Isomap Analysis

Isomap analysis was applied to PC1-4 of the 4D wave-wind feature space

Methods: PCA, Isomap, Local Bimodality Test

Storm Trajectories

Many storms approach a narrow transition region near Isomap Component   from different branches, pass through it, and then diverge into distinct regions of the manifold. This suggests the existence of a constrained intermediate regime shared across storms. Physically, this region may correspond to transitions between remotely forced and locally forced seas, increased wave–wind alignment, spectral narrowing, or onset of wave saturation.

Figure: Storm Trajectories Through Isomap Space (all 6)

The trajectories compress and overlap strongly near the manifold center, indicating reduced degrees of freedom and dynamically constrained evolution in this region.

Although the raw trajectories contain substantial point-to-point variability, smoothing reveals coherent low-frequency pathways. This suggests that the manifold represents an organizing geometry underlying noisy buoy-scale variability rather than deterministic observation-scale trajectories.

The large-scale manifold curvature is robust across storms and smoothing windows, indicating that storms evolve through preferred regions of state space while local observations fluctuate around those pathways.

Local Isomap Diagnostics

The local diagnostics reveal a strongly organized manifold centered around the narrow vertical region near Component  $1\approx 0$.

Central Manifold Core

The manifold center contains:

• the highest observation density,
• the longest residence times,
• and the slowest local trajectory speeds.

Together, these indicate that storms repeatedly evolve into and linger within a dynamically preferred wave–wind state rather than rapidly passing through it.

Figure: Isomap Local Diagnostics

Variance ellipses in this region are comparatively compact, indicating locally confined variability with weak directional preference.

Despite substantial variability in the raw trajectories, repeated convergence into this region across storms indicates persistent organization of the TC wave field under strong forcing.

Despite substantial noise in the unsmoothed buoy trajectories, the repeated convergence of multiple storms into this region indicates that the underlying wave wind system organizes into a persistent and physically preferred state under tropical cyclone forcing.

Outer Branches

The outer manifold branches exhibit lower densities, shorter residence times, stronger tangent coherence, and more elongated variance structure.

This indicates that once storms leave the manifold core, evolution becomes more directional and constrained along coherent local pathways rather than fluctuating isotropically.

The manifold core therefore behaves as a transition hub, while the outer branches support more organized directional evolution.

Together, these diagnostics suggest that the manifold contains both persistent preferred regimes and dynamically connected transition pathways corresponding to distinct modes of TC wave-field organization.

Isomap as a Function of Features

The Isomap embedding was colored by each of the four physical features defining the manifold to identify the dominant organizing gradients.

Peak Frequency

Peak frequency exhibits the clearest large-scale organization, increasing systematically along positive Component 1.

This indicates that the dominant manifold geometry is strongly controlled by spectral scale and wave-development state, ranging from mature long-period seas to short-period wind-sea-dominated conditions.

Mean Square Slope

MSS is maximized within the dense manifold core and decreases toward the outer branches, particularly along the lower-right extension. This suggests that storms preferentially organize into a recurrent energetic regime characterized by enhanced surface steepness.

Directional Spread

Directional spread varies primarily along Component 2. Lower regions of the manifold correspond to narrower directional structure, while upper regions correspond to broader directional distributions. This indicates that the vertical manifold structure primarily reflects directional organization and spectral coherence.

Tail Exponent

Tail exponent exhibits weak smooth organization across the manifold and instead varies locally within broader regimes defined by spectral scale, steepness, and directional structure. This suggests that high-frequency tail behavior is not the dominant organizing coordinate of the nonlinear manifold.

Figure: Isomap Colored by Physical Features

Isomap as a Function of Storm-Relative Coordinates

Projection of Hurricane Idalia observations into the all-storm Isomap embedding reveals strong storm-relative organization within the manifold.

Figure: Isomap Colored by Quadrant and Storm Distance (Idalia only)

The central and upper manifold regions are dominated by rear-quadrant and smaller-radius observations, while the lower-right branch is associated with larger-radius front-quadrant conditions.

The emergence of coherent storm-relative structure within an embedding constructed only from spectral variables strongly supports the physical relevance of the recovered manifold geometry.

Physical Manifold Analysis

The next phase of the analysis examined whether the manifold is locally single-valued or multivalued in directional-spread space.

Storm trajectories projected into physical feature space suggested that storms frequently revisited similar peak-frequency and MSS states while occupying substantially different directional-spread states. This raised the possibility that the manifold contains overlapping or folded sheets rather than a single-valued relationship between spectral scale, steepness, and directional organization.

Median Manifold Visualization

The Isomap embedding was discretized in Component-1/Component-2 space and mapped back into physical coordinates using median values of peak frequency, MSS, and directional spread within each bin.

The reconstructed manifold revealed a coherent low-dimensional surface with branching and folded geometry. However, storm trajectories frequently deviated systematically from the median surface, suggesting locally multivalued structure.

Figure: Binned Isomap Means Plotted in Physical Space (fp,mss,ds)

Bimodal Analysis

To test for multivalued structure quantitatively, directional-spread distributions were analyzed conditionally within $(f_p,\text{MSS})$  bins.

Each bin was fit with:

• a single-Gaussian model,
• and a two-component Gaussian mixture model (GMM). 

The single-Gaussian model represents locally single-sheet geometry, while the two-component model represents coexisting directional-spread populations occupying similar bulk wave states.

Model preference was evaluated using both AIC and BIC.

Large coherent regions favored the two-component model, with highly consistent AIC and BIC patterns. The improvement was spatially coherent rather than isolated to sparse regions, indicating that the multimodality reflects organized manifold geometry rather than sampling noise.

These results imply that directional spread is not uniquely determined by peak frequency and MSS alone.

The bimodal regions coincide with portions of the manifold where storm trajectories rise into elevated directional-spread states before returning through similar $(f_p,\text{MSS})$  coordinates. This behavior is consistent with folded manifold geometry composed of lower- and upper-directional-spread sheets.

Figure: Isomap Bimodal Analysis Results

Directional Spread Sheet Characterization

The manifold is not well described by a single-valued surface in $(f_p,\text{MSS},{\sigma }_{\theta })$  space. Instead, observations organize into partially overlapping directional-spread layers occupying similar peak-frequency and MSS ranges.

This implies that comparable bulk spectral scale and steepness can correspond to fundamentally different directional states.

When mapped back into storm-relative coordinates, sheet occupancy exhibits coherent quadrant dependence:

• the lower-spread sheet preferentially occupies the front-right quadrant,
• while the upper-spread sheet occurs more frequently in the left-front quadrant. This indicates that the sheets represent physically distinct forcing and wave-evolution environments rather than statistical partitions of the embedding.

Figure: Two Sheet Classification (physical space view) Figure: Isomap Colored by Sheet Membership Figure: Sheet Classification by Storm Quadrant (Idalia only)

Sheet Organization in Isomap Space

Projection of sheet membership back into Isomap space reveals coherent spatial organization within the manifold.

The lower sheet preferentially occupies the lower-right branch, while the upper sheet dominates the central and upper-left manifold structure.

Spectral composites demonstrate that the sheets correspond to physically distinct wave states.

Despite similar peak frequency, MSS, and tail exponent:

• the upper sheet exhibits broader directional structure and weaker concentration,
• while the lower sheet exhibits narrower directional organization and stronger wave–wind alignment. These differences indicate that directional organization evolves semi-independently from bulk spectral scale and steepness.

The sheet classification transforms the manifold from a geometric visualization into a quantifiable dynamical framework through which occupancy, transitions, residence times, and physical composites can be analyzed directly.

Figure: Sheet Member & Transition Spectra Figure: Sheet Diagnostic Bar Chart

Main Results

The central results of the analysis are:

• A low-dimensional nonlinear manifold emerges directly from the spectra without using storm metadata. 
• Storm-relative structure projects coherently onto the manifold afterward as an emergent property.
• The manifold is locally multivalued in directional spread relative to bulk wave-state variables.
• Storm trajectories evolve along preferred pathways and sheets rather than diffusely filling state space.
• Directional organization behaves semi-independently from bulk spectral scale and steepness

Traditional reduced wave-state descriptions implicitly assume that directional structure is subordinate to bulk energy scaling. The present analysis instead suggests that directional organization defines separate dynamically accessible wave states.

Methodological Strengths

A major strength of the framework is that coherent TC wave-state organization emerges directly from observations rather than being imposed through predefined storm categories or bulk metrics.

The combination of nonlinear dimensionality reduction and physically interpretable spectral features captures low-dimensional structure while retaining sensitivity to directional organization, spectral shape, and wave–wind coupling.

Importantly, coherent storm-relative organization emerges despite not being included in the manifold construction itself, providing strong validation that the recovered geometry reflects intrinsic physical structure rather than projection artifacts.

The framework also naturally supports analysis of:

• regime occupancy,
• transitions,
• trajectory evolution,
• and multivalued state behavior.

Directional Organization as an Independent Dynamical Degree of Freedom

A central result is that bulk wave-state metrics alone do not uniquely determine wave-field structure.

Observations with similar peak frequency, MSS and even tail slope can still occupy fundamentally different directional states.

The manifold analysis therefore suggests that directional structure behaves as an additional dynamical degree of freedom rather than a passive byproduct of bulk spectral properties.

Within matched $(f_p,\text{MSS})$  bins, the directional-spread sheets exhibit systematic differences in:

• high-frequency energy fraction,
• directional concentration,
• peak-tail alignment,
• and tail-derived stress proxies.

These differences persist after controlling for the primary bulk descriptors of the sea state, indicating that directional organization contains independent physical information not captured by conventional scalar metrics.

High Frequency Energetics Depend on Manifold Branch

The lower directional-spread sheet exhibits:

• larger high-frequency energy fraction,
• stronger directional concentration,
• smaller peak-tail directional separation,
• and larger tail-derived $u_{\ast }$ .

The upper sheet exhibits:

• broader directional spreading,
• weaker tail concentration,
• and greater directional decorrelation between the dominant wave system and the short-wave tail.

These results imply that high-frequency energetics are controlled not only by bulk wave geometry, but also by directional organization itself.

This has important implications for air–sea coupling physics because the high-frequency tail strongly influences aerodynamic roughness and wave-supported stress.

The results therefore support a framework in which directional organization acts as an active regulator of atmosphere–wave coupling rather than a secondary descriptor of wave geometry.