224 Stability of horizontal boundary layers
where x contains the discretized values of Ψ and U. It remains to invert B and
to compute the spectrum of B
−1
A.
Therefore if k, Re, and γ are given we can compute the spectrum of (9.1.2)
and (9.1.3) with an arbitrary solution. For a given N, we get of course only a finite
number of eigenvalues, but as N increases, a part of the eigenvalues concentrates
in a continuous spectrum and a part of the eigenvalues remains isolated and “true
eigenvalues”. If there exists an eigenvalue with positive imaginary part, the flow
is unstable, for these parameters, if not it is stable. To get the critical Reynolds
number it remains to find the smallest Reynolds number Re for which there exists
parameters k and γ and a corresponding eigenvalue c with ℑmc>0. Lilly has
found in [89] a critical Reynolds number Re
c
∼55 and has computed the most
unstable mode for various values of k and Re. Note that as Re
c
is moderately
high, we do not need many points N to discretize correctly the solution (N of
order 50 to 100 is sufficient).
9.2 Energy of a small perturbation
The aim of this section is to discuss the evolution of the energy of a perturba-
tion u, a solution of (LNSC
ε
).
For Re<Re
1
as seen at the end of Part I, the energy is decreasing. As the
flow is stable, it goes to 0.
For Re>Re
c
there exist exponentially increasing modes, therefore, in general
the energy of the perturbation will increase exponentially.
In the range [Re
1
,Re
c
] the situation is slightly different. First as the Reynolds
number is subcritical the energy of an arbitrary perturbation tends to 0 as time
goesto+∞. The question is then to know whether the energy is decreasing or if it
begins to increase before ultimately decreasing. Let Re
3
be the supremum of the
Reynolds numbers such that for every Re>Re
3
, the energy of any perturbation
decreases continuously. It is possible to compute numerically Re
3
, which is of
order 8. The final picture is the following:
• Re<Re
3
: the energy of any perturbation goes to 0 in a monotonic way;
• Re
3
<Re<Re
c
: the energy of any perturbation tends to 0 as time goes
to +∞, but may begin to increase, before its decay;
• Re>Re
c
: the energy of a general perturbation goes to +∞ as time goes
to +∞ (not always in a monotonic way).
The main consequence is that linear stability cannot be proved by energy
estimates in the range [Re
3
,Re
c
] since in this area we have only energy estim-
ates of the form ∂
t
u
2
L
2
≤Cu
2
L
2
/ε, which are useless in the limit ε →0. In
this range linear stability can only be proved by spectral arguments, using
refined pseudodifferential techniques. This has been done by G. M´etivier and
K. Zumbrun [97] in the case of the vanishing viscosity limit of parabolic systems,
leading to hyperbolic systems of conservation laws. As in rotating fluids, bound-
ary layers appear, which are stable under a smallness criterion. Simple energy