Etudes et évaluation de processus océaniques par des hiérarchies ...

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10.5. ENTRAINMENT 61
instability and also turbulence. This means that there are two competing phenomena and it is
key for vertical mixing to determine under which circumstances one of the processes dominates.
To this end we look at a stably stratified sheared flow, and consider the energy budget when to
equal volumes A and B, as shown in fig. 10.2, separated by a distance δz are exchanged. The
potential energy ∆E pot necessary to exchange the heavier and lower volume B with the lighter
and higher volume A is supposed to be provided by the kinetic energy ∆E kin in the shear. For
this to be possible it is clear, that ∆E total = ∆E kin + ∆E pot > 0 which are given by,
∆E kin = 2 ρV 2 ((δz/2)∂ zU) 2 (10.21)
∆E pot = −gV (δz) 2 ∂ z ρ. (10.22)
Indeed, E pot = gh∆m, and for our case h = δz and m = δzV ∂ z ρ is the mass difference between
volume B and A. ∆E total > 0 if the Richardson number ,
tel-00545911, version 1 - 13 Dec 2010
Ri =
or if we write δU = δz ∂ z U and δρ = δz ∂ z ρ we obtain,
Ri =
g ∂ zρ
ρ(∂ z U) 2 < 1 4 , (10.23)
g δρ δz
ρ(δU) 2 < 1 4 , (10.24)
Which means that using the kinetic energy of the volumes A and B it is possible to interchange
the volumes A and B when Ri < 1/4. Although that this calculation is very simple, only
comparing kinetic to potential energy, and does not tell us how the volumes A and B should
be exchanged, it is found in laboratory experiments that sheared stratified flow does indeed
become unstable around a critical Richardson number of one quarter.
The above, and more involved, calculations together with laboratory experiments and
oceanic observations have led to a variety of parametrisations of the vertical mixing based
on the Richardson number.
One of the simplest, and widely used, parametrisations for vertical mixing based on the
Richardson number was proposed by Philander and Pacanowski (1981):
ν eddy =
ν 0
(1 + αRi) n + ν b, (10.25)
where typical values of the parameters, used in today’s numerical models of the ocean dynamics,
are ν 0 = 10 −2 m 2 s −1 , ν b = 10 −4 m 2 s −1 , α = 5 and n = 2.
Exercise 60: Slippery Sea
10.5 Entrainment
Entrainment is the mixing of ambient (non or less turbulent) fluid into a turbulent current
so that the initially less turbulent fluid becomes part of the turbulent flow. Examples are:
a fluid jet that spreads and entrains ambient fluid with it, (ii) an avalanche that entrains
surrounding air and increases in size. The fluid flow is typically from the less turbulent fluid to
the more turbulent fluid. Entrainment is usually quantified by the entrainment velocity which
is the velocity with which the ambient fluid enters into the turbulent jet through the border
separating the two fluids. If the entrainment is negative on speaks of detrainment.