The Wee Spectrum That Grew: Tales o’ Gamma an’ the Growin’ Sea

By César Esparza

Since the beginning of modern ocean and coastal engineering, one of the key questions has been: “Can we mathematically describe the state of the sea?” We know waves don’t come alone or uniformly: they arrive in groups, with varying heights, frequencies, and directions. Spectral analysis allowed us to look at the sea as a sum of sinusoidal components. But what does that spectrum actually look like?

Throughout the 1950s and 60s, many researchers attempted to answer this. But it wasn’t until 1964 that Pierson and Moskowitz proposed a concrete spectral shape based on the similarity theory of Kitaigorodskii. Less than a decade later, Klaus Hasselmann and his team sharpened that curve with the now-famous JONSWAP experiment.

This blog-column takes you on that journey—explaining the foundations, the derivation of the PM spectrum, the need for JONSWAP, and the intriguing role of the peak enhancement factor: the famous $\gamma$.

Kitaigorodskii's Similarity Theory: Looking for a Universal Recipe

Imagine the sea as a recipe. In 1961, Kitaigorodskii asked: “Can we write a recipe that works anywhere in the world, regardless of wind strength or location?”

His answer was to apply dimensional analysis, much like finding the right proportions of ingredients whether you're cooking in grams or cups.

For a sea in equilibrium, he proposed that the wave spectrum should only depend on variables such as:

• the angular frequency $\omega$,

• gravity $g$,

• and eventually wind speed or turbulent parameters like $u_*$.

The result of this analysis was that, at high frequencies (the spectral tail), the spectrum should behave like:
$S(\omega) \sim \beta g^2 \omega^{-5}$

This result was reinforced by Phillips (1958), who introduced the concept of the equilibrium range: a region of the spectrum where energy is no longer controlled by the wind, but by wave breaking.

Phillips argued that in fully developed seas, crests become so steep they collapse. This physical condition generates a spectral tail of $\omega^{-5}$, independent of continued wind forcing.

In other words: the $\omega^{-5}$ shape isn’t arbitrary—it results from a balance between injected energy, nonlinear transport, and dissipation by breaking.

The Pierson-Moskowitz Spectrum: The Fully Grown Sea

Pierson and Moskowitz applied these ideas to real data from the North Atlantic. By curve-fitting, they proposed the spectrum:
$S(\omega) = \alpha g^2 \omega^{-5} \exp\left(-\beta \left(\frac{\omega_p}{\omega}\right)^4 \right)$

Where:

• $\alpha = 8.1 \times 10^{-3}$

• $\beta = 0.74$

• $\omega_p = \frac{g}{U}$

Now, where does this exponential form come from? Truth is, it wasn’t derived from first principles. It was a best-fit empirical model, based on Moskowitz’s measurements under fully developed sea conditions.

The $\omega^{-5}$ part was already justified by Phillips. But the exponential term:
$\exp\left(-\beta (\omega_p / \omega)^4 \right)$
was selected because it dropped sharply at low frequencies, mimicking the observed data and smoothing the spectrum before the peak.

$\beta$ controls how sharply the energy rises toward the peak—a higher value means a steeper slope; lower means flatter.
$\alpha$ simply scales the energy level, like a master volume knob.

In short:

• $\alpha$: controls total energy (scaling).

• $\beta$: shapes the spectrum before the peak.

• The exponential: not pure physics, but a reliable template.

How Did Engineers Trust It?

In the 1960s, wave measurement via buoys was new, but revolutionary. Spectra were built from surface records obtained using floating sensors.

Engineers trusted it because:

1. Data came from real ocean conditions with strong winds.

2. Spectra collapsed when scaled with $g$ and $U$, hinting at universality.

3. The PM model behaved conservatively—ideal for design.

It was like finding a curve that explained all the pancakes made in every kitchen across the North Atlantic. Even if the formula had some fitting, it worked—and it became coastal engineering law for a decade.

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The JONSWAP Experiment: The Sea Still in the Making

If the PM spectrum described an adult sea, fully grown, the JONSWAP experiment explored the teenage sea—still under wind’s influence and not yet matured.

In 1973, Klaus Hasselmann and colleagues conducted a massive field campaign in the North Sea, known as JONSWAP (Joint North Sea Wave Project). Over several weeks, they measured waves while the wind blew—capturing seas still in development. What they found was striking: real spectra had a much sharper peak than the PM curve.

This sharpening was modelled using a peak enhancement factor $\gamma$, leading to the JONSWAP spectrum:
$S(\omega) = \alpha g^2 \omega^{-5} \exp\left[-\beta\left(\frac{\omega_p}{\omega}\right)^4\right] \cdot \gamma^{\exp\left[-\frac{(\omega - \omega_p)^2}{2\sigma^2 \omega_p^2}\right]}$

This additional term, the gamma enhancement, amplifies the energy right at the spectral peak. And the physics behind it is fascinating.

Why Sharpen the Peak? Nonlinear Transfers and Energy Migration

JONSWAP isn’t just a prettier fit. It reflects deep physical truths: energy redistributes over time and space through nonlinear wave–wave interactions. Hasselmann emphasised this in 1973: nonlinear, wave-by-wave transfers—described by the wave action balance equation—are essential.

These interactions shift energy across frequencies. In young seas, energy piles up at the peak, creating a narrow, sharp spectrum. But over time, energy migrates to lower frequencies—this is how swells are born: the ocean’s mature waves.

So, as the sea develops, the peak smooths out. There's less energy concentrated at a single frequency, because nonlinear processes have spread it out.

Phillips, Breaking, and Dissipation

Phillips (1958) was the first to suggest that once waves steepen enough to break, generation halts and dissipation dominates.

This balance—between wind input, nonlinear redistribution, and breaking dissipation—produces the classic $\omega^{-5}$ tail. Phillips foresaw whitecapping as a “cutoff mechanism”: you can’t keep pumping energy into the same frequency forever.

In modern terms, whitecapping acts as a spectral safety valve, clipping excess energy and stabilising growth. Phillips introduced this as a condition of equilibrium; JONSWAP confirmed it with real data.

What Does $\gamma$ Represent? From Young to Old Seas

In JONSWAP—and in recent studies like Dematteis et al. (2019)—$\gamma$ isn’t just a number. It reflects the sea’s dynamic regime.

When $\gamma = 1$, we have a more linear spectrum, typical of swell, dominated by dispersion. Waves are orderly, with little energy at the peak. When $\gamma \approx 3.3$, we’re in an intermediate state where nonlinear interactions begin to matter. And when $\gamma \approx 6$, we enter strongly nonlinear territory—rogue wave regimes—where modulational instability and wave focusing become significant.

Conclusion

The JONSWAP spectrum is more than a formula. It’s a window into how the sea grows, evolves, and breaks. From Kitaigorodskii’s universal recipe, to Phillips’ equilibrium and the empirical clarity of Pierson and Moskowitz, through to Hasselmann’s peak sharpening: each step added depth to our understanding.

Today, when models and experiments tune $\gamma$ to represent different sea states, they’re not merely tweaking a parameter. They’re invoking an entire story—of energy, dissipation, and waves still rising out there.

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📚 References

Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright, D. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E., Kruseman, P., Meerburg, A., Müller, P., Olbers, D. J., Richter, K., Sell, W., & Walden, H. (1973). Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Ergänzungsheft zur Deutschen Hydrographischen Zeitschrift, Reihe A(8), Nr. 12.

Central reference introducing the JONSWAP spectrum, the $\gamma$ parameter, and the role of the wave action balance equation.

Pierson, W. J., & Moskowitz, L. (1964). A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii. Journal of Geophysical Research, 69(24), 5181–5190. https://doi.org/10.1029/JZ069i024p05181

Proposes the PM spectrum as an empirical fit under fully developed conditions.

Phillips, O. M. (1958). The equilibrium range in the spectrum of wind-generated waves. Journal of Fluid Mechanics, 4(4), 426–434. https://doi.org/10.1017/S0022112058000550

Introduces the wave-breaking-based energy dissipation mechanism and the $\omega^{-5}$ tail.

Kitaigorodskii, S. A. (1983). The Equilibrium Ranges in Wind–Wave Spectra. In B. Johns (Ed.), Marine Turbulence (pp. 35–60). Springer. https://doi.org/10.1007/978-1-4684-8980-4_2

A review of the dimensional similarity theory underlying spectral scaling.

Dematteis, G., Grafke, T., & Vanden-Eijnden, E. (2019). Experimental evidence of hydrodynamic instantons: The universal route to rogue waves. Physical Review X, 9(4), 041057. https://doi.org/10.1103/PhysRevX.9.041057

Links different $\gamma$ values to linear, intermediate, and strongly nonlinear sea states, including rogue wave conditions.