Undertow (water waves)

Oceanography

An "undertow" is a steady, offshore-directed compensation flow, which occurs below waves near the shore. Physically, nearshore, the wave-induced mass flux between wave crest and trough is onshore directed. This mass transport is localized in the upper part of the water column, i.e. above the wave troughs. To compensate for the amount of water being transported towards the shore, a second-order (i.e. proportional to the wave height squared), offshore-directed mean current takes place in the lower section of the water column. This flow – the undertow – affects the nearshore waves everywhere, unlike rip currents localized at certain positions along the shore.

The term undertow is used in scientific coastal oceanography papers.

Seaward mass flux

An exact relation for the mass flux of a nonlinear periodic wave on an inviscid fluid layer was established by Levi-Civita in 1924. In a frame of reference according to Stokes' first definition of wave celerity, the mass flux M_w of the wave is related to the wave's kinetic energy density E_k (integrated over depth and thereafter averaged over wavelength) and phase speed c through:

:M_w = \frac{2E_k}{c}.

Similarly, Longuet Higgins showed in 1975 that – for the common situation of zero mass flux towards the shore (i.e. Stokes' second definition of wave celerity) – normal-incident periodic waves produce a depth- and time-averaged undertow velocity:

:\bar{u} = - \frac{2 E_k}{\rho c h},

with h the mean water depth and \rho the fluid density. The positive flow direction of \bar{u} is in the wave propagation direction.

For small-amplitude waves, there is equipartition of kinetic (E_k) and potential energy (E_p):

:E_w = E_k + E_p \approx 2 E_k \approx 2 E_p,

with E_w the total energy density of the wave, integrated over depth and averaged over horizontal space. Since in general the potential energy E_p is much easier to measure than the kinetic energy, the wave energy is approximately {E_w\approx\tfrac18\rho g H^2} (with H the wave height). So

:\bar{u}\approx -\frac18 \frac{g H^2}{ c h }.

For irregular waves the required wave height is the root-mean-square wave height H_\text{rms}\approx\sqrt{8};\sigma, with \sigma the standard deviation of the free-surface elevation. The potential energy is E_p=\tfrac12\rho g \sigma^2 and E_w\approx\rho g \sigma^2.

The distribution of the undertow velocity over the water depth is a topic of ongoing research.

Confusion with rip currents

Main article: Rip current

In contrast to undertow, rip currents are responsible for the great majority of drownings close to beaches. When a swimmer enters a rip current, it starts to carry them offshore. The swimmer can exit the rip current by swimming at right angles to the flow, parallel to the shore, or by simply treading water or floating until the rip releases them. However, drowning can occur when swimmers exhaust themselves by trying unsuccessfully to swim directly against the flow of a rip.

On the United States Lifesaving Association website, it is noted that some uses of the word "undertow" are incorrect:

In some regions, rip currents are referred to by other, incorrect terms such as "rip tides" and "undertow". We encourage exclusive use of the correct term—rip currents. Use of other terms may confuse people and negatively impact public education efforts.}}

References

Notes

Other

References

1. Svendsen, I. A.. (1984). "Mass flux and undertow in a surf zone". [[Coastal Engineering Journal]].
2. "Rip Current Characteristics".
3. (2008). "Observations and a Model of Undertow over the Inner Continental Shelf". Journal of Physical Oceanography.
4. (2000). "Undertow over a barred beach". Journal of Geophysical Research.
5. (1994). "Vertical structure of mean cross-shore currents across a barred surf zone". Journal of Geophysical Research.
6. (2004). "Vertical flow structure during Sandy Duck: Observations and modeling". Coastal Engineering.
7. Longuet-Higgins, M.S.. (1983). "Wave set-up, percolation and undertow in the surf zone". Proceedings of the Royal Society of London A.
8. Levi-Civita, T.. (1924). "Questioni di meccanica classica e relativista". N. Zanichelli.
9. Longuet-Higgins, M.S.. (1975). "Integral properties of periodic gravity waves of finite amplitude". Proceedings of the Royal Society of London A.
10. (1985). "Calibration and verification of a dissipation model for random breaking waves". Journal of Geophysical Research.
11. "United States Lifesaving Association Rip Current Survival Guide". [[United States Lifesaving Association]].