7 Numerical Simulation of Generation, Distribution, and Impact …
91
initial parameters: I 0 = 0.7 MA, a 0 = 3.5 cm, R 0 = 6.5 cm, T e0 = T i0 = 10 eV,
n i0 = 8.3 × 10
17 cm
−3 , and U 0 = 3 × 10
7 cm/s.
For a more comprehensive understanding of TPB evolution during motion, the
posed problem in this work was gradually detailed and refined. At first, the behavior
of a homogeneous cylindrical layer in a one-dimensional cylindrical approximation
is considered. The gas is supposed to be ideal.
The calculation was carried out with Brode’s difference scheme [3] and von
Neumann artificial viscosity. As follows from the dimensionless equations and initial
conditions, the only parameter determining flow development is a 0 . The calculations
showed that at a zero value of the initial velocity u = u/u x , where u x =
√ γ kT 0 /m.
The ring-shaped structure quickly turns into an expanding solid disk in the absence of
any external forces and field inside the toroid for any initial a 0 , on which the collapse
time depends. However, as calculated studies have shown, if u(t = 0) ≥ 2, then the
internal rarefaction wave does not reach the center and the annular expansion will
continue in time. It should be noted, that this value u = 0 is less than the maximum
value u = 2(γ − 1) = 3, which according to the front of the plane rarefaction wave,
since in this case the movement is cylindrical and not self-similar.
7.3 Physico-mathematical Formulation of the Problem
of the Initial Stage of TPB Dynamics
TPB dynamics is described by variation in the two main parameters, R(t) and a(t).
The variation R(t) is determined by the radial tension created by the pressure of the
magnetic field inside the toroidal ring and is proportional to the squared current I
2
flowing inside the torus, as well as, by the action of the internal pressure inside TPB.
The expressions for both components of the force were obtained in studies [4]. Using
them, one can approximately write the equation for R(t) in the form of Eq. 7.2.
M 0
d
2 R
dt 2 = 2π
2 a
2 P +
I
2
2c 2
∂ L
∂ R
(7.2)
Here, P = nkT + n e kT e is the average plasma pressure inside TPB, and L =
2π R
ln
R
a
+ 0.25
is the torus inductance, c is the speed of light.
To calculate the variation a(t), we disregard the difference of pressures at the
inner and outer envelopes of the torus boundary and use the equation for the plasma
cylinder with the longitudinal current. For the velocity inside it, we have Eq. 7.3.
ρ
dU
dt
= −
∂ P
∂r
+
1
c
[j × B] z
(7.3)
Taking into account that U (t, r = 0) ∼ = 0 and the fact that the magnetic pressure at
the boundary decreases with time, we approximately assume that the velocity varies
linearly along the radius:
91
initial parameters: I 0 = 0.7 MA, a 0 = 3.5 cm, R 0 = 6.5 cm, T e0 = T i0 = 10 eV,
n i0 = 8.3 × 10
17 cm
−3 , and U 0 = 3 × 10
7 cm/s.
For a more comprehensive understanding of TPB evolution during motion, the
posed problem in this work was gradually detailed and refined. At first, the behavior
of a homogeneous cylindrical layer in a one-dimensional cylindrical approximation
is considered. The gas is supposed to be ideal.
The calculation was carried out with Brode’s difference scheme [3] and von
Neumann artificial viscosity. As follows from the dimensionless equations and initial
conditions, the only parameter determining flow development is a 0 . The calculations
showed that at a zero value of the initial velocity u = u/u x , where u x =
√ γ kT 0 /m.
The ring-shaped structure quickly turns into an expanding solid disk in the absence of
any external forces and field inside the toroid for any initial a 0 , on which the collapse
time depends. However, as calculated studies have shown, if u(t = 0) ≥ 2, then the
internal rarefaction wave does not reach the center and the annular expansion will
continue in time. It should be noted, that this value u = 0 is less than the maximum
value u = 2(γ − 1) = 3, which according to the front of the plane rarefaction wave,
since in this case the movement is cylindrical and not self-similar.
7.3 Physico-mathematical Formulation of the Problem
of the Initial Stage of TPB Dynamics
TPB dynamics is described by variation in the two main parameters, R(t) and a(t).
The variation R(t) is determined by the radial tension created by the pressure of the
magnetic field inside the toroidal ring and is proportional to the squared current I
2
flowing inside the torus, as well as, by the action of the internal pressure inside TPB.
The expressions for both components of the force were obtained in studies [4]. Using
them, one can approximately write the equation for R(t) in the form of Eq. 7.2.
M 0
d
2 R
dt 2 = 2π
2 a
2 P +
I
2
2c 2
∂ L
∂ R
(7.2)
Here, P = nkT + n e kT e is the average plasma pressure inside TPB, and L =
2π R
ln
R
a
+ 0.25
is the torus inductance, c is the speed of light.
To calculate the variation a(t), we disregard the difference of pressures at the
inner and outer envelopes of the torus boundary and use the equation for the plasma
cylinder with the longitudinal current. For the velocity inside it, we have Eq. 7.3.
ρ
dU
dt
= −
∂ P
∂r
+
1
c
[j × B] z
(7.3)
Taking into account that U (t, r = 0) ∼ = 0 and the fact that the magnetic pressure at
the boundary decreases with time, we approximately assume that the velocity varies
linearly along the radius: