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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>. These studies indicate that <math>g \, d \langle \zeta \rangle / dx</math> and <math>\partial \langle w_w u_w \rangle / \partial z</math> are the largest terms in the r.h.s. of Eq.(8) <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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− | 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=D91>Deigaard, R., Justeen, P. and Fredsoe, J. 1991. Modelling of undertow by a one-equation turbulence model. Coastal Eng. 15: 431-459</ref>. 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, <math>\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 \,</math>, where <math>f_w</math> is the wave bed friction coefficient<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>. Other symbols are <math>H=</math> wave height, <math>h=</math> depth.
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− | 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<ref name=DF>Deigaard, R. and Fredsøe, J. 1989. Shear stress distribution in dissipative water waves. Coast. Eng. 13: 357–378</ref> <math>\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 \, .</math> The second term in this expression represents energy dissipation due to wave breaking. Symbols are <math>k=\omega/c=</math> wavenumber, <math>c=</math> wave celerity, <math>T = 2 \pi / \omega=</math> wave period.
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− | Two boundary conditions are needed to solve the second order differential equation (8). 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 in the analytic model, the terms <math>g \, d \langle \zeta \rangle / dx</math> and <math>\partial \langle w_w u_w\rangle / \partial z </math> do not depend on <math>z</math>. 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=S2>Svendsen, I.A. 1984. Mass flux and undertow in a surf zone. Coast. Eng. 8: 347–365</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.
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