Three scores of the sea: PM, JONSWAP and TMA

The Pierson–Moskowitz discovery (1964): the universal melody of the mature sea

In 1964, Pierson and Moskowitz analysed wave records under strong, steady winds in the North Atlantic. They found that when the wind has blown long enough and over a large enough fetch, the sea reaches a state of full development in which the spectral shape becomes universal, independent of the details of the storm.

They proposed the now-famous formula:

\[ S_{\text{PM}}(\omega) = \alpha \,\frac{g^{2}}{\omega^{5}} \exp\!\left[-\beta \left(\frac{\omega_p}{\omega}\right)^{4}\right], \]

with \(\alpha \approx 8.1 \times 10^{-3}\) and \(\beta \approx 0.74\). The crucial feature is the \(\omega^{-5}\) decay at high frequency, the so-called equilibrium range: any attempt by the wind to pump more energy there leads to breaking and dissipation.

To compare seas of different sizes — for instance, one with 6-second waves and another with 20-second waves — they introduced dimensionless variables:

\[ f'=\frac{fU}{g}, \qquad S'(f)=\frac{S(f)g^2}{U^5}. \]

With this transformation, all spectra collapsed onto the same curve. It was like discovering that seas are the same melody, whether played on a flute or on a church organ. But the ocean rarely offers us fully developed seas.

---

The JONSWAP discovery (1973): sharpening the peak and the role of non-linearities

In the early 1970s, Klaus Hasselmann and colleagues carried out the Joint North Sea Wave Project (JONSWAP). They observed that under fetch- and duration-limited winds, the spectrum not only grew but did so in a very specific way.

Energy became more concentrated around the spectral peak than PM predicted, and the peak itself shifted slowly to lower frequencies. The explanation lay in non-linear wave–wave interactions: transfers of energy from slightly higher frequencies towards the peak, which enhanced it before equilibrium was reached.

To capture this, they proposed the JONSWAP spectrum:

\[ S_{\text{JONSWAP}}(f) = S_{\text{PM}}(f)\; \gamma^{\exp\!\left[-\frac{(f/f_p-1)^2}{2\sigma^2}\right]}. \]

When \(\gamma > 1\), the peak is sharper and higher: the signature of a young wind sea. As the sea ages into swell, non-linear transfers shift energy to longer frequencies and the peak loses height. If one compares a swell of 20 s with a PM spectrum of the same peak period, PM predicts a much higher peak. Fitting JONSWAP instead yields \(\gamma \approx 1\).

Thus:

• Seas: \(\gamma > 1\), more energy at the peak than PM would suggest.
• Swells: \(\gamma < 1\), peak lower than the PM equivalent.

This is why JONSWAP is more flexible: it can describe both young seas and swells, whereas PM systematically overestimates swell peaks.

---

The TMA discovery (1980s): when the seabed applies its own filter

Everything so far assumed deep water. But as waves approach the coast, the spectral tail can no longer follow the \(-5\) law. Short waves are dissipated and dispersion changes, so the decay softens to around \(-3\).

Bouws and colleagues proposed the TMA spectrum (Texel–MARSEN–ARSLOE) (Haha, yes, it does look odd at first sight! The TMA spectrum got its name from the three big coastal field experiments where the data came from Texel – an island off the Dutch coast, in the North Sea, Netherlands. MARSEN – short for MAin Research project on SEa waves and Non-linear effects, a German offshore field campaign in the 1970s. ARSLOE – Atlantic Remote Sensing Land–Ocean Experiment, a large US field study carried out on the North Carolina coast in the late 1970s. So, the name is basically an acronym pieced together from three different campaigns. Texel–MARSEN–ARSLOE (TMA). However, after all, it is essentially JONSWAP multiplied by a depth-dependent factor:

\[ S_{\text{TMA}}(f,h) = S_{\text{JONSWAP}}(f)\cdot\phi(2\pi f, h), \]

where \(\phi \to 1\) in deep water but reduces the high frequencies when \(kh \lesssim 1\). In this way, TMA retains JONSWAP’s ability to represent seas and swells, but also respects the finite-depth constraint.

---

Epilogue: three scores, one ocean

We may think of PM as the universal score of the mature sea: orderly, balanced, in full development. JONSWAP added the richness of real growth, showing how non-linear interactions sharpen the peak in young seas and how swells flatten it. TMA completed the picture by including the coastal filter: the seabed itself damping the high notes.

Behind the symbols — the \(\omega^{-5}\), the \(\gamma\), the \(\phi(\omega,h)\) — lies the same intuition: the spectrum is the fingerprint of the sea, and its shape tells us whether the sea is growing, travelling, or fading at the shore.