized dynamical restructuring rather than gradual drift.
Acceleration vectors align strongly along the manifold branches, suggesting that the manifold behaves as a low-dimensional dynamical phase space rather than a static statistical embedding.
Figure: Manifold Curvature and Acceleration Plots in Isomap Space
The dynamical analysis therefore suggests:
- the manifold contains preferred pathways,
- one axis governs spectral-scale evolution,
- another governs directional restructuring,
- transitions occur through localized corridors,
- those corridors coincide with elevated curvature, divergence, and rotational flow,
- and the resulting directional states exhibit distinct high-frequency energetics despite similar bulk sea states.
Together, these results suggest that directional organization is dynamically linked to how energy is partitioned within the high-frequency tail.
Transition Analysis
Transition populations were separated into:
- lower-sheet,
- upper-sheet,
- lower-to-upper transition,
- upper-to-lower transition states.
Figure: Difference Between Sheets Normalized by f_p and MSS bins
Two Sheet Interpretation
The lower sheet is characterized by:
- narrower directional structure,
- larger high-frequency energy fraction,
- stronger tail concentration,
- smaller peak-tail directional offsets,
- and larger inferred .
The upper sheet exhibits:
- broader directional structure,
- reduced high-frequency concentration,
- and larger peak-tail directional separation.
These differences emerge despite only minor differences in peak frequency, MSS, and bulk tail slope.
This indicates that the manifold branches represent distinct organizational pathways governing how energy is partitioned across the spectrum rather than simply different forcing magnitudes.
Transition State Interpretation
The transition diagnostics indicate that the manifold behaves as a dynamically connected system rather than a set of disconnected statistical regimes.
The transition populations are asymmetric:
- lower-to-upper transitions retain elevated high-frequency energy despite directional broadening,
- while upper-to-lower transitions recover coherent spectral structure before fully re-establishing lower-sheet directional organization.
This suggests that directional restructuring and high-frequency tail adjustment evolve on different effective timescales. The high-frequency tail therefore appears to retain memory of previous wave-field organization during transitions.
This behavior naturally produces hysteresis-like evolution in which storms revisit similar states while maintaining different directional and high-frequency organizations depending on prior pathway history.
Figure: Sheet Member & Transition Spectra Figure: Sheet Transition Comparison (path dependence)
Take Aways
The results support a hysteresis-like interpretation of the manifold geometry.
Wave states do not evolve uniquely as a function of peak frequency and MSS. Instead, storms traverse similar bulk sea states while occupying different directional branches associated with distinct high-frequency organizations.
The lower sheet represents a more directionally concentrated and tightly coupled state in which the high-frequency tail remains strongly connected to the dominant wave field.
The upper sheet represents a broader and more directionally decoupled regime.
Most importantly, the analysis suggests that directional organization is dynamically linked to how energy is partitioned into the high-frequency tail and therefore potentially to atmosphere–wave momentum exchange itself.