12 Oct 2026 · Climate & Oceans

How far do the waves carry you?

Waves don't just bob things up and down: they nudge water, plastic and people steadily downwind. This "Stokes drift" sets how ocean models mix heat and where spills and survivors end up. Explore how well wind speed alone can predict it.

Why it matters

Stokes drift steers oil spills, plastic and people lost at sea, and it deepens the ocean's mixed layer, which shapes how the ocean absorbs heat in climate and weather forecasts. This Naval Research Laboratory study shows that a one-line wind rule predicts surface drift well enough for routine forecasting, but warns that wind-only formulas get the drift's depth profile wrong, so a proper wave model is still needed below the surface.

The paper

Parametric wave spectrum models for prediction of Stokes drift profiles

W. Erick Rogers, Yalin Fan, Luis Zamudio, Elizabeth M. Douglass (Naval Research Laboratory and Florida State University). arXiv:2610.12752, submitted 8 Oct 2026. Read on arXiv · PDF

In plain language: the authors compared 15 variants of nine textbook formulas for the shape of ocean wave energy against averaged spectra from moored buoys, then computed the resulting drift. They found that simply taking 1% of the wind speed gives a fairly accurate surface drift, which matched drift from a big drifting-buoy dataset at about a quarter of the buoys' total motion. But no wind-only formula got the full depth profile right, and adding Stokes drift to the HYCOM global ocean model deepened its mixed layer and generally lowered its error against observations.

Try it

Wind speed at 10 m

Fetch (open water upwind)

Hours adrift

Surface drift, 1% wind rule

Surface drift, full PM spectrum

Drift left at 5 m depth (PM)

Rough total drift of a floater in that time

1% rule (surface only)PM spectrumPhillips tail from fetch peakSingle "peak" wave

PM energy spectrum E(f)Stokes contribution at the surface

The maths in this page

Each wave frequency f with energy E(f) adds drift that fades with depth z (negative downward) as e^2kz, where the deep-water wavenumber is k = ω²/g and ω = 2πf. The paper's deep-water formula, for waves all heading one way, is:

U(z) = 2 ∫ E(f) ω k e^2kz df

PM spectrum (Pierson–Moskowitz, fully developed sea, paper eq. 3): E(f) = αg² / ((2π)⁴f⁵) · exp(−1.25 (f_p/f)⁴) with α = 0.0081 and peak f_p = 0.13 g/U₁₀, integrated up to 0.729 Hz.
Phillips tail from a fetch peak (the Breivik et al. 2014 approach used in HYCOM, paper eq. 11): U(z) = 2 ∫_{ω_p}^ω_max α_P g ω⁻² e^2ω²z/g dω, with α_P = 0.0083 and ω_p ≈ 22 (g²/(U₁₀F))^1/3 from the JONSWAP fetch law, where F is fetch.
1% rule (Clarke and van Gorder 2018): surface drift = 0.01 U₁₀.
Single "peak" wave: the PM surface value decaying with only the peak wavenumber. The paper warns this representative-wavenumber shortcut underpredicts how sharply drift falls off near the surface, which you can see in the orange line.
Total drift: the paper's rule of thumb, about 3% of wind for the current plus about 1% for wind pushing on the object, so total drift ≈ 0.04 U₁₀. Buoy data showed large scatter around it (correlation 0.32), so treat it as an average, not a forecast.

How to use it

Drag the wind slider to see surface drift grow roughly in step with the wind while the deeper drift lags behind. Shorten the fetch to mimic a coastal or newly blowing wind and watch the fetch-based curve shift, and note the warning when that peak falls outside its valid range. Set the hours adrift to get a back-of-envelope distance for a floating object. This is a teaching tool built from the paper's formulas, not an operational drift forecast.