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In [[fluid dynamics]], '''Luke's variational principle''' is a [[Lagrangian]] [[Calculus of variations|variational]] description of the motion of [[ocean surface wave|surface waves]] on a [[fluid]] with a [[free surface]], under the action of [[Earth's gravity|gravity]]. This principle is named after J.C. Luke, who published it in 1967.<ref>
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{{cite journal
| author = J. C. Luke
| year = 1967
| title = A Variational Principle for a Fluid with a Free Surface
  | journal = [[Journal of Fluid Mechanics]]
| volume = 27 | issue = 2 | pages = 395–397
| doi = 10.1017/S0022112067000412
|bibcode = 1967JFM....27..395L }}</ref> This variational principle is for [[incompressible]] and [[inviscid]] [[potential flow]]s, and is used to derive approximate wave models like the so-called [[mild-slope equation]],<ref name=Ding1997>
{{cite book
| author = M. W. Dingemans
| year = 1997
| title = Water Wave Propagation Over Uneven Bottoms
| series = Advanced Series on Ocean Engineering
| volume = 13 |pages=271
| publisher = [[World Scientific]]
| location = Singapore
| isbn = 981-02-0427-2
}}</ref> or using the average-Lagrangian approach for wave propagation in inhomogeneous media.<ref name=Whit1974>
{{cite book
| author=G. B. Whitham
| author-link = Gerald B. Whitham
| year=1974
| title=Linear and Nonlinear Waves
| page=555
| publisher=[[Wiley-Interscience]]
| isbn = 0-471-94090-9
}}</ref>
 
Luke's Lagrangian formulation can also be recast into a [[Hamiltonian mechanics|Hamiltonian]] formulation in terms of the surface elevation and velocity potential at the free surface.<ref name=Zakharov1968>{{cite journal
| author = V. E. Zakharov
| year = 1968
| title = Stability of Periodic Waves of Finite Amplitude on the Surface of a Deep Fluid
| journal = [[Journal of Applied Mechanics and Technical Physics]]
| volume = 9 | issue = 2 | pages = 190–194
| doi = 10.1007/BF00913182
|bibcode = 1968JAMTP...9..190Z }} Originally appeared in ''[[Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki]]'' '''9'''(2): 86–94, 1968.</ref><ref name=Broer1974>
{{cite journal
| author = L. J. F. Broer
| year = 1974
| title = On the Hamiltonian Theory of Surface Waves
| journal = [[Applied Scientific Research]]
| volume = 29 | pages = 430–446
| doi = 10.1007/BF00384164
}}</ref><ref name=Miles1977>
{{cite journal
| author = J. W. Miles
| year = 1977
| title = On Hamilton's Principle for Surface Waves
| journal = [[Journal of Fluid Mechanics]]
| volume = 83 | issue = 1 | pages = 153–158
| doi = 10.1017/S0022112077001104
|bibcode = 1977JFM....83..153M }}</ref> This is often used when modelling the [[spectral density]] evolution of the free-surface in a [[sea state]], sometimes called [[wave turbulence]].
 
Both the Lagrangian and Hamiltonian formulations can be extended to include [[surface tension]] effects.
 
==Luke's Lagrangian==
 
Luke's [[Lagrangian]] formulation is for [[non-linear]] surface gravity waves on an—[[incompressible]], [[irrotational]] and [[viscosity|inviscid]]—[[potential flow]].
 
The relevant ingredients, needed in order to describe this flow, are:
*''Φ''('''''x''''',''z'',''t'') is the [[velocity potential]],
*''ρ'' is the fluid [[density]],
*''g'' is the acceleration by the [[Earth's gravity]],
*'''''x''''' is the horizontal coordinate vector with components ''x'' and ''y'',
*''x'' and ''y'' are the horizontal coordinates,
*''z'' is the vertical coordinate,
*''t'' is time, and
*&nabla; is the horizontal [[gradient]] operator, so &nabla;''Φ'' is the horizontal [[flow velocity]] consisting of &part;''Φ''/&part;''x'' and &part;''Φ''/&part;''y'',
*''V''(''t'') is the time-dependent fluid domain with free surface.
 
The Lagrangian <math>\mathcal{L}</math>, as given by Luke, is:
 
:<math>
  \mathcal{L} =
  -\int_{t_0}^{t_1} \left\{ \iiint_{V(t)} \rho
    \left[
      \frac{\partial\Phi}{\partial t}
      + \frac{1}{2} \left| \boldsymbol{\nabla}\Phi \right|^2
      + \frac{1}{2} \left( \frac{\partial\Phi}{\partial z} \right)^2
      + g\, z
    \right]\; \text{d}x\; \text{d}y\; \text{d}z\; \right\}\; \text{d}t.
</math>
 
From [[Bernoulli's principle]], this Lagrangian can be seen to be the [[integral]] of the fluid [[pressure]] over the whole time-dependent fluid domain ''V''(''t''). This is in agreement with the variational principles for inviscid flow without a free surface, found by [[Harry Bateman]].<ref name=Bateman1929>{{cite journal
| author = H. Bateman
| year = 1929
| title = Notes on a Differential Equation Which Occurs in the Two-Dimensional Motion of a Compressible Fluid and the Associated Variational Problems
| journal = [[Proceedings of the Royal Society of London A]]
| volume = 125 | issue = 799 | pages = 598–618
| doi = 10.1098/rspa.1929.0189
|bibcode = 1929RSPSA.125..598B }}</ref>
 
[[Calculus of variations|Variation]] with respect to the velocity potential ''Φ''('''''x''''',''z'',''t'') and free-moving surfaces like ''z''=''η''('''''x''''',''t'') results in the [[Laplace equation]] for the potential in the fluid interior and all required [[boundary conditions]]: [[kinematic]] boundary conditions on all fluid boundaries and [[dynamics (mechanics)|dynamic]] boundary conditions on free surfaces.<ref name=Whitham1974>
{{cite book
| author=G. W. Whitham
| year=1974
| title=Linear and Nonlinear Waves
  | pages=434–436
| publisher=[[John Wiley & Sons|Wiley]]
| location=New York
| isbn=0-471-94090-9
}}</ref> This may also include moving wavemaker walls and ship motion.
 
For the case of a horizontally unbounded domain with the free fluid surface at ''z''=''η''('''''x''''',''t'') and a fixed bed at ''z''=−''h''('''''x'''''), Luke's variational principle results in the Lagrangian:
 
:<math>
  \mathcal{L} =
  -\, \int_{t_0}^{t_1} \iint
  \left\{ \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \rho\,
    \left[
      \frac{\partial\Phi}{\partial t}
      +\, \frac{1}{2} \left| \boldsymbol{\nabla}\Phi \right|^2
      +\, \frac{1}{2} \left( \frac{\partial\Phi}{\partial z} \right)^2
    \right]\; \text{d}z\;
    +\, \frac{1}{2}\, \rho\, g\, \eta^2
  \right\}\; \text{d}\boldsymbol{x}\; \text{d}t.</math>
 
The bed-level term proportional to ''h''<sup>2</sup> in the potential energy has been neglected, since it is a constant and does not contribute in the variations.  
Below, Luke's variational principle is used to arrive at the flow equations for non-linear surface gravity waves on a potential flow.
 
===Derivation of the flow equations resulting from Luke's variational principle===
 
The variation <math>\delta\mathcal{L}=0</math> in the Lagrangian with respect to variations in the velocity potential ''Φ''('''''x''''',''z'',''t''), as well as with respect to the surface elevation ''η''('''''x''''',''t''), have to be zero. We consider both variations subsequently.
 
====Variation with respect to the velocity potential====
 
Consider a small variation ''δΦ'' in the velocity potential ''Φ''.<ref name=Whitham1974/> Then the resulting variation in the Lagrangian is:
 
:<math>\begin{align}
  \delta_\Phi\mathcal{L}\, &=\,
  \mathcal{L}(\Phi+\delta\Phi,\eta)\, -\, \mathcal{L}(\Phi,\eta) \\
  &=\, -\, \int_{t_0}^{t_1} \iint \left\{ \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)}
          \rho\, \left( \frac{\partial(\delta\Phi)}{\partial t}
                        +\, \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} (\delta\Phi)
                        +\, \frac{\partial\Phi}{\partial z}\,  \frac{\partial(\delta \Phi)}{\partial z}\,
                  \right)\; \text{d}z\, \right\}\; \text{d}\boldsymbol{x}\; \text{d}t.
\end{align}</math>
 
Using [[Leibniz integral rule]], this becomes, in case of constant density ''ρ'':<ref name=Whitham1974/>
 
:<math>\begin{align}
  \delta_\Phi\mathcal{L}\, =\,
  &-\, \rho\, \int_{t_0}^{t_1} \iint
    \left\{
      \frac{\partial}{\partial t} \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \delta\Phi\; \text{d}z\;
      +\, \boldsymbol{\nabla} \cdot \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \delta\Phi\, \boldsymbol{\nabla}\Phi\; \text{d}z\,
    \right\}\; \text{d}\boldsymbol{x}\; \text{d}t
  \\
  &+\, \rho\, \int_{t_0}^{t_1} \iint
    \left\{
      \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \delta\Phi\;
        \left( \boldsymbol{\nabla} \cdot \boldsymbol{\nabla}\Phi\, +\, \frac{\partial^2\Phi}{\partial z^2} \right)\; \text{d}z\,
    \right\}\; \text{d}\boldsymbol{x}\; \text{d}t
  \\
  &+\, \rho\, \int_{t_0}^{t_1} \iint
    \left[
      \left( \frac{\partial\eta}{\partial t}\, +\, \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} \eta\, -\, \frac{\partial\Phi}{\partial z} \right)\, \delta\Phi
    \right]_{z=\eta(\boldsymbol{x},t)}\; \text{d}\boldsymbol{x}\; \text{d}t
  \\
  &-\, \rho\, \int_{t_0}^{t_1} \iint
    \left[
      \left( \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} h\, +\, \frac{\partial\Phi}{\partial z} \right)\, \delta\Phi
    \right]_{z=-h(\boldsymbol{x})}\; \text{d}\boldsymbol{x}\; \text{d}t
  \\
  =\, &0.
\end{align}</math>
 
The first integral on the right-hand side integrates out to the boundaries, in '''''x''''' and ''t'', of the integration domain and is zero since the variations ''δΦ'' are taken to be zero at these boundaries. For variations ''δΦ'' which are zero at the free surface and the bed, the second integral remains, which is only zero for arbitrary ''δΦ'' in the fluid interior if there the [[Laplace equation]] holds:
 
:<math>\Delta \Phi\, =\, 0 \qquad \text{ for } -h(\boldsymbol{x})\, <\, z\, <\, \eta(\boldsymbol{x},t),</math>
 
with &Delta;=&nabla;&middot;&nabla; + &part;<sup>2</sup>/&part;''z''<sup>2</sup> the [[Laplace operator]].
 
If variations ''δΦ'' are considered which are only non-zero at the free surface, only the third integral remains, giving rise to the kinematic free-surface boundary condition:
 
:<math>
  \frac{\partial\eta}{\partial t}\, +\, \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} \eta\, -\, \frac{\partial\Phi}{\partial z}\, =\, 0.
  \qquad \text{ at } z\, =\, \eta(\boldsymbol{x},t).
</math>
 
Similarly, variations ''δΦ'' only non-zero at the bottom ''z'' = -''h'' result in the kinematic bed condition:
 
:<math>
  \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} h\, +\, \frac{\partial\Phi}{\partial z}\, =\, 0
  \qquad \text{ at } z\, =\, -h(\boldsymbol{x}).
</math>
 
====Variation with respect to the surface elevation====
 
Considering the variation of the Lagrangian with respect to small changes ''δη'' gives:
 
:<math>
  \delta_\eta\mathcal{L}\, =\,
  \mathcal{L}(\Phi,\eta+\delta\eta)\, -\, \mathcal{L}(\Phi,\eta)
  =\, -\, \int_{t_0}^{t_1} \iint
          \left[ \rho\, \delta\eta\,
            \left(
              \frac{\partial\Phi}{\partial t}
              +\, \frac12\, \left| \boldsymbol{\nabla}\Phi \right|^2\,
              +\, \frac12\, \left( \frac{\partial\Phi}{\partial z} \right)^2
              +\, g\, \eta 
            \right)\,
          \right]_{z=\eta(\boldsymbol{x},t)}\; \text{d}\boldsymbol{x}\; \text{d}t\,
  =\, 0.
</math>
 
This has to be zero for arbitrary ''δη'', giving rise to the dynamic boundary condition at the free surface:
 
:<math>
  \frac{\partial\Phi}{\partial t}
  +\, \frac12\, \left| \boldsymbol{\nabla}\Phi \right|^2\,
  +\, \frac12\, \left( \frac{\partial\Phi}{\partial z} \right)^2
  +\, g\, \eta\,
  =\, 0
  \qquad \text{ at } z\, =\, \eta(\boldsymbol{x},t).
</math>
 
This is the [[Bernoulli's principle|Bernoulli equation]] for unsteady potential flow, applied at the free surface, and with the pressure above the free surface being a constant — which constant pressure is taken equal to zero for simplicity.
 
==Hamiltonian formulation==
 
The [[Hamiltonian mechanics|Hamiltonian]] structure of surface gravity waves on a potential flow was discovered by [[Vladimir E. Zakharov]] in 1968, and rediscovered independently by [[Lambertus Johannes Folkert Broer|Bert Broer]] and [[John W. Miles|John Miles]]:<ref name=Zakharov1968/><ref name=Broer1974/><ref name=Miles1977/>
 
:<math>\begin{align}
  \rho\, \frac{\partial\eta}{\partial t}\, &=\, +\, \frac{\delta\mathcal{H}}{\delta\varphi},\\
  \rho\, \frac{\partial\varphi}{\partial t}\, &=\, -\, \frac{\delta\mathcal{H}}{\delta\eta},
\end{align}</math>
 
where the surface elevation ''η'' and surface potential ''φ'' — which is the potential ''Φ'' at the free surface ''z''=''η''('''''x''''',''t'') — are the [[canonical coordinates|canonical variables]]. The Hamiltonian <math>\mathcal{H}(\varphi,\eta)</math> is the sum of the [[kinetic energy|kinetic]] and [[potential energy]] of the fluid:
 
:<math>\mathcal{H}\, =\,
  \iint \left\{ 
      \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)}
        \frac12\, \rho\, \left[
                  \left| \boldsymbol{\nabla}\Phi \right|^2\,
              +\, \left( \frac{\partial\Phi}{\partial z} \right)^2
        \right]\, \text{d}z\,
    +\, \frac12\, \rho\, g\, \eta^2
  \right\}\; \text{d}\boldsymbol{x}.
</math>
 
The additional constraint is that the flow in the fluid domain has to satisfy [[Laplace's equation]] with appropriate boundary condition at the bottom ''z''=-''h''('''''x''''') and that the potential at the free surface ''z''=''η'' is equal to ''φ'': <math>\delta\mathcal{H}/\delta\Phi\,=\,0.</math>
 
===Relation with Lagrangian formulation===
 
The Hamiltonian formulation can be derived from Luke's Lagrangian description by using [[Leibniz integral rule]] on the integral of &part;''Φ''/&part;''t'':<ref name=Miles1977/>
 
:<math>\mathcal{L}_H = \int_{t_0}^{t_1} \iint \left\{ \varphi(\boldsymbol{x},t)\, \frac{\partial\eta(\boldsymbol{x},t)}{\partial t}\, -\, H(\varphi,\eta;\boldsymbol{x},t) \right\}\; \text{d}\boldsymbol{x}\; \text{d}t,</math>
 
with <math>\varphi(\boldsymbol{x},t)=\Phi(\boldsymbol{x},\eta(\boldsymbol{x},t),t)</math> the value of the velocity potential at the free surface, and <math>H(\varphi,\eta;\boldsymbol{x},t)</math> the Hamiltonian density — sum of the kinetic and potential energy density — and related to the Hamiltonian as:
 
:<math>\mathcal{H}(\varphi,\eta)\, =\, \iint H(\varphi,\eta;\boldsymbol{x},t)\; \text{d}\boldsymbol{x}.</math>
 
The Hamiltonian density is written in terms of the surface potential using [[Green's identities|Green's third identity]] on the kinetic energy:<ref name=Milder1977>
{{cite journal
| author = D. M. Milder
| year = 1977
| title = A note on: 'On Hamilton's principle for surface waves'
| journal = [[Journal of Fluid Mechanics]]
| volume = 83 | issue = 1 | pages = 159–161
| doi = 10.1017/S0022112077001116
|bibcode = 1977JFM....83..159M }}</ref>
 
:<math>
  H\, =\,
  \frac12\, \rho\, \sqrt{ 1\, +\, \left| \boldsymbol{\nabla} \eta \right|^2}\;\; \varphi\, \bigl( D(\eta)\; \varphi \bigr)\,
  +\, \frac12\, \rho\, g\, \eta^2,
</math>
 
where ''D''(''η'') ''φ'' is equal to the [[Surface normal|normal]] derivative of &part;''Φ''/&part;''n'' at the free surface. Because of the linearity of the Laplace equation — valid in the fluid interior and depending on the boundary condition at the bed ''z''=-''h'' and free surface ''z''=''η'' — the normal derivative &part;''Φ''/&part;''n'' is a ''linear'' function of the surface potential ''φ'', but depends non-linear on the surface elevation ''η''. This is expressed by the [[Poincaré–Steklov operator|Dirichlet-to-Neumann]] operator ''D''(''η''), acting linearly on ''φ''.
 
The Hamiltonian density can also be written as:<ref name=Miles1977/>
 
:<math>
  H\, =\,
  \frac12\, \rho\, \varphi\,
    \Bigl[
      w\, \left( 1\, +\, \left| \boldsymbol{\nabla} \eta \right|^2 \right)
      -\, \boldsymbol{\nabla}\eta \cdot \boldsymbol{\nabla}\, \varphi
    \Bigr]\,
  +\, \frac12\, \rho\, g\, \eta^2,
</math>
 
with ''w''('''''x''''',''t'') = &part;''Φ''/&part;''z'' the vertical velocity at the free surface ''z'' = ''η''. Also ''w'' is a ''linear'' function of the surface potential ''φ'' through the Laplace equation, but ''w'' depends non-linear on the surface elevation ''η'':<ref name=Milder1977/>
 
:<math>w\, =\, W(\eta)\, \varphi, </math>
 
with ''W'' operating linear on ''φ'', but being non-linear in ''η''. As a result, the Hamiltonian is a quadratic [[functional (mathematics)|functional]] of the surface potential ''φ''. Also the potential energy part of the Hamiltonian is quadratic. The source of non-linearity in surface gravity waves is through the kinetic energy depending non-linear on the free surface shape ''η''.<ref name=Milder1977/>
 
Further &nabla;''φ'' is not to be mistaken for the horizontal velocity &nabla;''Φ'' at the free surface:
 
:<math>
  \boldsymbol{\nabla}\varphi\, =\,
  \boldsymbol{\nabla} \Phi\bigl(\boldsymbol{x},\eta(\boldsymbol{x},t),t\bigr)\, =\,
  \left[ \boldsymbol{\nabla}\Phi\, +\, \frac{\partial\Phi}{\partial z}\, \boldsymbol{\nabla}\eta \right]_{z=\eta(\boldsymbol{x},t)}\, =\,
  \Bigl[ \boldsymbol{\nabla}\Phi \Bigr]_{z=\eta(\boldsymbol{x},t)}\, +\, w\, \boldsymbol{\nabla}\eta.
</math>
 
Taking the variations of the Lagrangian <math>\mathcal{L}_H</math> with respect to the canonical variables <math>\varphi(\boldsymbol{x},t)</math> and <math>\eta(\boldsymbol{x},t)</math> gives:
 
:<math>\begin{align}
  \rho\, \frac{\partial\eta}{\partial t}\, &=\, +\, \frac{\delta\mathcal{H}}{\delta\varphi},\\
  \rho\, \frac{\partial\varphi}{\partial t}\, &=\, -\, \frac{\delta\mathcal{H}}{\delta\eta},
\end{align}</math>
 
provided in the fluid interior ''Φ'' satisfies the Laplace equation, &Delta;''Φ''=0, as well as the bottom boundary condition at ''z''=-''h'' and ''Φ''=''φ'' at the free surface.
 
==References and notes==
{{reflist|2}}
 
{{physical oceanography}}
 
[[Category:Fluid dynamics]]

Revision as of 10:18, 1 March 2014

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