# Undertow (water waves)

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In physical oceanography, undertow is the under-current that is moving offshore when waves are approaching the shore. Undertow is a natural and universal feature for almost any large body of water: it is a return flow compensating for the onshore-directed average transport of water by the waves in the zone above the wave troughs. The undertow's flow velocities are generally strongest in the surf zone, where the water is shallow and the waves are high due to shoaling. [1]

Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters.

In fluid dynamics, wind waves, or wind-generated waves, are water surface waves that occur on the free surface of the oceans and other bodies. They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high.

In continuum mechanics the macroscopic velocity, also flow velocity in fluid dynamics or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar. It is also called velocity field; when evaluated along a line, it is called a velocity profile.

## Contents

In popular usage, the word "undertow" is often misapplied to rip currents. [2] An undertow occurs everywhere underneath shore-approaching waves, whereas rip currents are localized narrow offshore currents occurring at certain locations along the coast. Unlike undertow, rip currents are strong at the surface.

A rip current, often simply called a rip, is a specific kind of water current which can occur near beaches with breaking waves. A rip is a strong, localized, and narrow current of water which moves directly away from the shore, cutting through the lines of breaking waves like a river running out to sea, and is strongest near the surface of the water.

## 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. [3]

In physics and engineering, mass flux is the rate of mass flow per unit area, perfectly overlapping with the momentum density, the momentum per unit volume. The common symbols are j, J, q, Q, φ, or Φ, sometimes with subscript m to indicate mass is the flowing quantity. Its SI units are kg s−1 m−2. Mass flux can also refer to an alternate form of flux in Fick's law that includes the molecular mass, or in Darcy's law that includes the mass density.

A water column is a conceptual column of water from the surface of a sea, river or lake to the bottom sediment. Descriptively, the deep sea water column is divided into five parts—pelagic zones —from the surface to below the floor, as follows: epipelagic, from the surface to 200 meters below the surface; mesopelagic, from 200 to 1000 meters below the surface; bathypelagic, from 1000 to 4000 meters below the surface; abyssopelagic, from 4000 meters below the surface to the level sea floor; hadopelagic, depressions and crevices below the level sea floor.

In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighbouring trough. Wave height is a term used by mariners, as well as in coastal, ocean and naval engineering.

The term undertow is used in scientific coastal oceanography papers. [4] [5] [6] The distribution of flow velocities in the undertow over the water column is important as it strongly influences the on- or offshore transport of sediment. Outside the surf zone there is a near-bed onshore-directed sediment transport induced by Stokes drift and skewed-asymmetric wave transport. In the surf zone, strong undertow generates a near-bed offshore sediment transport. These antagonistic flows may lead to sand bar formation where the flows converge near the wave breaking point, or in the wave breaking zone. [4] [5] [6] [7]

Sediment transport is the movement of solid particles (sediment), typically due to a combination of gravity acting on the sediment, and/or the movement of the fluid in which the sediment is entrained. Sediment transport occurs in natural systems where the particles are clastic rocks, mud, or clay; the fluid is air, water, or ice; and the force of gravity acts to move the particles along the sloping surface on which they are resting. Sediment transport due to fluid motion occurs in rivers, oceans, lakes, seas, and other bodies of water due to currents and tides. Transport is also caused by glaciers as they flow, and on terrestrial surfaces under the influence of wind. Sediment transport due only to gravity can occur on sloping surfaces in general, including hillslopes, scarps, cliffs, and the continental shelf—continental slope boundary.

For a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the free surface of water waves, experiences a net Stokes drift velocity in the direction of wave propagation.

In fluid dynamics, a breaking wave is a wave whose amplitude reaches a critical level at which some process can suddenly start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy. At this point, simple physical models that describe wave dynamics often become invalid, particularly those that assume linear behaviour.

### 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. [8] In a frame of reference according to Stokes' first definition of wave celerity, the mass flux ${\displaystyle M_{w}}$ of the wave is related to the wave's kinetic energy density ${\displaystyle E_{k}}$ (integrated over depth and thereafter averaged over wavelength) and phase speed ${\displaystyle c}$ through:

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.

Tullio Levi-Civita, was an Italian mathematician, most famous for his work on absolute differential calculus and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus. His work included foundational papers in both pure and applied mathematics, celestial mechanics, analytic mechanics and hydrodynamics.

In physics, a frame of reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements.

${\displaystyle M_{w}={\frac {2E_{k}}{c}}.}$

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: [9]

${\displaystyle {\bar {u}}=-{\frac {2E_{k}}{\rho ch}},}$

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

For small-amplitude waves, there is equipartition of kinetic (${\displaystyle E_{k}}$) and potential energy (${\displaystyle E_{p}}$):

${\displaystyle E_{w}=E_{k}+E_{p}\approx 2E_{k}\approx 2E_{p},}$

with ${\displaystyle E_{w}}$ the total energy density of the wave, integrated over depth and averaged over horizontal space. Since in general the potential energy ${\displaystyle E_{p}}$ is much easier to measure than the kinetic energy, the wave energy is approximately ${\displaystyle {E_{w}\approx {\tfrac {1}{8}}\rho gH^{2}}}$ (with ${\displaystyle H}$ the wave height). So

${\displaystyle {\bar {u}}\approx -{\frac {1}{8}}{\frac {gH^{2}}{ch}}.}$

For irregular waves the required wave height is the root-mean-square wave height ${\displaystyle H_{\text{rms}}\approx {\sqrt {8}}\;\sigma ,}$ with ${\displaystyle \sigma }$ the standard deviation of the free-surface elevation. [10] The potential energy is ${\displaystyle E_{p}={\tfrac {1}{2}}\rho g\sigma ^{2}}$ and ${\displaystyle E_{w}\approx \rho g\sigma ^{2}.}$

The distribution of the undertow velocity over the water depth is a topic of ongoing research. [4] [5] [6]

## Confusion with rip currents

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. However, drowning may occur when swimmers exhaust themselves by trying unsuccessfully to swim directly against the flow.

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

A rip current is a horizontal current. Rip currents do not pull people under the water–-they pull people away from shore. Drowning deaths occur when people pulled offshore are unable to keep themselves afloat and swim to shore. This may be due to any combination of fear, panic, exhaustion, or lack of swimming skills.

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. [2]

• Longshore current   A current parallel to the shoreline caused by waves approaching at an angle to the shoreline

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## References

### Notes

1. Svendsen, I.A. (1984), "Mass flux and undertow in a surf zone", Coastal Engineering, 8 (4): 347–365, doi:10.1016/0378-3839(84)90030-9
2. United States Lifesaving Association Rip Current Survival Guide, United States Lifesaving Association, archived from the original on 2014-01-02, retrieved 2014-01-02
3. Lentz, S.J.; Fewings, M.; Howd, P.; Fredericks, J.; Hathaway, K. (2008), "Observations and a Model of Undertow over the Inner Continental Shelf", Journal of Physical Oceanography, 38 (11): 2341–2357, Bibcode:2008JPO....38.2341L, doi:10.1175/2008JPO3986.1, hdl:1912/4067
4. Garcez Faria, A.F.; Thornton, E.B.; Lippman, T.C.; Stanton, T.P. (2000), "Undertow over a barred beach", Journal of Geophysical Research, 105 (C7): 16, 999–17, 010, Bibcode:2000JGR...10516999F, doi:10.1029/2000JC900084
5. Haines, J.W.; Sallenger Jr., A.H. (1994), "Vertical structure of mean cross-shore currents across a barred surf zone", Journal of Geophysical Research, 99 (C7): 14, 223–14, 242, Bibcode:1994JGR....9914223H, doi:10.1029/94JC00427
6. Reniers, A.J.H.M.; Thornton, E.B.; Stanton, T.P.; Roelvink, J.A. (2004), "Vertical flow structure during Sandy Duck: Observations and modeling", Coastal Engineering, 51 (3): 237–260, doi:10.1016/j.coastaleng.2004.02.001
7. Longuet-Higgins, M.S. (1983), "Wave set-up, percolation and undertow in the surf zone", Proceedings of the Royal Society of London A, 390 (1799): 283–291, Bibcode:1983RSPSA.390..283L, doi:10.1098/rspa.1983.0132
8. Levi-Civita, T. (1924), Questioni di meccanica classica e relativista, Bologna: N. Zanichelli, OCLC   441220095, archived from the original on 2015-06-15
9. Longuet-Higgins, M.S. (1975), "Integral properties of periodic gravity waves of finite amplitude", Proceedings of the Royal Society of London A, 342 (1629): 157–174, Bibcode:1975RSPSA.342..157L, doi:10.1098/rspa.1975.0018
10. Battjes, J.A.; Stive, M.J.F. (1985), "Calibration and verification of a dissipation model for random breaking waves", Journal of Geophysical Research, 90 (C5): 9159–9167, Bibcode:1985JGR....90.9159B, doi:10.1029/JC090iC05p09159