Stokes drift

Mathematical description

The Lagrangian motion of a fluid parcel with position vector x = ξ(α,t) in the Eulerian coordinates is given by:

ξ ˙ = ∂ ξ ∂ t = u ( ξ , t ) ,

where ∂ξ / ∂t is the partial derivative of ξ(α,t) with respect to t, and

ξ(α,t) is the Lagrangian position vector of a fluid parcel, in meters, u(x,t) is the Eulerian velocity, in meters per second, x is the position vector in the Eulerian coordinate system, in meters, α is the position vector in the Lagrangian coordinate system, in meters, t is the time, in seconds.

Often, the Lagrangian coordinates α are chosen to coincide with the Eulerian coordinates x at the initial time t = t₀ :

ξ ( α , t 0 ) = α .

But also other ways of labeling the fluid parcels are possible.

If the average value of a quantity is denoted by an overbar, then the average Eulerian velocity vector ū_E and average Lagrangian velocity vector ū_L are:

u ¯ E = u ( x , t ) ¯ , u ¯ L = ξ ˙ ( α , t ) ¯ = ( ∂ ξ ( α , t ) ∂ t ) ¯ = u ( ξ ( α , t ) , t ) ¯ .

Different definitions of the average may be used, depending on the subject of study, see ergodic theory:

time average,

space average,

ensemble average and

phase average.

Now, the Stokes drift velocity ū_S equals

u ¯ S = u ¯ L − u ¯ 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 = u ^ sin ⁡ ( k x − ω t ) , one readily obtains by the perturbation theory – with k u ^ / ω as a small parameter – for the particle position x = ξ ( ξ 0 , t ) :

ξ ˙ = u ( ξ , t ) = u ^ sin ( k ξ − ω t ) , ξ ( ξ 0 , t ) ≈ ξ 0 + u ^ ω cos ⁡ ( k ξ 0 − ω t ) + k u ^ 2 2 ω 2 sin ⁡ 2 ( k ξ 0 − ω t ) + k u ^ 2 2 ω t .

Here the last term describes the Stokes drift 1 2 k u ^ 2 / ω .

Example: Deep water waves

The Stokes drift was formulated for water waves by George Gabriel Stokes in 1847. For simplicity, the case of infinite-deep water is considered, with linear wave propagation of a sinusoidal wave on the free surface of a fluid layer:

η = a cos ( k x − ω 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), and T is the wave period (seconds).

As derived below, the horizontal component ū_S(z) of the Stokes drift velocity for deep-water waves is approximately:

As can be seen, the Stokes drift velocity ū_S 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 quart wavelength, z = -¼ λ, 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

φ = ω k a e k z sin ( k x − ω t ) .

In order to have non-trivial solutions for this eigenvalue problem, the wave length and wave period may not be chosen arbitrarily, but must satisfy the deep-water dispersion relation:

ω 2 = g k .

with g the acceleration by gravity in (m / s²). Within the framework of linear theory, the horizontal and vertical components, ξ_x and ξ_z respectively, of the 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 ) .

The horizontal component ū_S of the Stokes drift velocity is estimated by using a Taylor expansion around x of the Eulerian horizontal-velocity component u_x = ∂ξ_x / ∂t at the 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 .