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8/13/2019 A Novel Set of Unified Maxwell Equations Describing Both Fluid and Electromagnetic Behavior
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A Novel Set of Unified Maxwell Equations Describing Both
Fluid and Electromagnetic Behavior
Richard J. Thompson and Trevor Moeller
University of Tennessee Space Institute, Tullahoma, TN, 37388, USA
This paper reviews a novel theoretical transformation of the two-fluid plasma equations into a set of Maxwells
equations, where new unified fields supplant the electric and magnetic fields, and contain both the fluid and
electromagnetic character of the plasma. The challenge to using this framework is that a knowledge of the
unified charge and unified current is presumed, which is a superposition of gasdynamic charge and current
and electromagnetic charge and current. While electromagnetic charge and current are a familiar physical
quantity, the idea of gasdynamic charge more foreign; therefore, this paper explores two numerical simulations
where the unified charge and current is postprocessed and examined; this reveals some preliminary knowledge
of the structure of the charge and current in the unified Maxwell equations.
Nomenclature
A Electromagnetic vector potential
(Subscript) species
B Magnetic induction field
E Electric field
0 Permittivity of free space
0 Permeability of free space
Electric scalar potential
Electrical conductivity
Unified charge
m Mass densityj Unified current
je Electric current
P Canonical momenta,P = U + (e/m)A
u Fluid velocity
Generalized vorticity, =P= + (e/m)B
Vorticity
Generalized Lamb vector, = ( P) u =
u= ( + (e/m)B) u
a Sonic speed
c0 Speed of light
e/m Charge-to-mass ratioh Enthalpy
I. Introduction
Previous work has revealeda novel theoretical framework wherein the equations of plasma dynamics (specifically,
the multifluid model including the full Maxwell equations) is shown to comprise a more general set of Maxwell
equations, where the new analogous electric and magnetic fields are composed of both the fluid and electromagnetic
behavior.1 This framework has been previously recognized and written about for incompressible fluid flow 2 and
compressible flow,3 and has recently been extended to plasmas.1 The major challenge imposed within this framework
is a conceptual problem: a generalization of the idea ofcharge and currentmust be introduced, since the solution
of the Maxwell equations insists on a knowledge of these charges and currents. Ideally, some intuitive, physical
understanding of these source terms must be developed in analogy to how a conceptual understanding of electric chargeand current was arrived at through nineteenth-century experimental science. The incompressible and compressible
Maxwell equations involve discovering a form offluid charge and fluid currentwhich drives the vorticity (magnetic
field) and fluid Lamb vector (electric field). A plasma broadens these source terms to plasma charge and plasma
current, which are superpositions of the fluid charges and currents and the electromagnetic charges and currents.
We previously constructed the unified Maxwell set specifically for two-fluid plasmas.1 In this paper, we introduce
the simplified Maxwell equations describing the unified behavior of the hydrodynamic and electrodynamic character of
the plasma under assumption of a strongly magnetized flow. This further simplifies the form of the unified charge and
Graduate Research Assistant, Dept. of Mechanical, Aerospace & Biomedical Engineering, 411 B. H. Goethert Pkwy, AIAA Student MemberAssistant Professor, Dept. of Mechanical, Aerospace & Biomedical Engineering, 411 B. H. Goethert Pkwy MS24, AIAA Associate Fellow
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43rd AIAA Plasmadynamics and Lasers Conference25 - 28 June 2012, New Orleans, Louisiana
AIAA 2012-329
Copyright 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
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unified current. We then present numerical simulations of compressible flow and MHD plasma to reveal the structure
of the unified source terms in the context of strongly magnetized plasmas and gasdynamics. It will be seen that the
unified charge density comprises a superposition of a few charges that propagate with the eigenvalues of the system,
which indicates that simple charge models can be constructed. The data further suggests that a first approximation to
the convective current might fashion a suitable model for the unified current in relation to the unified charge. In future
work, we hope to develop a numerical algorithm in which the Maxwell equations are solved with injected forms of
these charges and currents.
A. Magnetohydrodynamic and Two-fluid plasma models
The successful understanding, prediction and modeling of an engineering plasma relies on a sound physical model for
the coupled fluid dynamics and electrodynamics occurring in the plasma, as well as any additional behavior of impor-
tance (for example, radiative heat transfer, laser-plasma interactions or ablation physics). The coupling between fluids
and electrodynamics occurring in plasmas is known to be a very challenging mathematical problem, both analytically
and numerically.
Often, the physics of the plasma can be simplified by invoking the magnetohydrodynamic (MHD) approximation.
In the MHD framework, the electromagnetics are described by a limited subset of the Maxwell equations wherein only
diffusion behavior is permitted (i.e., no electromagnetic waves are permitted). Furthermore, an ad hoc assumption of
the macroscopic current is introduced (namely, that the conduction current dominates the convection current), and
usually the magnetic force is assumed to dominate over the electric force, since most of MHD concerns itself with
the study of strongly magnetized flows. MHD permits a variety of important waves to result (such as Alfvn and
magnetoacoustic modes), and allows for simplified analytical and computational solutions to be determined.
However, the MHD approximation is only valid for high-conductivity plasma. This can be seen by examining the
magnetic telegrapher equation, which describes the full behavior of the magnetic field,
1
c20
2B
t2 + 0
B
t+
2B= 0 (1)
Here we can see that the diffusive behavior of the magnetic field only dominates if the first time derivative quantity
vastly exceeds the second time derivative quantity. Otherwise, the magnetic field behavior will be inherently hy-
perbolic, no matter how small the second time derivative is. MHD fundamentally does away with this difficulty by
assuming that the limit of the diffusion behavior can be reached that is, that the conductivity is large enough to be
considered infinite. Of course, no true plasma possesses an infinite conductivity, but the approximation works rather
well for many laboratory and engineering plasmas.
There exists a wide variety of plasma that cannot be adequately described using the MHD model, even for engi-neering cases. The major deficiencies of the MHD model are as follows:
1. Although the MHD framework permits the existence of Alfvnic and magnetoacoustic waves, an abundance
of other modes are completely excluded; for example, the removal of the displacement current excludes the
possibility of electromagnetic waves. An example where this could be important is a plasma thruster operating
in vacuum; the vacuum region is a low-conductivity region, so the second time derivative in equation1becomes
significant, and wave propagation dominates the behavior of the electromagnetics in the vacuum region. Hence,
plasma propulsion thrusters involve transitions from high-conductivity to low-conductivity (vacuum) regions,
and the successful modeling of this disparity demands the capability to resolve both the diffusion and wave
limits of the plasma. Previous computational work has circumvented this problem by injecting field-carrying
fluid or by using experimental data in the vacuum region to correct for the wave behavior. 4, 5, 6
2. The MHD framework effectively reduces all phenomena to a single time scale. This removes unwanted micro-
scopic behavior specific only to certain species from playing any major role in the development of the overallplasma. However, engineering plasmas exist where this behavior actually may influence the overall plasma
behavior. Mathematically, this becomes a singular perturbation problem7 the MHD theory allows only a
limited part of the available phenomena to play a role, and hence cannot fully accommodate cases where the
plasma may see two-fluid effects develop.
3. MHD assumes a single-fluid behavior, which intrinsically limits the approximation to low-frequency plas-
mas. If higher-frequency phenomena is encountered, the MHD model will not provide sufficient fidelity to the
physics of when ions and electrons may react differently.
Seeking a new physical model that resolves the above mentioned deficiencies of the MHD model, we turn our at-
tention to the two-fluid (or, synonymously, multifluid) model of plasma dynamics. This model involves separate fluid
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models for each species (usually ions and electrons, although other species are permitted), which behave indepen-
dently of each other except for common electromagnetic fields, which are described using the full Maxwell equations.
This model does not suffer from the same above deficiencies as the magnetohydrodynamic model; however, the intro-
duction of the speed of light and the required timestep to properly resolve two-fluid waves incurs major computational
limitations, particularly due to the Lorentz force, which behaves as a source term in the fluid dynamic equations. This
drives the two-fluid equations to become very stiff, making it generally very difficult to solve numerically.
Hence, we can see that major problems are presented on both sides; the MHD model simplifies our treatment of
the problem, but sometimes at the expense of the true underlying physics, whereas the two-fluid model provides a
superior model for certain engineering plasmas, but the computational constraints imposed on solving the equations
produces a very stiffproblem that generally can only be solved with a great deal of difficulty, usually owing to the
presence of large source terms in the equations.
B. A unified approach to treating fluids and electromagnetics
In light of the deficiencies mentioned in the MHD model, and the present computational limitations of the two-fluid
model, it is natural to ask if another path to resolving the behavior of engineering plasmas in a general sense and yet
with computational simplicity can be achieved. In this paper, we review a new theoretical perspective of the two-fluid
equations, in which the equations can be written as a set of Maxwell equations; the electric and magnetic fields are
instead transplanted by new, more general field quantities that describe the evolution of both the electromagnetic and
fluid dynamic behavior of the plasma. Although the unified Maxwell equations lack closure, it is shown that some
limiting cases do result in an isomorphism to classical electrodynamics.
The major challenge imposed on this new set of Maxwell equations is pointed out by Jackson:8 Namely, that there
exist two limits in which the fields described by the Maxwell equations can be solved exactly, which correspond to
one in which the sources of charges and currents are specified and the resulting electromagnetic fields are calculated,
and the other in which external electromagnetic fields are specified and the motion of charged particles or currents is
calculated... . Occasionally... the two problems are combined. But the treatment is a stepwise one first the motion
of the charged particle in the external field is determined, neglecting the emission of radiation; then the radiation is
calculated from the trajectory as a given source distribution. It is evident that this manner of handling problems in
electrodynamics can be of only approximate validity. Hence, the new framework insists that if the unified fields are
to be determined, some knowledge of the source terms (the unified charge and unified current) must be possessed by
the investigator.
The implication of developing a knowledge of the source terms seems daunting. Extensive experimental testing
and numerical modeling could be demanded before analysis would yield a robust model that works in all applications.
In an effort to expose the nature of the source terms, this paper investigates the unified charge and current for two
test cases from compressible flow and MHD. While it remains our ultimate goal to develop models robust enough to
model two-fluid plasmas, developing some knowledge of the compressible flow and MHD charges and currents is a
necessary and more feasible step in revealing the nature of these source terms. Our results will show that the charges
for these systems is relatively simple, and could potentially be very feasible to model for a numerical scheme.
II. Theoretical framework of the unified Maxwell equations
A. Background
Often two physical theories can illustrate a remarkable degree of similarity in their mathematical structure. An example
is the wave equations of acoustics and electromagnetism; although these equations describe waves of totally different
physical character, the equations can be seen to be nearly identical in many cases. Another simple example is the
relationship between linear and angular kinematics; analogues may be constructed between each quantity in thesesystems. We borrow the language of Towne9 to refer to such a similarity as an isomorphism. The utility in constructing
an isomorphism between two physical theories is that well-established theorems and techniques of one field may be
correlated to the other. Another important result is that the physics of one theory may be illuminated in a novel and
insightful way by describing its analogous conditions in relation to another theory.
In this section, we expose the similarity between the two-fluid plasma model and the Maxwell equations of classical
electrodynamics. This allows us to reformulate the behavior of a plasma in terms of generalized electric and magnetic
fields, which unify the behavior of both the electromagnetics and fluid dynamics of the plasma. This results in a
set of equations remarkably similar to the Maxwell equations that describes the new unified fields, and introduces
generalized charge densities and current densities that include contributions from both the electromagnetics and fluid
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dynamics. In the limiting cases of strongly magnetized plasmas, magnetohydrodynamics, and gasdynamics, these
Maxwell equations become completely isomorphic.
B. The Maxwell equations for a plasma
Consider an isentropic, ideal plasma. Viscous and entropic terms can be added in later,1 which results in an analogous
polarization of the generalized electric field. Our derivation will largely follow that presented in Ref 1. The continuity
and momentum equations describing such a plasma are
tm,+
m,u
=
h
t+ u
h
+ a2 u =0 (2)
u
t+ u
u
= h
+
e
m
E + u
B
(3)
Isentropic thermodynamic relations have been used in equation 2to manipulate it in terms of species enthalpy h
,
species speed of sound squared,a2, and the divergence of the species velocity, u . Similar isentropic properties have
also been introduced in equation3to replace the pressure with the enthalpy.
To simplify equation3, we introduce the vector and scalar electromagnetic potentials, E = At
and B =
A. Next, we introduce the Lamb vector identity to break up the nonlinearity, u
u=u
+((1/2)u
u
),
and combine the kinetic energy gradient with the enthalpy gradient. This gives us
tu +
e
m
A + +e
m
B u = (H +e
m
) (4)
where H = h +(1/2)u u is the stagnation enthalpy. Recognizing that the first bracketed quantity is the mass-
specific canonical momenta, P= u
+
em
A, and the second bracketed term is the curl of the canonical momenta,
called the generalized vorticity, = +
em
B, and defining the mass-specific total energy = H +
em
, and finally
simplifying our notation by using =
u
, we have the following compact momentum equation:
P
t+
=
(5)
It is fruitful to consider the derivatives of equation5.We consider briefly the divergence, curl and time derivative
of this Euler equation in order to develop an analogue to classical electrodynamics:
Divergence
P
t
+
= 2
(6)
Curl
t+ =0 (7)
Time derivative2P
t2 +
t=
t
(8)
It can be shown (for details, see the appendix) that equation8can be rewritten as
t+j
a2 c2 (
e
m
B) = 0 (9)
with the vectorj
defined as
j
=
t(
1
2u
u
) +
2u
t2 a2
2u
u
h
+
1
0
e
m
je
(10)
If we replace equation 8with its modified form9, and include an equation exposing the divergencelessness of the
generalized vorticity, = P = 0, then our set of equations describing the derivatives of equation 5swells to
Div of
= 0 (11)
Divergence
P
t
+
= 2
(12)
Curl
t+
=0 (13)
Time derivative
t+j
a2 c2 (
e
m
B) = 0 (14)
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By simply rearranging equations 11through14, we end up with the following set of equations describing the
generalized quantities of the plasma:
Twofluid plasma unified Maxwell equations
No monopoles =0 (15)
Gauss law = =
2H
t P
(16)
Faraday law t
+ = 0 (17)
Ampere law
t+j
=a2 + c2 (
e
m
B) (18)
where j
is given in equation 10. These equations strongly resemble the Maxwell equations of classical electro-
dynamics. Here the generalized vorticity, = +
em
B, and the generalized Lamb vector, =
u
=
u+
em
B u
, supplant the usual magnetic and electric fields, respectively, and hence the new generalized
fields include both contributions from the fluid-dynamical and electrodynamical character of the plasma. Note that in
the context of our earlier discussion of an isomorphism, these equations are not rigorously isomorphic to the Maxwell
equations (they only resemble them). This is due to a disparity in the propagation speeds of the fluids and electromag-
netics; the true Maxwell equations expose a single speed of propagation. Therefore, equations15 through18 do not
constitute a rigorous isomorphism, but only a similarity. However, some limiting cases can be explored that expose
complete isomorphisms to the Maxwell equations, which we discuss next.
C. Limiting case: Magnetized plasma equations and Magnetohydrodynamic equations
Now that the full multifluid equations have been revealed in equations 15through18, some simplifications can be
achieved for certain cases which result in complete isomorphisms. One of the immediate simplifications we can make
is by assuming that the electric field term in the Lorentz force does not significantly affect the body force acting on
the plasma, or, synomously, the plasma is strongly magnetized. In such a case, only the je Bterm is retained in the
Lorentz force. Then equation5reduces to
u
t+
= H
=
h
+ k
(19)
If we follow the same process as outlined in the previous section, we arrive at the following simplified equations for a
strongly magnetized plasma:Strongly magnetized plasma unified Maxwell equations
No monopoles =0 (20)
Gauss law = =
2H
t
u
(21)
Faraday law
t+
=0 (22)
Ampere law
t+j
=a2 (23)
where the currentj
now takes the simplified form
j =
t(
1
2u u) +
2u
t2 a2
2
u u h
(24)
Here is the fluid vorticity, and =
u
= ( +
em
B) u
is still the generalized Lamb vector for the
plasma. Notice that we have not neglected the displacement current, E/t, here (no simplifications of EM Maxwells
equations were introduced; only the effect of the electric field on the fluid was neglected); therefore, this model still
admits a large number of waves and modes not seen in MHD. This Maxwell set is also fully isomorphic, with the
speed of sound now corresponding to the speed of light in the EM Maxwell set. These equations provide a simpler
formulation of the plasma in the special case that the electric field does not significantly influence the fluid behavior.
The magnetohydrodynamic equations may be further recovered from these equations if we take the limit of a single
species in the plasma (a single-fluid model), and fashion the electromagnetic current from an Ohms law.
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D. Limiting case: Gasdynamic equations
Removing both electric and magnetic fields reduces equation5back to the usual Euler equation for isentropic gasdy-
namics,
u
t+ u
= H
=
h
+ k
(25)
In this case, there is no electromagnetic contribution, and the usual compressible fluid flow limit is restored. We can
once again apply the process described above to determine the Maxwell equations describing gasdynamics,
Gasdynamic Maxwell equations
No monopoles =0 (26)
Gauss law ( u) = = 2H
t( u) (27)
Faraday law
t+ ( u) = 0 (28)
Ampere law
t( u) +j = a2 (29)
Notice that the charge in equation27and current in equation29are identical to those in equations21and24; we call
these thegasdynamic chargeandgasdynamic current. Since the electric charge is much more familiar, part of the task
of this paper is to expose the nature of the gasdynamic charge. Notice that we have reduced the number of species to a
single-fluid approximation as well. This set of equations describes the evolution of the fluid vorticity and the fluid
Lamb vector u. A very similar set of equations was introduced previously by Kambe.3
III. Analysis of unified source terms
The three Maxwell equation sets we introduced in the previous section reveal how magnetized and two-fluid
plasmas and gasdynamics may be evolved in time in terms of their unified fields. Two-fluid plasmas admit the greatest
number of modes, and the simpler case of a strongly magnetized plasma admits most of the two-fluid modes if multiple
species are retained. The MHD model may be recovered by reducing the strongly magnetized case to a single-fluid
case and neglecting the displacement current. Finally, removing all plasma modes and only retaining the gasdynamic
modes reduces the fields to just the vorticity and Lamb vector of the fluid.
In this research, we have taken advantage of the simplicity of the strongly magnetized form of the equations and
the gasdynamic form to study the unified charges and currents. To study the particular form of the source terms,two investigations were undertaken. In the first, finite volume numerical solutions of the Euler equations in a shock
tube were determined, and the fluid contributions of the gasdynamic charge and current were postprocessed. In the
second investigation, finite volume numerical solutions of the Brio and Wu electromagnetic plasma shock problem
were executed and the fluid and electromagnetic contributions to the source terms were determined. Although there
remains much more analysis that can be done (particularly for locating the difference in the unified charge by including
or excluding two-fluid effects), this provides an initial investigation into the structure of the charges. This analysis
provides a first step into exploring the nature of unified charges and currents for the above sets of Maxwell equations.
A. Gasdynamic charge and current
A shock-capturing finite volume solver was used to solve the approximate Riemann problem at each volume interface
in a one-dimensional mesh using an explicit Roe scheme. The Euler equations were solved in their conservation form,
t
u
E
+
x
u
uu + P
(E + P) u
=
0
0
0
(30)
The initial conditions were set up to imitate the Sod shock tube problem,10
u
P
Left
=
1
0
1
,
u
P
Right
=
1/8
0
1/10
(31)
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The initial discontinuity was centered in the computational mesh. The gas was assumed to be calorically perfect and
ideal, and the gasdynamic energy was taken as E = P/( 1)+ (1/2)mu2. The gas properties were taken as air at STP.
Simulations were run using a much lower Courant number than usual (CFL = 0.05), and used 500 volumes. This
provided a means of computing the gasdynamic charge as per equation 27. Using a much smaller Courant number
allowed us to construct movies of the charge for analysis.
0
0.2
0.4
0.6
0.8
1
-4 -2 0 2 4Density,
Velocity(X),Pressure(nondimensional)
Position (nondimensional)
Sod Gasdynamic Shock Tube Solution
DensityVelocity (X)Pressure
Figure 1. The numerical solution of the Sod shock tube problem.
Figure1shows the usual solution of the Sod shock
tube problem after a nondimensional time of t = 1.6.
Several nonlinear shock structures have resolved at dif-
ferent speeds, which correspond to the eigenvalues of
the system of conservation equations written in equa-
tion30. Figure2shows the results of the gasdynamic
charge at nondimensional time t= 1.6; individual con-
tributions from each term (enthalpic and spatiotemporal)
from equation26are visible. In Figure3,the left subfig-
ure shows a pseudocolor contour plot of the gasdynamic
charge against nondimensional position and nondimen-
sional time, and the right subfigure shows a detail of the
initial dispersion of the gasdynamic charge. Three pri-
mary charge structures are visible in the contour plots of
Figure3. There is a compression charge moving to the
right; this charge is narrow and strongly negative. The
contact discontinuity is associated with a smeared dipole
that is much wider than the compression charge. Fi-
nally, there is a rarefaction charge moving left corresponding to the expansion seen in the density profile (blue dashed
line in Figure2). The rarefaction charge appears as a series of small charges (much less in magnitude than the contact
discontinuity charge or the compression charge), shown in the inset in Figure 2.
Examination of the gasdynamic charge exposes a unique characteristic of its nature gasdynamic charge is asso-
ciated strongly with gasdynamic wave structure. This is anticipated by the analytical form of the gasdynamic charge
in equation27and the form of the current in equation 24.It is clearly visible in Figure2that the gasdynamic charges
follow the waves in the shock tube, and different waves correspond to different charge structure. The rarefaction wave
traveling to the left is comprised of a small-magnitude charge comb, or a series of charge teeth, and is shown in more
detail in the inset in Figure2.Figure4reveals more detail in the contact discontinuity dipole charge the compression
charge.
The structure of the shocks exposed by this analysis is encouraging. The entire shock tubes dynamics can becomposed of a superposition of a simple number of charges that march through the shock tube with known speeds
corresponding to the eigenvalues of the system of equations30,since their contributions will have nonzero magnitude
only near some changing structure in the flow. Since the eigenvalues are prerequisite to constructing a finite volume
solution of the equations anyway, the speeds of the charges propagating in the shock tube are already known. The
structure of each individual charge is less clear. The salient features of each charge must be determined as some
function of known parameters describing the problem at hand. Examination of equation 27 (the form of the gasdynamic
charge) also matches what we expect: magnitudes the enthalpic derivative 2Hand spatiotemporal derivativetu
only occur at the waves, so the results presented confirms our intuition that the charge follows the wave structure.
The gasdynamic current includes more contributions than the charge, as seen in equation 24, but still reveals
a simple form. Figure 5 shows the contributions of each term to the current, and the total gasdynamic current at
different simulation times, respectively. A significant aspect of solving the unified Maxwell equations involves having
a knowledge of the source terms in order to solve for the unified fields. A first approximation to the gasdynamic current
may be guessed as a convective term, j = u, where is the gasdynamic charge. Figure 5 exhibits a comparisonof this simple current model (the thin solid black line) to the actual gasdynamic current (the thick solid red line)
determined from the simulations. The magnitudes do not match for the compression and contact surface charges,
and the rarefaction current departs signifcantly from the convective model, but the general agreement in curve shape
suggests that this simple model might be a valuable first approach for modeling the current from the charge.
The solution of the gasdynamic equations30for the gasdynamic charge leads us to an important conclusion: that
the structure of the gasdynamic charge and current are coupled to the gasdynamic wave structure. Hence, we can
begin to start a foundation of knowledge for gasdynamic charge and current by thinking about the wave structure of
the gas. The presence of a gasdynamic charge or current inducesa wave structure in the unified Maxwell equations.
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-600
-400
-200
0
200
400
600
-4 -2 0 2 4
GasdynamicCharge(nondimension
al)
Position (nondimensional)
Gasdynamic Charge in Shock Tube
Density (Scaled)Enthalpic Term
Spatiotemporal TermTotal Gasdynamic Charge
-4
-2
0
2
4
-2 -1.5 -1 -0.5 0 0.5
Figure 2. Gasdynamic charge present in the Sod shock tube. The waves (visible in the density profile in blue dashes) correspond to the
charges in the system. A superposition of strong dipole charges due to separate terms constitutes the compression shock on the far right.
The contact surface moves with a dipole charge. The rarefaction charge has been shown in more detail in the inset figure. A series of charge
teeth are present, and covers the range over which the rarefaction wave occupies in the gas.
The structure of each charge varies for each wave.
B. MHD charges and currents in strongly magnetized flows
The previous section explored the case of gasdynamic charge and current. In the case of a plasma, Alfvn and
magnetoacoustic modes are frequently important. In order to analyze the unified charge and current associated with
these modes, the Brio and Wu electromagnetic plasma shock problem11 was considered. The equations of a single-
fluid plasma can be rewritten in a strong conservation term, which couples the electromagnetics to the fluid entirely
through the flux and conservation vector instead of introducing source terms.6 This allows to write:
t
u+ Semx /c
20
v+ Sem
y /c2
0w + S
emz /c
20
E+ uem
bx
by
bz
ex
ey
ez
+
x
ui
uu emxx + p
uv
emxy
uw emxz
(E+ p) u+ Semx
0
ez
ey
0
bz
by
=
0
0
00
0
0
0
0
jx
jy
jz
(32)
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Figure 3. (Left) A visualization of the time dependency of the charge. The far right line corresponds to the compression. The dipole travels
with a relatively constant strength. The rarefaction charge looks like ripples, and widens over time. (Right) A closer look at the initial onset
of the charges resolved in the computational domain.
-150
-100
-50
0
50
100
150
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5GasdynamicCharge(nondimension
al)
Position (nondimensional)
Structure of Contact Discontinuity Charge
-200
-150
-100
-50
0
50
100
2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4GasdynamicCharge(nondimension
al)
Position (nondimensional)
Structure of Compression Charge
Figure 4. A closer look at the charge structures in the Sod shock tube in Figure1. (Left) The contact surface charge. (Right) The
compression shock charge.
whereSem is the Poynting electromagnetic propagation vector, em is the Maxwell stress tensor, and uemis the electro-
magnetic energy density.6 Taking the limit asc0 will reduce this system of equations to a magnetohydrodynamic
simulation with magnetic diffusion. The initial conditions posed by Brio and Wu for a single species are
u
v
w
Pbx
by
bz
ex
ey
ez
Left
=
1
0
0
0
10.75
1
0
0
0
0
,
u
v
w
Pbx
by
bz
ex
ey
ez
Right
=
0.125
0
0
0
0.10.75
1
0
0
0
0
(33)
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-600
-400
-200
0
200
400
600
-4 -2 0 2 4
GasdynamicCurrent(nondimension
al)
Position (nondimensional)
Gasdynamic Current in Shock Tube
Density (Scaled)Term 1Term 2Term 3Term 4
Total Gasdynamic CurrentPredicted Current (rho*u)
-4
-2
0
2
4
-2 -1.5 -1 -0.5 0 0.5
Figure 5. Gasdynamic current in the Sod shock tube. The current closely follows the charge in structure. The black line indicates a
possible model for the current using a convective form, j = u, where is the gasdynamic charge. Although some scaling issues are clearly
visible, the convective model seems to resolve the structure in the correct locations. The current model departs for the rarefaction current.
-1.5
-1
-0.5
0
0.5
1
-0.4 -0.2 0 0.2 0.40
0.2
0.4
0.6
0.8
1
MagneticfieldinY(nondimensional)
Density(nondimensional)
Position (nondimensional)
Brio and Wu Plasma Shock Solution
FR
SC
CD
SS FR
Analytical DensityCalculated DensityAnalytical Pressure
Calculated Pressure
Figure 6. The solution to the Brio and Wu shock problem
after ten light transit times. The density and pressure profiles
are shown; they match with good agreement to the analytical
solution. The reason for the discrepancy is that the numerical
scheme solved the equations including the displacement cur-
rent, which MHD neglects.
The system of equations32 was solved using a finite vol-ume Roe scheme implemented in an explicit solver. For
Alfvnic and magnetoacoustic charge, a single species and the
full Maxwell equations were solved. Movies of the source
terms were postprocessed for analysis. Figure6 presents the
simulation results after ten light transit times in the computa-
tional domain. Figure 7 presents calculations of the unified
charge (equation21) at the same fixed time. Figure 8shows a
comparison of thex component of current (equation24) in the
shock tube at this time. Figure8also presents a comparison of
a convective current model and the actual current.
The same characteristic relationship between the charges
and waves are observed for the results here, except that more
waves are admitted to this sytem of equations, and so morecharges appear. In the single-fluid magnetohydrodynamic sys-
tem, we expect seven (two fast, two Alfvn, two slow, and a
gasdynamic) waves to be captured. The magnetoacoustic (fast
and slow) waves can manifest as shocks or as rarefactions in
the flow. In Figure6,we see (from left to right) a fast rarefac-
tion (FR), a slow compound wave (SC), a contact discontinuity (CD), a slow shock (SS), and a fast rarefaction wave
(FR) still propagating in the shock tube.
Figure7 makes clear the connection between the different waves captured at this time and their correlation to the
unified charges. Since the basic physics involved is fundamentally different, new structures for the charge emerge.
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-600
-400
-200
0
200
400
600
-0.4 -0.2 0 0.2 0.4
M
HD
Charge(nondimensional)
Position (nondimensional)
MHD Charge in Shock Tube
Density (Scaled)Enthalpic Term
Spatiotemporal TermTotal MHD Charge
-4
-2
0
2
4
-0.2 -0.15 -0.1 -0.05 0
Figure 7. The plasma charge in the computational domain at the same time as shown in Figure 6. The charge is considerably more
complicated than the gasdynamic form for the Sod shock tube. Still, similar charge structures are observed in the plasma case.
It is still clear that the charges are tied to the waves. The slow compound wave corresponds to a charge structure
shown in the inset of Figure7. While the charges and waves are different from the gasdynamic case, the correlation
between structure and wave remains similar. The unified current presented in Figure8 demonstrates similarity again
to the gasdynamic case, except that since the waves of the system are different, the resulting current is as well. An
immediate question is if the same simple convective model for the gasdynamic current, j = u, is still valid. Figure8
presents a comparison between the predicted plasma current using the simple model j = ufor the x direction. Strong
currents (contact discontinuity and shocks) matches the shape well, although not always the magnitude; the smaller
currents due to rarefaction waves (the two insets in Figure 8) depart from the simple convective model.
IV. Conclusions
A theoretical recasting of the equations for a two-fluid plasma reveal a set of equations remarkably similar to the
Maxwell equations. A mathematical difficulty presents itself here, since the equations given in15 through18are not
strictly isomorphic to the Maxwell equations. We hope to explore this issue in closer detail in the future. However,
simplifications of the plasma will reduce the set to a system of equations that is truly isomorphic to the Maxwell
equations. The primary simplified models we have endeavored to study in this paper are that of a strongly magnetized
plasma where the electric field effect on the fluid is small, and pure gasdynamics that is not influenced by the presence
of electromagnetic fields.
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-600
-400
-200
0
200
400
600
-0.4 -0.2 0 0.2 0.4
MHD
Current(nondimensional)
Position (nondimensional)
MHD Current in Shock Tube
Density (Scaled)Total MHD Current
Predicted (rho*u) Current
-4
-2
0
2
4
-0.2 -0.15 -0.1 -0.05 0
-4
-2
0
2
4
0.3 0.32 0.34 0.36 0.38 0.4
Figure 8. The plasma current in the computational domain after ten light transit times. The current features much of the same structure as
the charge here. The black solid line represents a simple convective model for the current, j = u, whereis the plasma charge. This simple
model shows some agreement to the current, although there are clearly scaling issue and the convective model departs for rarefaction cases.
For both strongly magnetized plasma and gasdynamics, the charge and current reduces to the same form. Twoparticular cases were investigated to reveal the structure of the charge and current: a gasdynamic Sod shock tube and a
Brio and Wu shock tube. The resulting solutions were postprocessed for the unified charge and current. An important
conclusion of this analysis is that the charges are directly tied to the waves in the system; therefore, they propagate as
single charges (or, in some cases like rarefaction waves, collections of similar charges) with the eigenvalues of the sys-
tem of equations. Since finite volume methods generally rely on a knowledge of the eigenvalues, this is encouraging,
because the charges could be modeled using the same information needed for finite volume solutions.
The structure of the charges for these two test cases has been illustrated and examined. Dipole charges correspond
to discontinuities, and rarefaction waves exhibited combs of charges. The current behaved in a very similar manner to
the charge. An initial test model of a convective current using the unified charge was compared to the actual current;
there is still more work needed in studying the magnitude of this model, but as a preliminary first step, it does seem to
indicate the proper structure of the current. A better understanding of scaling, and particularly the rarefaction charges,
is required before this model could sufficiently describe the current to an accurate degree.
We ultimately hope that this theoretical framework might allow a new approach to predicting plasmas for engineer-ing utility. A substantially better understanding of the charge and current is prerequisite to this goal. Robust models of
the charge and current which depend on functions of the known parameters of the problem must be developed before
a numerical solution of the unified Maxwell equations is attempted. If the isomorphism between the plasma equations
and Maxwell equations is rigorous, then solution techniques common in electromagnetism can be applied with equal
validity to the solution of the plasma equations. Particularly, since the eigenvalues of the Maxwell set will correspond
to the speed of wave propagation, finite volume methods can be implemented directly. Charges and currents can be
applied using empirical models. At present, these solution techniques have not been tested yet, since models of the
charge and current must be known; that is why this paper has focused on this development. Also, it should be pointed
out that equations15through18are not rigorously isomorphic; this problem needs to be solved as well before Maxwell
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solvers can be applied to these equations for full two-fluid plasmas.
Appendix: Derivation of the unified Ampere law
We have postponed the mathematical details of the derivation of the unified Ampere law until now. This derivation
reviews the approach given in Ref. 1.
Starting with equation8,we want to manipulate it to achieve the form of equation 9.To do so, we must re-introduce
the definitions of the canonical momenta, P
=u
+
e
m Aand the total energy, = H +
e
m . This yields
2u
t2 +
e
m
2A
t2 +
t=
tH
e
m
t (34)
t+
2u
t2 =
tH
+
e
m
t
2A
t2
(35)
where we have just shuffled the terms around in equation35.If we substitute the electric field,E, for 2At2
, then
the last term becomes (e/m)E/t; we can substitute the electromagnetic Ampere law in for the time derivative ofE
here to obtain
t+
2u
t2 =
tH
+
e
m
je
0+ c20 B
(36)
We have succeeded so far in recovering a curl of the magnetic field; now we are left with the task of introducing a curlof the vorticity in order to realize an effective Ampere law for both the vorticity and magnetic field. This can be done
by breaking up the gradient of the stagnation enthalpy, H=h
+ k
. We have
t+
2u
t2 =
th
tk
+
e
m
je
0+ c20 B
(37)
If we take the gradient of the continuity equation in the form given in equation2,we have the following identity:
th
=
u
h
a2
u
(38)
and applying the vector identity =u=
u
2u
, we recover the curl of the vorticity,
t
+2u
t
2 = u h + a
2
2u+ a2
t
k+
e
m
je
0
+ c20e
m
B (39)Simply moving the remaining terms to the left-hand side, we have
t+
2u
t2 a2
2u
u
h
+
tk
+
e
m
je
0
= a2 + c
20
e
m
B
(40)
or, more simply, just as
t+j
=a2 + c20
e
m
B
(41)
with
j
=2u
t2 a2
2u
u
h
+
tk
e
m
je
0(42)
Thus, we have equation9from some manipulation of equation8.It is important to realize that this is not rigorously an Ampere law, since two curls appear that cannot be combined to
yield the unified vorticity,
, due to the difference in propagation speeds,a and c0. A true set of Maxwell equations
would close this term to a curl of the unified vorticity,
, with a corresponding speed that would allow a
separate treatment per species. This is an important observation, because constructing a finite volume numerical
scheme capable of solving these equations demands a knowledge of the eigenvalues of the system. If the unified
Maxwell equations are not rigorously isomorphic to their electromagnetic form, then we cannot directly apply finite
volume techniques for the electromagnetic equations with equality on the unified set. We hope to explore this issue
in greater detail in a future paper. Notice that the system of equations20through23 and26through29 do constitute
rigorous isomorphisms to the Maxwell equations, since only a single speed is present.
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References
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