Even in perfectly
compressible (cold) or
incompressible limits this
configuration can only be
achieved by a radial
pressure gradient balancing
centripetal acceleration.
Figure 1. Sketch of the
axisymmetric case (m = 0) tube
structure. Twisted field lines
are shown. The inward force
associated with the field
curvature is balanced by the
combination of radial magnetic
field pressure and the
centripetal acceleration
associated with the rotation.
The inner cylinders illustrate
the contours of pressure.
  2 r  

d  B.b  B.br


dr  0 
0
d  B.b 


dr  0 
As in steady state
B.br
0
0
In fact, the situation represents the
column rotating exactly as if it were
a rigid body!
The only fluid effect is that the radial
pressure gradient balances
centripetal effects.
d
c
b
a
Figure 2. Cross-section of the
plasma column once steady
rotation has started, in the
axisymmetric (m = 0) case. The
magnetic field is assumed to be
into the page. (a) illustrates the
distribution of field aligned
current. A surface current on the
outer boundary shields the
exterior from the twisted field.
Current flows horizontally through
the conductor. The return field
aligned current is distributed
across the cross-section. The
overall current system is closed in
the travelling front which originally
brings the column into rotation.
(b) shows the flow.
(a)
(b)
m = 0 field aligned
current distribution
m = 0 flow
pattern
Figure 5. Illustration of a field line displaced by the
wave. The line in question lies on the boundary of
the cylinder. An unperturbed field line is illustrated
by the dotted straight line. The perturbed field line
passes through the points a, b, c, d, and e. From a
to c the line is on the near side of the cylinder.
From c to e the line is on the far side of the
cylinder and is shown dashed. In one wave length
(as illustrated) it does a complete circuit of the
tube. With a right handed rotation at the base, the
field makes a left handed spiral up the tube.
Because of the polarisation reversal across the
cylinder surface, the field lines outside make a
right hand spiral (not shown). On the distorted
column boundary where the spiral sense reverses
field aligned current flows.
Energy issues
Working to linear order in the amplitude of the disturbance on the background, the energy per unit length is second order,
W( m 0 )
 b||2
b2
 2 r 2 
  dS 



2

2

2
0
 0

(3)
Now there is equipartition between transverse components of velocity and field (i.e. b ) in (1) due to the Walen relation
2
2
a
a
 b||2




 
b
b
2 2
2 2
2 4
||
||
W( m 0 )   dS 
  r   2  rdr 
  r   2  rdr 
   a
 2 0

 2 0

 2 0  2
0
0

 br2

b2
 b2
 2 2
 2 2 
2


W( m 1)   dS 


 r     2  drr 

 
2
2
 2 0

0
 2 0 2 0

(3’)
(4)
There is equi-partition also in the m = 1, case (cf. Walen relation) so one may also write
W( m 1)  a 2  2 2
(4’)
As  can be vanishingly small, comparing 4’ and 3’, shows that the energy required to set up the m = 1 configuration is
less than that required to set up the m = 0 configuration as long as the displacement is much less than the column
radius, a. In fact, the size of  is determined by storage/transmission of angular momentum the boundary condition
along the field as the angular momentum stored is proportional to 2.
The m = 1 mode can be precluded if it is suppressed in some way. For example, if the boundary at r = a were rigid and
so radial motion there were impossible. Such a rigid boundary would impose a compressional m = 0 structure.
The rigid boundary example is an extreme case. The m = 0 mode is confined to the column whereas m = 1 includes a
thin exterior region.
Energy issues
Working to linear order in the amplitude of the disturbance on the background, the energy per unit length is second order,
W( m 0 )
 b||2
b2
 2 r 2 
  dS 



2

2

2
0
 0

(3)
Now there is equipartition between transverse components of velocity and field (i.e. b ) in (1) due to the Walen relation
W( m 0 )
2
2
a
a
 b||2




 
b
b
2 2
2 2
2 4
||
||
  dS 
  r   2  rdr 
  r   2  rdr 
   a
 2 0

 2 0

 2 0  2
0
0
W( m 0 ) 
W( m 1)

2
(3’)
(4)
 2 a 4

 br2

b2
 b2
 2 2
 2 2 
2
r     2  drr  2  2  
  dS 


2

2

2
0
 0

0
 0

There is equi-partition also in the m = 1, case (cf. Walen relation) so one may also write
W( m 1)  a 2  2 2
Instantaneous streamlines
Southwood and Cowley 2014
?
?
Figure 6. Comparison of the field aligned sheet
currents on the edge of the flux tube treading
the obstacle in a) the familiar obstacle in flow
case and b) the case of a rotating source
embedded in a plasma and emitting m = 1 kink
waves that is introduced in this paper. On the
surface reversed field aligned currents form on
each flank with strength varying sinusoidally in
azimuth. The sheet currents are indicated by the
positive (+z direction) and negative signs
(-z direction). Flow lines associated with the
wave in the frame of the obstacle are sketched in
each case. The patterns are similar. However, in
case b) everything rotates with angular velocity,
W, and the cross-section itself is displaced at
right angles to the transverse flow inside the
tube.
Figure 7. Alfvén mode wings and rotating kink modes
contrasted once more. Sketch (a) shows a crosssection in the (x, z) plane containing the field and flow
velocity. Sketch (b) shows the projection in the (y, z)
plane containing the field and transverse to the flow.
Sketches (c) and (d) show the equivalent
characteristics for the case of a steadily rotating
obstacle embedded in a stationary plasma. The
structure is stationary in the frame of the rotating
object. The current carrying structure spirals away
from the object and so the projections in the (x, z)
and (y, z) planes are in spatial quadrature (as
sketched).
Figure 8. Sketch of a rotating magnetised body
where the polar cap field lines are open. The
sense of rotation is indicated by the broad
arrows. The dotted boundary on the object
marks the open-closed field boundary on the
body. Open field lines at the boundary are
sketched. The field lines spiral and with the
same convention as Figure 4 continuous traces
show segments on the viewers side of the tube
and dashed lines indicate the trace behind. The
polar cap flux tube cross-section increases with
radial distance. The degree of kinking depends
on the angular momentum transfer. However,
independent of whether the internal field of the
body is axially symmetric or not, under all
circumstances kinking is necessary for
momentum transmission.
Answering the Krishan Khurana question (why the tail wags the dog)
N
Imagine a
steady
In flux of
solar wind
S
If the seasonal rotation differential were due to
differential illumination, in northern winter the
northern cap would rotate faster than the
southern.
NOT SEEN
In northern winter the northern cusp entry layer
presents a smaller cross-section to the solar
wind.
In northern winter, more solar wind particles
would access southern cusp/entry layer – adding
inertia to rotation of southern cap.
The northern winter, slower southern rotation is
than the northern is likely due to the differential
access to and then loading of cusp/entry layer
Equinoctial Oscillations
N
Imagine a
steady
In flux of
solar wind
S
Equinoctial Oscillations
N
Imagine a
steady
In flux of
solar wind
S
Equinoctial Oscillations
N
Imagine a
steady
In flux of
solar wind
S
Equinoctial Oscillations
N
S
Imagine a
steady
In flux of
solar wind
Equinoctial Oscillations
N
S
Imagine a
steady
In flux of
solar wind
Equinoctial Oscillations
N
S
Imagine a
steady
In flux of
solar wind
Equinoctial Oscillations
N
Imagine a
steady
In flux of
solar wind
S
Equinoctial Oscillations
N
Imagine a
steady
In flux of
solar wind
S
Equinoctial Oscillations
N
Imagine a
steady
In flux of
solar wind
S
Equinoctial Oscillations
N
Imagine a
steady
In flux of
solar wind
S
Equinoctial Oscillations
N
Imagine a
steady
In flux of
solar wind
S
Equinoctial Oscillations
N
S
Imagine a
steady
In flux of
solar wind
Equinoctial Oscillations
N
S
Imagine a
steady
In flux of
solar wind
Equinoctial Oscillations
N
S
Imagine a
steady
In flux of
solar wind
Equinoctial Oscillations
Imagine a
steady
In flux of
solar wind
At equinox, periods north and south
should be similar
- however
recall the offset dipole and so
N
northern cusp is closer to pole at actual
equinox – south should dominate
S
- Immediately
post-equinox, can envisage
chaotic switching
- Once southern winter in full swing
northern should dominate and be
slower – Prediction!!
13226-365rtpksmmod
Bph
4.0
0.0
-4.0
-288 hrs.
30.0
LT, R
20.0
10.0
0.0
-288 hrs.
-10.0
100
Br, Bt
50
0
-50
-100
DOY: 47
2013-Feb-16
InvLat
054-00
061-00
068-00
075-00
082-00
82.75
78.23
77.71
80.33
78.19
C:\Restore\cass-saturn\data\2013\ksmrtpmod\13226-365rtpksmmod -- March 22, 2017 11:16
Return flow of accelerated
residual solar wind material
Vasyliunas regime
(confined near equator
slowly rotating)
Outflow down tail
(“planetary” wind)
06-001-225modSLS4S1025
B
0.0
-5.0
-10.0
-15.0
4.0
B
2.0
0.0
-2.0
-4.0
DOY: 106
2006-Apr-16
R
Lshell
InvLat
mlat
LT
109-00
115-00
121-00
127-00
21.70
21.70
77.60
0.26
23.28
45.20
45.20
81.45
0.22
1.27
2006 DoY:Hr
56.92
56.92
82.38
0.02
3.87
34.38
34.38
80.18
-0.13
5.08
C:\Restore\cass-saturn\data\2006new\sls4scsv\06-001-225modSLS4S1025 -- March 23, 2017 15:18
comb13001-200modph
Bth, dBth
40
20
0
Bph
8
0
-8
DOY: 59 00:00
2013-Feb-28
R
mlat
Lshell
InvLat
LT
22.95
-12.26
24.04
78.23
13.83
00:00
00:00
00:00
00:00
00:00
00:00
00:00
00:00
00:00
00:00
00:00
12.46
-41.29
22.08
77.71
23.00
25.42
-32.12
35.44
80.33
15.34
11.05
47.11
23.86
78.19
10.51
23.52
-48.47
53.50
82.14
17.61
20.36
-1.29
20.37
77.20
13.11
7.96
61.53
35.05
80.28
8.80
17.15
-53.48
48.40
81.74
17.85
22.59
-23.17
26.73
78.85
14.22
18.11
9.48
18.61
76.60
12.59
6.73
-10.76
6.98
67.75
0.54
19.72
-44.20
38.38
80.71
16.21