Luke's variational principle

In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967.[1] This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the so-called mild-slope equation,[2] or using the averaged Lagrangian approach for wave propagation in inhomogeneous media.[3]

Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.[4][5][6] 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, and by using Clebsch potentials to include vorticity.[1]

Luke's Lagrangian

Luke's Lagrangian formulation is for non-linear surface gravity waves on an—incompressible, irrotational and inviscidpotential 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
  • ∇ is the horizontal gradient operator, so ∇Φ is the horizontal flow velocity consisting of ∂Φ/∂x and ∂Φ/∂y,
  • V(t) is the time-dependent fluid domain with free surface.

The Lagrangian , as given by Luke, is:

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.[7]

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 dynamic 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 at z=η(x,t) and a fixed bed at z=−h(x), Luke's variational principle results in the Lagrangian:

The bed-level term proportional to h2 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 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.[8] Then the resulting variation in the Lagrangian is:

Using Leibniz integral rule, this becomes, in case of constant density ρ:[8]

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:

with Δ=∇·∇ + ∂2/∂z2 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:

Similarly, variations δΦ only non-zero at the bottom z = -h result in the kinematic bed condition:

Variation with respect to the surface elevation

Considering the variation of the Lagrangian with respect to small changes δη gives:

This has to be zero for arbitrary δη, giving rise to the dynamic boundary condition at the free surface:

This is the 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 structure of surface gravity waves on a potential flow was discovered by Vladimir E. Zakharov in 1968, and rediscovered independently by Bert Broer and John Miles:[4][5][6]

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 variables. The Hamiltonian is the sum of the kinetic and potential energy of the fluid:

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:

Relation with Lagrangian formulation

The Hamiltonian formulation can be derived from Luke's Lagrangian description by using Leibniz integral rule on the integral of ∂Φ/∂t:[6]

with the value of the velocity potential at the free surface, and the Hamiltonian density — sum of the kinetic and potential energy density — and related to the Hamiltonian as:

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

where D(η) φ is equal to the normal 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 bed z=-h and free surface z=η — the normal derivative ∂Φ/∂n is a linear function of the surface potential φ, but depends non-linear on the surface elevation η. This is expressed by the Dirichlet-to-Neumann operator D(η), acting linearly on φ.

The Hamiltonian density can also be written as:[6]

with w(x,t) = ∂Φ/∂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:[9]

with W operating linear on φ, but being non-linear in η. As a result, the Hamiltonian is a quadratic 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.[9]

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

Taking the variations of the Lagrangian with respect to the canonical variables and gives:

provided in the fluid interior Φ satisfies the Laplace equation, ΔΦ=0, as well as the bottom boundary condition at z=-h and Φ=φ at the free surface.

References and notes

  1. ^ a b J. C. Luke (1967). "A Variational Principle for a Fluid with a Free Surface". Journal of Fluid Mechanics. 27 (2): 395–397. Bibcode:1967JFM....27..395L. doi:10.1017/S0022112067000412.
  2. ^ M. W. Dingemans (1997). Water Wave Propagation Over Uneven Bottoms. Advanced Series on Ocean Engineering. 13. Singapore: World Scientific. p. 271. ISBN 981-02-0427-2.
  3. ^ G. B. Whitham (1974). Linear and Nonlinear Waves. Wiley-Interscience. p. 555. ISBN 0-471-94090-9.
  4. ^ a b V. E. Zakharov (1968). "Stability of Periodic Waves of Finite Amplitude on the Surface of a Deep Fluid". Journal of Applied Mechanics and Technical Physics. 9 (2): 190–194. Bibcode:1968JAMTP...9..190Z. doi:10.1007/BF00913182. Originally appeared in Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki 9(2): 86–94, 1968.
  5. ^ a b L. J. F. Broer (1974). "On the Hamiltonian Theory of Surface Waves". Applied Scientific Research. 29: 430–446. doi:10.1007/BF00384164.
  6. ^ a b c d J. W. Miles (1977). "On Hamilton's Principle for Surface Waves". Journal of Fluid Mechanics. 83 (1): 153–158. Bibcode:1977JFM....83..153M. doi:10.1017/S0022112077001104.
  7. ^ H. Bateman (1929). "Notes on a Differential Equation Which Occurs in the Two-Dimensional Motion of a Compressible Fluid and the Associated Variational Problems". Proceedings of the Royal Society of London A. 125 (799): 598–618. Bibcode:1929RSPSA.125..598B. doi:10.1098/rspa.1929.0189.
  8. ^ a b c G. W. Whitham (1974). Linear and Nonlinear Waves. New York: Wiley. pp. 434–436. ISBN 0-471-94090-9.
  9. ^ a b c D. M. Milder (1977). "A note on: 'On Hamilton's principle for surface waves'". Journal of Fluid Mechanics. 83 (1): 159–161. Bibcode:1977JFM....83..159M. doi:10.1017/S0022112077001116.
Bahama Banks

The Bahama Banks are the submerged carbonate platforms that make up much of the Bahama Archipelago. The term is usually applied in referring to either the Great Bahama Bank around Andros Island, or the Little Bahama Bank of Grand Bahama Island and Great Abaco, which are the largest of the platforms, and the Cay Sal Bank north of Cuba. The islands of these banks are politically part of the Bahamas. Other banks are the three banks of the Turks and Caicos Islands, namely the Caicos Bank of the Caicos Islands, the bank of the Turks Islands, and wholly submerged Mouchoir Bank. Further southeast are the equally wholly submerged Silver Bank and Navidad Bank north of the Dominican Republic.

Bernoulli's principle

In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler who derived Bernoulli's equation in its usual form in 1752. The principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g. heat radiation) are small and can be neglected.

Bernoulli's principle can be applied to various types of fluid flow, resulting in various forms of Bernoulli's equation; there are different forms of Bernoulli's equation for different types of flow. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).

Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy, potential energy and internal energy remains constant. Thus an increase in the speed of the fluid – implying an increase in its kinetic energy (dynamic pressure) – occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ρ g h) is the same everywhere.Bernoulli's principle can also be derived directly from Isaac Newton's Second Law of Motion. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.

Calculus of variations

Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions

and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

Carbonate platform

A carbonate platform is a sedimentary body which possesses topographic relief, and is composed of autochthonic calcareous deposits. Platform growth is mediated by sessile organisms whose skeletons build up the reef or by organisms (usually microbes) which induce carbonate precipitation through their metabolism. Therefore, carbonate platforms can not grow up everywhere: they are not present in places where limiting factors to the life of reef-building organisms exist. Such limiting factors are, among others: light, water temperature, transparency and pH-Value. For example, carbonate sedimentation along the Atlantic South American coasts takes place everywhere but at the mouth of the Amazon River, because of the intense turbidity of the water there. Spectacular examples of present-day carbonate platforms are the Bahama Banks under which the platform is roughly 8 km thick, the Yucatan Peninsula which is up to 2 km thick, the Florida platform, the platform on which the Great Barrier Reef is growing, and the Maldive atolls. All these carbonate platforms and their associated reefs are confined to tropical latitudes. Today's reefs are built mainly by scleractinian corals, but in the distant past other organisms, like archaeocyatha (during the Cambrian) or extinct cnidaria (tabulata and rugosa) were important reef builders.

Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field is:

where the scalar fields and are known as Clebsch potentials or Monge potentials, named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and is the gradient operator.

Dispersion (water waves)

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water interface and gravity as the only force restoring it to flatness – propagate faster with increasing wavelength. On the other hand, for a given (fixed) wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves (i.e. only forced by surface tension) propagate faster for shorter wavelengths.

Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves.

Hamiltonian fluid mechanics

Hamiltonian fluid mechanics is the application of Hamiltonian methods to fluid mechanics. Note that this formalism only applies to nondissipative fluids.

Index of physics articles (L)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

List of submarine volcanoes

A list of active and extinct submarine volcanoes and seamounts located under the world's oceans. There are estimated to be 40,000 to 55,000 seamounts in the global oceans. Almost all are not well-mapped and many may not have been identified at all. Most are unnamed and unexplored. This list is therefore confined to seamounts that are notable enough to have been named and/or explored.

List of variational topics

This is a list of variational topics in from mathematics and physics. See calculus of variations for a general introduction.

Action (physics)

Averaged Lagrangian

Brachistochrone curve

Calculus of variations



Dirichlet principle

Euler–Lagrange equation cf. Action (physics)

Fermat's principle

Functional (mathematics)

Functional derivative

Functional integral




Lagrangian mechanics

Legendre transformation

Luke's variational principle

Minimal surface

Morse theory

Noether's theorem

Path integral formulation

Plateau's problem

Prime geodesic

Principle of least action

Soap bubble

Soap film

Tautochrone curve

Mild-slope equation

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

The mild-slope equation models the propagation and transformation of water waves, as they travel through waters of varying depth and interact with lateral boundaries such as cliffs, beaches, seawalls and breakwaters. As a result, it describes the variations in wave amplitude, or equivalently wave height. From the wave amplitude, the amplitude of the flow velocity oscillations underneath the water surface can also be computed. These quantities—wave amplitude and flow-velocity amplitude—may subsequently be used to determine the wave effects on coastal and offshore structures, ships and other floating objects, sediment transport and resulting geomorphology changes of the sea bed and coastline, mean flow fields and mass transfer of dissolved and floating materials. Most often, the mild-slope equation is solved by computer using methods from numerical analysis.

A first form of the mild-slope equation was developed by Eckart in 1952, and an improved version—the mild-slope equation in its classical formulation—has been derived independently by Juri Berkhoff in 1972. Thereafter, many modified and extended forms have been proposed, to include the effects of, for instance: wave–current interaction, wave nonlinearity, steeper sea-bed slopes, bed friction and wave breaking. Also parabolic approximations to the mild-slope equation are often used, in order to reduce the computational cost.

In case of a constant depth, the mild-slope equation reduces to the Helmholtz equation for wave diffraction.

Oceanic plateau

An oceanic or submarine plateau is a large, relatively flat elevation that is higher than the surrounding relief with one or more relatively steep sides.There are 184 oceanic plateaus covering an area of 18,486,600 km2 (7,137,700 sq mi), or about 5.11% of the oceans. The South Pacific region around Australia and New Zealand contains the greatest number of oceanic plateaus (see map).

Oceanic plateaus produced by large igneous provinces are often associated with hotspots, mantle plumes, and volcanic islands — such as Iceland, Hawaii, Cape Verde, and Kerguelen. The three largest plateaus, the Caribbean, Ontong Java, and Mid-Pacific Mountains, are located on thermal swells. Other oceanic plateaus, however, are made of rifted continental crust, for example Falkland Plateau, Lord Howe Rise, and parts of Kerguelen, Seychelles, and Arctic ridges.

Plateaus formed by large igneous provinces were formed by the equivalent of continental flood basalts such as the Deccan Traps in India and the Snake River Plain in the United States.

In contrast to continental flood basalts, most igneous oceanic plateaus erupt through young and thin (6–7 km (3.7–4.3 mi)) mafic or ultra-mafic crust and are therefore uncontaminated by felsic crust and representative for their mantle sources.

These plateaus often rise 2–3 km (1.2–1.9 mi) above the surrounding ocean floor and are more buoyant than oceanic crust. They therefore tend to withstand subduction, more-so when thick and when reaching subduction zones shortly after their formations. As a consequence, they tend to "dock" to continental margins and be preserved as accreted terranes. Such terranes are often better preserved than the exposed parts of continental flood basalts and are therefore a better record of large-scale volcanic eruptions throughout Earth's history. This "docking" also means that oceanic plateaus are important contributors to the growth of continental crust. Their formations often had a dramatic impact on global climate, such as the most recent plateaus formed, the three, large, Cretaceous oceanic plateaus in the Pacific and Indian Ocean: Ontong Java, Kerguelen, and Caribbean.

Outline of oceanography

The following outline is provided as an overview of and introduction to Oceanography.

Physical oceanography

Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters.

Physical oceanography is one of several sub-domains into which oceanography is divided. Others include biological, chemical and geological oceanography.

Physical oceanography may be subdivided into descriptive and dynamical physical oceanography.Descriptive physical oceanography seeks to research the ocean through observations and complex numerical models, which describe the fluid motions as precisely as possible.

Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and numerical models. These are part of the large field of Geophysical Fluid Dynamics (GFD) that is shared together with meteorology. GFD is a sub field of Fluid dynamics describing flows occurring on spatial and temporal scales that are greatly influenced by the Coriolis force.

Undersea mountain range

Undersea mountain ranges are mountain ranges that are mostly or entirely underwater, and specifically under the surface of an ocean. If originated from current tectonic forces, they are often referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known undersea mountain range is a mid-ocean ridge, the Mid-Atlantic Ridge. It has been observed that, "similar to those on land, the undersea mountain ranges are the loci of frequent volcanic and earthquake activity".

Wave base

The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.

Wind wave

In fluid dynamics, wind waves, or wind-generated waves, are water surface waves that occur on the free surface of the oceans and other bodies (like lakes, rivers, canals, puddles or ponds). They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high.When directly generated and affected by local waters, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells. More generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago. Wind waves in the ocean are called ocean surface waves.

Wind waves have a certain amount of randomness: subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a stochastic process, in combination with the physics governing their generation, growth, propagation, and decay—as well as governing the interdependence between flow quantities such as: the water surface movements, flow velocities and water pressure. The key statistics of wind waves (both seas and swells) in evolving sea states can be predicted with wind wave models.

Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves.

Ocean zones
Sea level

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