Ecole doctorale de Physique de la région Parisienne (ED107)

or
4.2 Equations and numbers 113
Tv ∼
1
2 max [Es, Eb] Ω ,
where Es and Eb are, respectively, the Ekman number for shear and bulk viscosities.
In other words, the Ekman number must be interpreted as characteristic of the ratio
between the period and the viscous time. This number is typically less than 10 −7 and
the viscous time is more than 7 orders of magnitude greater than the period [see Cutler
& Lindblom (1987) for more precise values]. A flow will be said “rotation dominated”
if the above Ekman and Rossby numbers are small compared to the unity. Note that
another usual hydrodynamical number appears with them : the Reynolds number, prodrome
of turbulence. In a rotating fluid, it is defined by the ratio between Rossby and
Ekman numbers, or in any fluid by the ratio between the nonlinear and the viscous terms.
The last typical time to evaluate here is the instability rising time Tg associated with
the RR force. To get an idea of its value, we come back to the dimensionless RR acceleration,
formulas (4.4) and definitions (4.6). Analytical calculation with V i being the l = m
linear r-mode (with time-dependent amplitude)
or in the spherical orthonormal basis
V = A[τ]
V = A[τ] 1
m R
ξ m
ξ ∧ ∇Ymm
0
ξ m (sin ϑ) m−1 sin[mϕ]
ξ m (sin ϑ) m−1 cos[ϑ] cos[mϕ]
(4.11)
(4.12)
(where R means the real part of the complex function), gives in dimensioned variables at
the lowest order for the m = 2 r-modes
∂tA[t] = GR7 Ω 6 n0
c 7
216 π
38 52 1
A[t] = A[t]. (4.13)
7 Tg
For a typical spherical NS with R = 10 km, M = 1.4 M⊙ and Ω ∼ (2 π)200 Hz, Tg is
something like 10 8 periods of the NS. Thus, depending on the viscosity given by the EOS,
it will or will not be larger than Tv, and then, the inertial instability will or will not be
relevant. At this step, a new typical number seems natural to introduce. We shall propose
to call “Chandra number” 1 the ratio between the viscous time Tv and the rising time Tg.
In the same spirit as the Rossby and Ekman numbers, it should also quantify the ratio
between the viscous and RR forces :
Ch = 216 π
38 52 G n0
7
2 R9 Ω6 c7 max[η, ζ]
. (4.14)
1 Chandrasekhar was the first who studied the gravitational radiation driven instability for the l =
m = 2 fundamental modes of uniform density MacLaurin spheroid in 1970. See Chandrasekhar (1970).