Turbulence in rotating stars
Paramita Barai
Physics & Astronomy, Georgia State University, Atlanta, GA
Abstract. I review some analytical studies on the turbulent transport of matter and angular
momentum in the radiative zones of differentially rotating stars. Two prescriptions for the
horizontal transport caused by the shear turbulence are outlined in detail. Both of them give
higher values for the coefficient of diffusion as compared to the earlier accepted result. The
results of the new prescriptions implemented within the evolution models of rotating stars are
compared with each other.
1. Introduction
In rotating stars the centrifugal force breaks the radiative equilibrium resulting in a slow
but widespread circulation, which transports mass and angular momentum. Such circulation is
generated locally near a differentially rotating convection zone, causing radiation zones to rotate
non-uniformly. Hence instabilities of various forms generate turbulence in the star. As turbulence
generally departs from ideal conditions; i.e. homogeneity, isotropy and stationarity, it is
extremely difficult to formulate the models. Here I review some efforts to analytically formulate
turbulence produced by shear instabilities, which will probably be the most powerful generator
of turbulence in a non-magnetic star.
The Reynolds number of a fluid flow, measuring the relative strength of advection and
viscous transport, is given as: Re = VL/, where L is the typical size of the flow, V is the velocity
contrast across L and  is the kinematic viscosity. When the Reynolds number in a stellar
radiation zone exceeds a critical value (Rec), the zone becomes turbulent. Other important factors
that come into play are the buoyancy force in a stable stratification and the Coriolis force for fast
1
rotation, which prevent the shearing flow from becoming unstable. This makes the turbulence
anisotropic, with stronger transport in the horizontal than in the vertical direction.
Several papers have been devoted to the formulation of such shear turbulence, the recent
relevant ones being Zahn (1992), Maeder (1995, 2003), Talon & Zahn (1997) and Mathis et. al.
(2004). In section 2 reviews of the various results are outlined. In the subsequent sections, two
new prescriptions for the horizontal turbulence from Maeder (2003) and Mathis, Palacios &
Zahn (2004), are outlined in detail. The results of these two new calculations differ significantly
from the previous ones, but agree with each other to a significant degree.
2. Turbulence caused by vertical shear
If z is the vertical coordinate and V is the amplitude of the horizontal velocity, stable
entropy stratification tends to lessen the vertical shear, V(z). In the adiabatic case (i.e. when there
are no thermal leakage and no viscous dissipation), shear instability occurs when the Richardson
2
 dV 
criterion is met, i.e. N 2  
 Ri c .
 dz 
(1)
In words, the shear rate dV/dz must exceed the stabilizing effect of the buoyancy force, measured
by the Brunt-Väisälä frequency (or buoyancy frequency), N given as,
N 2  N T2  N 2 
g
g
( ad  ) 
 .
HP
HP
(2)
Ric is the critical Richardson number, which is of the order of unity for typical flow profiles. In
the following Ric = ¼ is adopted. In the above equations usual notation of symbols is used;
namely, temperature T, density , pressure P, mean molecular weight , gravitational
acceleration g, pressure scale height HP, the logarithmic gradients:  = d ln T/d ln P and  = d
ln /d ln P, adiabatic temperature gradient, and ad = (  ln T/  ln P)ad. The following
2
coefficients are derived from the equation of state:  = -(  ln /  ln T)P, and  = (  ln /  ln
)P,T , which are both unity for an ideal gas.
When heat exchange with the surroundings is allowed in the turbulent eddies, the
stabilizing effect of buoyancy is reduced. In an optically thick medium, this reduction is
determined by the Péclet number characterizing the turbulence: Pe 
velocity and size of the largest turbulent elements, K 
vl
, where v and l are the
K
16 T 3
is the thermal diffusivity,
3 C P  2
with  the Stefan Boltzmann constant,  the Rosseland mean opacity and Cp the heat capacity at
constant pressure. When vl << K, thermal diffusion proceeds faster than advection, and in the
absence of composition gradient, this Richardson criterion for instability is modified to (Zahn
2
v 2
 dV 
N T  Ri C 
1974, eq. 6):
 .
K
 dz 
(3)
Now the largest vl satisfying this inequality is the turbulent diffusivity, Dt, hence given as
2
Ri  dV 
Dt  K C2 
 .
NT  dz 
(4)
If one also considers the vertical composition gradient, only the thermal component of the
buoyancy force is weakened by radiative diffusion, which makes the instability criterion (Maeder
2
1995 eq. 2.3 & eq. 2.8):
v 2
 dV 
N T  N 2  Ri C 
 .
K
 dz 
(5)
From Mathis et. al. 2004 (see references given there) this condition is too severe, and does not
allow for mixing that is observed in massive stars.
But the -component of the buoyancy can also be reduced by the turbulence, particularly
when it is anisotropic with stronger transport in the horizontal than the vertical direction. Then v
and l characterize the largest isotropic eddies, and the horizontal component of the turbulent
3
diffusivity is much larger than the vertical one: Dh >> vl. It was reasoned by Talon and Zahn
(1997) that such turbulence erodes the fluctuations of chemical composition which provide the
restoring force, and hence the Richardson criterion takes the form:
2
v
v 2
 dV 
NT2 
N   RiC 
 .
K  Dh 
Dh
 dz 
(6)
The rotation rate  does not appear here, as the buoyancy force is stronger than the Coriolis
force, N >> . From (6), the vertical component of the turbulent diffusivity (Dv = vl) is
 d 
Dv 
r
 .
2 
2
N   dr 
NT

K  Dh  Dh
Ri c
2
(7)
Here the shear rate is written in spherical polar coordinates (i.e.
dV
d
 r sin 
), and a
dz
d
spherical average is done. The critical Richardson number has been renormalized; the canonical
value now used is Ric = 1/6.
3. Turbulence caused by horizontal shear
Differential rotation in latitude () becomes linearly unstable when one of the following
conditions is met. The specific angular momentum (r sin )2 has to decrease from the pole to
the equator, on a level surface, according to the Høiland-Solberg criterion (which is highly
improbable in stars). Or, the velocity profile  (sin2  )/(sin   ) must have a maximum in ,
a criterion analogous to the inflexion point theorem, which can be transposed to spherical
geometry (for detailed references see Mathis et. al. 2004). Probably this requirement too is not
satisfied in stars, except perhaps in a tachocline.
Since in stars the Reynolds number characterizing the differential rotation is very large, it
may seem that a slight shear, although linearly stable, is prone to non–linear instabilities.
4
However, the Coriolis force has a stabilizing effect on shear instabilities. In laboratory
experiments, turbulence tends to suppress the cause of the instability, namely the differential
rotation, when the turbulence is generated by the shear (Richard & Zahn, 1999). It is assumed
that a similar situation exists in stellar radiation zones, and the main role of turbulent viscosity is
to reduce the horizontal shear produced by the advection of angular momentum through the
meridional circulation.
The coefficient of viscosity due to horizontal turbulence is denoted by h and the
coefficient of horizontal diffusion by Dh (which are of same order). The differential rotation at
ˆ r ,  ,
colatitude  is written as (r , )  r   
(8)
ˆ r ,    r P cos  ,
where 
2
2
with P2 cos  being the Legendre Polynomial of second order. Here (r, ) is the angular
ˆ ( r , ) .
velocity, which is composed of the average value (r ) , and the differential part 
With the stationary approximation, advection and diffusion balance each other i.e.,
r2V r   U r   5vh  2 .
(9)
Here r is the eulerian coordinate of the isobar considered. Apart from the extreme rotation case,
the parameter r is close to the average radius of an isobar, which is the radius at which
P2 cos   0 , namely for  = 54.7 degrees. U(r) and V(r) are the vertical and horizontal
components of the meridional circulation velocity, and  
1 d ln r 2 
.
2 d ln r
For the lack of a better idea! it was proposed in Zahn (1992, eq. (2.27)) that h would be
determined from the condition that the amount of differential rotation 2/  is a small
(compared to unity) postulated value. Hence:
h 
1
r 2V r   U r  .
ch
(10)
5
Here ch is a parameter of order unity or smaller (ch <=1). The horizontal turbulent diffusivity, Dh
has a similar expression, i.e. Dh   h . But there are several shortcomings in this calculation as
given in the next section.
If the turbulence is anisotropic then the meridional circulation and horizontal diffusion
1 rU r 
, and this is comparable
30 Dh
2
results in a vertical diffusion given by the coefficient: Deff 
to Dv   v , caused by the vertical shear.
4. Motivation for a new estimate
Some of the difficulties with the existing derivation of Zahn (1992), (the most popular
description of the horizontal turbulence before the new works) and the related works rooted on
Zahn’s formulation are the following:
1. In Zahn’s calculation the differential rotation comes out to be constant with r, namely,
 2 r  ch
 , but this need not be the case always.
5
r 
2. This crude prescription allows for too large horizontal fluctuations of molecular weight,
which tend to stop the meridional circulation.
3. The resulting vertical diffusivity, Dv (eq. 7) is insufficient to account for mixing observed
in massive stars.
Some numerical models (Maeder & Meynet 2001) indicate that the diffusion coefficient
of horizontal turbulence Dh is not much larger than the coefficient of vertical diffusion, Dv by
shear, especially at low Z (metalicity) where U(r) is small. In other words the condition Dh >> Dv
is not guaranteed, but to get the effective diffusivity Deff, the large Dh condition must be
assumed. Also for ‘shellular rotation’ to be valid Dh must be larger.
6
5. New result of Maeder (2003): Dissipation & feeding of turbulent energy
A. Maeder (2003) proposes the following new prescription for the horizontal shear viscosity.
First consider the rate of dissipation of turbulent energy. From Zahn (1992, eq. 2.17), the
rate of viscous dissipation of energy present in the differential zonal motions on an isobar is
2
ˆ



t (r , )   h  sin 
  ,
 

(11)
per unit mass and time and for an interval of latitude .
Using eq. (8),
ˆ
P cos  

 2 2
, and from (11),


t (r , )  9 h  22 r sin 4  cos 2  2 .
(12)
This differential rotation brings about an excess of energy on an isobar. For an interval of
latitude , the difference of rotational velocity is given as
ˆ r ,   r sin  r 
vr ,   r sin 
2
dP2 cos 
  3r sin 2  cos 2 r  .
d
Here only the velocity difference due to the shear on the equipotential is expressed. The excess
of energy over interval  due to the differential rotation in latitude is hence given as,
Ediff r ,   v 2 r ,   r 2 sin 4  cos 2  22 r  2 .
1
4
9
4
(13)
This small excess of energy over an interval  is smeared out in a dynamical timescale tdiff.
On an isobar, the differential rotation due to 2 produces a shift in longitude () for two
fluid elements located at different colatitude. Considering the dynamical time as the time taken
by the differential motion in  to perform n axial rotations, (Maeder 2003, eq. 15) the
characteristic timescale can be written as, t diff
 5n r
 
 3  2V
7



1 2
.
(14)
The ratio of the excess energy (eq. 13) and the rate of viscous dissipation (eq. 12) is of
the order of this timescale, so
Ediff r , 
 t diff . Using eqns (12), (13) & (14), the coefficient of
t r , 
1/ 2
 3

r 3  2V 
horizontal turbulence is obtained as,  h  
 80n

.
(15)
Now h can also be determined by dividing the square of a typical lengthscale (of order r)
by the diffusion timescale given by eq. (14). From conservation of angular momentum, by taking
into account the horizontal variations of rotation 2, the following expression can be derived
(Zahn 1992, eq (2.11 b)),
 h  2 r   r r 2V  U  ,
1
5
(16)
with  as described in eq. (9). Now eliminating 2 between the equations (15) & (16), we get
coefficient of viscosity due to horizontal turbulence as following,


 h  Ar r (r )V 2V  U 
1/ 3
 3 
, with A  

 400n 
1/ 3
.
(17)
For n = 1, 3 or 5, A = 0.134, 0.0927, 0.0782, respectively.
This expression can be written in the usual form h=(1/3) l.v for a viscosity where the velocity, v
is a geometric mean of 3 relevant velocities: 1st, (2V-U) as the given by Zahn (1992, eq. 2.27);
2nd, the horizontal component, V of the meridional circulation; and 3rd, the average local
rotational velocity r  (r). This rotational velocity is usually much larger than U(r) or V(r),
typically by 6 to 8 orders of magnitude in an upper Main Sequence star rotating with the average
velocity.
6. Formulation of turbulence from laboratory experiment (Mathis et. al.)
8
The following is another method of finding horizontal turbulence given by Mathis, Palacios &
Zahn (2004). It is derived from laboratory experiment results.
Cylindrical Couette–Taylor flow has been found to be turbulent in the laboratory when
the rotation rate  increases with the distance s from the axis. On the other hand linear stability
theory predicts that the flow should be stable. Experiments performed with the inner cylinder at
rest, and varying the size s of the gap between the cylinders, have shown that when the relative
gap, s/s is larger than about 1/20, turbulence sets in whenever the gradient Reynolds number
s3(d/ds)/ exceeds some critical value.
From torque measurements, the turbulent viscosity is found to scale as:  t   s 3
d
,
ds
with  ~ 1.510–5 (Richard & Zahn 1999, eq. 7 & eq. 9). This is termed ‘viscosity’ and has
been applied to accretion disks by Huré et. al. (2001), where it yields somewhat different results
compared to the classical viscosity introduced by Shakura & Sunyaev (1973).
This formulation has been established for maximal differential rotation; it remains to be
verified if it is valid for milder shear rates (more typical of stellar interiors) too. Writing t in
spherical geometry,
 t   r sin  3
dP
d
 r 2  2 sin 3  2 ,
rd
d
(18)
and averaging over the sphere the turbulent diffusivity is obtained as  h 
1 2
r  2 .
2
Inserting this value of 2 in terms of h in eq. (9) (the eq. governing balance of advection and
diffusion), in the stationary limit, the horizontal turbulent viscosity is obtained as,

h   
 10 
1/ 2
r  r 2V r   U r  
2
1/ 2
1/ 2
.
(19)
The horizontal turbulent diffusivity, Dh is considered to have same value, i.e. Dh = h.
9
7. Results of different works
7.1. Comparison of Maeder & Zahn’s work
Comparing the values of h calculated by Maeder (2003) and that given by Zahn (1992) the
following results are obtained. Obtaining the ratio from eqs. (10) and (17),
1

3
 h ( Maeder )
r V

 Ach 
2 
 h ( Zahn)


2
V


U


(20)
The quantities V and U are of the same order of magnitude. Numerical models show that
typically V  U / 3 and (2VU)  V, which gives the following order of magnitude relation,
1
 r  3
 h ( Maeder )
 .
 Ach 

 h ( Zahn)
V


(21)
Hence the ratio of the diffusion coefficients by the two methods goes as (1/3)rd power of the ratio
of the local rotational velocity to the horizontal velocity of meridional circulation at a certain
level. For an illustration, we will consider a 20 solar mass star, which has an average rotation
velocity of 220 km s–1 at the surface. At the middle of the main sequence phase, the vertical
component of the meridional circulation lies between 310–4 and 310–3 m s–1 as shown by
stellar structure models in Maeder (2003). Thus, typically, h(Maeder) /h(Zahn) is of the order
of 102. Hence the estimate of the diffusion coefficient of the horizontal turbulence given by
Maeder (2003, eq. 19) is much larger than that given by Zahn (1992, eq. (2.27)).
The degree of differential rotation corresponding to this value of h is now estimated.
Using equations (16) and (17) we can write,
2
1  2V  U 


(r ) 5 A 
r V
2
1
3

 .


(22)
10
There is no coefficient ch in this ratio. Numerically, this is 1/5 of the inverse of the ratio given by
Eq. (20), where ch = 1 would be used. Hence from these estimates, we get a ratio of
 2 / r  =210–3. This small degree of differential rotation on an isobar is logical as the value
of h here is much larger. This value of coefficient h reinforces Zahn’s hypothesis of shellular
rotation. Also the ratio  2 / r  is larger for slowly rotating stars. As h grows with the velocity
of rotation, this is a consistent feature.
In Maeder (2003), stellar models are calculated for a 20 solar mass star with composition
X=0.705 and Z=0.02 and using same physics as Maeder & Meynet (2001). The initial rotation
velocity is 300 km s–1, which corresponds to an average rotation during the MS phase of about
240 km s–1. Figs 1, 2 & 3 in Maeder (2003) give the following key results. Here the coefficient
Dshear describes diffusion by shear instabilities. Deff expresses the contribution of the meridional
circulation and horizontal turbulence to the diffusion of the elements (as already introduced in
sec. 3).
A larger horizontal turbulence makes Dshear larger and Deff smaller. But the relative
importance of these two coefficients is not the same throughout the star. Dshear always dominates
at some distance from the stellar core, while Deff tends to dominate near the core, especially if Dh
is small. In addition, the ratio of the two coefficients changes during stellar evolution, making the
situation complicated. In brief, a larger Dh tends to reduce the size of the core by reducing Deff,
which is important close to the core. At the same time the spread of the processed elements (He
and N) up to the surface is larger.
A larger horizontal turbulence Dh reduces the horizontal gradients thus limiting the
importance of the currents.
The horizontal turbulence also helps to smear out the temperature and density
fluctuations on an equipotential, which reduces the effects driving the meridional circulation.
11
7.2. Comparison of Mathis et. al. & Zahn’s work
Comparing the values of h calculated by Mathis et. al. (2004) and that given by Zahn (1992) the
following results are obtained. Obtaining ratio from eq. (10) and (19),
1/ 2


 h ( Mathis _ et.al.)   
r

   ch 



 h ( Zahn)
2
V


U
 10 


1/ 2
(23)
As the quantities V and U are of the same order from previous subsection, we get the following
order of magnitude relation,
1/ 2
 r 
 h (Mathis _ et.al.)   
 .
   ch 

 h (Zahn)
 10 
V 
1/ 2
(24)
Hence the ratio of the diffusion coefficients by these two methods goes as (1/2) power of the
ratio of the local rotational velocity to the horizontal velocity of meridional circulation at a
certain level.
7.3. Comparison of all three works by stellar model
The different diffusivity prescriptions were compared in Mathis et. al (2004), from where
I take the following numerical results. These three different turbulence formulations are
implemented in the stellar structure and evolution code STAREVOL, running a 1.5 solar mass
Population I star (X=0.705, Y=0.275, Z=0.020).
The transport of chemicals and angular
momentum was treated according to Zahn (1992). The initial equatorial velocity was taken as
110 km s-1, which was assumed to reduce to 90 – 100 km s-1, at the age of the Hyades cluster.
The results for two ages of the star: 0.368 Gyr and 1.368 Gyr are as shown in the Figures 1–3. In
the figures the curve labeled “this work” is the result from Mathis et. al. 2004.
12
Fig. 1. – Dh, horizontal component of the turbulent diffusivity generated by horizontal shear in a
1.5 Msun rotating star, at two evolutionary stages in its life; according to the prescriptions of Zahn
(1992), Maeder (2003) & Mathis et. al. (2004) (labeled “this work” in the plots) [All the figs
used here have been taken from Mathis et. al. 2004].
Fig. 2. – Deff, the effective diffusivity in the vertical direction resulting from the erosion of the
advective transport through the horizontal diffusivity shown in Fig 1.
13
Fig. 3. – Dv vertical component of turbulent diffusivity generated by the vertical shear. Same
format as Figs 1 & 2.
Fig. 1 compares the different prescriptions for the horizontal diffusivity generated by the
latitudinal shear. The recent formulations (eq. 19 of Maeder 2003 & eq. 19 of Mathis et. al 2004)
yield values greater than the old one (eq. 2.27 of Zahn 1992) by 3 to 4 orders of magnitude. The
enhancement of Dh has several effects. Firstly, the differential rotation in latitude is maintained at
a low level, validating the assumption of ‘shellular rotation’ made by Zahn (1992) to calculate
the meridional circulation. Secondly, this reduces the latitudinal fluctuations of chemical
composition, which otherwise tend to reduce the meridional circulation. The circulation speed U2
increases, which compensates for the effect of Dh. Hence the effective diffusivity is less affected
by the increase of Dh, as clear from fig. 2. The larger Dh increases the turbulent diffusivity Dv in
the regions where it is reduced by a molecular weight gradient. This is clear from fig. 3, where
Dv is everywhere at least four orders of magnitude smaller than Dh.
14
8. Conclusions
Two new forms of expression for the coefficient of diffusion (Dh) by the horizontal turbulence
in rotating stars are derived. One is accomplished by equating the energy dissipated by horizontal
turbulence and the excess energy present in the differential rotation on an equipotential, which can be
dissipated in a dynamical timescale. Another is derived from Couette-Taylor lab experiments. Both of
these new forms of coefficients are larger than that proposed by Zahn (1992) by factor of 10 2–103. The
change of Dh has been studied numerically on stellar structure with the following results: it increases
the vertical transport, enhancing the mixing as the star evolves and a molecular weight gradient is
building up. The higher horizontal turbulence reduces the importance of the -currents and also, to a
smaller extent, the driving of the meridional circulation. The larger Dh tends to contain the size of the
core and at the same time to favor the spread of processed elements up to stellar surface.
The differential rotation on an equipotential is found to be much smaller. This reinforces the
hypothesis of ‘shellular rotation’ by Zahn (1992).
But there are still uncertainties about the resulting accuracy when apply these formulations for
real stellar interiors, because of several simplifying assumptions made. To deal with full-fledged
realistic turbulence, more efficient 3-dimensional numerical techniques have to be developed.
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15
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