Stokes Drift - Mathematical Description

Mathematical Description

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

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 :

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:

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

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 mathematical 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).