# Langmuir circulation

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Langmuir circulation
White streaks in this lagoon are due to the Langmuir circulation.
These lines of sargassum can stretch for miles along the surface. The clumps of floating algae are often concentrated by the strong winds and wave action associated with the Gulf Stream.

In physical oceanography, Langmuir circulation consists of a series of shallow, slow, counter-rotating vortices at the ocean's surface aligned with the wind. These circulations are developed when wind blows steadily over the sea surface. Irving Langmuir discovered this phenomenon after observing windrows of seaweed in the Sargasso Sea in 1927.[1] Langmuir circulations circulate within the mixed layer, there is some uncertainty as to how strongly they can cause mixing at the base of the mixed layer. [2]

## Theory

The driving force of these circulations is an interaction of the mean flow with wave averaged flows of the surface waves. Stokes drift velocity of the waves stretches and tilts the vorticity of the flow near the surface. The production of vorticity in the upper ocean is balanced by downward (often turbulent) diffusion $\nu_T$. For a flow driven by a wind $\tau$ characterized by friction velocity $u_*$ the ratio of vorticity diffusion and production defines the Langmuir number [2]

$\mathrm{La} = \sqrt{\frac{\nu^3_Tk^6}{\sigma a^2u^2_*k^4}} \ \mathrm{or}\ \sqrt{\frac{\nu_T^3\beta^6}{u^2_*S_0\beta^3}}$

where the first definition is for a monochromatic wave field of amplitude $a$, frequency $\sigma$, and wavenumber $k$ and the second uses a generic inverse length scale $\beta$, and Stokes velocity scale $S_0$. This is exemplified by the Craik-Leibovich equations[3] which are an approximation of the Lagrangian mean [4] .[5] In the Boussinesq approximation the governing equations can be written

$\frac{\partial u_i}{\partial t} +u_j\nabla_ju_i = \begin{array}{l} -2\epsilon_{ijk}\Omega_j\left(u^s_k+u_k\right) -\nabla_i\left(\frac{P}{\rho_0}+\frac{1}{2}u^s_ju^s_j+u^s_ju_j\right)\\ +\epsilon_{ijk}u^s_j\epsilon_{klm}\nabla_lu_m +g_i\frac{\rho}{\rho_0} +\nabla_j\nu\nabla_ju_i \end{array}$
$\nabla_iu_i=0$
$\frac{\partial \rho}{\partial t} +u_j\nabla_j\rho = \nabla_i\kappa\nabla_i\rho$

where $u_i$ is the fluid velocity, $\Omega$ is planetary rotation, $u^s_i$ is the stokes drift velocity of the surface wave field, $P$ is the pressure, $g_i$ is the acceleration due to gravity, $\rho$ is the density, $\rho_0$ is the reference density, $\nu$ is the viscosity, and $\kappa$ is the diffusivity.

In the open ocean conditions where there may not be a dominant length scale controlling the scale of the Langmuir cells the concept of Langmuir Turbulence is advanced. [6]

## Observations

The circulation has been observed to be between 0°-20° to the right of the wind in the northern hemisphere [7] and the helix forming bands of divergence and convergence at the surface. At the convergence zones, there are commonly concentrations of floating seaweed, foam and debris along these bands. Along these divergent zones, the ocean surface is typically clear of debris since diverging currents force material out of this zone and into adjacent converging zones. At the surface the circulation will set a current from the divergence zone to the convergence zone and the spacing between these zones are of the order of 1–300 m (3–1,000 ft). Below convergence zones narrow jets of downward flow form and the magnitude of the current will be comparable to the horizontal flow. The downward propagation will typically be in the order of meters or tenths of meters and will not penetrate the pycnocline. The upwelling is less intense and takes place over a wider band under the divergence zone. In wind speeds ranging from 2–12 m/s (6.6–39.4 ft/s) the maximum vertical velocity ranged from 2–10 cm/s (0.79–3.94 in/s) with a ratio of down-welling to wind velocities ranging from -0.0025 to -0.0085. [8]

## References

1. ^ Open University (2001), Ocean Circulation (2nd ed.), Butterworth-Heinemann, ISBN 9780750652780
2. ^ a b Thorpe, S.A. (2004), "Langmuir circulation", Annual Reviews Fluid Mechanics 36: 55–79, Bibcode:2004AnRFM..36...55T, doi:10.1146/annurev.fluid.36.052203.071431
3. ^ Craik, A.D.D.; Leibovich, S. (1976), "A Rational model for Langmuir circulations", Journal of Fluid Mechanics 73: 401–426, Bibcode:1976JFM....73..401C, doi:10.1017/S0022112076001420
4. ^ Andrews, D.G.; McIntyre, M.E. (1978), "An exact theory of nonlinear waves on a Lagrangian-mean flow", Journal of Fluid Mechanics 89: 609–646, Bibcode:1978JFM....89..609A, doi:10.1017/S0022112078002773
5. ^ Leibovich, S. (1980), "On wave-current interactions theories of Langmuir circulations", Journal of Fluid Mechanics 99: 715–724, Bibcode:1980JFM....99..715L, doi:10.1017/S0022112080000857
6. ^ McWilliams, J.; Sullivan, P.; Moeng, C. (1997), "Langmuir turbulence in the ocean", Journal of Fluid Mechanics 334: 1–30, doi:10.1017/S0022112096004375
7. ^ Stewart, Robert H. (2002), Introduction To Physical Oceanography (Fall 2002 ed.)
8. ^ Leibovich, S. (1983), "The form and dynamics of Langmuir circulations", Annual Reviews of Fluid Mechanis 15: 391–427, Bibcode:1983AnRFM..15..391L, doi:10.1146/annurev.fl.15.010183.002135