Stokes Drift
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For a pure
wave In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...
in
fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, the Stokes drift velocity is the
average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
when following a specific
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
parcel as it travels with the
fluid flow In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
. For instance, a particle floating at the
free surface In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
of
water waves In fluid dynamics, a wind wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result of the wind blowing over the water's surface. The contact distance in the direction of the wind is k ...
, experiences a net Stokes drift velocity in the direction of
wave propagation In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. '' Periodic waves'' oscillate repeatedly about an equilibrium (resting) value at some f ...
. More generally, the Stokes drift velocity is the difference between the
average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
Lagrangian
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
of a fluid parcel, and the average Eulerian
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
of the
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
at a fixed position. This
nonlinear In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
phenomenon is named after
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish mathematician and physicist. Born in County Sligo, Ireland, Stokes spent his entire career at the University of Cambridge, where he served as the Lucasi ...
, who derived expressions for this drift in his 1847 study of
water waves In fluid dynamics, a wind wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result of the wind blowing over the water's surface. The contact distance in the direction of the wind is k ...
. The Stokes drift is the difference in end positions, after a predefined amount of time (usually one wave period), as derived from a description in the
Lagrangian and Eulerian coordinates Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
. The end position in the Lagrangian description is obtained by following a specific fluid parcel during the time interval. The corresponding end position in the Eulerian description is obtained by integrating the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
at a fixed position—equal to the initial position in the Lagrangian description—during the same time interval. The Stokes drift velocity equals the Stokes drift divided by the considered time interval. Often, the Stokes drift velocity is loosely referred to as Stokes drift. Stokes drift may occur in all instances of oscillatory flow which are
inhomogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
in space. For instance in
water waves In fluid dynamics, a wind wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result of the wind blowing over the water's surface. The contact distance in the direction of the wind is k ...
,
tide Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another. Tide tables ...
s and atmospheric waves. In the Lagrangian description, fluid parcels may drift far from their initial positions. As a result, the unambiguous definition of an
average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
Lagrangian velocity and Stokes drift velocity, which can be attributed to a certain fixed position, is by no means a trivial task. However, such an unambiguous description is provided by the '' Generalized Lagrangian Mean'' (GLM) theory of Andrews and McIntyre in 1978. The Stokes drift is important for the
mass transfer Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction, or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtra ...
of various kinds of material and organisms by oscillatory flows. It plays a crucial role in the generation of Langmuir circulations. For nonlinear and periodic water waves, accurate results on the Stokes drift have been computed and tabulated.


Mathematical description

The Lagrangian motion of a fluid parcel with
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
''x = ξ(α, t)'' in the Eulerian coordinates is given bySee Phillips (1977), page 43. : \dot = \frac = \mathbf\big(\boldsymbol(\boldsymbol, t), t\big), where : ∂''ξ''/∂''t'' is the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
of ''ξ''(''α'', ''t'') with respect to ''t'', : ''ξ''(''α'', ''t'') is the Lagrangian position vector of a fluid parcel, : u(x, ''t'') is the Eulerian
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
, : x is the position vector in the Eulerian coordinate system, : ''α'' is the position vector in the Lagrangian coordinate system, : ''t'' is
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
. Often, the Lagrangian coordinates ''α'' are chosen to coincide with the Eulerian coordinates x at the initial time ''t'' = ''t''0: : \boldsymbol(\boldsymbol, t_0) = \boldsymbol. If the
average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
value of a quantity is denoted by an overbar, then the average Eulerian velocity vector ūE and average Lagrangian velocity vector ūL are : \begin \bar\mathbf_\text &= \overline, \\ \bar\mathbf_\text &= \overline = \overline = \overline. \end Different definitions of the
average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
may be used, depending on the subject of study (see
ergodic theory Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...
): *
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
average, *
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
average, * ensemble average, * phase average. The Stokes drift velocity ūS is defined as the difference between the average Eulerian velocity and the average Lagrangian velocity: : \bar\mathbf_\text = \bar\mathbf_\text - \bar\mathbf_\text. In many situations, the mapping of average quantities from some Eulerian position x to a corresponding Lagrangian position ''α'' forms a problem. Since a fluid parcel with label ''α'' traverses along a
path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desir ...
of many different Eulerian positions x, it is not possible to assign ''α'' to a unique x. A mathematically sound basis for an unambiguous mapping between average Lagrangian and Eulerian quantities is provided by the theory of the generalized Lagrangian mean (GLM) by Andrews and McIntyre (1978).


Example: A one-dimensional compressible flow

For the Eulerian velocity as a monochromatic wave of any nature in a continuous medium: u = \hat \sin(kx - \omega t), one readily obtains by the
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
with k\hat/\omega as a small parameter for the particle position : \dot\xi = u(\xi, t) = \hat \sin(k\xi - \omega t), : \xi(\xi_0, t) \approx \xi_0 + \frac \cos(k\xi_0 - \omega t) - \frac14 \frac \sin 2(k\xi_0 - \omega t) + \frac12 \frac t. Here the last term describes the Stokes drift velocity \tfrac12 k\hat^2/\omega.


Example: Deep water waves

The Stokes drift was formulated for
water waves In fluid dynamics, a wind wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result of the wind blowing over the water's surface. The contact distance in the direction of the wind is k ...
by
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish mathematician and physicist. Born in County Sligo, Ireland, Stokes spent his entire career at the University of Cambridge, where he served as the Lucasi ...
in 1847. For simplicity, the case of infinitely deep water is considered, with
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
wave propagation In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. '' Periodic waves'' oscillate repeatedly about an equilibrium (resting) value at some f ...
of a
sinusoidal A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is '' simple harmonic motion''; as rotation, it correspond ...
wave on the
free surface In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
of a fluid layer:See e.g. Phillips (1977), page 37. : \eta = a \cos(kx - \omega t), where : ''η'' is the
elevation The elevation of a geographic location (geography), ''location'' is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational equipotenti ...
of the
free surface In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
in the ''z'' direction (meters), : ''a'' is the wave
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
(meters), : ''k'' is the
wave number In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of r ...
: ''k'' = 2''π''/''λ'' (
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s per meter), : ''ω'' is the
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
: ''ω'' = 2''π''/''T'' (
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s per
second The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...
), : ''x'' is the horizontal
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
and the wave propagation direction (meters), : ''z'' is the vertical
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
, with the positive ''z'' direction pointing out of the fluid layer (meters), : ''λ'' is the wave length (meters), : ''T'' is the wave period (
second The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...
s). As derived below, the horizontal component ''ū''S(''z'') of the Stokes drift velocity for deep-water waves is approximately:See Phillips (1977), page 44. Or Craik (1985), page 110. : \bar_\text \approx \omega k a^2 \text^ = \frac \text^. As can be seen, the Stokes drift velocity ''ū''S is a nonlinear quantity in terms of the wave
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
''a''. Further, the Stokes drift velocity decays exponentially with depth: at a depth of a quarter wavelength, ''z'' = −''λ''/4, it is about 4% of its value at the mean
free surface In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
, ''z'' = 0.


Derivation

It is assumed that the waves are of
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
and the
free surface In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
oscillates around the
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
level ''z'' = 0. The waves propagate under the action of gravity, with a constant
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
by
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
(pointing downward in the negative ''z'' direction). Further the fluid is assumed to be inviscidViscosity has a pronounced effect on the mean Eulerian velocity and mean Lagrangian (or mass transport) velocity, but much less on their difference: the Stokes drift outside the
boundary layers In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condi ...
near bed and free surface, see for instance Longuet-Higgins (1953). Or Phillips (1977), pages 53–58.
and
incompressible Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...
, with a constant
mass density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter rho), although the Latin letter ''D'' (or ''d'') ...
. The fluid flow is
irrotational In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not chan ...
. At infinite depth, the fluid is taken to be at
rest REST (Representational State Transfer) is a software architectural style that was created to describe the design and guide the development of the architecture for the World Wide Web. REST defines a set of constraints for how the architecture of ...
. Now the flow may be represented by a
velocity potential A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788. It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a ca ...
''φ'', satisfying the
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delt ...
and : \varphi = \frac a \text^ \sin(kx - \omega t). In order to have
non-trivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group, topological space). The noun triviality usual ...
solutions for this
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
problem, the wave length and wave period may not be chosen arbitrarily, but must satisfy the deep-water dispersion relation:See ''e.g.'' Phillips (1977), page 38. : \omega^2 = gk with ''g'' the
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
by
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
in (m/s2). Within the framework of
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
theory, the horizontal and vertical components, ''ξx'' and ''ξz'' respectively, of the Lagrangian position ''ξ'' are : \begin \xi_x &= x + \int \frac\, \textt = x - a \text^ \sin(kx - \omega t), \\ \xi_z &= z + \int \frac\, \textt = z + a \text^ \cos(kx - \omega t). \end The horizontal component ''ū''S of the Stokes drift velocity is estimated by using a
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
around x of the Eulerian horizontal velocity component ''ux'' = ∂''ξx'' / ∂''t'' at the position ''ξ'': : \begin \bar_\text &= \overline - \overline \\ &= \overline - \overline \\ &\approx \overline + \overline \\ &= \overline \\ &+ \overline \\ &= \overline \\ &= \omega ka^2 \text^. \end


See also

* Coriolis–Stokes force * Darwin drift *
Lagrangian and Eulerian coordinates Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
*
Material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material de ...


References


Historical

* *
Reprinted in:


Other

* * * * * *


Notes

{{physical oceanography Fluid dynamics Water waves