Analytical Model of Fully Ionized Plasma Flow in

Axisymmetric Magnetic Nozzle

Yoshiki Takama ∗ and Kojiro Suzuki †

The University of Tokyo, Kashiwa, Chiba, 277-8561, Japan

The analytical model of the axisymmetric magnetic nozzle flow is proposed on the basisof magnetohydrodynamics(MHD) in the present work. The developed model is describedin quasi one dimensional form for simplicity of the analysis. This model implies that thegeometric throat of the channel is not always identical to the point where Mach number isone. Then, the comparison with the experimental results is made. The present analyticalmodel successfully describes the acceleration of the plasma flow from subsonic to supersonic,though qualitative agreement with the experimental results is not satisfactory. We useJ = σ(E + u ×B) as Ohm’ s law in place of the ideal MHD assumption. The accelerationprocess in the present magnetic nozzle model shows the same tendency as that in idealMHD, when we consider perfectly-ionized plasma and extremely large electric conductivity,because such situation is very close to that of the ideal MHD. The present model is expectedto be quite useful in the case of weakly-ionized plasma flows, which are often seen in plasmawind tunnels.

Nomenclature

a Sound velocityB Magnetic field vectords Linear element vectordS Surface element vectorE Electric field vectorh Height of the streamtubehe Hall parameterJ Current density vectorM Mach numberne Electron number densitynj j-component of normal vectorPij Stress tensor including magnetic pressurep PressureT Temperatureu Axial velocityvj j-component of velocity vectorx Axial coordinatey Radial coordinateγ Specific heat ratioθ Angle between x-direction and magnetic flux vectorµ Permeabilityρ Densityσ Electric conductivity

∗Graduate student, Department of Advanced Energy, The University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba, 277-8561, Japan, Student Member AIAA

†Associate Professor, Department of Advanced Energy, The University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba,277-8561, Japan, Member AIAA

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φ Angle between x-direction and fluid velocity vectorSubscriptw Channel boundary

I. Introduction

Magnetic nozzle is defined as the flow channel which accelerates the plasma flow by magnetic field. Itmakes the boundary of the flow channel nonintrusively by its magnetic pressure. Compared with the

acceleration by a solid nozzle, such nonintrusive acceleration leads to the decrease in the heat loss. Anotheradvantage is flexibility. The control of the magnetic field makes it possible to change the properties of theplasma flow at the exit. Therefore, higher thermal efficiency and wider operation range are expected whenit is applied to the plasma thrusters and the plasma wind tunnels.

Inutake et al. applied the convergent-divergent(C-D) magnetic field as shown in Fig.1(b) to the magne-toplasmadynamic(MPD) thruster and investigated the flow characteristics.1 As a result, they successfullygenerated the supersonic plasma flow. However, this phenomenon is so complicated and careful considera-tion is required for its understanding. The phenomenon varies completely, depending on whether the plasmabehaves as a particle or as a fluid, as described below.

Figure 1. Particle or fluid entering convergent-divergent magnetic field

First, let us consider the situation where only one charged particle enters C-D magnetic field, as shownin Fig.1(a). If magnetic field is strong enough, the charged particle winds around the magnetic field line,and its guiding center moves along the magnetic field line. In this case, C-D magnetic field is equivalent toa magnetic mirror,2 where both the particle energy and the magnetic moment are conserved:

v2‖ + v2

⊥ = c1, (1)

v2⊥B

= c2, (2)

where c1 and c2 are positive constants. v‖ and v⊥ are the velocity component parallel and perpendicularto the magnetic field line, along which the guiding center of the charged particle moves, respectively. UsingEqs.(1) and (2), the following equation is derived:

v2‖ = c1 − c2B. (3)

Equation (3) shows that v‖ decreases when B increases, and vice versa. The velocity component parallelto the magnetic field line varies by the magnetic field strength. Therefore, when the particle enters C-D magnetic field, v‖ decreases toward the position E shown in Fig.1(a), reaches the minimum there, andincreases again, if the charged particle has enough energy to pass the position E, c1 > c2BE . Consequently,

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it is just that the velocity component parallel to the magnetic field line is symmetrical to the position E,and it is not accelerated through C-D magnetic field. If c1 < c2BE , the particle cannot pass through C-Dmagnetic field.

The above-mentioned description based on particle motion cannot explain the experimental results byInutake et al., that the plasma is accelerated through C-D magnetic field. Next, we consider the situationwhere the plasma fluid enters C-D magnetic field as shown in Fig.1(b), instead of a charged particle. In thiscase, the behavior of the plasma flow is described by magnetohydrodynamics(MHD). According to severalanalytical researches,3–7 the plasma flow is accelerated through C-D magnetic field if u//B. Intuitively insuch a case, it is as if the plasma fluid moved in the C-D solid wall channel. As a result, the flow would beaccelerated on the basis of the nozzle theory described by the compressible fluid dynamics. Therefore, C-Dmagnetic field is regarded as magnetic nozzle. Whether a charged particle or a fluid enters C-D magneticfield changes the phenomena completely. Experimental results by Inutake et al. seem to be explained byMHD.

Then, where does the assumption u//B come from? To clarify it, Ohm’s law is first given below:

J = σ(E + u×B). (4)

If the plasma can be treated as ideal MHD plasma (σ = ∞) and E = 0,

u×B = 0. (5)

This equation means u//B. Thus, it is found that the condition u//B assumes ideal MHD fluid withoutelectric field, which cannot be applied to general cases.

It is of much interest how the plasma fluid behaves in magnetic nozzle, if these strict assumptions arenot satisfied. However, there have been no analytical researches so far. Thus, we use eq.(4) as Ohm’s law,and consider analytically the plasma acceleration by magnetic nozzle, Considering the applications to theplasma thrusters and the plasma wind tunnels, it is meaningful to investigate analytically the behavior of thenon-ideal MHD plasma fluid in magnetic nozzle. Our final goal is to clarify the phenomena in the magneticnozzle by MHD, and predict to what extent the flow will be accelerated. As the first step in the presentstudy, the analytical modeling of the axisymmetric magnetic nozzle flow into the quasi one dimensional formis proposed. Then, the comparison with the experimental data is made.

II. Governing Equations

A. Axisymmetric Two-Dimensional Description

The flow studied here is an axisymmetric magnetic nozzle flow. The coordinates are cylindrical ones asshown in Fig.2. The direction in the flow axis is set as x, and the direction normal to x is set as y. Everyvariable is a function of x and y in this subsection, except h which is a streamtube height at x. In Fig.2,the flow streamtube at x is TP, and that at x + dx is RQ. The boundary of the flow streamtube between xand x + dx is TR. It is noted that TR is not a solid boundary. The magnetic field at T is Bw, which has adifferent direction from that of the flow velocity.

The governing equations are steady state MHD equations. The continuum, momentum, energy, and stateequations are as follows:

∫

S

ρvndS = 0, (6)∫

S

(Pijnj + ρvivn) dS = 0, (7)

p

ργ= const, (8)

p = ρRT, (9)

Pij =

(p− B2

2µ cos 2θ −B2

2µ sin 2θ

−B2

2µ sin 2θ p + B2

2µ cos 2θ

), (10)

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and Maxwell equations and Ohm’s law without hall and ion slip effects:

divB = 0, (11)rotB = µJ , (12)rotE = 0, (13)divE = 0, (14)

J = σ(E + u×B). (15)

Equation (7) means the conservation of the i-direction momentum. i = 1 and 2 corresponds to x and y,respectively. vn is the velocity component normal to the surface element, and it is positive outward from thesurface. Equations (11), (13), and (14) are unnecessary to solve since they must be automatically satisfied.

Figure 2. Coordinate system

When Eq.(6) is applied to the closed surface PQRT,[∫

S

ρvndS

]

PT

+[∫

S

ρvndS

]

TR

+[∫

S

ρvndS

]

RQ

= 0. (16)

Considering no flux (vn = 0) through the streamtube boundary TR,[∫

S

ρvndS

]

PT

+[∫

S

ρvndS

]

RQ

= 0. (17)

Since we can write:[∫

S

ρvndS

]

PT

=∫ h

0

ρ(−u) · 2πydy, (18)

[∫

S

ρvndS

]

RQ

=∫ h

0

ρu · 2πydy + d

[∫ h

0

ρu · 2πydy

], (19)

Eq.(6) is described finally as follows:

d

[∫ h

0

ρu · 2πydy

]= 0. (20)

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Likewise when Eq.(7) is applied to the closed surface PQRT, the following equation is obtained consideringno flux through the streamtube boundary TR and the relation dSTR sin φw = 2πhdh:

d

[∫ h

0

(p− B2

2µcos 2θ + ρu2

)· 2πydy

]+

[(pw − B2

w

2µcos 2θw

)(− sin φw)− B2

w

2µsin 2θw cosφw

]2πhdh

sin φw= 0.

(21)

Since the y-direction momentum is conserved because of symmetry, it is not necessary to consider.As an energy equation, the adiabatic relation is used, though the energy is actually dissipated through

viscosity and thermal conductivity, and increased by Joule heating. For analytical simplicity, those contri-butions are assumed to be negligible.

Unlike a solid nozzle flow, an equation to determine the channel boundary in the magnetic nozzle isnecessary. Equations (12), (13), and (15) give the following one:

rot(u×B) +1

µσ52 B = 0. (22)

Equation (22) means the relationship between the direction of magnetic field vector and that of fluid velocityvector. When Eq.(22) is integrated on the cross section of the streamtube (a circle with the radius h) atposition x as shown in Fig.3,

∫ ∫

Sα

rot(u×B) · dSα +∫ ∫

Sα

1µσ

52 B · dSα = 0. (23)

By applying Stokes’s theorem to the first term, and considering the vector dSα in the second term has only

Figure 3. Direction of uw and Bw

x-component,∫

sα

(u×B) · dsα +∫ ∫

Sα

1µσ

[52B]x

dSα = 0, (24)

... uwBw sin(θw − φw) · 2πh +∫ h

0

1µσ

[∂2Bx

∂y2+

1y

∂Bx

∂y+

∂2Bx

∂x2

]· 2πydy = 0, (25)

... φw = θw − sin−1

[− 1

2πhuwBw

∫ h

0

1µσ

(∂2Bx

∂y2+

1y

∂Bx

∂y+

∂2Bx

∂x2

)· 2πydy

]. (26)

Assuming the magnetic field induced by the plasma current is negligible, the magnetic field can be determinedonly by the configuration of the external coils and the electric current.

Finally, the assumptions used in this subsection are shown again. They are no viscosity, adiabatic process,thermal equilibrium, a constant specific heat ratio, and no induced electric current.

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B. Approximation into Quasi One Dimensional Form

All the variables are nondimensionalized by their reference values denoted by the subscript 0. Afterwardswe form the equations using these nondimensional variables. For example, x is nondimensionalized by x0,and we write x/x0 as just x below.

For analytical simplicity, the magnetic field is approximated to satisfy divB = 0 as follows:

Bx = e−mx2, By = mxye−mx2

, (27)

B2 = (1 + m2x2y2)e−2mx2, θ = tan−1(mxy), (28)

cos θ =1√

1 + m2x2y2, sin θ =

mxy√1 + m2x2y2

, (29)

cos 2θ =1−m2x2y2

1 + m2x2y2, sin 2θ =

2mxy

1 + m2x2y2. (30)

m is a parameter selected to reproduce the true magnetic field.The other variables are assumed to be a function of only x. Thus, Eqs.(20) and (21) are reduced as

follows:

dρ

ρ+

du

u+ 2

dh

h= 0, (31)

dp = −ρudu + Kdx, (32)

K =B2

0

µ0ρ0u20

12µ

m2h2x(2mx2 − 1)e−2mx2. (33)

By the definition of the sound velocity of gas dynamics,

dρ

ρ=

dρ

dp

dp

ρ=

dp

ρa2=−ρudu + Kdx

ρa2. (34)

Substituting Eq.(31) into Eq.(34), the following equation is finally obtained:

du

dx=

u

M2 − 1

(2h

dh

dx+

K

ρa2

), (35)

where M = u/a. If K is equal to zero, Eq.(35) is identical to the equation led by compressible flow dynamics.Equation (35) suggests that Mach number does not always correspond to one at the geometric throat wheredhdx = 0. Also, Eq.(26) yields the form:

φw = tan−1(mxh)− sin−1

[− x0

µ0σ0u0

mh

µσu

2mx2 − 1√1 + m2x2h2

]. (36)

C. Singularity Analysis

Since the above equations are ordinary differential ones with respect to x, they can be solved numerically inthe x-direction by four-staged Runge-Kutta method. The equations in the quasi one dimensional form are

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shown below:

du

dx=

u

M2 − 1

(2h

dh

dx+

K

γp

), (37)

dp

dx= −ρu

du

dx+ K, (38)

dρ

dx=

ρ

γp

dp

dx, (39)

dh

dx= tan φw, (40)

K =B2

0

µ0ρ0u20

12µ

m2h2x(2mx2 − 1)e−2mx2, (41)

φw = tan−1(mxh)− sin−1

[− x0

µ0σ0u0

mh

µσu

2mx2 − 1√1 + m2x2h2

]. (42)

Since M = 1 is a singular point, it is necessary to expand the governing equations around the singular point.In the vicinity of the singular point, we write u, p, ρ, and h as the first approximation for the small value ofthe parameter x = x− xc:

u = uc + u1x, (43)p = pc + p1x, (44)ρ = ρc + ρ1x, (45)h = hc + h1x, (46)

where the subscripts c and 1 denote the singular point and the spatial derivative with respect to x at thesingular point, respectively. At the singular point, the following equations must be formed:

u2c = γ

pc

ρc, (47)

2hc

h1 +Kc

γpc= 0, (48)

ρcucπh2 = const, (49)(12ρcu

2c +

γ

γ − 1pc

)ucπh2

c = const, (50)

pc

ργc

= const. (51)

Truncating the terms of higher than the second order in x, the substitution of the approximations Eqs.(43)-(46) into Eqs.(37)-(41) give:

u1 = u3c

−2(

h1hc

)2

+ Kc

γpc

(K1Kc− p1

pc

)

2ucu1 − γ pc

ρc

(p1pc− ρ1

ρc

)[1 +

(u1

uc+

p1

pc− ρ1

ρc

)x

], (52)

p1 = −ρcucu1 + Kc + [−(ρcu1 + ρ1uc)u1 + K1] x, (53)

ρ1 =ρc

γpcp1

[1 +

(ρ1

ρc− p1

pc

)x

], (54)

h1 = tan−1(mxchc)− sin−1

[− x0

µ0σ0u0

mhc

µσcuc

2mx2c − 1√

1 + m2x2ch

2c

], (55)

K1 =B2

0

µ0ρ0u20

12µ

m2e−2mx2c[2xchch1(2mx2

c − 1) + (−8mx4c + 10mx2

c − 1)h2c

]. (56)

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Figure 4. Approximated magnetic field

Figure 5. Comparison with approximated and true magnetic field strength on the x axis

III. Comparison with Experimental Data

The results by our analytical model are compared with the experimental results obtained by Inutake etal.1 The plasma flow is argon and is assumed to be perfectly ionized. The electric conductivity is calculatedby Spitzer-Harm formula:

σ =1.56× 10−2 · T 1.5

ln(1.23× 104 · T 1.5n−0.5e )

[1/(Ω ·m)]. (57)

First, the quantities at the singular point are calculated. Next, the space-marching calculation proceeds fromthe singular point both upstream and downstream. In the vicinity of the singular point where |M | < 0.01,Eqs.(43)-(46), (52)-(56) are used.

The parameters B0 and m are set as 0.22 and 0.0024, respectively. The magnetic field approximated byEq.(27) is shown in Fig.4. On the x axis, the approximated magnetic field strength is compared with thatin the experiment, as shown in Fig.5. This approximation gives the fair agreement.

The profiles of temperature, Mach number, and streamtube height are compared in Fig.6. Regarding h,the calculated results by our model is in good agreement with the experimental results. However regardingT and M , the calculated results are partly in agreement, and partly not. The present analytical modelsuccessfully describes the acceleration of the plasma flow from subsonic to supersonic, though qualitativeagreement with the experimental results is not satisfactory, which suggests the possible improvement of ourmodel. The singular point is located at x = 0.22 cm, which is very close to x = 0, where the magnetic fieldis the strongest.

In Fig.7, the angle between x-direction and magnetic flux vector, θw and the angle between x-directionand fluid velocity vector, φw are shown. They agree with each other almost completely. Though we don’tuse the assumption of ideal MHD, our model results in the situation where the plasma flow is parallel to themagnetic field. This is expected to be because the electric conductivity is extremely large, and as a result thesituation becomes close to that described by ideal MHD. Hypothetically, we decreased the value of electricconductivity by a factor of 0.01 and the other conditions are the same as the above case. The results of θw

and φw are shown in Fig.8 It shows that the condition u//B is not necessarily satisfied. Therefore, whenwe consider weakly-ionized plasma flow which has less electric conductivity, our way of treating Ohm’s law

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Figure 6. Comparison with experimental data

Figure 7. θw and φw

will be quite useful. However, we need some modifications of the pressure tensor term, since the neutral gasdoes not feel magnetic pressure.

Finally, we estimate the ratio of the magnetic field strength by the plasma current to that by the externalcoil, Binduced/Bexternal, and the hall parameter he. Binduced/Bexternal is estimated as follows. The plasmacurrent jp is induced by the external magnetic field.

jp = σ(E + u×B) ≈ (0, 0, σuBy). (58)

jp makes the self-induced magnetic field Bp:

rotBp = µjp, (59)

...∣∣∣∣∂Bp,y

∂x

∣∣∣∣ ≈ µσuBy. (60)

Choosing the flow channel length xR, 30 cm as a representative scale and estimating Binduced/Bexternal atx = 0,

Binduced

Bexternal≈ Bp,y

By≈ µ(σu)x=0xR = 0.11. (61)

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Figure 8. φw and θw in the hypothetical situation

Figure 9. The effect of induced current and hall parameter

If Binduced/Bexternal is much less than one, the influence of the induced magnetic field is negligible. In ourcase, the influence is not so large, however it should be taken into account in a stricter analysis. On theother hand, Hall parameter is defined as:

he =σB

nee. (62)

For Ohm’s law to have the form Eq.(15), hall parameter has to be much less than one. he, which is calculatedfrom the results shown in Fig.6, is shown in Fig.9. It is found that the hall effect has to be taken into accountin the future.

IV. Conclusion

Magnetic nozzle is a promising device for plasma thrusters and plasma wind tunnels. In the presentwork, the analytical model of the axisymmetric magnetic nozzle flow is proposed on the basis of MHD Thedeveloped model is described in quasi one dimensional form for simplicity of the analysis. This model impliesthat the geometric throat of the channel is not always identical to the point where Mach number is one.Then, the comparison with the experimental results is made. The present analytical model successfullydescribes the acceleration of the plasma flow from subsonic to supersonic, though qualitative agreement with

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the experimental results is not satisfactory, which suggests the possible improvement of our model. We useJ = σ(E + u ×B) as Ohm’ s law in place of the ideal MHD assumption. The acceleration process in thepresent magnetic nozzle model shows the same tendency as that in ideal MHD, when we consider perfectly-ionized plasma and extremely large electric conductivity, because such situation is very close to that of theideal MHD. The present model is expected to be quite useful in the case of weakly-ionized plasma flows,which are often seen in plasma thrusters and plasma wind tunnels.

V. Acknowledgment

This work is supported by Grand-in-Aid for Scientific Research No.18·11672 of Japan Society for thePromotion of Science. The primary author is supported by Research Fellowships of the Japan Society forthe Promotion of Science for Young Scientists.

References

1Inutake, M. Hosokawa, Y., Sato, R., Ando, A., Tobari, H., and Hattori, K., “Improvements of Flow Characteristics foran Advanced Plasma Thruster,” Transactions of Fusion Science and Technology, Vol. 47, No.1T, 2005, pp. 191-196

2Gurnett, D. A. and Bhattacharjee, A., Introduction to Plasma Physics with Space and Laboratory Applications, Cam-bridge University Press, 2005, pp. 39-45

3Chubb, D., “Fully Ionized Quasi-One-Dimensional Magnetic Nozzle Flow,” AIAA Journal, Vol. 10, No. 2, 1972, pp.113-114

4Liffman, K., “An Analytic Flow Solution for YSO Jets,” Publications of the Astronomical Society of Australia, Vol. 15,No. 2, 1997, pp. 259-264

5Schoenberg, K. F., Gerwin, R. A., Moses Jr., R. W., Scheuer, J. T., and Wagner, H. P., “Magnetohydrodynamic flowof magnetically nozzled plasma accelerators with applications to advanced manufacturing,” Physics of Plasmas, Vol. 5, No. 5,1998, pp. 2090-2104

6Wang, Z. and Barnes, C. W., “Exact solutions to magnetized flow,” Physics of Plasmas, Vol. 8, No. 3, 2001, pp. 957-9637Arefiev, A. V. and Breizman, B. N., “MHD scenario of plasma detachment in a magnetic nozzle,” Physics of Plasmas,

Vol. 12, 2005, 043405

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