Abstract
The Stokes velocity uS, defined approximately by Stokes (1847, Trans. Camb. Philos. Soc., 8, 441–455), and exactly via the Generalized Lagrangian Mean,
is divergent even in an incompressible fluid. We show that the Stokes velocity can be naturally decomposed into a solenoidal component, uS sol, and
a remainder that is small for waves with slowly varying amplitudes. We further show that uS sol arises as the sole Stokes velocity when the Lagrangian
mean flow is suitably redefined to ensure its exact incompressibility. The construction is an application of Soward & Roberts’s glmtheory (2010, J. FluidMech., 661, 45–72) which we specialise to surface gravity
waves and implement effectively using a Lie series expansion. We further show that the corresponding Lagrangian-mean momentum equation is formally
identical to the Craik–Leibovich equation with uS sol replacing uS, and we discuss the form of the Stokes pumping associated with both uS and uS
sol.
is divergent even in an incompressible fluid. We show that the Stokes velocity can be naturally decomposed into a solenoidal component, uS sol, and
a remainder that is small for waves with slowly varying amplitudes. We further show that uS sol arises as the sole Stokes velocity when the Lagrangian
mean flow is suitably redefined to ensure its exact incompressibility. The construction is an application of Soward & Roberts’s glmtheory (2010, J. FluidMech., 661, 45–72) which we specialise to surface gravity
waves and implement effectively using a Lie series expansion. We further show that the corresponding Lagrangian-mean momentum equation is formally
identical to the Craik–Leibovich equation with uS sol replacing uS, and we discuss the form of the Stokes pumping associated with both uS and uS
sol.
| Original language | English |
|---|---|
| Article number | 20210032 |
| Number of pages | 15 |
| Journal | Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |
| Volume | 380 |
| Issue number | 2225 |
| Early online date | 25 Apr 2022 |
| DOIs | |
| Publication status | Published - 13 Jun 2022 |