Differential Rotation and Angular Momentum
G J J Botha
E A Evangelidis
University of Leeds, United Kingdom
Demokritos University of Thrace, Greece
Introduction
Drift velocity
In the analysis of motion of a charged particle on a magnetic field line, Alfvén showed the existence
of a force
mu2⊥κ
N,
f =−
(1)
2
with N the first normal of the co-moving trihedral given by (T, N, b) and κ the
curvature. Here T is the unit vector along the magnetic field and b the binormal
_
of the orthogonal system. The subscripts indicate direction with respect to T. In
Ω
N
_
CC
a reference frame located at the centre of curvature (CC) and rotating with an
P
angular velocity Ω, a particle at point P and moving in a circular orbit, develops
b
_
a centrifugal force −mΩ2ρN = −(mu2k/ρ)N with ρ the distance from P to the
origin of the reference frame. When combined with the force described by (1), this
gives the total force
!
2
muk
u2⊥
mu2⊥κ
N−
N = −mκ
+ u2k N.
(2)
ft = −
2
ρ
2
The motion of a particle is governed by the equation of motion (18), where fΩ is described by (9) and
f∇Ω by (10). From (17) the force due to the differential rotation is written as
f∇Ω = mρ2 φ̇ + Ω ∇Ω
(19)
T
The force acting on a charge at point P produces the well known drift velocity
!
ft × B
1
mκ u2⊥
2 b.
vd =
=
f
×
T
=
+
u
t
k
qB
qB
2
qB 2
in plane polar coordinates. The terms in fΩ are well known in the literature, with first term the Coriolis
force and the second term the centrifugal force. The latter can be written as ρΩ2êρ and for our purpose
it suffices to remark that this is the centrifugal term (vk2/ρ)(−N) already included in (2). Retaining the
contribution from the Coriolis force, fΩ becomes
(20)
fΩ = −2mΩρ̇êφ + mρΩ 2φ̇ + Ω êρ
in plane polar coordinates. We would like to emphasize the inability of the Coriolis force to do work,
in contradistinction to the centrifugal force and f∇Ω . The equation of motion (18) can be written as
dp
= fL = −2mΩρ̇êφ + mρΩ 2φ̇ + Ω êρ + mρ2 φ̇ + Ω ∇Ω − ∇U.
dt
(3)
In the derivation of expression (2) it was tacitly assumed that the angular velocity Ω is constant and
therefore, the possible influence of the rotation of the reference system on the particle moving along
a fiducial magnetic line is that of the centrifugal force only. Experimental evidence in the laboratory,
however, especially during heating of plasmas in tokamaks with neutral beams and ion cyclotron
resonance, shows that the plasma rotates in the toroidal direction with a varying angular rotation
[4]. In accretion disks the Keplerian profile of the angular velocity is well established. With these
measurements and observations in mind, we consider a reference system at the origin of a rotating
disk, with its angular velocity a function of the radial direction.
Rotating reference frame
(21)
When the particle carries a charge, the Lorentz force comes into play and one should add (1) to this
expression. If we assume rotational symmetry, then ∇Ω = (∂Ω/∂ρ)êρ, so that (21) becomes
∂Ω fL = −2mΩρ̇êφ + mρ Ω 2φ̇ + Ω + ρ φ̇ + Ω
êρ − ∇U.
(22)
∂ρ
Neglecting ∇U , the guiding centre drift in a magnetic field due to this force is
mρ
2mρ̇Ω
∂Ω
−Bz , 0, Bρ +
0, −Bz , Bφ ,
vd =
2φ̇ + Ω Ω + ρ φ̇ + Ω
2
2
∂ρ
qB
qB
(23)
with q the charge of the particle under consideration. In tokamaks Bφ is dominant, so that (23)
reduces to motion out of the plane of differential rotation. In the case of accretion disks, a dominant
axial magnetic field (Bz ) reduces (23) to motion mainly in the rotational plane. Apart from drift due
to particle motion (i.e. the ρ̇ and φ̇ terms) there exists azimuthal drift due to the differential rotation of
the reference system.
In a rotating reference frame the Lagrangian of a moving particle takes on the form [2]
1 2
1
L = mv + mv · Ω × r + m(Ω × r)2 − U,
2
2
(4)
Differential rotation profiles
with the velocity in an inertial reference frame v0 = v + Ω × r composed of the particle velocity v
and the rotational velocity Ω × r of the reference frame. The particle has mass m and position vector
r with U the potential energy. To obtain the equation of motion
d ∂L
∂L
(5)
=
dt ∂v
∂r
In a tokamak the magnetic field is dominated by its toroidal component, i.e. Bφ in cylindrical coordinates. For particles accelerated due to preferential heating, the average value over many rotations is
expected to approach the value of the rotation of the background medium, so that φ̇ = Ω. The drift
velocity (23) becomes
∂Ω
ρΩ
vd =
3Ω + 2ρ
êz ,
(24)
ω
∂ρ
we need to calculate the derivatives with respect to the velocity and position vectors. Here we differ
from the usual analysis in that the angular velocity varies with the radial distance, that is, the system
is sheared. This is a common phenomenon in galaxies and laboratory plasmas. Thus,
where ω = qBφ/m is the gyrofrequency. The profile of a tokamak plasma spun toroidally by neutral
beam injection [4] is presented below, together with the size of the associated drift velocity (24) for
ions, using ωp = 10MHz. The drift velocity for electrons has the opposite sign and with typical values
of 60GHz < ωe < 600GHz, is O(103) smaller.
∂L
= m (v + Ω × r) ,
∂v
∂L
= m [(∇Ω) × r] · (v + Ω × r) + m (v × Ω) + mΩ × (r × Ω) − ∇U.
∂r
(6)
(7)
The equation of motion (5) becomes
dv
m = fΩ + f∇Ω + mr × Ω̇ − ∇U,
dt
(8)
fΩ = m [ 2v × Ω + Ω × (r × Ω) ] ,
f∇Ω = m [(∇Ω) × r] · (v + Ω × r) .
(9)
(10)
with
The two terms in fΩ are the Coriolis and centrifugal forces. The last term in (8) is the force field. The
term containing Ω̇ is zero if the rotation is constant (which we assume to be the case). The existence
of f∇Ω is owed to the gradient in the angular velocity of the rotating system. In order to analyze f∇Ω ,
we write it in component form. Let (êx, êy , êz ) be a system of orthogonal unit vectors attached to the
rotating reference frame, with Ω = Ωêz . It follows that
[(∇Ω) × r] · v = xvy − yvx ∇Ω,
(11)
[(∇Ω) × r] · (Ω × r) = Ω x2 + y 2 ∇Ω,
(12)
where we have used Ω = |Ω|. Transforming to plane polar coordinates (ρ, φ), the above results can be
written as
[(∇Ω) × r] · v = ρ2φ̇∇Ω,
[(∇Ω) × r] · (Ω × r) = ρ2Ω∇Ω.
(13)
(14)
If we assume that Bz is the main magnetic field component in accretion disks, then (23) reduces to
Ω
ρ
∂Ω
vd = −2ρ̇ êρ −
2φ̇ + Ω Ω + ρ φ̇ + Ω
êφ.
(25)
ω
ω
∂ρ
The further assumption that the plasma is frozen into the reference frame (ρ̇ = 0 and φ̇ = Ω) simplifies
the above expression to
∂Ω
ρΩ
vd = −
3Ω + 2ρ
êφ.
(26)
ω
∂ρ
An accretion disc around a young star typically rotates once every 10 days at its innermost radius.
We assume a profile [3] that is uniform at short distances (R R0) and Keplerian at R0 R. A
magnetic field equal to the solar interplanetary value (5nT) is used. The associated drift velocity for
ions (ωp = 7.5 × 10−2 Hz) is presented below. Electrons move in the opposite azimuthal direction and
with ωe = 1.4 × 102 Hz [1] their drift velocity is O(104) slower.
From the definition of angular momentum M = mr × ṙ it follows that its z component in the inertial
reference frame can be written as
h
i
Mz = m xvy − yvx + x2 + y 2 Ω
(15)
in Cartesian coordinates, or equally
Mz = mρ2 φ̇ + Ω
(16)
f∇Ω = Mz ∇Ω,
(17)
in plane polar coordinates. Therefore,
so that equation (8) becomes
dp
= fΩ + Mz ∇Ω − ∇U
(18)
dt
where p is the linear momentum of the moving particle. We observe that a radial force term appears
due to the coupling of the angular momentum and the gradient of the differential rotation. We conclude
that this force is a property of the Ω field – independent of the particle motion – as can be seen by
eliminating the angular motion of some test particle (φ̇ = 0).
References
[1] Baumjohann W., Treumann R.A., 1997, Basic Space Plasma Physics, (Imperial College Press,
London), App. A.3.
[2] Landau L.D., Lifshitz E.M., 1975, The Classical Theory of Fields, (Butterworth-Heinemann Ltd.,
Oxford), par. 22.
[3] Rüdiger G., Kitchatinov L.L., 2005, A&A, 434, 629.
[4] Wesson J., 2004, Tokamaks, (Oxford Science Publications, Oxford), par. 3.13.