Capítulo 2 Obstáculos epistemológicos de los números irracionales
1. Obstáculos relacionados con la representación
i
of n . As before we search for the value of CO ’I ■| at which the series converge, this being the value at | g
which the instability occurs.
I
1
i
Results ,6 4 J' In this problem we consider only the case yS ~ | Obut, as shown in section 3.2, it gives the same growth rate as model B in which ^ and are both constant, andyg takes any value.
The variation of w with k is the same as in the 6 ^ " ^ case (Figure 25) page 160, except that increasing
moves the whole curve down without changing it in any other way. The curves thus have a maximum frequency
which we draw in Figure (29) page 182, as a function of B t ; f o r B <= lo'^. We see that for a given S , when is small enough (much less than S ^) tu^^^^is constant, o3o say, and we find that
-- , V Ç. O • b
Csl = 2.1 S , 1 & 7 G )
the same variation that was found when 8.^ - o ^
in section 3.1 .
I
An order of magnitude calculation indicates that the tearing
mode instability is unaffected if Bj is small enough. The
4
effect of is to allow the transport of energy across 1
the sheet by Alfven waves, so if the growth rate of the ] tearing mode instability is very much greater than the growth rate of the transport of energy across the sheet then B. |
■I
7
6
5 = 1 0 5 32
4 3-2
-1Figure (29). The variation of 6Jmax the maximum growth rate of the tearing mode instability in a neutral current sheet
with a component of magnetic field, Bj., transverse to the neutral line, as a function of where the constant, B n » the
'a -'Q
value of the magnetic field strength at the edge of the neutral current sheet.
The curves are drawn for several values of
S
, the ratio of the diffusion time scale to the Alfvenic time scale.The time scale for energy transport by Alfven waves across the sheet isT At 'At^ and the time scale for the tearing mode instability with o , is, from Furth et al (1963)
"C ~ T 1 where tr_ is the diffusion time scale and. T a
hv 0 Ao'
the Alfven time scale when - o „ The effect of is ignorable if
V
aoV V
AOThis order of magnitude calculation is confirmed in
Figure (29), page 182, where the tearing mode instability
is unaffected when ^ S
We did not calculate the variation of w^^with 6 1
for ^ ^10% because then & I so that
is not very much greater than « and the diffusion of the equilibrium magnetic field becomes important.
As is increased we find that w falls in value
until to — Moot where the method breaks down because the T.M.I. time scale is approximately equal to the diffusion time scale of the equilibrium magnetic field. When
Targe enough, decreases linearly with B^/ on the log-log plot. Further, the linear
curve that is approached is independent of S ; it has the equation
( f D. /n \ ]
) U - 7 7 1
■= Z Ç ^
1
6(x. '3.^/,
and ( 8 t / i - = o -1 4- from the intersection of the curve in Figure (29), page 182 with the log^^ % $
3.4.: T.M.I. IK A CURRENT SHLET WITH A TAANSVÜRSE MAGÜÜTIC FIBLD | (Contd.)
As 0>i. / a is increased, so co falls in value and the question arises as to what value of cr’vb say, inhibits
the tearing mode. The answer depends on whether we want the value of Bf/ which reduces T3 to a certain fixed value co , say, or the value which reduces c^^o a certain fraction of say W o I S)/ n '
In the first case, suppose to, is less than 10 , then the graph in Figure (29) is linear, independent of S , and so from (3.77) the minimum value of 8.^ which inhibits the tearing mode is given
by -72')
Or, in the limit as oo, approaches I, OKit / B <x “ O l d - .
In the second case, we require the value of 0,^ which makes unequal to '7^ , or from (3.76),
Fi ' (3.79)
Now provided n > 3 we can use (3.77) to estimate co,r*\0/>c SO (3.79) reduces to
—
O lh
" z3 2 o
Again, when exceeds the tearing mode can be said to be significantly inhibited by the presence of the transverse field
3,5.: ANALYTIC TkhAT:'/j:;NT OF THL TEARING MODE INSTABILITY AlTH A'TRÀNÔVEdSL MAGNETIC FI^LD CüüPLüüNT PR&SENT,
U - 2 1)
K + 'è. J ^ > a
V f 8 E ■*■ C > ^ o E 1
where B and are constants. This magnetic field satisfies the equilibrium condition that
£ a L C 2 - S o ' l K S o l •= O
in a piecewise manner. The equilibrium velocity q is assumed to be zero.
The linearized equations, from (3.11) and curl (3.12)^ for small perturbations V., and j are
- 2 -T , C.B-SO
fc|,C2^y,r3^2A[(,'2.A8,VS,^-UA8>Usl , l.B-23')
Bi ~ Z.y.\ ~ o .
If we next assume V_, and B\ to be proportional to cOt: 4. L k y.
the (^component (3.82) gives The method used in this section is based on the
analysis of the T.M.I, by Priest (1969) in which the growth rate is calculated for an equilibrium magnetic field that has no transverse field component.
We approximate the equilibrium current sheet magnetic % field by
while the ^ component of (3.8 3) reduces (after using
(2.8 5) and (3.8 4)) to
u - '
f
%B u f è} W (PIT %y^o\
-b tvTTCof ^A " dTn- " — ; In terms of the diniensionless variables
(3.86)
B j Vu^ " ~ Lkt,p, ^ k - ^ uj = u) t:(%. ICm ToJ-/,^ , ^ I
the two equations (3.85) and (3.86) for and I
become _ / I
1''
:: 1=» + k U + -k A ^ ^ l3 i L h. o * 7 ^ 11 + - I " V ^ ( | + q . - ) - c l S ’- i ; i7
1
?
-
Bio) - A L ) (3.8 8)where a dash denotes differentiation with respect to , Equations (3*87) and (3.88) can be solved for
and for given values of w , (z and S « Solutions
are obtained in two regions, an outer region ( ^ ^ ) ;J and an inner region ( o c c 6.^ ) and then matched | at the boundary ( "^ %: 6^ ) . In the outer region the
resistive terms are neglected so that the magnetic field is ’’frozen into” the plasma, whiJe on the inner region diffusion is important.
It is because of diffusion that the magnetic field can 4
reconnect across the neutral sheet and allow the T.M.I. I
to occur, so that it is in the inner region that the 3 small transverse field component, 8^^ , will have most -I influence on the instability* In order to simplify | the analysis, while still retaining the essential y elements, we shall assume that B^o Ts so small that
it does not influence the dynamics in the outer region 1
but still remains important in the inner region, % Outer Region: ( > <Lo )
'I
In this region where diffusion is unimportant we | neglect terras of order VS, in (3,87) and (3.88), which become
B'i ■ Czv I - + v,^ 6 ^ 0 + B y ^
A li Z “ o k B y "
o) OJ) — — f 7
- Ü. [v^ 1 B y A •
]
In general B<^ and are complex, L say, I so rewriting (3,89) and (3.90) in their real and imaginary parts, assuming o3 and k are re a l (i.e. we are looking for ! a purely growing i n s t a b i l i t y which is susceptible to a 4
perturbation-with an oscillatory X variation), we find g , (3.91) I