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

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54 CHAPTER 9. ABYSSAL AND OVERTURNING CIRCULATION
˙ S 2 = |q|(S 1 − S 2 ) − P (9.6)
q = kα(T 2 − T 1 ) + kβ(S 2 − S 1 ) (9.7)
Where α > 0 and β < 0 are the expansion coefficients of temperature and salinity, respectively,
and k is a coefficient that connects q to the density difference, and P > 0. For simplicity of the
mathematics, and as we are only interested in qualitative results, we fix α = 1, β = −1 and
k = −1. The actual values can be adjusted based on observations. We then define ∆S = S 2 −S 1
and ∆T = T 2 − T 1 and note that ∆T < 0 (and ∆S < 0 if P > 0)!
1
∆Ṡ = −|q|∆S − P, (9.8)
2
with q = ∆S − ∆T. Looking for stationary states (∆Ṡ = 0) we obtain:
|∆S − ∆T |∆S + P = 0. (9.9)
tel-00545911, version 1 - 13 Dec 2010
We will call a THC with q > 0 forward and with q < 0 reverse. Solving these equations we
obtain the following stationary states:
∆S = 1 2 (∆T ± √ (∆T) 2 − 4P) if q > 0 (9.10)
∆S = 1 2 (∆T − √ (∆T) 2 + 4P) if q < 0 (9.11)
A fourth solution contradicts the q < 0 condition. We can now distinguish several cases (see
also fig. 9.2):
(1) for P < 0 an unrealistic forcing, there is only one solution which is a strong forward
THC, as salinity and temperature favor a positive q.
(2) 0 < P < (∆T) 2 /4 and we have three solutions, one unstable and two stable. The two
stable solutions are
∆S = 1 2 (∆T + √ (∆T) 2 − 4P) and q = 1 2 (−∆T + √ (∆T) 2 − 4P) > 0 (9.12)
∆S = 1 2 (∆T − √ (∆T) 2 + 4P) and q = 1 2 (−∆T − √ (∆T) 2 + 4P) < 0 (9.13)
What is the physics of these two stationary solutions? The first is the usual fast and
forward thermohaline circulation, this means that the THC is so fast that precipitation has
no time to act and temperature effects dominate over salinity. The second solution is slower
and reversed, the circulation is slow so precipitation can do its job and salinity dominates
temperature differences.
(3) P > (∆T) 2 /4 that is strong precipitation and we have only one stationary solution
which is dominated by salinity and is an inverse THC (perhaps the Pacific Ocean and the
North Atlantic at the end of glacial periods).
We have thus seen that using mixed boundary conditions for temperature and salinity, we
can have two solutions for the same forcing! A nonlinear equation can have several solutions
for the same set of parameters and boundary conditions.
Another important point is that such ocean model exhibits a hysteresis behaviour as a
function of a control variable as for example the precipitation. When small perturbations are
added, such model can give rise to abrupt changes between the two stable states followed by
periods of stability of arbitrary length. The observed break down of the thermohaline circulation
in the North Atlantic is often explained by such kind of model and multiple equilibria.
Exercise 57: We have written all the equations in non-dimensional form. Perform the
calculations for a concrete example (for example: volume of the boxes 1m 3 , ...).