Ecole doctorale de Physique de la région Parisienne (ED107)

E(t) / E 0
1.15
1.1
1.05
1
4.4 Test and calibration of the code 123
Time evolution of energy with reaction force
0 20 40 60
t Ω
Without correction
With correction
Figure 4.6 – Time evolution of the ratio between energy and initial energy in an inviscid
and incompressible rigidly rotating fluid with an RR force corresponding to what should
exist in a NS with M = 1.4M⊙, R = 10 km and Ω (2π) 500 Hz. The first curve
corresponds to the basic code and the second one to the code with improved conservation
of energy. The expected final value is about 1.057. The difference between this analytical
calculation and the corrected numerical result is less than 10 −3 .
happen with physical values. As expected, there is an axial mode (no radial velocity)
driven to instability with exactly the frequency (w = 4Ω)
and coefficients (l = 2) of the
3
r-mode. Moreover, a single look at the Figure 4.9 shows that even when the velocity is
mainly noisy, the tensor that plays the key-role in the instability is quite smooth. In this
figure, we also drew a zoom corresponding to tΩ < 50 and the associated power spectrum.
It shows two frequencies in this component of the tensor. One of them is the unstable
r-mode with 3w/Ω = 4 and the other (with 3w/Ω ∼ 2.70) disappears with a longer run, as
the scale is adapted to the growing mode. Yet, even in the spectrum of the full evolution,
a trace of it can still be seen. We achieved exactly the same features with others noisy
initial conditions, for instance a Dirac kick into the NS1 . We also did the same calculations
with other spatial lattices (up to 64 × 48 × 4), and this result did not change at all.
4.4.3 Test of the anelastic approximation
The last calculations we did in a rigid and Newtonian background were to test the
effect of the anelastic approximation. The idea was to compare the previous results obtained
with the divergence-free approximation and a rigid crust BC with results coming
from the same initial conditions but with the anelastic approximation and the free surface
1 We mean an initial velocity equal to 0 anywhere except in an arbitrary point.