Wave and Current Interaction in the Ocean - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 332

7.1.1 Damped waves in deep water.
Let us apply these equations to water waves. In Chapter 3, we replicated a
well known hydrodynamic solution, and showed that gravity-capillary waves in
deep quiescent water propagate according to
r (k) = (gk + Tk 3 / ) (7.4)
where g, and T represent gravitational acceleration, mass density and surface
tension. The velocity potential solution, which is nontrivial, was contingent
upon the existence of nondissipative inviscid flow. Now suppose that the
frequency-dependent effects of viscosity must be estimated. In separate studies,
it is known that water waves damp like e

it , with

i (k) = -2 k 2

(7.5)

where is the kinematic viscosity (Lamb, 1945). We now wish to determine
how a wave will propagate and damp in space and time, using the separate
results in Equations 7.4 and 7.5, and next, more ambitiously, the additional
effect of a variable background mean flow (e.g., an ocean current).

7.1.1.1 Effect of low-order dissipation.
Following the ideas introduced in Chapter 2, the first objective is
straightforwardly accomplished. Since the wave kinematics, to leading order, is
unaffected by dissipation, the wavenumber field k(x,t) and position x(t) must
satisfy dk/dt = 0 and dx/dt = r k (k). But energy is affected by viscosity: the
governing differential equation is obtained by “plugging into” the general result
in Equation 7.2; thus, E/ t + ( r k (k)E)/ x = - 4 k 2 E since r t (k,x,t) vanishes
identically. The solution to the general initial value problem can be determined
along the lines described in Chapter 2. Similar comments apply to wave
momentum.

7.1.1.2 Effect of variable background flow.
Now consider waves propagating parallel to any background current
U(x,t), where variations with respect to x and t are weak. Using variational
arguments similar to those in Whitham (1974), Chin (1976, 1980a) showed that
E(x,t) and M(x,t) now satisfy

E/ t + ((U(x,t) +

r k (k,x,t))E)/ x =

= E{ 2

i + { r t (k,x,t) + U r x (k,x,t) - k r k (k,x,t) U x }/ r } (7.6)

M/ t + ((U(x,t) +

r k (k,x,t))M)/ x = M{2 i - r x (k,x,t)/k - U x } (7.7)

The wave and current superposition is nontrivial, because dynamical
interactions occur through a so-called “radiation” or “Reynolds stress tensor”
(Longuet-Higgins and Stewart, 1964). Equations 7.6 and 7.7, of course, apply to