Undertow

Definition of Undertow: Undertow is the current flowing offshore near the seabed in the surf zone, driven by the vertical imbalance of the opposing gradients in radiation stress and cross-shore wave set-up pressure. 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.

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 (radiation stress), which is much 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 offshore-directed pressure gradient related to the slope of the mean water surface, the wave set-up.

The undertow current compensates for the onshore mass transport in the upper layer 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

Shoreface profile

Shallow-water wave theory

Currents

Wave set-up

Appendix: Wave-averaged momentum balance

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) and cyclic wave motion. 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 averaged momentum balance in the propagation direction is

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

Averaging over the wave cycle and integration over the depth gives

$0 = \Big\langle \Large\frac{\partial}{\partial t}\normalsize \int_{-d}^{\zeta} u dz \Big\rangle = \Big\langle u (\zeta) \Large\frac{\partial \zeta}{\partial t}\normalsize + \int_{-d}^{\zeta} \Large\frac{\partial u}{\partial t}\normalsize dz \Big\rangle \, , \qquad (3)$

$\Big\langle \Large\frac{\partial}{\partial x}\normalsize \int_{-d}^{\zeta} u^2 dz \Big\rangle = \Big\langle u^2 (\zeta) \Large\frac{\partial \zeta}{\partial x}\normalsize + \int_{-d}^{\zeta} 2 u \Large\frac{\partial u}{\partial x}\normalsize dz\Big\rangle \, , \qquad (4)$

$\Big\langle \Large\frac{\partial}{\partial z}\normalsize \int_{-d}^{\zeta} u w dz \Big\rangle = \Big\langle \int_{-d}^{\zeta} \Big( - u \Large\frac{\partial u}{\partial x}\normalsize + w \Large\frac{\partial u}{\partial z}\normalsize \Big) dz \Big\rangle = \lt u(\zeta) w(\zeta) \gt + \lt \tau_0 \gt - \lt \tau_b \gt = \Big\langle u(\zeta) \Big( \Large\frac{\partial \zeta}{\partial t}\normalsize + u(\zeta) \Large\frac{\partial \zeta}{\partial x}\normalsize \Big) \Big\rangle + \lt \tau_0 \gt - \lt \tau_b \gt \, . \qquad (5)$

Here we have used the continuity equation $\Large\frac{\partial u}{\partial x}\normalsize = - \Large\frac{\partial w}{\partial z}\normalsize \qquad (6)$

and the boundary conditions at the surface $z=\zeta$ and bottom $z=-d$,

$w(\zeta) = \Large\frac{\partial \zeta}{\partial t}\normalsize + u(\zeta) \Large\frac{\partial \zeta}{\partial x}\normalsize + w'(\zeta) \, , \quad w_0(-d)=0 \, , \quad \lt u'(\zeta)w'(\zeta)\gt =\lt \tau_0\gt \, , \quad \lt u_w(-d) w_w(-d) + u'(-d)w'(-d)\gt = \lt \tau_b\gt \, . \qquad (7)$

Summing Eqs. (3, 4, 5) and using Eq. (7) we get

$\Big\langle \Large\frac{\partial}{\partial x}\normalsize \int_{-d}^{\zeta} u^2 dz \Big\rangle = \Big\langle \int_{-d}^{\zeta} \Big( \Large\frac{\partial u}{\partial t}\normalsize + u \Large\frac{\partial u}{\partial x}\normalsize + w \Large\frac{\partial u}{\partial z}\normalsize \Big) dz \Big\rangle - \lt \tau_0\gt + \lt \tau_b\gt \, . \qquad (8)$

Combining with Eqs. (1) and (2) gives

$\Big\langle \Large\frac{\partial}{\partial x}\normalsize \int_{-d}^{\zeta} u^2 dz \Big\rangle = \Big\langle \Large\frac{\partial}{\partial x}\normalsize \int_{-d}^{\zeta} (u_0^2 + \lt u_w^2\gt + \lt u'^2\gt ) dz \Big\rangle = - \Large\frac{1}{\rho}\frac{\partial}{\partial x}\normalsize \Big\langle \int_{-d}^{\zeta} p dz \Big\rangle + \lt \tau_0\gt - \lt \tau_b\gt \, . \qquad (9)$

The radiation stress is given by

$S_{xx} = \Big\langle \int_{-d}^{\zeta} \big( \rho u_w^2 + \rho u'^2 + p \big) \, dz \Big\rangle - \large\frac{1}{2}\normalsize \rho \, g \, h^2 \, . \qquad (10)$

Because $\Big\langle \int_{-d}^{\zeta_0} (\zeta-z) \, dz \Big\rangle - \large\frac{1}{2}\normalsize h^2 =0 \, , \;$

a major contribution of the pressure gradient to the radiation stress is provided by the pressure gradient above the still water level.

Using $\Large\frac{\partial}{\partial x}\normalsize \lt h^2\gt = 2h \Large\frac{\partial \zeta_0}{\partial x}\normalsize$ we finally have

$\Big\langle \Large\frac{\partial}{\partial x}\normalsize \int_{-d}^{\zeta} u_0^2 dz \Big\rangle [1] = \rho g h \Large\frac{\partial \zeta_0}{\partial x}\normalsize [2] - \Large\frac{\partial S_{xx}}{\partial x}\normalsize [3] + \lt \tau_0\gt [4] - \lt \tau_b\gt [5] \, . \qquad (11)$

The gradient in the depth-integrated flux of wave-averaged momentum [1] is balanced by the wave-averaged contributions [2] + [3] + [4|+ [5], where [2] = pressure gradient due to wave set-up, [3] = radiation stress gradient, [4] = surface stress produced by the roller of the spilling wave that follows the wave as it propagates shoreward, [5] = bed shear stress.

There is no net onshore or offshore mass transfer, thus

$\Big\langle \int_{-d}^{\zeta} \rho \, u(z,t) \, dz \Big\rangle = \rho h u_0 + \Big\langle \int_{-d}^{\zeta} \rho \, u_w(z,t) \, dz \Big\rangle = 0 \, . \qquad(12)$

The term $\Big\langle \int_{-d}^{\zeta} \rho \, u_w(z,t) \, dz \Big\rangle$ should include the mass transport by the surface roller.. To solve these equations, boundary conditions have to be specified and (empirical) expressions must be provided for Reynolds stresses, bed shear stress and the contributions of the roller to the horizontal mass and momentum fluxes.

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 13-03-2025] 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