Influid dynamics,Luke's variational principle is aLagrangianvariational description of the motion ofsurface waves on afluid with afree surface, under the action ofgravity. This principle is named after J.C. Luke, who published it in 1967.^[1] This variational principle is forincompressible andinviscidpotential flows, and is used to derive approximate wave models like themild-slope equation,^[2] or using theaveraged Lagrangian approach for wave propagation in inhomogeneous media.^[3]

Luke's Lagrangian formulation can also be recast into aHamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.^[4][5][6] This is often used when modelling thespectral density evolution of the free-surface in asea state, sometimes calledwave turbulence.

Both the Lagrangian and Hamiltonian formulations can be extended to includesurface tension effects, and by usingClebsch potentials to includevorticity.^[1]

Luke'sLagrangian formulation is fornon-linear surface gravity waves on an—incompressible,irrotational andinviscid—potential flow.

The relevant ingredients, needed in order to describe this flow, are:

• Φ(x,z,t) is thevelocity potential,
• ρ is the fluiddensity,
• g is the acceleration by theEarth's gravity,
• x is the horizontal coordinate vector with componentsx andy,
• x andy are the horizontal coordinates,
• z is the vertical coordinate,
• t is time, and
• ∇ is the horizontalgradient operator, so∇Φ is the horizontalflow velocity consisting of∂Φ/∂x and∂Φ/∂y,
• V(t) is the time-dependent fluid domain with free surface.

The Lagrangian${\displaystyle \mathcal{L}}$, as given by Luke, is:$${\displaystyle \mathcal{L}=-{\int }_{t_0}^{t_1}\left\{{\iiint }_{V(t)}\rho \left[\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }t}+\frac{1}{2}{\left|\mathbf{\nabla }\mathrm{\Phi }\right|}^2+\frac{1}{2}{\left(\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}\right)}^2+gz\right]\mathrm{d}x\mathrm{d}y\mathrm{d}z\right\}\mathrm{d}t.}$$

FromBernoulli's principle, this Lagrangian can be seen to be theintegral of the fluidpressure over the whole time-dependent fluid domainV(t). This is in agreement with the variational principles for inviscid flow without a free surface, found byHarry Bateman.^[7]

Variation with respect to the velocity potentialΦ(x,z,t) and free-moving surfaces likez =η(x,t) results in theLaplace equation for the potential in the fluid interior and all requiredboundary conditions:kinematic boundary conditions on all fluid boundaries anddynamic boundary conditions on free surfaces.^[8] This may also include moving wavemaker walls and ship motion.

For the case of a horizontally unbounded domain with the free fluid surface atz =η(x,t) and a fixed bed atz = −h(x), Luke's variational principle results in the Lagrangian:$${\displaystyle \mathcal{L}=-{\int }_{t_0}^{t_1}\iint \left\{{\int }_{-h(\mathit{x})}^{\eta (\mathit{x},t)}\rho \left[\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }t}+\frac{1}{2}{\left|\mathbf{\nabla }\mathrm{\Phi }\right|}^2+\frac{1}{2}{\left(\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}\right)}^2\right]\mathrm{d}z+\frac{1}{2}\rho g{\eta }^2\right\}\mathrm{d}\mathit{x}\mathrm{d}t.}$$

The bed-level term proportional toh² 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.

The variation${\displaystyle \delta \mathcal{L}=0}$ in the Lagrangian with respect to variations in the velocity potential Φ(x,z,t), as well as with respect to the surface elevationη(x,t), have to be zero. We consider both variations subsequently.

Consider a small variationδΦ in the velocity potentialΦ.^[8] Then the resulting variation in the Lagrangian is:

UsingLeibniz integral rule, this becomes, in case of constant densityρ:^[8]

The first integral on the right-hand side integrates out to the boundaries, inx andt, of the integration domain and is zero since the variationsδΦ are taken to be zero at these boundaries. For variationsδΦ which are zero at the free surface and the bed, the second integral remains, which is only zero for arbitraryδΦ in the fluid interior if there theLaplace equation holds:withΔ = ∇ ⋅ ∇ + ∂²/∂z² theLaplace operator.

If variationsδΦ 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:$${\displaystyle \frac{\mathrm{\partial }\eta }{\mathrm{\partial }t}+\mathbf{\nabla }\mathrm{\Phi }\cdot \mathbf{\nabla }\eta -\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}=0.\text{ at }z=\eta (\mathit{x},t).}$$

Similarly, variationsδΦ only non-zero at the bottomz = −h result in the kinematic bed condition:$${\displaystyle \mathbf{\nabla }\mathrm{\Phi }\cdot \mathbf{\nabla }h+\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}=0\text{ at }z=-h(\mathit{x}).}$$

Considering the variation of the Lagrangian with respect to small changesδη gives:$${\displaystyle {\delta }_{\eta }\mathcal{L}=\mathcal{L}(\mathrm{\Phi },\eta +\delta \eta )-\mathcal{L}(\mathrm{\Phi },\eta )=-{\int }_{t_0}^{t_1}\iint {\left[\rho \delta \eta \left(\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }t}+\frac{1}{2}{\left|\mathbf{\nabla }\mathrm{\Phi }\right|}^2+\frac{1}{2}{\left(\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}\right)}^2+g\eta \right)\right]}_{z=\eta (\mathit{x},t)}\mathrm{d}\mathit{x}\mathrm{d}t=0.}$$

This has to be zero for arbitraryδη, giving rise to the dynamic boundary condition at the free surface:$${\displaystyle \frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }t}+\frac{1}{2}{\left|\mathbf{\nabla }\mathrm{\Phi }\right|}^2+\frac{1}{2}{\left(\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}\right)}^2+g\eta =0\text{ at }z=\eta (\mathit{x},t).}$$

This is theBernoulli 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.

TheHamiltonian structure of surface gravity waves on a potential flow was discovered byVladimir E. Zakharov in 1968, and rediscovered independently byBert Broer andJohn Miles:^[4][5][6]where the surface elevationη and surface potentialφ — which is the potentialΦ at the free surfacez =η(x,t) — are thecanonical variables. The Hamiltonian${\displaystyle \mathcal{H}(\phi ,\eta )}$ is the sum of thekinetic andpotential energy of the fluid:$${\displaystyle \mathcal{H}=\iint \left\{{\int }_{-h(\mathit{x})}^{\eta (\mathit{x},t)}\frac{1}{2}\rho \left[{\left|\mathbf{\nabla }\mathrm{\Phi }\right|}^2+{\left(\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}\right)}^2\right]\mathrm{d}z+\frac{1}{2}\rho g{\eta }^2\right\}\mathrm{d}\mathit{x}.}$$

The additional constraint is that the flow in the fluid domain has to satisfyLaplace's equation with appropriate boundary condition at the bottomz = −h(x) and that the potential at the free surfacez =η is equal toφ:${\displaystyle \delta \mathcal{H}/\delta \mathrm{\Phi }=0.}$

The Hamiltonian formulation can be derived from Luke's Lagrangian description by usingLeibniz integral rule on the integral of∂Φ/∂t:^[6]$${\displaystyle {\mathcal{L}}_H={\int }_{t_0}^{t_1}\iint \left\{\phi (\mathit{x},t)\frac{\mathrm{\partial }\eta (\mathit{x},t)}{\mathrm{\partial }t}-H(\phi ,\eta ;\mathit{x},t)\right\}\mathrm{d}\mathit{x}\mathrm{d}t,}$$with${\displaystyle \phi (\mathit{x},t)=\mathrm{\Phi }(\mathit{x},\eta (\mathit{x},t),t)}$ the value of the velocity potential at the free surface, and${\displaystyle H(\phi ,\eta ;\mathit{x},t)}$ the Hamiltonian density — sum of the kinetic and potential energy density — and related to the Hamiltonian as:$${\displaystyle \mathcal{H}(\phi ,\eta )=\iint H(\phi ,\eta ;\mathit{x},t)\mathrm{d}\mathit{x}.}$$

The Hamiltonian density is written in terms of the surface potential usingGreen's third identity on the kinetic energy:^[9]

$${\displaystyle H=\frac{1}{2}\rho \sqrt{1+{\left|\mathbf{\nabla }\eta \right|}^2}\phi (D(\eta )\phi )+\frac{1}{2}\rho g{\eta }^2,}$$

whereD(η)φ is equal to thenormal derivative of∂Φ/∂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 bedz = −h and free surfacez =η — the normal derivative∂Φ/∂n is alinear function of the surface potentialφ, but depends non-linear on the surface elevationη. This is expressed by theDirichlet-to-Neumann operatorD(η), acting linearly onφ.

The Hamiltonian density can also be written as:^[6]$${\displaystyle H=\frac{1}{2}\rho \phi [w\left(1+{\left|\mathbf{\nabla }\eta \right|}^2\right)-\mathbf{\nabla }\eta \cdot \mathbf{\nabla }\phi ]+\frac{1}{2}\rho g{\eta }^2,}$$withw(x,t) = ∂Φ/∂z the vertical velocity at the free surfacez =η. Alsow is alinear function of the surface potentialφ through the Laplace equation, butw depends non-linear on the surface elevationη:^[9]$${\displaystyle w=W(\eta )\phi ,}$$withW operating linear onφ, but being non-linear inη. As a result, the Hamiltonian is a quadraticfunctional of the surface potentialφ. 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 shapeη.^[9]

Further∇φ is not to be mistaken for the horizontal velocity∇Φ at the free surface:

$${\displaystyle \mathbf{\nabla }\phi =\mathbf{\nabla }\mathrm{\Phi }(\mathit{x},\eta (\mathit{x},t),t)={\left[\mathbf{\nabla }\mathrm{\Phi }+\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}\mathbf{\nabla }\eta \right]}_{z=\eta (\mathit{x},t)}=[\mathbf{\nabla }\mathrm{\Phi }{]}_{z=\eta (\mathit{x},t)}+w\mathbf{\nabla }\eta .}$$

Taking the variations of the Lagrangian${\displaystyle {\mathcal{L}}_H}$ with respect to the canonical variables${\displaystyle \phi (\mathit{x},t)}$ and${\displaystyle \eta (\mathit{x},t)}$ gives:provided in the fluid interiorΦ satisfies the Laplace equation,ΔΦ = 0, as well as the bottom boundary condition atz = −h andΦ =φ at the free surface.

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