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Astron. Astrophys. 362, 756-761 (2000)

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4. Angular momentum transport

Our turbulence can only model the situation in accretion disks if it transports the angular momentum outwards, i.e. if the stress [FORMULA] is positive. Additionally, we know from observations the value of the normalized angular momentum transport,

[EQUATION]

being of order [FORMULA] so that - if we find a relation between both the alphas - the dynamo-[FORMULA] can be estimated. For historical reasons the quantity (31) is called the viscosity-[FORMULA]. The notation arises from the Boussinesq relation postulating a direct correspondence between stress and strain. In Paper I we have shown that even for rigid rotation a finite (negative) value for the angular momentum (31) exists which clearly can not be due to an `eddy viscosity'.

For the correlation tensor of the magnetic-forced turbulence the complex expression

[EQUATION]

is obtained with

[EQUATION]

and

[EQUATION]

It follows

[EQUATION]

Note that in (35) a basic viscosity term 3 exists which does not vanish for [FORMULA].

One could discuss the sign and the value of (35) in full generality for a series of turbulence models which, however, is not in the scope of the present paper. All we need is to find whether the `viscosity' term in (35) may dominate the non-viscosity term so that the sign of the angular momentum transport could be changed from minus to plus.

For our magnetic-driven turbulence and for a magnetic field with dominating toroidal component [FORMULA] we find

[EQUATION]

for the Reynolds stress at the equator. It does not vanish for rigid rotation so that the term `viscosity' here indeed makes no real sense. The rigid-rotation term in (36) reflects the [FORMULA]-effect of rotating turbulence fields which is responsible for the maintenance of differential rotation in stellar convection zones (see Paper I). Here it is negative. The total angular momentum transport, however, for a Kepler flow is positive as then

[EQUATION]

results if [FORMULA] as the radial pressure scale. In Kepler flows the Reynolds stress is positive, hence the angular momentum is always transported outwards.

For the Maxwell stress we simply obtain

[EQUATION]

which leads to the magnetic-induced angular momentum transport

[EQUATION]

It is a viscosity-term and it survives the limit [FORMULA]. For (31) we then arrive at the positive value

[EQUATION]

with the amplitude

[EQUATION]

where the Coriolis number is [FORMULA]. If the magnetic fields are in equipartition with the turbulence then we find

[EQUATION]

as an estimate. The maximal value of (42) might be [FORMULA], which in accretion disks itself hardly exceeds the order of unity. The viscosity-[FORMULA], therefore, proves to be smaller than unity for subsonic turbulence. On the other hand, for equipartition of the magnetic energy with the thermal energy ([FORMULA]) the value of (42) should not be much smaller than unity. The desired order of magnitude of [FORMULA] for the viscosity-[FORMULA] might be well reproduced by the presented theory.

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© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000
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