Abyssal Mixing in the Laboratory
225
where the index r refers to the reflected wave field. Its kinetic energy density corresponds to E c,r =
1
2
̄
𝜌k
2
r |𝜓 0,r |
2 . The complete wave field is therefore 𝜓 = 𝜓 i + 𝜓 r .
As the flow does not penetrate the sloping boundary, the total stream function
field must vanish at the boundary defined by x = −z tan 𝛼. In order to simplify
the boundary condition, one usually defines the coordinates attached to the slope
(x s , z s ), as shown in Fig. 1. The velocity fields in the slope coordinate system are
(u s , w s ) = (−𝜕𝜓∕𝜕z s , 𝜕𝜓∕𝜕x s ) and the wave number is noted 𝐤 s = (k x s , k z s ). The nonpenetration condition can be expressed as u s = 0 at x s = 0 and for all z s and time t.
On the total stream function field, this becomes
k z s ,i 𝜓 0,i exp
[
i
( 𝜔 i t − 𝐤 s,i ⋅ 𝐫 s
)]
+ k z s ,r 𝜓 0,r exp
[
i
(
𝜔 r t − 𝐤 s,r ⋅ 𝐫 s
)]
= 0,
(5)
at x s = 0 and for all z s and time t. This leads to 𝜔 i = 𝜔 r ≡ 𝜔, k z s ,i = k z s ,r and 𝜓 0,i =
𝜓 0,r ≡ 𝜓 0 . Thus, the frequency and the wave vector component parallel to the sloping
boundary are conserved during the reflection. The normal component of the wave
vector can be determined using geometrical construction and the dispersion relation:
one gets k x s ,i = k z s ,i tan(𝜃 − 𝛼) and k x s ,r = k z s ,r tan(𝜃 + 𝛼). Thus, the ratio between the
norms of the two wave vectors is given by
k r
k i
=
|
|
|
|
cos(𝜃 − 𝛼)
cos(𝜃 + 𝛼)
|
|
|
|
≡ 𝛾.
(6)
This defines the focusing parameter 𝛾. Equation (6) confirms immediately that there
is neither focusing, nor defocusing when the wall is vertical (𝛼 = 0) or horizontal
(𝛼 = 𝜋∕2): both cases lead indeed to 𝛾 = 1. One recovers indeed the case of the
Descartes reflection, since keeping the angle with respect to the gravity for the internal waves, does correspond to keep the angle with respect to the normal of the wall
(that is orthogonal or parallel to the gravity!).
However as soon as one considers a sloping boundary (𝛼 ≠ 0) or (𝛼 ≠ 𝜋∕2), the
focusing parameter is different from unity: it is for example greater than 1 in Fig. 1.
The width of the reflected beam is thus reduced by the factor 𝛾. This is a focusing
reflection, the energy in the incident beam being concentrated. Indeed, Eq. (6) leads
to E c,r = 𝛾 2 E c,i showing that the energy density is increased by a factor 𝛾 2 > 1.
Interestingly, this parameter diverges when 𝜃 + 𝛼 tends to 90
◦ . This corresponds
to the case where the waves have a propagation angle very close to the slope of the
wall. This situation is called critical reflection. Indeed, it is critical because 𝛾 diverges
and thus, the wave length of the reflected wave tends to 0: nonlinear and dissipation
effects cannot be overlooked and should be treated carefully. Using a weakly nonlinear theory, it has been shown [11] how to heal this singularity using matched
asymptotic expansion. This is however not the case under study in the remainder of
this work.