Tropical-Cyclone Wave Normalization Framework

The normalization should separate four distinct aspects of the problem:

1. overall storm size;
2. azimuthal deformation of the wind field;
3. storm intensity and translation;
4. wave propagation relative to the evolving storm.

No single radius or velocity scale can represent all four. The approach is therefore to use one stable storm-scale coordinate system while retaining local directional structure and wave-relative propagation as separate dimensionless variables.

1. Storm-relative position

For a wave-ray component at position ${\mathbf{x}}_r(t)$, define the storm-relative displacement

$\mathbf{r}(t)={\mathbf{x}}_r(t)-{\mathbf{x}}_{TC}(t)$,

with radial distance

$r(t)=|\mathbf{r}(t)|$

and storm-relative azimuth

$\varphi (t)=atan2[r_y(t),r_x(t)]$.

This is calculated at every time step along the backward wave ray.

It answers:

Where was this wave component relative to the storm center at each point in its history?

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2. Directional $R_{34}$ boundary

Estimate the radius of the 34-kt wind contour at 16 azimuths,

$R_{34}({\varphi }_j,t),\Delta \varphi =22.5^{\circ }$.

At each azimuth:

1. sample the radial 10-m wind profile;
2. locate the 34-kt crossing by radial interpolation;
3. flag missing or ambiguous crossings;
4. fit a smooth periodic representation.

A low-order harmonic model is

$R_{34}(\varphi ,t)=a_0(t)+{\sum }_{n=1}^N\left[a_n(t)\cos (n\varphi )+b_n(t)\sin (n\varphi )\right]$,

with N=2 or 3 as a reasonable initial choice.

The harmonic terms represent:

$n=0:\text{mean storm size}$,

$n=1:\text{one-sided asymmetry}$,

$n=2:\text{ellipticity or two-sided deformation}$,

$n=3:\text{broader localized asymmetry}$.

The fitted coefficients should be weakly smoothed in time to prevent grid-scale changes in the wind threshold from creating artificial motion in the normalized coordinates.

The directional boundary answers:

How large was the actual gale-force wind envelope in the direction occupied by the wave ray?

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3. Area-equivalent storm radius

Use the directional boundary to define one scalar storm-size scale:

$R_{34,\mathrm{eq}}(t)=\sqrt{\frac{1}{2\pi }{\int }_0^{2\pi }{R}_{34}^2(\varphi ,t)d\varphi }$.

This is the radius of a circle with the same area as the fitted 34-kt wind envelope.

Use $R_{34,\mathrm{eq}}$ as the primary characteristic length scale:

$L_s(t)=R_{34,\mathrm{eq}}(t)$.

It answers:

What is the overall size of the storm’s gale-force wind footprint, independent of its azimuthal deformation?

This is preferable to the arithmetic mean of the directional radii because it preserves the approximate enclosed area.

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4. Two complementary normalized radial coordinates

Overall storm-size coordinate

$r_{\mathrm{eq}}^{\ast }(t)=\frac{r(t)}{R_{34,\mathrm{eq}}(t)}$.

This measures position relative to the overall storm size.

It answers:

How far from the center was the wave, measured in characteristic storm radii?

This is the preferred coordinate for interstorm comparison and self-similarity tests.

Local forcing-envelope coordinate

$r_{\mathrm{dir}}^{\ast }(t)=\frac{r(t)}{R_{34}[\varphi (t),t]}$.

This measures position relative to the local directional wind boundary.

Its interpretation is direct:

$r_{\mathrm{dir}}^{\ast }<1:\text{inside the local 34-kt envelope}$,

$r_{\mathrm{dir}}^{\ast }=1:\text{on the local 34-kt boundary}$,

$r_{\mathrm{dir}}^{\ast }>1:\text{outside the local 34-kt envelope}$.

It answers:

Was the wave component inside or outside the significant wind-forcing region on that side of the storm?

The two coordinates should both be retained because they answer different questions.

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5. Directional size anomaly

Separate storm size from azimuthal deformation using

$A_R(\varphi ,t)=\frac{R_{34}(\varphi ,t)}{R_{34,\mathrm{eq}}(t)}$.

Then

$r_{\mathrm{dir}}^{\ast }=\frac{r_{\mathrm{eq}}^{\ast }}{A_R}$.

Interpretation:

$A_R>1:\text{the wind envelope is broader than average in that direction}$,

$A_R<1:\text{the wind envelope is narrower than average}$.

It answers:

Is this wave ray located in an expanded or contracted part of the storm wind field?

This prevents storm asymmetry from being hidden entirely inside the radial normalization.

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6. Inner-core compactness

Retain the radius of maximum wind through the dimensionless compactness parameter

${\lambda }_R(t)=\frac{R_{\mathrm{mw}}(t)}{R_{34,\mathrm{eq}}(t)}$.

A smaller ${\lambda }_R$ indicates a more compact inner core relative to the outer wind envelope. A larger value indicates that the maximum-wind radius occupies a greater fraction of the storm.

It answers:

How concentrated is the dynamically active inner core relative to the total forcing footprint?

Use $R_{\mathrm{mw}}$ as a structural parameter rather than as the primary storm length scale because it can vary rapidly and may be less robustly resolved.

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7. Atmospheric velocity scale

Use

$U_s(t)=V_{\max }(t)$

as the primary storm-scale velocity.

This is appropriate for normalizing:

• storm translation;
• wave group velocity;
• currents;
• local wind relative to storm intensity.

It answers:

How fast is a process compared with the characteristic circulation speed of the storm?

However, $V_{\max }$ should not be treated as the actual forcing experienced by the wave. Retain the local or history-weighted wind separately.

Define

$U_{\mathrm{loc}}^{\ast }(t)=\frac{U_{10}[{\mathbf{x}}_r(t),t]}{V_{\max }(t)}$

or

$U_{\mathrm{eff}}^{\ast }=\frac{U_{\mathrm{eff}}}{V_{\max }}$.

This answers:

How strong was the wind acting on the wave relative to the storm’s maximum intensity?

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8. Characteristic storm timescale

Using the characteristic storm length and velocity,

$T_s(t)=\frac{R_{34,\mathrm{eq}}(t)}{V_{\max }(t)}$.

This is the atmospheric advective timescale associated with crossing one characteristic storm radius at the maximum-wind velocity.

It answers:

How rapidly does the storm circulation act relative to its own size?

For evolving storm scales, normalized elapsed time should be accumulated as

$d{\tau }^{\ast }=\frac{V_{\max }(t)}{R_{34,\mathrm{eq}}(t)}dt$,

so that backward look-back time is

${\tau }^{\ast }(t)={\int }_t^{t_o}\frac{V_{\max }(t^{\prime })}{R_{34,\mathrm{eq}}(t^{\prime })}d{t}^{\prime }$.

In discrete form,

$\Delta {\tau }_i^{\ast }=\Delta t_i\frac{V_{\max }(t_i)}{R_{34,\mathrm{eq}}(t_i)}$,

${\tau }_n^{\ast }={\sum }_{i=1}^n\Delta {\tau }_i^{\ast }$.

It answers:

How many instantaneous storm advective times separate an earlier ray position from the observation?

This is more physically consistent than simply dividing the total elapsed time by the observation-time value of $R_{34}/V_{\max }$.

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9. Storm translation scale

Define the nondimensional translation speed

$\mathcal{T}(t)=\frac{V_{TC}(t)}{V_{\max }(t)}$.

It answers:

How rapidly does the forcing pattern move compared with the storm circulation?

A small value indicates a slowly translating storm with potentially long local forcing duration. A larger value indicates stronger translation effects and greater front–back asymmetry in effective fetch and wave residence time.

A separate storm-passage timescale is

$T_{\mathrm{pass}}(t)=\frac{R_{34,\mathrm{eq}}(t)}{V_{TC}(t)}$.

It answers:

How long does the broad storm wind envelope take to pass a fixed location?

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10. Wave and current propagation scale

For each spectral ray component, calculate its storm-relative propagation velocity:

$\mathbf{q}(t)={\mathbf{c}}_g(t)+{\mathbf{U}}_c(t)-{\mathbf{V}}_{TC}(t)$.

Normalize it by the storm velocity scale:

${\mathbf{q}}^{\ast }(t)=\frac{{\mathbf{c}}_g(t)+{\mathbf{U}}_c(t)-{\mathbf{V}}_{TC}(t)}{V_{\max }(t)}$.

This is the central wave-propagation variable.

It answers:

How quickly and in what direction does the wave packet move through the storm-relative wind field compared with the storm circulation speed?

The radial component is

$q_r^{\ast }(t)={\mathbf{q}}^{\ast }(t)\cdot \hat{\mathbf{r}}(t)$.

The tangential component is

$q_{\varphi }^{\ast }(t)={\mathbf{q}}^{\ast }(t)\cdot \hat{\mathbf{\varphi }}(t)$.

The radial component measures escape from or approach toward the storm center, while the tangential component measures motion around the storm.

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11. Wave residence or trapping parameter

A characteristic residence time across one storm radius is

$T_{\mathrm{res}}\sim \frac{R_{34,\mathrm{eq}}}{|\mathbf{q}\cdot \hat{\mathbf{n}}|}$,

where $\hat{\mathbf{n}}$ is the direction across the relevant forcing region.

Normalize this by the storm advective time:

${\Pi }_{\mathrm{res}}=\frac{T_{\mathrm{res}}}{T_s}=\frac{V_{\max }}{|\mathbf{q}\cdot \hat{\mathbf{n}}|}$.

Equivalently, define an escape parameter

$\chi =\frac{|\mathbf{q}\cdot \hat{\mathbf{n}}|}{V_{\max }}$,

with

${\Pi }_{\mathrm{res}}=\frac{1}{\chi }$.

Interpretation:

$\chi \ll 1\text{or}{\Pi }_{\mathrm{res}}\gg 1:\text{slow storm-relative escape and prolonged forcing}$,

$\chi \sim 1:\text{wave propagation and storm advection occur on similar timescales}$,

$\chi \gg 1:\text{rapid passage through the forcing region}$.

It answers:

How long can this spectral wave component remain exposed to approximately coherent tropical-cyclone forcing?

This should be calculated separately for each frequency and direction because ${\mathbf{c}}_g$ is spectral.

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12. Current importance

Two current parameters are useful.

Current relative to wave propagation

$\mathcal{C}(t)=\frac{|{\mathbf{U}}_c(t)|}{|{\mathbf{c}}_g(t)|}$.

It answers:

Is current advection large enough to materially alter the wave-ray path?

Current-gradient or refraction parameter

${\mathcal{S}}_c=\frac{R_{34,\mathrm{eq}}}{|{\mathbf{c}}_g|}|\nabla {\mathbf{U}}_c|$.

It answers:

Can current gradients substantially refract or change the wavenumber of the wave during its passage across the storm?

The first parameter measures bulk advection; the second measures accumulated effects of spatial current gradients.

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13. Effective wave forcing and wave age

For wave development, use the wind acting along the wave direction rather than $V_{\max }$.

For a lagged directional wind history,

$U_{\parallel ,\mathrm{eff}}={\sum }_i{K}_i{U}_i\cos \Delta {\theta }_i$,

$U_{\perp ,\mathrm{eff}}={\sum }_i{K}_i{U}_i\sin \Delta {\theta }_i$.

Then define effective wave age as

$A_{\mathrm{eff}}=\frac{c_p}{U_{\parallel ,\mathrm{eff}}}$.

It answers:

How developed is the wave relative to the component of the recent wind history that was actually aligned with it?

This is more physically relevant than $c_p/V_{\max }$, which compares the wave with storm intensity but not with its experienced forcing.

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14. Instantaneous versus fixed-reference treatment

The primary analysis should use instantaneous storm scales:

$r_{\mathrm{eq}}^{\ast }(t)=\frac{r(t)}{R_{34,\mathrm{eq}}(t)}$,

${\mathbf{q}}^{\ast }(t)=\frac{\mathbf{q}(t)}{V_{\max }(t)}$,

$d{\tau }^{\ast }=\frac{V_{\max }(t)}{R_{34,\mathrm{eq}}(t)}dt$.

This describes the wave relative to the storm as it existed at each earlier time.

A fixed observation-time version may be retained as a sensitivity test:

$R_o=R_{34,\mathrm{eq}}(t_o),U_o=V_{\max }(t_o)$,

$r_o^{\ast }(t)=\frac{r(t)}{R_o}$,

${\mathbf{q}}_o^{\ast }(t)=\frac{\mathbf{q}(t)}{U_o}$,

${\tau }_o^{\ast }(t)=(t_o-t)\frac{U_o}{R_o}$.

This describes the full history in one coordinate system defined by the storm at the observation time.

The instantaneous formulation is preferable for physical self-similarity. The fixed-reference formulation is useful for determining how much the inferred history depends on storm evolution.

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15. Minimum variable set

For each time step along each backward wave ray, retain:

$r_{\mathrm{eq}}^{\ast }=\frac{r}{R_{34,\mathrm{eq}}}$,

$r_{\mathrm{dir}}^{\ast }=\frac{r}{R_{34}(\varphi )}$,

$A_R=\frac{R_{34}(\varphi )}{R_{34,\mathrm{eq}}}$,

${\lambda }_R=\frac{R_{\mathrm{mw}}}{R_{34,\mathrm{eq}}}$,

$\mathcal{T}=\frac{V_{TC}}{V_{\max }}$,

${\mathbf{q}}^{\ast }=\frac{{\mathbf{c}}_g+{\mathbf{U}}_c-{\mathbf{V}}_{TC}}{V_{\max }}$,

${\tau }^{\ast }=\int \frac{V_{\max }}{R_{34,\mathrm{eq}}}dt$,

$U_{\mathrm{eff}}^{\ast }=\frac{U_{\mathrm{eff}}}{V_{\max }}$,

and

$A_{\mathrm{eff}}=\frac{c_p}{U_{\parallel ,\mathrm{eff}}}$.

Together, these answer:

• where the wave was relative to overall storm size;
• whether it was inside the local forcing envelope;
• whether that side of the storm was expanded or contracted;
• how compact the inner core was;
• how important storm translation was;
• whether the wave propagated with, against, or nearly with the storm;
• how long it remained exposed to forcing;
• how strong the experienced wind was relative to storm intensity;
• how developed the wave was relative to its directional forcing history.

The purpose of this framework is not to force every storm onto one universal curve. It is to express the principal interstorm differences through a small set of physically interpretable dimensionless controls.