Example: Deep water waves Stokes drift
stokes drift under periodic waves in deep water, period t = 5 s , mean water depth of 25 m. left: instantaneous horizontal flow velocities. right: average flow velocities. black solid line: average eulerian velocity; red dashed line: average lagrangian velocity, derived generalized lagrangian mean (glm).
the stokes drift formulated water waves george gabriel stokes in 1847. simplicity, case of infinite-deep water considered, linear wave propagation of sinusoidal wave on free surface of fluid layer:
η
=
a
cos
(
k
x
−
ω
t
)
,
{\displaystyle \eta \,=\,a\,\cos \,\left(kx-\omega t\right),}
where
η elevation of free surface in z-direction (meters),
a wave amplitude (meters),
k wave number: k = 2π / λ (radians per meter),
ω angular frequency: ω = 2π / t (radians per second),
x horizontal coordinate , wave propagation direction (meters),
z vertical coordinate, positive z direction pointing out of fluid layer (meters),
λ wave length (meters), and
t wave period (seconds).
as derived below, horizontal component ūs(z) of stokes drift velocity deep-water waves approximately:
as can seen, stokes drift velocity ūs nonlinear quantity in terms of wave amplitude a. further, stokes drift velocity decays exponentially depth: @ depth of quart wavelength, z = -¼ λ, 4% of value @ mean free surface, z = 0.
derivation
it assumed waves of infinitesimal amplitude , free surface oscillates around mean level z = 0. waves propagate under action of gravity, constant acceleration vector gravity (pointing downward in negative z-direction). further fluid assumed inviscid , incompressible, constant mass density. fluid flow irrotational. @ infinite depth, fluid taken @ rest.
now flow may represented velocity potential φ, satisfying laplace equation and
φ
=
ω
k
a
e
k
z
sin
(
k
x
−
ω
t
)
.
{\displaystyle \varphi \,=\,{\frac {\omega }{k}}\,a\;{\text{e}}^{kz}\,\sin \,\left(kx-\omega t\right).}
in order have non-trivial solutions eigenvalue problem, wave length , wave period may not chosen arbitrarily, must satisfy deep-water dispersion relation:
ω
2
=
g
k
.
{\displaystyle \omega ^{2}\,=\,g\,k.}
with g acceleration gravity in (m / s). within framework of linear theory, horizontal , vertical components, ξx , ξz respectively, of lagrangian position ξ are:
ξ
x
=
x
+
∫
∂
φ
∂
x
d
t
=
x
−
a
e
k
z
sin
(
k
x
−
ω
t
)
,
ξ
z
=
z
+
∫
∂
φ
∂
z
d
t
=
z
+
a
e
k
z
cos
(
k
x
−
ω
t
)
.
{\displaystyle {\begin{aligned}\xi _{x}\,&=\,x\,+\,\int \,{\frac {\partial \varphi }{\partial x}}\;{\text{d}}t\,=\,x\,-\,a\,{\text{e}}^{kz}\,\sin \,\left(kx-\omega t\right),\\\xi _{z}\,&=\,z\,+\,\int \,{\frac {\partial \varphi }{\partial z}}\;{\text{d}}t\,=\,z\,+\,a\,{\text{e}}^{kz}\,\cos \,\left(kx-\omega t\right).\end{aligned}}}
the horizontal component ūs of stokes drift velocity estimated using taylor expansion around x of eulerian horizontal-velocity component ux = ∂ξx / ∂t @ position ξ :
u
¯
s
=
u
x
(
ξ
,
t
)
¯
−
u
x
(
x
,
t
)
¯
=
[
u
x
(
x
,
t
)
+
(
ξ
x
−
x
)
∂
u
x
(
x
,
t
)
∂
x
+
(
ξ
z
−
z
)
∂
u
x
(
x
,
t
)
∂
z
+
⋯
]
¯
−
u
x
(
x
,
t
)
¯
≈
(
ξ
x
−
x
)
∂
2
ξ
x
∂
x
∂
t
¯
+
(
ξ
z
−
z
)
∂
2
ξ
x
∂
z
∂
t
¯
=
[
−
a
e
k
z
sin
(
k
x
−
ω
t
)
]
[
−
ω
k
a
e
k
z
sin
(
k
x
−
ω
t
)
]
¯
+
[
a
e
k
z
cos
(
k
x
−
ω
t
)
]
[
ω
k
a
e
k
z
cos
(
k
x
−
ω
t
)
]
¯
=
ω
k
a
2
e
2
k
z
[
sin
2
(
k
x
−
ω
t
)
+
cos
2
(
k
x
−
ω
t
)
]
¯
=
ω
k
a
2
e
2
k
z
.
{\displaystyle {\begin{aligned}{\overline {u}}_{s}\,&=\,{\overline {u_{x}({\boldsymbol {\xi }},t)}}\,-\,{\overline {u_{x}({\boldsymbol {x}},t)}}\,\\&=\,{\overline {\left[u_{x}({\boldsymbol {x}},t)\,+\,\left(\xi _{x}-x\right)\,{\frac {\partial u_{x}({\boldsymbol {x}},t)}{\partial x}}\,+\,\left(\xi _{z}-z\right)\,{\frac {\partial u_{x}({\boldsymbol {x}},t)}{\partial z}}\,+\,\cdots \right]}}-\,{\overline {u_{x}({\boldsymbol {x}},t)}}\\&\approx \,{\overline {\left(\xi _{x}-x\right)\,{\frac {\partial ^{2}\xi _{x}}{\partial x\,\partial t}}}}\,+\,{\overline {\left(\xi _{z}-z\right)\,{\frac {\partial ^{2}\xi _{x}}{\partial z\,\partial t}}}}\\&=\,{\overline {{\bigg [}-a\,{\text{e}}^{kz}\,\sin \,\left(kx-\omega t\right){\bigg ]}\,{\bigg [}-\omega \,k\,a\,{\text{e}}^{kz}\,\sin \,\left(kx-\omega t\right){\bigg ]}}}\,\\&+\,{\overline {{\bigg [}a\,{\text{e}}^{kz}\,\cos \,\left(kx-\omega t\right){\bigg ]}\,{\bigg [}\omega \,k\,a\,{\text{e}}^{kz}\,\cos \,\left(kx-\omega t\right){\bigg ]}}}\,\\&=\,{\overline {\omega \,k\,a^{2}\,{\text{e}}^{2kz}\,{\bigg [}\sin ^{2}\,\left(kx-\omega t\right)+\cos ^{2}\,\left(kx-\omega t\right){\bigg ]}}}\\&=\,\omega \,k\,a^{2}\,{\text{e}}^{2kz}.\end{aligned}}}
^ see e.g. phillips (1977), page 37.
^ see phillips (1977), page 44. or craik (1985), page 110.
^ viscosity has pronounced effect on mean eulerian velocity , mean lagrangian (or mass transport) velocity, less on difference: stokes drift outside boundary layers near bed , free surface, see instance longuet-higgins (1953). or phillips (1977), pages 53–58.
^ see e.g. phillips (1977), page 38.
^ cite error: named reference phil1977p43 invoked never defined (see page).
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