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SPHEROMAK SHAPE PARAMETER

By Charles Rhodes, P.Eng., Ph.D.

INTRODUCTION:
The spheromak shape parameter So is defined by:
So^2 = (Rs / Rc)
where:
Rs = spheromak maximum outside radius
and
Rc = spheromak minimum inside radius
is one of the most important spheromak characterization parameters. Plasma spheromaks can be photographed and the photos used to determine So^2. The parameter So^2 can be used to calculate the field parameter A.

The spheromak boundary condition developed on the web page titled: ELECTROMAGNETIC SPHEROMAK is:
Nr^2 + Kc^2 B^2 = 4 A^4 / Pi^2
where:
Nr = (Np / Nt)
and
Kc = (ellipse perimeter length) / (contained circle perimeter length)
and
B = (So^2 - 1) / (So^2 + 1)
and
So^2 = (Rs / Rc)
and
A = 2 Zf / (Rc - Rc)
and
Zf = maximum spheromak height above its equatorial plane
and
Pi = 3.14159265

Rearrangement of the equation:
B = (So^2 - 1) / (So^2 + 1)
gives:
So^2 = [(1 + B) / (1 - B)]

The spheromak boundary condition has an important special case. If:
Nr^2 << Kc^2 B^2
then:
Kc^2 B^2 ~ 4 A^4 / Pi^2
or
Kc B ~ 2 A^2 / Pi
or
B ~ 2 A^2 / (Pi Kc)

For a spheromak in which the ellipse is close to being circular ellipse geometry shows that:
A^2 / Kc ~ 1
which gives:
B ~ (2 / Pi)
= 0.6366

Hence:
So^2 = (1 + B) / (1 - B)
~ (1 + (2 / Pi)) / (1 - (2 / Pi))
= (Pi + 2) / ( Pi - 2)
= (5.14 / 1.14)
= 4.5

By comparison, photographs of plasma spheromaks show typical So^2 values of about:
So^2 = 4.2

This experimental observation suggests that the actual value of B for plasma spheromaks is closer to about:
B = (So^2 - 1) / (So^2 + 1)
= (4.2 - 1) / (4.2 + 1) = 3.2 / 5.2
= 0.615

A precise theoretical analysis, which properly takes into account ellipse geometry and the magnitude of Nr^2, indicates that for an atomic spheromak:
B = 0.600
which corresponds to:
So^2 = 4.000

FIX FROM HERE ONWARD

Then application of the condition:
M dNp = - dNt
gives:
Nr = (1 / 2 M){-1 + [1 + (16 M^2 A^4 / Pi^2)]^0.5}
and
[Kc B] = _______{-1 + [1 + (16 M^2 A^4 / Pi^2)]^0.5}
where:
for Solution Family #1
M = 2
and for Solution Family #2
M = (1 / 2).

Rearrangement of the above equation:
B = (So^2 - 1) / (So^2 + 1)
shows that:
So^2 = [(1 + B) / (1 - B)]
and rearangement of the boundary condition shows that:
B^2 =[1 / Kc]^2 [(4 A^4 / Pi^2) - Nr^2].
or

B = [4 A^4 / Pi^2 - Nr^2]^0.5 [1 / Kc]

CONSIDER A SIMPLE SPHEROMAK IN WHICH A = 1.00000
If:
A = 1.00000
then:
Kc = 1.00000
then
B = 0.4640298321
giving:
So^2 = 1.4640298321 / 0.5359701679
= 2.73155097

CONSIDER A REALISTIC SPHEROMAK
in which:
So^2 = 4.0
then:
B = [(So^2 - 1) / (So^2 + 1)
= 0.60

Nr^2 + Kc^2 B^2 = [4 A^4 / Pi^2]
or
A^4 = [Pi^2 / 4][Kc^2 B^2 + Nr^2]
__________________________ = [Pi^2 / 4][Kc^2 (.36) + .1899610504]

However, the relationship between A and Kc is governed by ellipse geometry.
Kc = [(A + 1) / 2] Kh
where:
Kh =

Hence in physical situations where So^2 can be measured, as is the case with plasma spheromaks, A can be calculated.

SPHEROMAK COLLAPSE:
Recall that:
Np / Nt = (- 1 / 4) + [(1 / 16) + (4 A^4 / Pi^2)]^0.5
or
[(Np / Nt) + (1 / 4)]^2 = [(1 / 16) + (4 A^4 / Pi^2)]

At (Np / Nt) = (1 / 2)
LHS = 9 / 16
4 A^4 / Pi^2 = 8 / 16 = (1 / 2) or A^4 = Pi^2 / 8
or
A^2 = Pi / 2^1.5 = 1.110720733
or
A = 1.053907365

This A value is prohibited because it leads to spheromak collapse.

There is another spheromak collapse point at:
(Np / Nt) = (3 / 4)
where:
4 A^4 / Pi^2 = 15 / 16
A^4 = Pi^2 15 / 64
A = 1.233254638

Note that the A value for the plasma spheromak is well separated from the nearest A value that leads to spheromak collapse.

ATOMIC PARTICLE SPHEROMAKS:
For atomic particle spheromaks a different method of spheromak analysis is required.

CONSIDER THE CASE OF A^4 = 7 Pi^2 / 64
A^2 = [7 Pi^2 / 64]^0.5
= [7^0.5 Pi / 8]
= 1.038984109
or
A = 1.019305699

(Np / Nt) = {-1 + [1 + (4 A^4 / Pi^2)]^0.5}
= {-1 + [1 + (4 (7) / 64)]^0.5}
= {-1 + [1 + (7 / 16)]^0.5}
= 0.1989578808

Kc = [(1 + A) / 2] {1 + (1 / 4)[(A - 1) / (A + 1)]^2
= [(2.019305699) / 2] {1 + (1 / 4)[(.019305699) / (2.019305699)]^2
= [1.00965285] {1 + .00002285109033}
= [1.00965285] [1.00002285109033]
= 1.009675922

FIND B FOR A^4 = 7 Pi^2 / 64:

Recall that:
B = {1 - [Pi^2 / 4 A^4]({-1 + [1 + (64 A^4 / Pi^2)]^0.5} / 4)^2}^0.5
[2 A^2 / Kc Pi]

= {1 - [16 / 7]({-1 + [1 + (7)]^0.5} / 4)^2}^0.5
[2 (7^0.5) / 8 Kc]

= {1 - [16 / 7]({(8)]^0.5 - 1} / 4)^2}^0.5 [2 (7^0.5) / 8 Kc]

= {1 - [16 / 7](0.4571067812)^2}^0.5 [2 (7^0.5) / 8 Kc]

= {1 - [16 / 7](0.2089466094)}^0.5 [2 (7^0.5) / 8 Kc]

= {0.5224077499}^0.5 [2 (7^0.5) / 8 Kc]

= 0.4780725788 / Kc
= 0.4780725788 / 1.009675922
= 0.4748628762 = B

(1 / Alpha) = [Nt / 2 A]{1 - [(1 - B^2)^0.5 / 2]}
= [Nt / 2 A]{0.5599701008}
= [Nt / 2 (1.019305699)]{0.5599701008}
= [Nt (0.2746821201)]
= 137.03599915 Nt = 498.8894039

Thus for modest values of A of about:
A^4 = 7 Pi^2 / 64 = 1.079487979
or
A = 1.019305699
we can expect Nt values of about 500. This value is consistent with atomic particle experimental data. Note that this A value is far from the nearest spheromak collapse A value of 1.053907365.

APPROXIMATE SOLUTION FOR A PLASMA:
(Np / Nt) = N / [P - 2 N]

Confirm the important constraint condition that for a stable spheromak in which Np and Nt share no common factors:
dNp / dN = 1
dNt / dN = - 2
2 dNp = - dNt

Then:
[2 Np dNp (So^2 + 1)^2] + [2 Nt dNt (So^2 - 1)^2 Kc^2]}

= [2 Np (- dNt / 2) (So^2 + 1)^2] + [2 Nt dNt (So^2 - 1)^2 Kc^2]}

= dNt [- Np (So^2 + 1)^2 + 2 Nt (So^2 - 1)^2 Kc^2]}

Near the point of stability where dSo / dN ~ 0:
Np / Nt ~ 2 (So^2 - 1)^2 Kc^2 / (So^2 + 1)^2
or
(So^2 - 1)^2 Kc^2 / (So^2 + 1)^2 = (Np / Nt)

or
B^4 Kc^4 = Nr^2

The web page titled ELECTROMAGNETIC SPHEROMAK shows that the spheromak boundary condition is:
Nr^2 + Kc^2 B^2 = [4 A^4 / Pi^2]
and
Nr = Kc^2 B^2

Hence:
Kc^4 B^4 + Kc^2 B^2 - [4 A^4 / Pi^2] = 0
or
Kc^2 B^2 = {- 1 + [1 + 4 (1)4 A^4 / Pi^2]^0.5} / 2
= {- 1 + [1 + 16 A^4 / Pi^2]^0.5} / 2

At A = 1.00000, Kc = 1.0000
Kc^2 B^2 = 0.3094965939
giving:
B = 0.5563241806

So^2 = (1 + B) / (1 - B)
= 1.5563241806 / 0.4436758194
= 3.507795811

Photographs of plasma spheromaks indicate that So^2 ~ 4.1

This web page last updated May 17, 2019.