Representing Wind Forcing History for Wave Observations

Motivation

The Spotter wave spectra are available at hourly intervals, whereas the colocated model wind fields are available at higher temporal resolutions. The conventional approach of averaging the wind over each one-hour sampling interval implicitly assumes that the wave field responds to a simple mean of the preceding forcing. Physically, however, the observed spectrum represents the integrated response of the wave field to a continuously evolving wind history, with different spectral properties likely responding over different timescales.

The objective of this methodology and documentation is to explore alternative representations of the preceding wind forcing that preserve information contained in the higher-frequency model output. Rather than reducing the forcing to a single hourly average, these methods seek to characterize the temporal evolution, persistence, variability, and cumulative effects of the wind prior to each wave observation. Such representations may improve the interpretation of wave-state variability and provide insight into the characteristic response times of different wave diagnostics, including MSS, high-frequency energy fraction, directional organization, and wind–wave alignment.

Quantifying the Forcing History of Tropical-Cyclone Wave Fields

Fundamental framework - as informed by previous work

1. Conceptual definition of wave history

The instantaneous wave spectrum observed at location ${\mathbf{x}}_o$ and time $t_o$ is not determined solely by the simultaneous local wind. It reflects the cumulative evolution of spectral wave components as they propagate through a spatially and temporally varying wind field. Under tropical cyclones, this evolution depends on wind field characteristics, storm size, storm translation, wave propagation, pre-existing sea state, nonlinear spectral transfers, dissipation, currents, and—in some circumstances—rain-induced modification of the short-wave field.

We therefore define the history of a wave component as the cumulative forcing and redistribution experienced along the propagation pathway of that component before it reaches the observation point. In its most general form, wave history is frequency- and direction-dependent:

$$\mathcal{H}=\mathcal{H}\left(f,\theta ,{\mathbf{x}}_o,t_o\right).$$

This definition follows naturally from the spectral action-balance framework used in third-generation wave models and from the wave-ray interpretation of tropical-cyclone wave development introduced by Kudryavtsev et al. Wave trains generated in different storm sectors may follow different trajectories, remain under the storm wind field for different durations, or leave the storm as swell. Consequently, the forcing history associated with an observation is generally nonlocal and cannot be represented exactly by a fixed-duration average of wind at the observation location. (AGU Publications)

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2. Governing spectral framework

2.1 Spectral action balance

Let

$$E(\mathbf{x},k,\theta ,t)$$

denote the directional variance spectrum, where $k$ is wavenumber and $\theta$ is the wave propagation direction. The corresponding action-density spectrum is

$$N(\mathbf{x},k,\theta ,t)=\frac{E(\mathbf{x},k,\theta ,t)}{\sigma }$$

where $\sigma$ is the intrinsic angular frequency. Wave action, rather than energy, is conserved during propagation through slowly varying currents.

The governing action-balance equation may be written as

$$\frac{\partial N}{\partial t}+{\nabla }_{\mathbf{x}}\cdot \left[\left({\mathbf{c}}_g+{\mathbf{U}}_c\right)N\right]+\frac{\partial }{\partial k}\left(\dot{k}N\right)+\frac{\partial }{\partial \theta }\left(\dot{\theta }N\right)=\frac{S_{\mathrm{tot}}}{\sigma }\text{\hspace{1em}(1)}$$

where

• ${\mathbf{c}}_g$ is the intrinsic group-velocity vector
• ${\mathbf{U}}_c$ is the surface-current velocity
• $\dot{k}$ represents changes in wavenumber caused by depth or current gradients
• $\dot{\theta }$ represents spectral refraction
• $S_{\mathrm{tot}}$ is the net spectral source term

The net source term is conventionally decomposed as

$$S_{\mathrm{tot}}=S_{\mathrm{in}}+S_{\mathrm{nl}}+S_{\mathrm{ds}}+S_{\mathrm{bot}}+S_{\mathrm{br}}+S_{\mathrm{ice}}+\cdots ,\text{\hspace{1em}(2)}$$

where $S_{\mathrm{in}}$ is atmospheric wind input, $S_{\mathrm{nl}}$ represents nonlinear wave–wave interactions, and $S_{\mathrm{ds}}$ represents whitecapping and other deep-water dissipation. Additional terms account for bottom friction, depth-induced breaking, ice, or other processes when relevant. Equations (1)–(2) are the governing equations used by spectral models such as WAVEWATCH III. (polar.ncep.noaa.gov)

For deep-water gravity waves,

$${\sigma }^2=gk\text{\hspace{1em}(3)}$$

and the phase and group velocities are

$$c_p=\frac{\sigma }{k}=\sqrt{\frac{g}{k}}\text{\hspace{1em}(4)}$$ $$c_g=\frac{\partial \sigma }{\partial k}=\frac{1}{2}{c}_p=\frac{g}{2\sigma }=\frac{g}{4\pi f}\text{\hspace{1em}(5)}$$

Equation (5) is central to the history problem because different frequencies propagate at different speeds. Long peak waves can carry memory from distant regions and earlier storm stages, whereas high-frequency waves propagate slowly and generally reflect more local forcing.

2.2 Wave-following form

Along a spectral characteristic or wave ray, Equation (1) reduces schematically to

$$\frac{DN}{Dt}=\frac{S_{\mathrm{tot}}}{\sigma }\text{\hspace{1em}(6)}$$

where the derivative follows the path of the wave component:

$$\frac{D}{Dt}=\frac{\partial }{\partial t}+\left({\mathbf{c}}_g+{\mathbf{U}}_c\right)\cdot {\nabla }_{\mathbf{x}}+\dot{k}\frac{\partial }{\partial k}+\dot{\theta }\frac{\partial }{\partial \theta }.\text{\hspace{1em}(7)}$$

Integrating backward from an observation at $({\mathbf{x}}_o,t_o)$ gives

$$N_o=N_i+{\int }_{t_i}^{t_o}\frac{S_{\mathrm{in}}+S_{\mathrm{nl}}+S_{\mathrm{ds}}+\cdots }{\sigma }d{t}^{\prime }\text{\hspace{1em}(8)}$$

where the source terms are evaluated along the wave ray

$${\mathbf{x}}_r\left(t^{\prime };f,\theta \right)$$

Equation (8) provides the fundamental physical definition of wave history. The observed spectrum consists of an initial or incoming spectrum $N_i$, plus the accumulated effects of wind input, nonlinear redistribution, dissipation, and other processes along the propagation pathway.

This also establishes an important interpretive limitation: wind-history metrics alone estimate the forcing component of wave history, principally $S_{\mathrm{in}}$. They do not fully separate wind input from nonlinear interactions or dissipation unless source-term output from a spectral model is available.

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3. Wave-ray reconstruction

3.1 Backward propagation

For a fixed observation point, the simplest deep-water approximation to the backward trajectory of a spectral component is

$${\mathbf{x}}_r(t_o-\tau ;f,\theta )={\mathbf{x}}_o-{\int }_{t_o-\tau }^{t_o}\left[{\mathbf{c}}_g(f,\theta ,t^{\prime })+{\mathbf{U}}_c(t^{\prime })\right]d{t}^{\prime }\text{\hspace{1em}(9)}$$

If currents and spectral refraction are neglected over the integration interval,

$${\mathbf{x}}_r(t_o-\tau ;f,\theta )\approx {\mathbf{x}}_o-{\mathbf{c}}_g(f,\theta )\tau \text{\hspace{1em}(10)}$$

The forcing wind sampled along this ray is

$${\mathbf{U}}_r\left(\tau ;f,\theta \right)={\mathbf{U}}_{10}\left[{\mathbf{x}}_r(t_o-\tau ;f,\theta ),t_o-\tau \right]\text{\hspace{1em}(11)}$$

Equations (9)–(11) convert the model wind field into a frequency- and direction-specific forcing history.

The ray approximation is most reliable when spatial gradients in depth and current occur over scales much larger than the wavelength. It neglects diffraction and phase-resolving interactions but is consistent with the geometrical-optics approximation used by spectral wave models. (polar.ncep.noaa.gov)

3.2 Storm-relative wave rays

Let the tropical-cyclone center be located at

$${\mathbf{x}}_{TC}(t)$$

with translation velocity

$${\mathbf{V}}_{TC}(t)=\frac{d{\mathbf{x}}_{TC}}{dt}\text{\hspace{1em}(12)}$$

The storm-relative location of a wave component is

$${\mathbf{r}}_r(t)={\mathbf{x}}_r(t)-{\mathbf{x}}_{TC}(t)\text{\hspace{1em}(13)}$$

The storm-relative propagation velocity is therefore

$${\mathbf{c}}_{g,\mathrm{rel}}={\mathbf{c}}_g+{\mathbf{U}}_c-{\mathbf{V}}_{TC}\text{\hspace{1em}(14)}$$

Its component in the storm-motion direction,

$$c_{g,\parallel }^{\mathrm{rel}}={\mathbf{c}}_{g,\mathrm{rel}}\cdot {\hat{\mathbf{V}}}_{TC}\text{\hspace{1em}(15)}$$

determines how rapidly the wave packet moves through or escapes the translating wind field.

When

$${\mathbf{c}}_g+{\mathbf{U}}_c\approx {\mathbf{V}}_{TC}\text{\hspace{1em}(16)}$$

the relative propagation speed becomes small and the wave packet may remain beneath favorable winds for an extended period. This is the trapped-fetch or group-velocity quasi-resonance mechanism. It is expected to be strongest for waves propagating approximately in the direction of storm motion and is one reason for enhanced wave development in the right-front sector of Northern Hemisphere tropical cyclones. (American Meteorological Society Journals)

A useful scalar trapped-fetch indicator is

$$Q_{\mathrm{trap}}\left(f,\theta ,t\right)=\frac{\left|{\mathbf{c}}_g+{\mathbf{U}}_c-{\mathbf{V}}_{TC}\right|}{\left|{\mathbf{c}}_g+{\mathbf{U}}_c\right|+\left|{\mathbf{V}}_{TC}\right|}\text{\hspace{1em}(17)}$$

where small values indicate near-matching of wave and storm velocities. This is a diagnostic introduced for the present analysis rather than a standard published index.

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4. Storm-scale normalization

4.1 Characteristic storm size

Oh et al. demonstrated that the radius of 34-kt winds, $R_{34}$, is more closely related to storm-induced maximum significant wave height than the radius of maximum wind, $R_{\mathrm{MW}}$. Their simulations gave correlations of approximately $0.84$ – $0.95$ between $R_{34}$ and maximum $H_s$, compared with approximately $0.33$ – $0.58$ for $R_{\mathrm{MW}}$. They attributed this result to the much stronger relationship between $R_{34}$ and the accumulated wind field aligned with storm propagation. The result indicates that broad regions of moderately strong winds can be more important to cumulative wave development than the narrow region containing maximum winds. (Frontiers)

Storm-relative radius should therefore be normalized primarily by the quadrant-specific $R_{34}$:

$$r^{\ast }=\frac{r}{R_{34,q}}\text{\hspace{1em}(18)}$$

where $q$ denotes the relevant storm quadrant. If quadrant-specific wind radii are unavailable, an azimuthal mean $R_{34}$ may be used, but this suppresses physically important storm asymmetry.

Additional normalized coordinates may be retained for comparison:

$$r_{\mathrm{MW}}^{\ast }=\frac{r}{R_{\mathrm{MW}}},r_{50}^{\ast }=\frac{r}{R_{50}}\text{\hspace{1em}(19)}$$

4.2 Storm crossing timescale

A first-order Eulerian storm timescale is

$$T_{34}^{TC}=\frac{R_{34}}{V_{TC}}\text{\hspace{1em}(20)}$$

representing the time required for the translating storm to move one $R_{34}$.

A wave-following exposure timescale is

$$T_{34}^w\left(f,\theta \right)=\frac{L_{34}}{\left|{\mathbf{c}}_g(f,\theta )+{\mathbf{U}}_c-{\mathbf{V}}_{TC}\right|}\text{\hspace{1em}(21)}$$

where $L_{34}$ is the path length of the relevant wave ray through the region enclosed by the 34-kt wind contour. In the absence of a reconstructed contour intersection, $L_{34}$ may initially be approximated as $R_{34}$ or $2R_{34}$, depending on whether the ray originates within or crosses the full wind footprint.

A dimensionless elapsed history is then

$${\tau }_{34}^{\ast }=\frac{\tau }{T_{34}^{TC}}=\frac{\tau V_{TC}}{R_{34}}\text{\hspace{1em}(22)}$$

or, for a wave-following formulation,

$${\tau }_w^{\ast }=\frac{\tau \left|{\mathbf{c}}_g+\mathbf{U}c-\mathbf{V}TC\right|}{L_{34}}\text{\hspace{1em}(23)}$$

These normalized histories permit comparison across storms of different sizes and translation speeds.

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5. Wind-input representation

5.1 Physical wind-input source term

Wind input to a spectral component is commonly represented as

$$S_{\mathrm{in}}\left(f,\theta \right)=\beta \left(f,\theta \right)\sigma E\left(f,\theta \right)\text{\hspace{1em}(24)}$$

where $\beta$ is a nondimensional growth rate dependent on wind–wave relative speed, alignment, atmospheric stability, wave steepness, and the selected source-term parameterization.

A generic directional dependence can be expressed through

$$\chi =\frac{U_{\mathrm{eff}}}{c_p}\cos \left({\theta }_U-\theta \right)\text{\hspace{1em}(25)}$$

where ${\theta }_U$ is the wind-to direction and $U_{\mathrm{eff}}$ is either $U_{10}$, a friction-velocity-scaled wind, or another effective wind used by the source-term package.

For example, the ST6 formulation implemented in WAVEWATCH III includes a favorable-wind factor of the form

$$W_{+}=\stackrel{2}{\max }\left[0,\frac{U_{\mathrm{eff}}}{c_p}\cos (\theta -{\theta }_U)-1\right]\text{\hspace{1em}(26)}$$

with separate treatment of adverse winds. The full ST6 input also depends on spectral saturation and directional narrowness. (polar.ncep.noaa.gov)

Equation (24) cannot generally be reconstructed exactly from wind observations alone because it depends on the evolving spectrum. Nevertheless, it motivates reduced exposure metrics that retain wind magnitude, wave age, and directional alignment.

5.2 Wave-supported stress

The momentum transferred from the atmosphere into the wave field is

$${\tau }_w={\rho }_{w}g{\int }_0^{2\pi }\int \frac{\mathbf{k}}{\sigma }{S}_{\mathrm{in}}\left(k,\theta \right)dkd\theta \text{\hspace{1em}(27)}$$

where ${\rho }_w$ is seawater density. The total atmospheric stress is partitioned into viscous or turbulent stress and wave-supported stress:

$${\tau }_{\mathrm{tot}}={\tau }_v+{\tau }_w\text{\hspace{1em}(28)}$$

This establishes the direct connection between spectral forcing history and drag: the relevant history is not merely accumulated wind speed, but the accumulated, directionally resolved momentum input to the evolving spectrum. (polar.ncep.noaa.gov)

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6. Reduced wind-history metrics

Because complete source-term information is not available from observations alone, a hierarchy of reduced history metrics should be evaluated.

6.1 Fixed-window wind statistics

For a history window (T), the time-weighted mean wind is

$${\overline{U}}_T(t_o)=\frac{1}{T}{\int }_{t_o-T}^{t_o}{U}_{10}(t^{\prime })d{t}^{\prime }\text{\hspace{1em}(29)}$$

Additional statistics include

$$U_{\max ,T}=\underset{t_o-T\le t^{\prime }\le t_o}{\max }{U}_{10}(t^{\prime })\text{\hspace{1em}(30)}$$ $${\sigma }_{U,T}={\left[\frac{1}{T}{\int }_{t_o-T}^{t_o}{\left(U_{10}(t^{\prime })-{\overline{U}}_T\right)}^{2}d{t}^{\prime }\right]}^{1/2}\text{\hspace{1em}(31)}$$

and the wind tendency

$$G_{U,T}=\frac{U_{10}(t_o)-U_{10}(t_o-T)}{T}\text{\hspace{1em}(32)}$$

These metrics distinguish steady from rapidly increasing or decreasing forcing but do not preserve the full timing or direction of prior forcing.

6.2 Cumulative wind exposure

A generalized cumulative exposure is

$$I_p(T)={\int }_{t_o-T}^{t_o}{U}_{10}(t^{\prime }{)}^p,d{t}^{\prime },\text{\hspace{1em}(33)}$$

where $p$ controls the emphasis placed on high wind speeds.

Useful candidates are

$$I_1=\int U_{10}dt\text{\hspace{1em}(34)}$$ $$I_2=\int U_{10}^{2}dt\text{\hspace{1em}(35)}$$ $$I_3=\int U_{10}^{3}dt\text{\hspace{1em}(36)}$$

These metrics are empirical proxies rather than direct source terms. $I_1$ measures total wind-duration exposure; $I_2$ emphasizes kinetic-energy-like scaling; and $I_3$ provides a wind-power-like weighting. None should be interpreted directly as wave-energy input without accounting for the spectrum and wind–wave relative speed.

To compare windows of unequal length, define

$${\tilde{I}}_p(T)={\left[\frac{1}{T}{I}_p(T)\right]}^{1/p}\text{\hspace{1em}(37)}$$

This has units of wind speed and may be interpreted as a generalized history-weighted effective wind.

6.3 Thresholded exposure

The influence of extended moderate winds may be isolated using

$$I_{p,U_c}(T)={\int }_{t_o-T}^{t_o}{\left[\max \left(U_{10}(t^{\prime })-U_c,0\right)\right]}^{p}d{t}^{\prime }.\text{\hspace{1em}(38)}$$

Candidate thresholds should include

$$U_c\in 10,15,17.5,20,25\mathrm{m},{\mathrm{s}}^{-1}\text{\hspace{1em}(39)}$$

The choice $U_c=17.5\mathrm{m},{\mathrm{s}}^{-1}$ approximately corresponds to 34 kt and provides a local analogue to the $R_{34}$-based storm-size result of Oh et al.

A duration-above-threshold metric is

$$D_{U_c}(T)={\int }_{t_o-T}^{t_o}\mathbb{I}\left[U_{10}(t^{\prime })>U_c\right]d{t}^{\prime },\text{\hspace{1em}(40)}$$

where $\mathbb{I}$ is an indicator function.

6.4 Exponentially weighted history

A finite-memory forcing metric can be defined as

$$I_{p,\lambda }(t_o)={\int }_{-\infty }^{t_o}{U}_{10}(t^{\prime }{)}^p\exp \left[-\frac{t_o-t^{\prime }}{\lambda }\right]d{t}^{\prime }\text{\hspace{1em}(41)}$$

where $\lambda$ is an e-folding memory timescale.

The normalized form is

$${\overline{I}}_{p,\lambda }=\frac{1}{\lambda }{I}_{p,\lambda }\text{\hspace{1em}(42)}$$

Equation (41) avoids the abrupt cutoff of a boxcar window and is equivalent to the prognostic equation

$$\frac{d{I}_{p,\lambda }}{dt}=U_{10}^p-\frac{I_{p,\lambda }}{\lambda }\text{\hspace{1em}(43)}$$

Multiple values of $\lambda$ should be tested because the appropriate memory is expected to differ between the high-frequency tail and the spectral peak.

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7. Directional forcing history

7.1 Wind vectors

Using wind-to convention, define

$${\mathbf{U}}_{10}=U_{10}\left[\begin{array}{c}\cos {\theta }_U\\ \sin {\theta }_U\end{array}\right]\text{\hspace{1em}(44)}$$

For a target wave direction ${\theta }_w$ define the along-wave and cross-wave wind components:

$$U_{\parallel }=U_{10}\cos \left({\theta }_U-{\theta }_w\right)\text{\hspace{1em}(45)}$$ $$U_{\perp }=U_{10}\sin \left({\theta }_U-{\theta }_w\right)\text{\hspace{1em}(46)}$$

The wave direction may be defined separately for

• the spectral peak, ${\theta }_p$
• the frequency-integrated mean, $\overline{\theta }$
• the high-frequency band, ${\theta }_{\mathrm{HF}}$
• each individual spectral frequency, $\theta (f)$.

7.2 Favorable aligned exposure

A positive aligned-exposure metric is

$$I_{\parallel ,p}^{+}(T)={\int }_{t_o-T}^{t_o}{U}_{10}(t^{\prime }{)}^p\max \left[\cos \left({\theta }_U(t^{\prime })-{\theta }_w\right),0\right]d{t}^{\prime }\text{\hspace{1em}(47)}$$

A more strongly directional metric is

$$I_{\parallel ,p}^{(m)}(T)={\int }_{t_o-T}^{t_o}{U}_{10}(t^{\prime }{)}^p\max {\left[\cos \left({\theta }_U(t^{\prime })-{\theta }_w\right),0\right]}^{m}d{t}^{\prime }\text{\hspace{1em}(48)}$$

where $m=1,2,$ or $4$ controls directional selectivity.

The signed aligned history is

$$I_{\parallel ,p}^{\mathrm{signed}}(T)={\int }_{t_o-T}^{t_o}{U}_{10}(t^{\prime }{)}^p\cos \left({\theta }_U(t^{\prime })-{\theta }_w\right)d{t}^{\prime }\text{\hspace{1em}(49)}$$

Unlike Equation (47), Equation (49) permits adverse winds to subtract from favorable forcing.

The cross-wave history is

$$I_{\perp ,p}(T)={\int }_{t_o-T}^{t_o}{U}_{10}(t^{\prime }{)}^p\left|\sin \left({\theta }_U(t^{\prime })-{\theta }_w\right)\right|d{t}^{\prime }\text{\hspace{1em}(50)}$$

A normalized directional forcing ratio is

$$A_p(T)=\frac{I_{\parallel ,p}^{+}}{I_{\parallel ,p}^{+}+I_{\perp ,p}}\text{\hspace{1em}(51)}$$

Values approaching one indicate persistently along-wave forcing; smaller values indicate substantial cross-wave forcing.

7.3 Vector-integrated wind history

The vector exposure is

$${\mathbf{I}}_p(T)={\int }_{t_o-T}^{t_o}{U}_{10}(t^{\prime }{)}^{p-1}{\mathbf{U}}_{10}(t^{\prime }),d{t}^{\prime }\text{\hspace{1em}(52)}$$

Its direction is

$${\theta }_{\mathcal{H}}=atan2\left(I_{p,y},I_{p,x}\right)\text{\hspace{1em}(53)}$$

and its magnitude is

$$I_{p,\mathrm{vec}}=\left|{\mathbf{I}}_p\right|\text{\hspace{1em}(54)}$$

A directional persistence index is

$$P_p(T)=\frac{\left|{\mathbf{I}}_p(T)\right|}{{\int }_{t_o-T}^{t_o}{U}_{10}(t^{\prime }{)}^{p}d{t}^{\prime }}\text{\hspace{1em}(55)}$$

This metric satisfies

$$0\le P_p\le 1$$

A value near one indicates a nearly constant forcing direction, whereas a low value indicates substantial wind rotation or cancellation during the history window.

The alignment between the accumulated wind direction and the observed wave direction is

$$\Delta {\theta }_{\mathcal{H},w}=\mathrm{wrap}\left({\theta }_{\mathcal{H}}-{\theta }_w\right)\text{\hspace{1em}(56)}$$

Equations (55)–(56) are particularly relevant to high-frequency directional spread because they separate the magnitude of accumulated forcing from the directional coherence of that forcing.

7.4 Circular directional variability

The wind-direction concentration is

$$R_{\theta }(T)=\left|\frac{{\int }_{t_o-T}^{t_o}w(t^{\prime })e^{i{\theta }_U(t^{\prime })}d{t}^{\prime }}{{\int }_{t_o-T}^{t_o}w(t^{\prime })d{t}^{\prime }}\right|\text{\hspace{1em}(57)}$$

where $w(t^{\prime })$ may equal $1$, $U_{10}$, $U_{10}^2$, or a thresholded wind weight.

The circular standard deviation is

$${\sigma }_{\theta ,U}=\sqrt{-2\ln R_{\theta }}\text{\hspace{1em}(58)}$$

These quantities measure whether the wave field has recently experienced a stable or rotating wind direction without the discontinuity associated with conventional angular variance.

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8. Wave-age-dependent forcing history

Wind input depends not only on wind speed but also on the relative wind–wave speed. For a spectral component,

$$\mathrm{WA}(f)=\frac{c_p(f)}{U_{10}}\text{\hspace{1em}(59)}$$

or, using friction velocity,

$${\mathrm{WA}}_{\ast }(f)=\frac{c_p(f)}{u_{\ast }}\text{\hspace{1em}(60)}$$

The inverse wave age is

$$\mathrm{IWA}(f)=\frac{U_{10}}{c_p(f)}\text{\hspace{1em}(61)}$$

A directionally projected inverse wave age is

$${\mathrm{IWA}}_{\parallel }=\left(f,\theta ,t\right)\frac{U_{10}(t)}{c_p(f)}\cos \left[{\theta }_U(t)-\theta \right]\text{\hspace{1em}(62)}$$

Wind input is favorable when the projected wind exceeds the phase speed. A simple excess-speed proxy is

$$X(f,\theta ,t)=\max \left[\frac{U_{10}(t)}{c_p(f)}\cos \left({\theta }_U-\theta \right)-1,0\right]\text{\hspace{1em}(63)}$$

The corresponding accumulated forcing proxy is

$$I_X(f,\theta ,T)={\int }_{t_o-T}^{t_o}X(f,\theta ,t^{\prime }{)}^m{U}_{10}(t^{\prime }{)}^{p}d{t}^{\prime }.\text{\hspace{1em}(64)}$$

Equation (64) is more physically specific than $I_p$ because forcing contributes only when the wind component in the wave direction exceeds the wave phase speed. Its structure is motivated by spectral wind-input formulations such as ST6, although it should not be interpreted as the exact ST6 source term. (polar.ncep.noaa.gov)

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9. Frequency-dependent wave history

Because phase speed and group velocity depend on frequency, the forcing history should be evaluated separately for spectral bands.

For a frequency band $B=[f_1,f_2]$, define its energy-weighted direction as

$${\theta }_B=atan2\left[{\int }_{f_1}^{f_2}\int E(f,\theta )\sin \theta d\theta df,{\int }_{f_1}^{f_2}\int E(f,\theta )\cos \theta d\theta df\right]\text{\hspace{1em}(65)}$$

The representative group velocity may be defined as

$${\overline{\mathbf{c}}}_{g,B}=\frac{{\int }_{f_1}^{f_2}\int {\mathbf{c}}_g(f,\theta )E(f,\theta )d\theta df}{{\int }_{f_1}^{f_2}\int E(f,\theta )d\theta df}\text{\hspace{1em}(66)}$$

At minimum, histories should be calculated separately for:

$$B_p:\text{spectral peak neighborhood}$$ $$B_m:\text{intermediate-frequency wind sea}$$ $$B_{\mathrm{HF}}:0.2\text{–}0.5\mathrm{Hz}$$

The band-specific ray history is

$${\mathcal{H}}_B(T)={\int }_{t_o-T}^{t_o}\mathcal{F}\left[{\mathbf{U}}_{10}\left({\mathbf{x}}_{r,B}(t^{\prime }),t^{\prime }\right),E_B(t^{\prime }),{\theta }_B(t^{\prime })\right]d{t}^{\prime },\text{\hspace{1em}(67)}$$

where $\mathcal{F}$ denotes the selected wind-input proxy.

High-frequency waves have small $c_g$, remain comparatively local, and should generally be tested with shorter memory windows. Peak waves have larger $c_g$ and may contain forcing history from earlier times and different storm sectors.

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10. Effective fetch and duration

Classical wind-wave development can be represented using dimensionless wave height, peak frequency, fetch, and duration. Following the formulation summarized by Hwang and Fan, define

$${\eta }^{\ast }=\frac{H_s^2{g}^2}{16U_{10}^4}\text{\hspace{1em}(68)}$$ $${\omega }^{\ast }=\frac{2\pi U_{10}}{T_{p}g}\text{\hspace{1em}(69)}$$ $$x^{\ast }=\frac{x_{f}g}{U_{10}^2}\text{\hspace{1em}(70)}$$ $$t^{\ast }=\frac{t_{d}g}{U_{10}}\text{\hspace{1em}(71)}$$

Their fetch-limited growth relations are

$$\frac{H_s^2{g}^2}{16U_{10}^4}=6.19\times 10^{-7}{\left(\frac{x_{f}g}{U_{10}^2}\right)}^{0.81}\text{\hspace{1em}(72)}$$ $$\frac{2\pi U_{10}}{T_{p}g}=11.86{\left(\frac{x_{f}g}{U_{10}^2}\right)}^{-0.24}\text{\hspace{1em}(73)}$$

Their duration-limited relations are

$$\frac{H_s^2{g}^2}{16U_{10}^4}=1.27\times 10^{-8}{\left(\frac{t_{d}g}{U_{10}}\right)}^{1.06}\text{\hspace{1em}(74)}$$ $$\frac{2\pi U_{10}}{T_{p}g}=2.94{\left(\frac{t_{d}g}{U_{10}}\right)}^{-0.34}\text{\hspace{1em}(75)}$$

These relations show that effective fetch $x_f$ and effective duration $t_d$ can be interpreted as compressed measures of prior wave forcing. Hwang and Fan found that retrieved effective fetch and duration varied approximately linearly with storm-relative radius, with slope and intercept varying systematically with azimuth relative to storm heading. (www7320.nrlssc.navy.mil)

Effective fetch inferred from significant wave height is

$$x_{f,H}=4.24\times 10^7{U}_{10}^{-2.93}{H}_s^{2.47}\text{\hspace{1em}(76)}$$

and that inferred from peak period is

$$x_{f,T}=2.29\times 10^4{U}_{10}^{-2.22}{T}_p^{4.22}\text{\hspace{1em}(77)}$$

Similarly, effective duration estimates are

$$t_{d,H}=1.75\times 10^4{U}_{10}^{-2.77}{H}_s^{3.77}\text{\hspace{1em}(78)}$$ $$t_{d,T}=4.81\times 10^4{U}_{10}^{-2.22}{T}_p^{3.22}\text{\hspace{1em}(79)}$$

These inversions provide observation-derived history diagnostics. Differences such as

$$\Delta x_f=x_{f,H}-x_{f,T}\text{\hspace{1em}(80)}$$

or

$$\Delta t_d=t_{d,H}-t_{d,T}\text{\hspace{1em}(81)}$$

may indicate mixed seas, nonstationary forcing, swell contamination, or departure from classical one-dimensional wind-wave growth.

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11. Self-similar tropical-cyclone coordinates

Kudryavtsev et al. showed that tropical-cyclone wave development can be represented in a self-similar storm-following framework in which wave rays and spectral properties are scaled by characteristic storm size, wind intensity, and translation speed. The precise analytical solution depends on the prescribed radial wind profile, but the relevant dimensionless variables may be written generally as

$${\mathbf{r}}^{\ast }=\frac{\mathbf{x}-{\mathbf{x}}_{TC}}{R_s}\text{\hspace{1em}(82)}$$ $${\mathbf{V}}^{\ast }=\frac{{\mathbf{V}}_{TC}}{U_{\max }}\text{\hspace{1em}(83)}$$ $${\mathbf{c}}_g^{\ast }=\frac{{\mathbf{c}}_g}{U_{\max }}\text{\hspace{1em}(84)}$$ $$E^{\ast }=\frac{g^{2}E}{U_{\max }^4}\text{\hspace{1em}(85)}$$ $$f^{\ast }=\frac{f{U}_{\max }}{g}\text{\hspace{1em}(86)}$$

where ${\mathbf{x}}_{TC}$ is the storm-center position, ${\mathbf{V}}_{TC}$ is the storm translation velocity, ${\mathbf{c}}_g$ is the intrinsic wave group velocity, and $U_{\max }$ is the maximum storm wind speed. The maximum wind speed is retained as the characteristic velocity scale in all cases because it represents the amplitude of the storm forcing and preserves the intensity dependence of the nondimensional wave energy, frequency, translation speed, and group velocity.

Two alternative choices for the characteristic radial scale $R_s$ are considered. The first is the radius of maximum wind,

$$R_s=R_{\mathrm{MW}}$$

which gives the inner-core coordinate

$${\mathbf{r}}_{\mathrm{MW}}^{\ast }=\frac{\mathbf{x}-{\mathbf{x}}_{TC}}{R_{\mathrm{MW}}}\text{\hspace{1em}(87)}$$

This scaling is the most direct analogue of the self-similar framework used by Kudryavtsev et al., because $R_{\mathrm{MW}}$ and $U_{\max }$ jointly describe the location and magnitude of the wind maximum in the prescribed radial wind profile. Analysis in ${\mathbf{r}}_{\mathrm{MW}}^{\ast }$ therefore tests whether wave development collapses according to inner-core storm structure. Successful collapse under this scaling would indicate that the dominant organization of the wave field is controlled by distance relative to the wind maximum and by storm intensity.

The second choice is the radius of 34-kt winds,

$$R_s=R_{34},$$

which gives the outer-forcing coordinate

$${\mathbf{r}}_{34}^{\ast }=\frac{\mathbf{x}-{\mathbf{x}}_{TC}}{R_{34}}\text{\hspace{1em}(88)}$$

This coordinate measures position relative to the broader tropical-storm-force wind footprint rather than relative to the inner-core wind maximum. It is therefore expected to be more directly relevant to cumulative wave exposure, because $R_{34}$ characterizes the radial extent over which appreciable wind forcing can act on the wave field. A stronger collapse in ${\mathbf{r}}_{34}^{\ast }$ would support the interpretation that wave history is governed primarily by the spatial extent and residence time of the forcing footprint rather than by inner-core structure alone.

The use of $R_{34}$ does not require replacing $U_{\max }$ with the 34-kt threshold speed. A fixed velocity scale of approximately $17.5\mathrm{m},{\mathrm{s}}^{-1}$ would describe only the contour used to define $R_{34}$ and would remove much of the storm-to-storm intensity dependence from the nondimensionalization. The hybrid scaling

$$R_s=R_{34},U_s=U_{\max }$$

instead separates storm size from storm intensity: $R_{34}$ measures the outer forcing footprint, while $U_{\max }$ measures the amplitude of the forcing.

The storm-relative propagation velocity of a wave packet is

$${\mathbf{c}}_{g,\mathrm{rel}}={\mathbf{c}}_g+{\mathbf{U}}_c-{\mathbf{V}}_{TC}\text{\hspace{1em}(89)}$$

where ${\mathbf{U}}_c$ is the ocean-current velocity. Its nondimensional form is

$${\mathbf{c}}_{g,\mathrm{rel}}^{\ast }=\frac{{\mathbf{c}}_g+\mathbf{U}c-\mathbf{V}TC}{U_{\max }}\text{\hspace{1em}(90)}$$

This quantity describes how rapidly and in what direction wave energy moves through the storm-relative forcing field. Small values of $\left|{\mathbf{c}}_{g,\mathrm{rel}}\right|$ indicate that a wave packet remains near the same location relative to the translating storm and may therefore experience prolonged forcing. Large values indicate that the wave packet either escapes from or is overtaken by the storm more rapidly.

A complementary scalar trapping parameter may be defined using the component of the wave group velocity along the storm-translation direction:

$$\Gamma (f,\theta )=\frac{\left({\mathbf{c}}_g(f,\theta )+\mathbf{U}c\right)\cdot \hat{\mathbf{V}}TC}{\left|{\mathbf{V}}_{TC}\right|}\text{\hspace{1em}(91)}$$

where

$${\hat{\mathbf{V}}}_{TC}=\frac{{\mathbf{V}}_{TC}}{\left|{\mathbf{V}}_{TC}\right|}$$

is the unit vector in the storm-translation direction. The parameter $\Gamma$ compares the Earth-relative propagation speed of the wave packet along the storm track with the storm translation speed.

Its interpretation is

$$\Gamma <1:\text{the storm overtakes the wave packet},$$ $$\Gamma \approx 1:\text{the wave packet and storm translate at similar speeds},$$ $$\Gamma >1:\text{the wave packet propagates ahead of the storm}.$$

The regime $\Gamma \approx 1$ corresponds to near-synchronous propagation and provides a kinematic measure of potential wave trapping. In this regime, wave energy remains within a similar storm-relative forcing environment for an extended period, increasing the opportunity for cumulative wind input and asymmetric wave growth. However, $\Gamma$ alone does not guarantee sustained forcing, because trapping also depends on propagation direction, the finite spatial extent of the wind field, storm curvature, and temporal changes in the wind structure.

For peak-wave diagnostics, the trapping parameter may be evaluated as

$${\Gamma }_p=\frac{\left({\mathbf{c}}_{g,p}+\mathbf{U}c\right)\cdot \hat{\mathbf{V}}TC}{\left|{\mathbf{V}}_{TC}\right|}\text{\hspace{1em}(92)}$$

where ${\mathbf{c}}_{g,p}$ is the group velocity at the spectral peak. For deep-water waves,

$$c_{g,p}=\frac{g{T}_p}{4\pi }\text{\hspace{1em}(93)}$$

Because $\Gamma$ is frequency- and direction-dependent, a storm cannot be classified as universally slow or fast relative to the entire wave field. The same storm may overtake short, young waves while moving synchronously with longer peak waves or being outrun by mature swell. Consequently, storm translation should be retained as a continuous control through

$$\frac{\left|{\mathbf{V}}_{TC}\right|}{U_{\max }}$$

while trapping should be evaluated separately through $\Gamma (f,\theta )$ or ${\Gamma }_p$.

The two radial scalings therefore test distinct physical hypotheses. The $R_{\mathrm{MW}}$ scaling evaluates whether wave development is organized by inner-core structure and the location of maximum forcing. The $R_{34}$ scaling evaluates whether wave development is organized by the broader forcing footprint and cumulative exposure. In both cases, the trapping parameter determines whether a given wave component remains within that scaled forcing region long enough to accumulate substantial wind input.

The central practical implication is that wave history should be compared across storms using storm-scaled trajectory, exposure, and propagation coordinates rather than raw elapsed time and distance alone. A complete comparison should therefore retain both

$$\frac{\mathbf{x}-{\mathbf{x}}_{TC}}{R_{\mathrm{MW}}}\text{and}\frac{\mathbf{x}-{\mathbf{x}}_{TC}}{R_{34}}$$

together with

$$\frac{{\mathbf{V}}_{TC}}{U_{\max }},{\mathbf{c}}_{g,\mathrm{rel}}^{\ast },{\Gamma }_p$$

Comparing the degree of cross-storm collapse obtained under the two radial scalings provides a direct test of whether observed wave evolution is governed more strongly by inner-core similarity, outer-storm exposure, or wave-relative residence within the translating wind field.

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12. Separating local wind sea from advected swell

A wave component should not be interpreted as locally forced solely because it is observed within the tropical cyclone. A first-order wind-sea classification may use the projected wave age:

$$\frac{c_p(f)}{U_{10}\cos ({\theta }_U-\theta )}\text{\hspace{1em}(88)}$$

Components for which

$$U_{10}\cos ({\theta }_U-\theta )c_p(f)\text{\hspace{1em}(89)}$$

are candidates for active local wind input. Components with

$$U_{10}\cos ({\theta }_U-\theta )\le c_p(f)\text{\hspace{1em}(90)}$$

are likely weakly forced, mature, or swell-like relative to the current wind.

For each observation, define an actively forced energy fraction:

$$F_{\mathrm{active}}=\frac{\int {\int }_{\mathcal{A}}E(f,\theta )d\theta df}{\int \int E(f,\theta )d\theta df}\text{\hspace{1em}(91)}$$

where

$$\mathcal{A}=\left\{(f,\theta ):U_{10}\cos ({\theta }_U-\theta )>c_p(f)\right\}.\text{\hspace{1em}(92)}$$

Similarly, the swell-like fraction is

$$F_{\mathrm{swell}}=1-F_{\mathrm{active}}\text{\hspace{1em}(93)}$$

These classifications are approximate because spectral input does not transition discontinuously at one threshold, but they help distinguish locally responsive high-frequency energy from older propagating systems.

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13. Memory kernels estimated from observations

Rather than prescribing a history window, the effective memory kernel may be estimated statistically. Let $Y(t)$ be a wave-state diagnostic such as

$$Y\in R_{\mathrm{HF}},\mathrm{MSS},\mathrm{HFDS},\text{alignment},f_p,H_s$$

A distributed-lag model is

$$Y(t)=\alpha +{\int }_0^{T_{\max }}{K}_Y(\tau )F(t-\tau )d\tau +ϵ(t)\text{\hspace{1em}(94)}$$

where $F$ is a forcing metric and $K_Y(\tau )$ is the response kernel.

For discrete data,

$$Y_i=\alpha +\sum _{j=0}^J{K}_j{F}_{i-j}+{ϵ}_i\text{\hspace{1em}(95)}$$

Because neighboring lag coefficients are strongly collinear, $K_j$ should be estimated using regularization or a low-dimensional basis:

$$K_Y(\tau )=\sum _{m=1}^M{a}_m{B}_m(\tau )\text{\hspace{1em}(96)}$$

where $B_m$ may be spline, exponential, or Gaussian basis functions.

A normalized kernel is

$${\tilde{K}}_Y(\tau )=\frac{K_Y(\tau )}{{\int }_0^{T_{\max }}|K_Y(s)|ds}\text{\hspace{1em}(97)}$$

The kernel centroid provides an effective lag:

$${\tau }_{\mathrm{eff},Y}=\frac{{\int }_0^{T_{\max }}\tau |K_Y(\tau )|d\tau }{{\int }_0^{T_{\max }}|K_Y(\tau )|d\tau }\text{\hspace{1em}(98)}$$

and its width provides a memory duration:

$${\sigma }_{\tau ,Y}={\left[\frac{\int (\tau -{\tau }_{\mathrm{eff},Y}{)}^2|K_Y(\tau )|d\tau }{\int |K_Y(\tau )|d\tau }\right]}^{1/2}\text{\hspace{1em}(99)}$$

This approach allows the HF tail, spectral peak, and bulk wave field to exhibit different empirically inferred memory timescales.

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14. Source-term attribution

Wind-history metrics cannot by themselves uniquely identify the physical process responsible for a spectral change. From Equation (8),

$$\Delta N=\Delta N_{\mathrm{in}}+\Delta N_{\mathrm{nl}}+\Delta N_{\mathrm{ds}}+\Delta N_{\mathrm{prop}}+\cdots \text{\hspace{1em}(100)}$$

If spectral-model source terms are available, process-specific accumulated histories can be calculated as

$${\mathcal{H}}_{\mathrm{in}}={\int }_{t_i}^{t_o}\frac{S_{\mathrm{in}}}{\sigma }dt\text{\hspace{1em}(101)}$$ $${\mathcal{H}}_{\mathrm{nl}}={\int }_{t_i}^{t_o}\frac{S_{\mathrm{nl}}}{\sigma }dt\text{\hspace{1em}(102)}$$ $${\mathcal{H}}_{\mathrm{ds}}={\int }_{t_i}^{t_o}\frac{S_{\mathrm{ds}}}{\sigma }dt\text{\hspace{1em}(103)}$$

Band-integrated contributions are

$${\mathcal{H}}_{j,B}={\int }_{t_i}^{t_o}{\int }_B\int \frac{S_j(f,\theta ,t)}{\sigma }d\theta dfdt,\text{\hspace{1em}(104)}$$

where $j$ denotes wind input, nonlinear interactions, or dissipation.

A normalized process contribution is

$$C_{j,B}=\frac{{\mathcal{H}}_{j,B}}{{\sum }_l\left|{\mathcal{H}}_{l,B}\right|}\text{\hspace{1em}(105)}$$

These diagnostics provide the cleanest method for distinguishing wind forcing, nonlinear redistribution, and dissipation. Without source-term output, such attribution remains inferential.

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15. Current-modified history

Currents modify wave history through both advection and the intrinsic frequency:

$$\omega =\sigma +\mathbf{k}\cdot {\mathbf{U}}_c\text{\hspace{1em}(106)}$$

where $\omega$ is the absolute angular frequency.

Current gradients alter wavenumber and direction along the ray, while current velocity changes the propagation pathway through Equation (9). A current-relative wind may also be defined as

$${\mathbf{U}}_{\mathrm{rel}}={\mathbf{U}}_{10}-{\mathbf{U}}_c\text{\hspace{1em}(107)}$$

Wind-input proxies should then use

$$U_{\mathrm{rel}}=|{\mathbf{U}}_{\mathrm{rel}}|\text{\hspace{1em}(108)}$$

and

$${\theta }_{\mathrm{rel}}=\arg \left({\mathbf{U}}_{\mathrm{rel}}\right)\text{\hspace{1em}(109)}$$

in place of earth-relative wind. Comparisons between histories constructed with ${\mathbf{U}}_{10}$ and ${\mathbf{U}}_{\mathrm{rel}}$ provide a first-order test of current influence.

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16. Rain exposure

Rain may affect short waves through direct momentum transfer, enhanced turbulence, splash-generated disturbances, and damping or modification of short-scale roughness. Because the dominant mechanism and scaling are uncertain, rain should initially be represented using empirical exposure metrics rather than folded directly into the wind-input term.

For rain rate $P(t)$,

$$I_P(T)={\int }_{t_o-T}^{t_o}P(t^{\prime })d{t}^{\prime },\text{\hspace{1em}(110)}$$ $$I_{P,q}(T)={\int }_{t_o-T}^{t_o}P(t^{\prime }{)}^{q}d{t}^{\prime },\text{\hspace{1em}(111)}$$

and

$$D_{P_c}(T)={\int }_{t_o-T}^{t_o}\mathbb{I}\left[P(t^{\prime })>P_c\right]d{t}^{\prime }\text{\hspace{1em}(112)}$$

A conditional wind-history model may be written as

$$Y=\alpha +{\beta }_H{\mathcal{H}}_U+{\beta }_{P}P+{\beta }_{HP}{\mathcal{H}}_{U}P+ϵ\text{\hspace{1em}(113)}$$

where the interaction term tests whether the relationship between wind history and the wave field changes under heavy rain.

Rain-related conclusions should remain cautious because precipitation observations and modeled rain rates can have substantial spatial and temporal uncertainty.

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17. Discrete implementation

For model output at irregular or storm-dependent time intervals, all history integrals should be evaluated with explicit timestep weighting. For a generic function $F_j$,

$$\mathcal{H}(t_i)\approx \sum _{j=1}^{n-1}\frac{F_j+F_{j+1}}{2}\Delta t_j\text{\hspace{1em}(114)}$$

where

$$\Delta t_j=t_{j+1}-t_j$$

This permits direct comparison of histories calculated from 15-min and hourly model snapshots, provided the underlying process varies sufficiently slowly relative to the sampling interval.

For an exposure metric,

$$I_p(t_i)\approx \sum _j\frac{U_j^p+U_{j+1}^p}{2}\Delta t_j\text{\hspace{1em}(115)}$$

For vector exposure,

$${\mathbf{I}}_p(t_i)\approx \sum _j\frac{U_j^{p-1}\mathbf{U}j+U_{j+1}^{p-1}{\mathbf{U}}_{j+1}}{2}\Delta t_j\text{\hspace{1em}(116)}$$

The hourly Ian winds can therefore be included in histories longer than approximately 1–2 h, but they cannot resolve subhourly wind rotation, eyewall transitions, or short HF-tail adjustment. Interpolating the hourly snapshots to 15-min resolution may facilitate numerical integration but does not add independent information and should not be interpreted as resolving subhourly variability.

17.1 History completeness

For a requested window $T$, define the available historical duration

$$T_{\mathrm{avail}}=t_o-t_{\mathrm{first}}\text{\hspace{1em}(117)}$$

The history-completeness fraction is

$$C_T=\min \left(\frac{T_{\mathrm{avail}}}{T},1\right)\text{\hspace{1em}(118)}$$

Only observations with

$$C_T=1$$

should be used in the primary analysis for that window. Incomplete observations may be retained in sensitivity tests but should not be normalized upward as though the missing forcing were known.

For wave rays, completeness should additionally account for whether the traced pathway remains inside the model domain and time interval.

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18. Recommended hierarchy of history representations

The following sequence should be used to determine how much physical complexity is required.

Level 0: instantaneous forcing

$$U_{10}(t_o),{\theta }_U(t_o),\frac{c_p}{U_{10}},{\theta }_U-{\theta }_w\text{\hspace{1em}(119)}$$

Level 1: Eulerian scalar history

$${\overline{U}}_T,I_p(T),I_{p,U_c}(T),G_{U,T}\text{\hspace{1em}(120)}$$

Level 2: Eulerian directional history

$$I_{\parallel ,p}^{+},I_{\perp ,p},P_p,\Delta {\theta }_{\mathcal{H},w}\text{\hspace{1em}(121)}$$

Level 3: storm-normalized history

$$r/R_{34},\tau V_{TC}/R_{34},Q_{\mathrm{trap}}\text{\hspace{1em}(122)}$$

Level 4: wave-following history

$$\mathcal{H}\left(f,\theta \right)=\int \mathcal{F}\left[{\mathbf{U}}_{10}\left({\mathbf{x}}_r(t^{\prime }),t^{\prime }\right),f,\theta \right]d{t}^{\prime }\text{\hspace{1em}(123)}$$

Level 5: full process history

$${\mathcal{H}}_{\mathrm{in}},{\mathcal{H}}_{\mathrm{nl}},{\mathcal{H}}_{\mathrm{ds}},{\mathcal{H}}_{\mathrm{current}},{\mathcal{H}}_{\mathrm{rain}}\text{\hspace{1em}(124)}$$

Each level should be evaluated against the same wave-state outcomes. Increased complexity is justified only if it improves explanatory or predictive performance under cross-validation and produces physically coherent relationships across storms and storm sectors.

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19. Primary diagnostics proposed for this study

The primary reduced diagnostics are:

$$\boxed{I_{2,17.5}(T)={\int }_{t_o-T}^{t_o}\max {\left[U_{10}(t^{\prime })-17.5,0\right]}^{2}d{t}^{\prime }}\text{\hspace{1em}(125)}$$

for cumulative exposure above approximately gale-force winds;

$$\boxed{I_{\parallel ,2}^{+}(T)={\int }_{t_o-T}^{t_o}{U}_{10}(t^{\prime }{)}^2\max \left[\cos \left({\theta }_U(t^{\prime })-{\theta }_{\mathrm{HF}}\right),0\right]d{t}^{\prime }}\text{\hspace{1em}(126)}$$

for cumulative forcing aligned with the high-frequency wave field;

$$\boxed{P_2(T)=\frac{\left|\int U_{10}(t^{\prime }){\mathbf{U}}_{10}(t^{\prime })d{t}^{\prime }\right|}{\int U_{10}(t^{\prime }{)}^{2}d{t}^{\prime }}}\text{\hspace{1em}(127)}$$

for wind-direction persistence;

$$\boxed{{\tau }_{34}^{\ast }=\frac{T{V}_{TC}}{R_{34}}}\text{\hspace{1em}(128)}$$

for storm-size-normalized history duration;

and

$$\boxed{Q_{\mathrm{trap}}=\frac{\left|{\mathbf{c}}_g+{\mathbf{U}}_c{\mathbf{V}}_{TC}\right|}{\left|{\mathbf{c}}_g+{\mathbf{U}}_c\right|+V_{TC}}}\text{\hspace{1em}(129)}$$

for wave–storm propagation matching.

These diagnostics should initially be computed for

$T=0.5,1,2,3,6,12\mathrm{h}$

subject to model resolution and history completeness. The 0.5-h window should be restricted to storms with 15-min winds; hourly Ian data should primarily be used for $T\ge 2$ h.

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20. Evaluation strategy

For each wave diagnostic (Y), compare nested models:

$$M_0:Y\sim U_{10}\text{\hspace{1em}(130)}$$ $$M_1:Y\sim U_{10}+{\mathcal{H}}_{\mathrm{scalar}}\text{\hspace{1em}(131)}$$ $$M_2:Y\sim U_{10}+{\mathcal{H}}_{\mathrm{scalar}}+{\mathcal{H}}_{\mathrm{directional}}\text{\hspace{1em}(132)}$$ $$M_3:Y\sim U_{10}+{\mathcal{H}}_{\mathrm{directional}}+r/R_{34}+{\tau }_{34}^{\ast }\text{\hspace{1em}(133)}$$ $$M_4:Y\sim {\mathcal{H}}_{\mathrm{ray}}+Q_{\mathrm{trap}}+\text{storm sector}.\text{\hspace{1em}(134)}$$

Models should include storm- and buoy-level effects where appropriate:

$$Y_{s,b,t}=\alpha +u_s+v_{s,b}+{\mathbf{\beta }}^T{\mathbf{X}}_{s,b,t}+{ϵ}_{s,b,t},\text{\hspace{1em}(135)}$$

where $u_s$ is a storm-level random intercept and $v_{s,b}$ is a buoy-within-storm effect.

Evaluation should use leave-one-storm-out cross-validation. This prevents a history metric from appearing successful merely because it encodes the temporal evolution of one storm.

The contribution of history is established if it:

1. improves held-out predictive skill relative to instantaneous wind
2. reduces unexplained trajectory spread
3. produces consistent coefficient signs across storms
4. retains physical ordering across storm quadrants
5. predicts future wave-state changes rather than merely covarying with the current state

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21. Interpretation

Under this framework, “wave history” is not a single variable. It comprises at least four distinguishable elements:

$$\text{wave history}=\text{forcing intensity}+\text{forcing duration}+\text{directional coherence}+\text{propagation pathway}.\text{\hspace{1em}(136)}$$

Storm size determines the spatial extent over which forcing can accumulate; storm translation modifies the relative residence time of wave packets beneath the wind field; group velocity makes that residence time frequency-dependent; and spectral source terms determine how the accumulated forcing is partitioned between growth, redistribution, and dissipation.

Accordingly, the central working hypothesis is:

The evolving tropical-cyclone wave state is controlled by cumulative, directionally coherent wind exposure along frequency-dependent propagation pathways through the translating storm wind field. Instantaneous local wind represents only the terminal value of this history.

The reduced history metrics provide testable approximations to this physical definition, while storm-relative wave-ray integration represents the closest observationally constrained approximation to the full action-balance history.

Goal: We approximate the unresolved Lagrangian forcing history of the observed wave field using a hierarchy of local, storm-normalized, and wave-ray-based wind-exposure metrics. These metrics are not intended to reproduce the complete spectral source-term balance, but to test whether cumulative and propagation-aware wind forcing explains wave-state variability beyond instantaneous local conditions.