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Forcing of the mean flow through wave–current interaction
In
fluid dynamics , the Craik–Leibovich (CL) vortex force describes a
forcing of the
mean flow through
wave–current interaction , specifically between the
Stokes drift velocity and the mean-flow
vorticity . The CL vortex force is used to explain the generation of
Langmuir circulations by an
instability mechanism . The CL vortex-force mechanism was derived and studied by
Sidney Leibovich and Alex D. D. Craik in the 1970s and 80s, in their studies of Langmuir circulations (discovered by
Irving Langmuir in the 1930s).
Description
The CL vortex force is
ρ
u
S
×
ω
,
{\displaystyle \rho \,{\boldsymbol {u}}_{S}\times {\boldsymbol {\omega }},}
with
u
S
{\displaystyle {\boldsymbol {u}}_{S}}
the (
Lagrangian ) Stokes drift
velocity and vorticity
ω
=
∇
×
u
{\displaystyle {\boldsymbol {\omega }}=\nabla \times {\boldsymbol {u}}}
(i.e. the
curl of the
Eulerian mean-flow velocity
u
{\displaystyle {\boldsymbol {u}}}
). Further
ρ
{\displaystyle \rho }
is the fluid
density and
∇
×
{\displaystyle \nabla \times }
is the curl operator.
The CL vortex force finds its origins in the appearance of the Stokes drift in the
convective acceleration terms in the mean momentum equation of the
Euler equations or
Navier–Stokes equations . For constant density, the momentum equation (divided by the density
ρ
{\displaystyle \rho }
) is:
[1]
∂
t
u
⏟
(a)
+
u
⋅
∇
u
⏟
(b)
+
2
Ω
×
u
⏟
(c)
+
2
Ω
×
u
S
⏟
(d)
+
∇
(
π
+
u
⋅
u
S
)
⏟
(e)
=
u
S
×
(
∇
×
u
)
⏟
(f)
+
ν
∇
⋅
∇
u
⏟
(g)
,
{\displaystyle \underbrace {\partial _{t}{\boldsymbol {u}}} _{\text{(a)}}+\underbrace {{\boldsymbol {u}}\cdot \nabla {\boldsymbol {u}}} _{\text{(b)}}+\underbrace {2{\boldsymbol {\Omega }}\times {\boldsymbol {u}}} _{\text{(c)}}+\underbrace {2{\boldsymbol {\Omega }}\times {\boldsymbol {u}}_{S}} _{\text{(d)}}+\underbrace {\nabla (\pi +{\boldsymbol {u}}\cdot {\boldsymbol {u}}_{S})} _{\text{(e)}}=\underbrace {{\boldsymbol {u}}_{S}\times (\nabla \times {\boldsymbol {u}})} _{\text{(f)}}+\underbrace {\nu \,\nabla \cdot \nabla {\boldsymbol {u}}} _{\text{(g)}},}
with
The CL vortex force can be obtained by several means. Originally, Craik and Leibovich used
perturbation theory . An easy way to derive it is through the
generalized Lagrangian mean theory.
[1] It can also be derived through a
Hamiltonian mechanics description.
[2]
Notes
References
Craik, A.D.D. (1990), Wave interactions and fluid flows , Cambridge University Press, pp. 113–122,
ISBN
0-521-36829-4 ,
LCCN
lc85007803
Holm, D.D. (1996), "The ideal Craik–Leibovich equations", Physica D , 98 (2): 415–441,
Bibcode :
1996PhyD...98..415H ,
doi :
10.1016/0167-2789(96)00105-4
Leibovich, S. (1980), "On wave–current interaction theories of Langmuir circulations", Journal of Fluid Mechanics , 99 (4): 715–724,
Bibcode :
1980JFM....99..715L ,
doi :
10.1017/S0022112080000857 ,
S2CID
14996095
Leibovich, S. (1983), "The form and dynamics of Langmuir circulations",
Annual Review of Fluid Mechanics , 15 : 391–427,
Bibcode :
1983AnRFM..15..391L ,
doi :
10.1146/annurev.fl.15.010183.002135
Sullivan, P.P.; McWilliams, J.C. (2010), "Dynamics of winds and currents coupled to surface waves",
Annual Review of Fluid Mechanics , 42 : 19–42,
Bibcode :
2010AnRFM..42...19S ,
doi :
10.1146/annurev-fluid-121108-145541
Thorpe, S.A. (2004), "Langmuir circulation",
Annual Review of Fluid Mechanics , 36 : 55–79,
Bibcode :
2004AnRFM..36...55T ,
doi :
10.1146/annurev.fluid.36.052203.071431