Undertow

Definition of Undertow: Undertow is the current flowing offshore near the seabed in the surf zone, mainly driven by wave set-up at the shoreline, and compensating for onshore mass transport by wave crests and wave bores. This is the common definition for Undertow, other definitions can be discussed in the article

Notes

The undertow is a net circulation in the cross-shore vertical plane representing a mechanism for maintaining the mass balance in the surf zone. Other possible mechanisms in the nearshore circulation is the three dimensional pattern of rip currents and the pattern of longshore currents in the case of oblique wave incidence. When standing just seaward of the shoreline in the surf zone, one can clearly feel the onshore surface current as a wave crest arrives, and the seaward current near the bottom that occurs beneath the next wave trough.

There is no generally applicable formula for the undertow velocity, as it depends on the particular shoreface morphology. Driving forces for the undertow are (a) the gradient in the net onshore momentum flux (local radiation stress), which is stronger near the surface than near the bottom; (b) the net wave- and turbulence-induced vertical momentum flux towards the wave boundary layer (which is responsible for momentum dissipation and near-bed forward streaming); (c) the momentum flux associated with the surface roller of the spilling wave bore; (d) the pressure gradient related to the onshore slope of the mean water surface, the wave set-up.

The undertow current compensates for the onshore mass transport in the upper part of the vertical between wave trough and crest (Stokes drift and roller transport). The turbulent frictional dissipation of momentum by the undertow current is dynamically related to radiation stress decay.

Related articles

Wave set-up

Breaker index

Radiation stress

Wave transformation

Shallow-water wave theory

Shoreface profile

Currents

Appendix: Undertow equations

Fig. 1. Definition sketch for the momentum balance equations.

This appendix reproduces the shallow-water equations from which the undertow can be determined. The equations refer to shore-normal wave incidence on a uniform coast (no longshore current). The driving force is a surface wave incident from the far field,

$\zeta_w (x,t) = \dfrac{H}{2} \cos(\omega t – k x)$.

Symbols are defined in Fig. 1. Other symbols: $\big\langle … \big\rangle \, =$wave-averaged value (averaged over one or more wave cycles, encompassing the turbulence time scale), $\; u(x,z,t), \, w(x,z,t) \,=$ horizontal, vertical velocity; $\; u_0 = \lt u\gt , \, w_0=\lt w\gt$, $\; u_w, \, w_w \,=$ horizontal, vertical wave orbital velocities, $\; u', \, w' \, =$ turbulent velocity fluctuations.

The velocities $u, \, w$ and surface elevation $\zeta$ are decomposed as

$u = u_0 + u_w+u' \, , \; w = w_0 + w_w +w' \, , \; \zeta = \zeta_0 + \zeta_w \, . \qquad (1)$

The momentum balance equations in the propagation direction and in the vertical direction are ($g$ is the gravitational acceleration)

$\dfrac{\partial u}{\partial t} + \dfrac{\partial u^2}{\partial x} + \dfrac{\partial w u}{\partial z} + \dfrac{1}{\rho}\dfrac{\partial p}{\partial x} = 0 \, .\qquad (2)$

$\dfrac{\partial w}{\partial t} + \dfrac{\partial u w}{\partial x} + \dfrac{\partial w^2}{\partial z} + \dfrac{1}{\rho}\dfrac{\partial p}{\partial z} = -g \, .\qquad (3)$

Averaging both equations over the wave cycle eliminates the time derivatives of $u$ and $w$. As vertical scales are much smaller than horizontal scales the term $\partial u w / \partial x$ can be ignored relative to $\partial w^2 /\partial z$. The Eqs.(2,3) then become

$\Big\langle \dfrac{\partial u^2}{\partial x} + \dfrac{\partial w u}{\partial z} + \dfrac{1}{\rho}\dfrac{\partial p}{\partial x} \Big\rangle= 0 \, .\qquad (4)$

$\Big\langle \dfrac{\partial w^2}{\partial z} + \dfrac{1}{\rho}\dfrac{\partial p}{\partial z} \Big\rangle = -g \, .\qquad (5)$

Eq. (5) can be integrated yielding $\langle p \rangle = \rho g (\langle \zeta \rangle -z) + \rho \langle w^2(z=0) -w^2(z) \rangle \, . \;$ Substitution in Eq. (4) gives

$\dfrac{\partial}{\partial x}\langle u^2 – w^2 + g \zeta \rangle + \dfrac{\partial \langle w u\rangle }{\partial z} = 0 \, .\qquad (6)$

In this equation the term $\partial w^2 / \partial x$ can be ignored relative to $\partial u^2 / \partial z$. The term $\langle w u \rangle$ has two components, a wave-induced vertical transport of momentum $\langle w_w u_w \rangle$ and a turbulent momentum transport $\langle w' u' \rangle$. The latter term represents a net turbulent shear stress that diffuses momentum from the net circulation $u_0(x,z)$ over the vertical. This can represented to a first approximation by a gradient-type diffusion with an eddy-viscosity coefficient $K(x,z)$,

$\langle u'w' \rangle = - K(x,z) \dfrac{\partial u_0}{\partial z} \, , \qquad (7)$

The net circulation $u_0$ can now be obtained from the momentum balance (Eq. 6), which is rewritten as

$\dfrac{\partial}{\partial z}K(x,z) \dfrac{\partial u_0}{\partial z} = \dfrac{\partial}{\partial x}\langle u^2 + g \zeta \rangle + \dfrac{\partial \langle w_w u_w\rangle }{\partial z} \, .\qquad (8)$

To solve this equation, the eddy-viscosity coefficient $K(x,z)$ and the functions $\partial \langle u^2 \rangle / \partial x$, $g \, d \langle \zeta \rangle / dx$ and $\partial \langle w_w u_w\rangle / \partial z$ must be known. These functions can be determined by numerically solving the Eqs. (2,3) or they can be determined from field or laboratory measurements. These studies indicate that $g \, d \langle \zeta \rangle / dx$ and $\partial \langle w_w u_w \rangle / \partial z$ are the largest terms in the r.h.s. of Eq.(8) .

Approximate analytical expressions have been derived using shallow-water wave theory outside the near-bed wave boundary layer and assuming that bed slope effects can be neglected. The wave set-up $d \langle \zeta \rangle / dx$ is related to the radiation stress $S_{xx}$ induced by wave dissipation, $\dfrac{d \langle \zeta \rangle}{dx} = - \dfrac{1}{\rho g h} \dfrac{d}{dx} S_{xx} \approx \dfrac{3 f_w}{64} \Big( \dfrac{H}{h} \Big)^3 \,$, where $f_w$ is the wave bed friction coefficient. Other symbols are $H=$ wave height, $h=$ depth.

Frictional effects in the wave boundary layer modify the phase relationship between the horizontal and vertical wave orbital velocities. A simplified analytical model, assuming that waves are only weakly decaying, yields the expression $\dfrac{\partial \langle w_w u_w\rangle }{\partial z} \approx -\dfrac{f_w g}{64} \Big( \dfrac{H}{h} \Big)^3 - \dfrac{g kh}{16 \pi} \Big( \dfrac{H}{h} \Big)^3 \, .$ The second term in this expression represents energy dissipation due to wave breaking. Symbols are $k=\omega/c=$ wavenumber, $c=$ wave celerity, $T = 2 \pi / \omega=$ wave period.

Two boundary conditions are needed to solve the second order differential equation (8). At the seabed, $z=-h$, the undertow velocity vanishes, $u_0(z=-h)=0$. The second condition is the overall mass balance represented by the equation $\int_{-h}^0 u_0(z) dz \approx - \langle (h+\zeta)u_w \rangle - \dfrac{A}{T} \approx - \dfrac{c H^2}{8h} - \dfrac{A}{T}$. This expression includes the mass transport by the roller, representing a water volume $A$ (volume per longshore meter) which is transported onshore with the wave bore (crest of the broken wave, moving with celerity $c$). The volume $A$ is of the order of $H^2$ (wave height squared).

A qualitative impression of the undertow velocity profile can be derived from the vertical distribution of the eddy viscosity $K(z)$. If the eddy viscosity is assumed uniform over the vertical, the undertow velocity $u_0(z)$ has a parabolic profile, because in the analytic model, the terms $g \, d \langle \zeta \rangle / dx$ and $\partial \langle w_w u_w\rangle / \partial z$ do not depend on $z$. However, as most turbulence is generated by wave breaking near the water surface, the eddy viscosity is more likely a decreasing function with depth. In this case, the strongest variation of the undertow (greatest gradient) is close to the seabed, as sketched in Fig.1. This undertow profile, with maximum offshore velocity in the lower part of the vertical, best matches observations.

References

The main author of this article is Job Dronkers Please note that others may also have edited the contents of this article. Citation: Job Dronkers (2025): Undertow. Available from http://www.coastalwiki.org/wiki/Undertow [accessed on 7-05-2025] For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Dronkers J