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In
fluid dynamics, the radiation stress is the depth-integrated – and thereafter
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
-
average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
d – excess
momentum flux caused by the presence of the
surface gravity waves, which is exerted on the
mean flow In fluid dynamics, the fluid flow is often decomposed into a mean flow and deviations from the mean. The averaging can be done either in space or in time, or by ensemble averaging
In machine learning, particularly in the creation of artificial ...
. The radiation stresses behave as a second-order
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
.
The radiation stress tensor describes the additional
forcing due to the presence of the waves, which changes the mean depth-integrated horizontal
momentum in the fluid layer. As a result, varying radiation stresses induce changes in the mean surface elevation (
wave setup
In fluid dynamics, wave setup is the increase in mean water level due to the presence of breaking waves. Similarly, wave setdown is a wave-induced decrease of the mean water level before the waves break (during the shoaling process). For short, th ...
) and the mean flow (wave-induced currents).
For the mean
energy density in the
oscillatory part of the fluid motion, the radiation stress tensor is important for its
dynamics, in case of an
inhomogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
mean-flow
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
The radiation stress tensor, as well as several of its implications on the physics of surface gravity waves and mean flows, were formulated in a series of papers by
Longuet-Higgins and Stewart in 1960–1964.
Radiation stress derives its name from the analogous effect of
radiation pressure
Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
for
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
.
Physical significance
The radiation stress – mean excess momentum-flux due to the presence of the waves – plays an important role in the explanation and modeling of various coastal processes:
* ''Wave setup'' and ''setdown'' – the radiation stress consists in part of a
radiation pressure
Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
, exerted 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 ...
elevation of the mean flow. If the radiation stress varies spatially, as it does in the
surf zone
As ocean surface waves approach shore, they get taller and break, forming the foamy, bubbly surface called ''surf''. The region of breaking waves defines the surf zone, or breaker zone. After breaking in the surf zone, the waves (now reduced i ...
where the
wave height
In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighboring trough. ''Wave height'' is a term used by mariners, as well as in coastal, ocean and naval engineering.
At sea, the te ...
reduces by
wave breaking
In fluid dynamics, a breaking wave or breaker is a wave whose amplitude reaches a critical level at which large amounts of wave energy transform into turbulent kinetic energy. At this point, simple physical models that describe wave dynamics ...
, this results in changes of the mean surface elevation called wave setup (in case of an increased level) and setdown (for a decreased water level);
* ''Wave-driven current'', especially a ''longshore current'' in the surf zone – for oblique incidence of waves on a beach, the reduction in wave height inside the surf zone (by breaking) introduces a variation of the shear-stress component ''S''
''xy'' of the radiation stress over the width of the surf zone. This provides the forcing of a wave-driven longshore current, which is of importance for
sediment transport (
longshore drift) and the resulting coastal
morphology
Morphology, from the Greek and meaning "study of shape", may refer to:
Disciplines
* Morphology (archaeology), study of the shapes or forms of artifacts
* Morphology (astronomy), study of the shape of astronomical objects such as nebulae, galaxies ...
;
* ''Bound long waves'' or ''forced long waves'', part of the
infragravity wave
Infragravity waves are surface gravity waves with frequencies lower than the wind waves – consisting of both wind sea and swell – thus corresponding with the part of the wave spectrum lower than the frequencies directly generated by forc ...
s – for
wave groups the radiation stress varies along the group. As a result, a non-linear long wave propagates together with the group, at the
group velocity of the modulated short waves within the group. While, according to the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
, a long wave of this length should propagate at its own – higher –
phase velocity. 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 ...
of this bound long wave varies with the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of the wave height, and is only significant in shallow water;
* ''
Wave–current interaction
In fluid dynamics, wave–current interaction is the interaction between surface gravity waves and a mean flow. The interaction implies an exchange of energy, so after the start of the interaction both the waves and the mean flow are affected.
For ...
'' – in varying
mean-flow fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
, the energy exchanges between the waves and the mean flow, as well as the mean-flow forcing, can be modeled by means of the radiation stress.
Definitions and values derived from linear wave theory
One-dimensional wave propagation
For uni-directional wave propagation – say in the ''x''-coordinate direction – the component of the radiation stress tensor of
dynamical importance is ''S''
xx. It is defined as:
[Mei (2003), p. 457.]
:
where ''p''(''x'',''z'',''t'') is the fluid
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
is the horizontal ''x''-component of the
oscillatory part of 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 ...
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, ''z'' is the vertical coordinate, ''t'' is time, ''z'' = −''h''(''x'') is the bed elevation of the fluid layer, and ''z'' = ''η''(''x'',''t'') is the surface elevation. Further ''ρ'' is the fluid
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and ''g'' is the
acceleration by gravity, while an overbar denotes
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
averaging
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
. The last term on the right-hand side, ½''ρg''(''h''+')
2, is the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
of the
hydrostatic pressure
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body " fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imm ...
over the still-water depth.
To lowest (second) order, the radiation stress ''S''
xx for traveling
periodic wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s can be determined from the properties of surface gravity waves according to
Airy wave theory
In fluid dynamics, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mea ...
:
[Mei (2003), p. 97.]
:
where ''c''
p is the
phase speed
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
and ''c''
g is the
group speed of the waves. Further ''E'' is the mean depth-integrated wave energy density (the sum of the
kinetic
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to i ...
and
potential energy) per unit of horizontal area. From the results of Airy wave theory, to second order, the mean energy density ''E'' equals:
:
with ''a'' 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 ...
and ''H'' = 2''a'' the
wave height
In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighboring trough. ''Wave height'' is a term used by mariners, as well as in coastal, ocean and naval engineering.
At sea, the te ...
. Note this equation is for periodic waves: in
random waves the
root-mean-square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
wave height ''H''
rms should be used with ''H''
rms = ''H''
m0 / , where ''H''
m0 is the
significant wave height
In physical oceanography, the significant wave height (SWH, HTSGW or ''H''s)
is defined traditionally as the mean ''wave height'' ( trough to crest) of the highest third of the waves (''H''1/3). Nowadays it is usually defined as four times the ...
. Then ''E'' = ''ρgH''
m02.
Two-dimensional wave propagation
For wave propagation in two horizontal dimensions the radiation stress
is a second-order
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
with components:
:
With, in a
Cartesian coordinate system (''x'',''y'',''z''):
:
where
and
are the horizontal ''x''- and ''y''-components of the oscillatory part
of the flow velocity vector.
To second order – in wave amplitude ''a'' – the components of the radiation stress tensor for progressive periodic waves are:
:
where ''k''
x and ''k''
y are the ''x''- and ''y''-components of the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
vector ''k'', with length ''k'' = , ''k'', = and the vector ''k'' perpendicular to the wave
crests. The phase and group speeds, ''c''
p and ''c''
g respectively, are the lengths of the phase and group velocity vectors: ''c''
p = , ''c''
p, and ''c''
g = , ''c''
g, .
Dynamical significance
The radiation stress tensor is an important quantity in the description of the phase-averaged dynamical interaction between waves and mean flows. Here, the depth-integrated dynamical conservation equations are given, but – in order to model three-dimensional mean flows forced by or interacting with surface waves – a three-dimensional description of the radiation stress over the fluid layer is needed.
Mass transport velocity
Propagating waves induce a – relatively small – mean
mass transport
Public transport (also known as public transportation, public transit, mass transit, or simply transit) is a system of transport for passengers by group travel systems available for use by the general public unlike private transport, typica ...
in the wave propagation direction, also called the wave (pseudo)
momentum. To lowest order, the wave momentum M
w is, per unit of horizontal area:
:
which is exact for progressive waves of permanent form in
irrotational flow
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 any path between two points does not ...
. Above, ''c''
p is the phase speed relative to the mean flow:
:
with ''σ'' the ''intrinsic angular frequency'', as seen by an observer moving with the mean horizontal flow-velocity while ''ω'' is the ''apparent angular frequency'' of an observer at rest (with respect to 'Earth'). The difference ''k''⋅ is the
Doppler shift.
The mean horizontal momentum ''M'', also per unit of horizontal area, is the mean value of the integral of momentum over depth:
:
with v(''x'',''y'',''z'',''t'') the total flow velocity at any point below the free surface ''z'' = ''η''(''x'',''y'',''t''). The mean horizontal momentum ''M'' is also the mean of the depth-integrated horizontal mass flux, and consists of two contributions: one by the mean current and the other (''M''
w) is due to the waves.
Now the mass transport velocity is defined as:
:
Observe that first the depth-integrated horizontal momentum is averaged, before the division by the mean water depth (''h''+') is made.
Mass and momentum conservation
Vector notation
The equation of mean mass conservation is, in
vector notation
In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space.
For representing a vector, the common typographic convention is l ...
:
[Phillips (1977), pp. 61–63.]
:
with including the contribution of the wave momentum ''M''
w.
The equation for the conservation of horizontal mean momentum is:
:
where ⊗ denotes the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of with itself, and τ
w is the mean wind
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
at the free surface, while τ
b is the bed shear stress. Further I is the identity tensor, with components given by the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
δ
ij. Note that the
right hand side of the momentum equation provides the non-conservative contributions of the bed slope ∇''h'', as well the forcing by the wind and the bed friction.
In terms of the horizontal momentum ''M'' the above equations become:
:
Component form in Cartesian coordinates
In a
Cartesian coordinate system, the mass conservation equation becomes:
:
with
x and
y respectively the ''x'' and ''y'' components of the mass transport velocity .
The horizontal momentum equations are:
:
Energy conservation
For an
inviscid flow
In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...
the mean
mechanical energy
In physical sciences, mechanical energy is the sum of potential energy and kinetic energy. The principle of conservation of mechanical energy states that if an isolated system is subject only to conservative forces, then the mechanical energy is ...
of the total flow – that is the sum of the energy of the mean flow and the fluctuating motion – is conserved. However, the mean energy of the fluctuating motion itself is not conserved, nor is the energy of the mean flow. The mean energy ''E'' of the fluctuating motion (the sum of the
kinetic
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to i ...
and
potential energies satisfies:
[Phillips (1977), pp. 65–66.]
:
where ":" denotes the
double-dot product, and ''ε'' denotes the dissipation of mean mechanical energy (for instance by
wave breaking
In fluid dynamics, a breaking wave or breaker is a wave whose amplitude reaches a critical level at which large amounts of wave energy transform into turbulent kinetic energy. At this point, simple physical models that describe wave dynamics ...
). The term
is the exchange of energy with the mean motion, due to
wave–current interaction
In fluid dynamics, wave–current interaction is the interaction between surface gravity waves and a mean flow. The interaction implies an exchange of energy, so after the start of the interaction both the waves and the mean flow are affected.
For ...
. The mean horizontal wave-energy transport ( + c
g) ''E'' consists of two contributions:
* ''E'' : the transport of wave energy by the mean flow, and
* ''c''
g ''E'' : the mean energy transport by the waves themselves, with the
group velocity ''c''
g as the wave-energy transport velocity.
In a Cartesian coordinate system, the above equation for the mean energy ''E'' of the flow fluctuations becomes:
:
So the radiation stress changes the wave energy ''E'' only in case of a spatial-
inhomogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
current field (
x,
y).
Notes
References
;Primary sources
*
*
*
*
;Further reading
*
*
{{Physical oceanography
Physical oceanography
Water waves