Stokes drift

The Lagrangian motion of a fluid parcel with position vector x = ξ(α, t) in the Eulerian coordinates is given by : \dot{\boldsymbol{\xi}} = \frac{\partial \boldsymbol{\xi}}{\partial t} = \mathbf{u}\big(\boldsymbol{\xi}(\boldsymbol{\alpha}, t), t\big), where : ∂ξ/∂t is the partial derivative 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, : x is the position vector in the Eulerian coordinate system, : α is the position vector in the Lagrangian coordinate system, : t is time. Often, the Lagrangian coordinates α are chosen to coincide with the Eulerian coordinates x at the initial time t = t0: : \bar\mathbf{u}_\text{S} = \bar\mathbf{u}_\text{L} - \bar\mathbf{u}_\text{E}. 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 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{u} \sin(kx - \omega t), one readily obtains by the perturbation theory with k\hat{u}/\omega as a small parameter for the particle position : \dot\xi = u(\xi, t) = \hat{u} \sin(k\xi - \omega t), : \xi(\xi_0, t) \approx \xi_0 + \frac{\hat{u}}{\omega} \cos(k\xi_0 - \omega t) - \frac14 \frac{k\hat{u}^2}{\omega^2} \sin 2(k\xi_0 - \omega t) + \frac12 \frac{k\hat{u}^2}{\omega} t. Here the last term describes the Stokes drift velocity \tfrac12 k\hat{u}^2/\omega. ==Example: Deep water waves==

T = 5 s and a 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, as derived from the Generalized Lagrangian Mean (GLM). The Stokes drift was formulated for water waves by George Gabriel Stokes in 1847. For simplicity, the case of infinitely deep water is considered, with linear wave propagation of a sinusoidal wave on the free surface of a fluid layer: : \eta = a \cos(kx - \omega t), where : η is the elevation of the free surface in the z direction (meters), : a is the wave amplitude (meters), : k is the wave number: k = 2π/λ (radians per meter), : ω is the angular frequency: ω = 2π/T (radians per second), : x is the horizontal coordinate and the wave propagation direction (meters), : z is the vertical coordinate, with the positive z direction pointing out of the fluid layer (meters), : λ is the wave length (meters), : T is the wave period (seconds). As derived below, the horizontal component ūS(z) of the Stokes drift velocity for deep-water waves is approximately: : \bar{u}_\text{S} \approx \omega k a^2 \text{e}^{2kz} = \frac{4\pi^2 a^2}{\lambda T} \text{e}^{4\pi z / \lambda}. As can be seen, the Stokes drift velocity ūS is a nonlinear quantity in terms of the wave amplitude 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, z = 0. Derivation It is assumed that the waves are of infinitesimal amplitude and the free surface oscillates around the mean level z = 0. The waves propagate under the action of gravity, with a constant acceleration vector by gravity (pointing downward in the negative z direction). Further the fluid is assumed to be inviscid and incompressible, with a constant mass density. The fluid flow is irrotational. At infinite depth, the fluid is taken to be at rest. Now the flow may be represented by a velocity potential φ, satisfying the Laplace equation and : \omega^2 = gk with g the acceleration by gravity in (m/s2). Within the framework of linear theory, the horizontal and vertical components, ξx and ξz respectively, of the Lagrangian position ξ are : \begin{align} \xi_x &= x + \int \frac{\partial \varphi}{\partial x}\, \text{d}t = x - a \text{e}^{kz} \sin(kx - \omega t), \\ \xi_z &= z + \int \frac{\partial \varphi}{\partial z}\, \text{d}t = z + a \text{e}^{kz} \cos(kx - \omega t). \end{align} The horizontal component ūS of the Stokes drift velocity is estimated by using a Taylor expansion around x of the Eulerian horizontal velocity component ux = ∂ξx / ∂t at the position ξ: : \begin{align} \bar{u}_\text{S} &= \overline{u_x(\boldsymbol{\xi}, t)} - \overline{u_x(\mathbf{x}, t)} \\ &= \overline{\left[ u_x(\mathbf{x}, t) + (\xi_x - x) \frac{\partial u_x(\mathbf{x}, t)}{\partial x} + (\xi_z - z) \frac{\partial u_x(\mathbf{x}, t)}{\partial z} + \cdots \right]} - \overline{u_x(\mathbf{x} ,t)} \\ &\approx \overline{(\xi_x - x) \frac{\partial^2 \xi_x}{\partial x\, \partial t}} + \overline{(\xi_z - z) \frac{\partial^2 \xi_x}{\partial z\, \partial t}} \\ &= \overline{\left[-a \text{e}^{kz} \sin(kx - \omega t)\right] \left[-\omega ka \text{e}^{kz} \sin(kx - \omega t)\right]} \\ &+ \overline{\left[a \text{e}^{kz} \cos(kx - \omega t)\right] \left[\omega ka \text{e}^{kz} \cos(kx - \omega t)\right] } \\ &= \overline{\omega ka^2 \text{e}^{2kz} \left[\sin^2(kx - \omega t) + \cos^2(kx - \omega t)\right]} \\ &= \omega ka^2 \text{e}^{2kz}. \end{align} ==See also==