Difference between revisions of "Undertow"
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| + | ==Appendix: Undertow equations== | ||
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| + | [[File:UndertowSymbols.jpg|thumb|right|400px|Fig. 1. Definition sketch for the momentum balance equations.]] | ||
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| + | 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, | ||
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| + | <math>\zeta_w (x,t) = \dfrac{H}{2} \cos(\omega t – k x)</math>. | ||
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| + | Symbols are defined in Fig. 1. Other symbols: <math>\big\langle … \big\rangle \, =</math>wave-averaged value (averaged over one or more wave cycles, encompassing the turbulence time scale), <math>\; u(x,z,t), \, w(x,z,t) \,=</math> horizontal, vertical velocity; <math>\; u_0 = <u>, \, w_0=<w></math>, <math>\; u_w, \, w_w \,=</math> horizontal, vertical wave orbital velocities, <math>\; u', \, w' \, =</math> turbulent velocity fluctuations. | ||
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| + | The velocities <math>u, \, w</math> and surface elevation <math>\zeta</math> are decomposed as | ||
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| + | <math>u = u_0 + u_w+u' \, , \; w = w_0 + w_w +w' \, , \; \zeta = \zeta_0 + \zeta_w \, . \qquad (1)</math> | ||
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| + | The momentum balance equations in the propagation direction and in the vertical direction are (<math>g</math> is the gravitational acceleration) | ||
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| + | <math>\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)</math> | ||
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| + | <math>\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)</math> | ||
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| + | Averaging both equations over the wave cycle eliminates the time derivatives of <math>u</math> and <math>w</math>. As vertical scales are much smaller than horizontal scales the term <math>\partial u w / \partial x</math> can be ignored relative to <math>\partial w^2 /\partial z</math>. The Eqs.(2,3) then become | ||
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| + | <math>\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)</math> | ||
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| + | <math>\Big\langle \dfrac{\partial w^2}{\partial z} + \dfrac{1}{\rho}\dfrac{\partial p}{\partial z} \Big\rangle = -g \, .\qquad (5)</math> | ||
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| + | Eq. (5) can be integrated yielding <math> \langle p \rangle = \rho g (\langle \zeta \rangle -z) + \rho \langle w^2(z=0) -w^2(z) \rangle \, . \;</math> Substitution in Eq. (4) gives | ||
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| + | <math>\dfrac{\partial}{\partial x}\langle u^2 – w^2 + g \zeta \rangle + \dfrac{\partial \langle w u\rangle }{\partial z} = 0 \, .\qquad (6)</math> | ||
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| + | In this equation the term <math>\partial w^2 / \partial x</math> can be ignored relative to <math> \partial u^2 / \partial z</math>. The term <math> \langle w u \rangle </math> has two components, a wave-induced vertical transport of momentum <math>\langle w_w u_w \rangle</math> and a turbulent momentum transport <math>\langle w' u' \rangle </math>. The latter term represents a net turbulent shear stress that diffuses momentum from the net circulation <math>u_0(x,z)</math> over the vertical. This can represented to a first approximation by a gradient-type diffusion with an eddy-viscosity coefficient <math>K(x,z)</math>, | ||
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| + | <math>\langle u'w' \rangle = - K(x,z) \dfrac{\partial u_0}{\partial z} \, , \qquad (7)</math> | ||
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| + | The net circulation <math>u_0</math> can now be obtained from the momentum balance (Eq. 6), which is rewritten as | ||
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| + | <math> \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)</math> | ||
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| + | To solve this equation, the eddy-viscosity coefficient <math>K(x,z)</math> and the functions <math>\partial \langle u^2 \rangle / \partial x</math>, <math>g \, d \langle \zeta \rangle / dx</math> and <math>\partial \langle w_w u_w\rangle / \partial z </math> 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<ref name=SW>Stive, M. J. F. and Wind, H.G. 1986. Cross-shore mean flow in the surf zone. Coastal Eng. 10: 325– 340</ref><ref name=W17>van der Werf, J., Ribberink, J., Kranenburg, W., Neessen, K. and Boers, M. 2017. Contributions to the wave-mean momentum balance in the surf zone. Coastal Engineering 121: 212–220</ref>. | ||
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| + | 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<ref name=S1>Svendsen, I.A. 1984. Wave heights and set-up in a surf zone. Coast. Eng. 8: 303–329</ref><ref name=S2>Svendsen, I.A. 1984. Mass flux and undertow in a surf zone. Coast. Eng. 8: 347–365</ref><ref name=D89>Deigaard, R. and Fredsoe, J. 1989. Shear Stress Distribution in Dissipative Water Waves. Coastal Eng. 13: 357-378</ref>. | ||
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| + | According to shallow-water wave theory, the wave energy <math>E=\rho g H^2 /8</math> and <math> \langle u^2 \rangle \approx E /(\rho h)</math>. Assuming that wave energy is mainly lost through depth-induced wave breaking (see [[Breaker index]]), | ||
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| + | <math>\dfrac{dE}{dx} \approx - \dfrac{2 H E}{ h c T} = - \dfrac{\rho c h}{4 T} \Big( \dfrac{H}{h} \Big)^3</math> and <math>\dfrac{\partial \langle u^2 \rangle }{ \partial x } \approx - \dfrac{c}{4 T} \Big( \dfrac{H}{h} \Big)^3 \, . \qquad (9)</math> | ||
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| + | Symbols are: <math>H=</math> wave height, <math>h=</math> depth, <math>k=\omega/c=</math> wavenumber (<math>kh <<1</math>), <math>c=\sqrt{gh}=</math> wave celerity, <math>T = 2 \pi / \omega=</math> wave period. | ||
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| + | The term <math>\langle u_w v_w \rangle \ne 0 </math> because friction in the wave boundary modifies the phase relationship between the horizontal and vertical wave orbital velocities. Bottom friction generates vorticity <math>\Omega = \partial w_w / \partial x - \partial u_w / \partial z</math> in the wave boundary layer. Using the continuity equation, <math>\partial u_w / \partial x + \partial w_w / \partial z =0</math>, one finds the relationship<ref>Rivero, F. J. and Arcilla, A.S. 1995. On the vertical-distribution of <uw>. Coastal Eng. 25: 137– 152</ref> | ||
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| + | <math>\dfrac{\partial}{\partial z} (u_w w_w) = \Omega w_w - \dfrac{1}{2} \dfrac{\partial}{\partial x} \Big( u_w^2 - w_w^2 \Big) \, .</math> | ||
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| + | As the wave motion above the boundary layer is irrotational (<math>\Omega=0</math>) we have | ||
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| + | <math>\dfrac{\partial}{\partial z} \langle u_w w_w \rangle = - \dfrac{1}{2} \dfrac{\partial}{\partial x} \Big( \langle u_w^2 \rangle - \langle w_w^2 \rangle \Big) \approx \dfrac{c}{8 T} \Big( \dfrac{H}{h} \Big)^3\, . \qquad (10)</math> | ||
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| + | The [[wave set-up]] <math>d \langle \zeta \rangle / dx</math> is related to the [[radiation stress]] <math>S_{xx}</math> induced by wave dissipation<ref name=S2>Svendsen, I.A. 1984. Mass flux and undertow in a surf zone. Coast. Eng. 8: 347–365</ref>, | ||
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| + | <math> g \dfrac{d \langle \zeta \rangle}{dx} = - \dfrac{1}{\rho h} \dfrac{d}{dx} S_{xx} \approx - \dfrac{3}{2 \rho h} \dfrac{dE}{dx} \approx \dfrac{3c}{8T} \Big( \dfrac{H}{h} \Big)^3 \, . \qquad (11)</math> | ||
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| + | Collecting the different terms gives the approximate analytical undertow equation <math>\quad \dfrac{\partial}{\partial z}K(x,z) \dfrac{\partial u_0}{\partial z} = \dfrac{c}{4T} \Big( \dfrac{H}{h} \Big)^3 \, . \qquad (12)</math> | ||
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| + | Zou et al. (2006<ref name=Z6>Zou, Q., Bowen, A.J. and Hay, A.E. 2006. Vertical distribution of wave shear stress in variable water depth: theory and observations. J. Geophys. Res. 111: 1–17</ref>) give more elaborate analytical expressions that include the effect of a seabed slope. The bed slope effect appears to be important when comparing results with field observations. | ||
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| + | Two boundary conditions are needed to solve the second order differential equation (12). At the seabed, <math>z=-h</math>, the undertow velocity vanishes, <math>u_0(z=-h)=0</math>. The second condition is the overall mass balance represented by the equation <math>\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} </math>. This expression includes the mass transport by the roller, representing a water volume <math>A</math> (volume per longshore meter) which is transported onshore with the wave bore (crest of the broken wave, moving with celerity <math>c</math>).<ref name=S1>Svendsen, I.A. 1984. Wave heights and set-up in a surf zone. Coast. Eng. 8: 303–329</ref> The volume <math>A</math> is of the order of <math>H^2</math> (wave height squared). | ||
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| + | A qualitative impression of the undertow velocity profile can be derived from the vertical distribution of the eddy viscosity <math>K(z)</math>. If the eddy viscosity is assumed uniform over the vertical, the undertow velocity <math>u_0(z)</math> has a parabolic profile, because the r.h.s. of Eq. (12) does not depend on <math>z</math> in the analytic model. However, as most turbulence is generated by wave breaking near the water surface, the eddy viscosity is more likely a decreasing function with depth<ref name=D91>Deigaard, R., Justesen, P. and Fredsoe, J. 1991. Modelling of undertow by a one-equation turbulence model. Coast. Eng. 15: 431-458</ref>. 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. | ||
Revision as of 21:02, 13 April 2025
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
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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 [1][2] (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
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,
[math]\zeta_w (x,t) = \dfrac{H}{2} \cos(\omega t – k x)[/math].
Symbols are defined in Fig. 1. Other symbols: [math]\big\langle … \big\rangle \, =[/math]wave-averaged value (averaged over one or more wave cycles, encompassing the turbulence time scale), [math]\; u(x,z,t), \, w(x,z,t) \,=[/math] horizontal, vertical velocity; [math]\; u_0 = \lt u\gt , \, w_0=\lt w\gt [/math], [math]\; u_w, \, w_w \,=[/math] horizontal, vertical wave orbital velocities, [math]\; u', \, w' \, =[/math] turbulent velocity fluctuations.
The velocities [math]u, \, w[/math] and surface elevation [math]\zeta[/math] are decomposed as
[math]u = u_0 + u_w+u' \, , \; w = w_0 + w_w +w' \, , \; \zeta = \zeta_0 + \zeta_w \, . \qquad (1)[/math]
The momentum balance equations in the propagation direction and in the vertical direction are ([math]g[/math] is the gravitational acceleration)
[math]\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)[/math]
[math]\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)[/math]
Averaging both equations over the wave cycle eliminates the time derivatives of [math]u[/math] and [math]w[/math]. As vertical scales are much smaller than horizontal scales the term [math]\partial u w / \partial x[/math] can be ignored relative to [math]\partial w^2 /\partial z[/math]. The Eqs.(2,3) then become
[math]\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)[/math]
[math]\Big\langle \dfrac{\partial w^2}{\partial z} + \dfrac{1}{\rho}\dfrac{\partial p}{\partial z} \Big\rangle = -g \, .\qquad (5)[/math]
Eq. (5) can be integrated yielding [math] \langle p \rangle = \rho g (\langle \zeta \rangle -z) + \rho \langle w^2(z=0) -w^2(z) \rangle \, . \;[/math] Substitution in Eq. (4) gives
[math]\dfrac{\partial}{\partial x}\langle u^2 – w^2 + g \zeta \rangle + \dfrac{\partial \langle w u\rangle }{\partial z} = 0 \, .\qquad (6)[/math]
In this equation the term [math]\partial w^2 / \partial x[/math] can be ignored relative to [math] \partial u^2 / \partial z[/math]. The term [math] \langle w u \rangle [/math] has two components, a wave-induced vertical transport of momentum [math]\langle w_w u_w \rangle[/math] and a turbulent momentum transport [math]\langle w' u' \rangle [/math]. The latter term represents a net turbulent shear stress that diffuses momentum from the net circulation [math]u_0(x,z)[/math] over the vertical. This can represented to a first approximation by a gradient-type diffusion with an eddy-viscosity coefficient [math]K(x,z)[/math],
[math]\langle u'w' \rangle = - K(x,z) \dfrac{\partial u_0}{\partial z} \, , \qquad (7)[/math]
The net circulation [math]u_0[/math] can now be obtained from the momentum balance (Eq. 6), which is rewritten as
[math] \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)[/math]
To solve this equation, the eddy-viscosity coefficient [math]K(x,z)[/math] and the functions [math]\partial \langle u^2 \rangle / \partial x[/math], [math]g \, d \langle \zeta \rangle / dx[/math] and [math]\partial \langle w_w u_w\rangle / \partial z [/math] 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[3][4].
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[5][6][7].
According to shallow-water wave theory, the wave energy [math]E=\rho g H^2 /8[/math] and [math] \langle u^2 \rangle \approx E /(\rho h)[/math]. Assuming that wave energy is mainly lost through depth-induced wave breaking (see Breaker index),
[math]\dfrac{dE}{dx} \approx - \dfrac{2 H E}{ h c T} = - \dfrac{\rho c h}{4 T} \Big( \dfrac{H}{h} \Big)^3[/math] and [math]\dfrac{\partial \langle u^2 \rangle }{ \partial x } \approx - \dfrac{c}{4 T} \Big( \dfrac{H}{h} \Big)^3 \, . \qquad (9)[/math]
Symbols are: [math]H=[/math] wave height, [math]h=[/math] depth, [math]k=\omega/c=[/math] wavenumber ([math]kh \lt \lt 1[/math]), [math]c=\sqrt{gh}=[/math] wave celerity, [math]T = 2 \pi / \omega=[/math] wave period.
The term [math]\langle u_w v_w \rangle \ne 0 [/math] because friction in the wave boundary modifies the phase relationship between the horizontal and vertical wave orbital velocities. Bottom friction generates vorticity [math]\Omega = \partial w_w / \partial x - \partial u_w / \partial z[/math] in the wave boundary layer. Using the continuity equation, [math]\partial u_w / \partial x + \partial w_w / \partial z =0[/math], one finds the relationship[8]
[math]\dfrac{\partial}{\partial z} (u_w w_w) = \Omega w_w - \dfrac{1}{2} \dfrac{\partial}{\partial x} \Big( u_w^2 - w_w^2 \Big) \, .[/math]
As the wave motion above the boundary layer is irrotational ([math]\Omega=0[/math]) we have
[math]\dfrac{\partial}{\partial z} \langle u_w w_w \rangle = - \dfrac{1}{2} \dfrac{\partial}{\partial x} \Big( \langle u_w^2 \rangle - \langle w_w^2 \rangle \Big) \approx \dfrac{c}{8 T} \Big( \dfrac{H}{h} \Big)^3\, . \qquad (10)[/math]
The wave set-up [math]d \langle \zeta \rangle / dx[/math] is related to the radiation stress [math]S_{xx}[/math] induced by wave dissipation[6],
[math] g \dfrac{d \langle \zeta \rangle}{dx} = - \dfrac{1}{\rho h} \dfrac{d}{dx} S_{xx} \approx - \dfrac{3}{2 \rho h} \dfrac{dE}{dx} \approx \dfrac{3c}{8T} \Big( \dfrac{H}{h} \Big)^3 \, . \qquad (11)[/math]
Collecting the different terms gives the approximate analytical undertow equation [math]\quad \dfrac{\partial}{\partial z}K(x,z) \dfrac{\partial u_0}{\partial z} = \dfrac{c}{4T} \Big( \dfrac{H}{h} \Big)^3 \, . \qquad (12)[/math]
Zou et al. (2006[2]) give more elaborate analytical expressions that include the effect of a seabed slope. The bed slope effect appears to be important when comparing results with field observations.
Two boundary conditions are needed to solve the second order differential equation (12). At the seabed, [math]z=-h[/math], the undertow velocity vanishes, [math]u_0(z=-h)=0[/math]. The second condition is the overall mass balance represented by the equation [math]\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} [/math]. This expression includes the mass transport by the roller, representing a water volume [math]A[/math] (volume per longshore meter) which is transported onshore with the wave bore (crest of the broken wave, moving with celerity [math]c[/math]).[5] The volume [math]A[/math] is of the order of [math]H^2[/math] (wave height squared).
A qualitative impression of the undertow velocity profile can be derived from the vertical distribution of the eddy viscosity [math]K(z)[/math]. If the eddy viscosity is assumed uniform over the vertical, the undertow velocity [math]u_0(z)[/math] has a parabolic profile, because the r.h.s. of Eq. (12) does not depend on [math]z[/math] in the analytic model. However, as most turbulence is generated by wave breaking near the water surface, the eddy viscosity is more likely a decreasing function with depth[1]. 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
- ↑ Jump up to: 1.0 1.1 Deigaard, R., Justeen, P. and Fredsoe, J. 1991. Modelling of undertow by a one-equation turbulence model. Coastal Eng. 15: 431-459 Cite error: Invalid
<ref>tag; name "D91" defined multiple times with different content - ↑ Jump up to: 2.0 2.1 Zou, Q., Bowen, A.J. and Hay, A.E. 2006. Vertical distribution of wave shear stress in variable water depth: theory and observations. J. Geophys. Res. 111: 1–17
- Jump up ↑ Stive, M. J. F. and Wind, H.G. 1986. Cross-shore mean flow in the surf zone. Coastal Eng. 10: 325– 340
- Jump up ↑ van der Werf, J., Ribberink, J., Kranenburg, W., Neessen, K. and Boers, M. 2017. Contributions to the wave-mean momentum balance in the surf zone. Coastal Engineering 121: 212–220
- ↑ Jump up to: 5.0 5.1 Svendsen, I.A. 1984. Wave heights and set-up in a surf zone. Coast. Eng. 8: 303–329
- ↑ Jump up to: 6.0 6.1 Svendsen, I.A. 1984. Mass flux and undertow in a surf zone. Coast. Eng. 8: 347–365
- Jump up ↑ Deigaard, R. and Fredsoe, J. 1989. Shear Stress Distribution in Dissipative Water Waves. Coastal Eng. 13: 357-378
- Jump up ↑ Rivero, F. J. and Arcilla, A.S. 1995. On the vertical-distribution of <uw>. Coastal Eng. 25: 137– 152
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