a novel set of unified maxwell equations describing both fluid and electromagnetic behavior

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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American Institute of Aeronautics and Astronautics

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


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


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)


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)


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)


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)


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)


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)


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)


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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pp. 010702.2Marmanis, H., Analogy between the Navier-Stokes equations and Maxwells equations: Application to turbulence, Physics of Fluids,

Vol. 10, No. 6, 1998, pp. 14281437.3Kambe, T., A new formulation of equations of compressible fluids by analogy with Maxwells equations, Fluid Dynamics Research,

Vol. 42, 2010, pp. 455502.4Cramer Kenneth, R. and Pai, S.-I.,Magnetofluid Dynamics for Engineers and Applied Physicists, Scripta, 1973.5

Rooney, D., Moeller, T., Keefer, D., Rhodes, R., and Merkle, C., Experimental and Computer Simulation Studies of a Pulsed PlasmaAccelerator, AIAA Joint Propulsion Conference, 2007.

6Li, D., Merkle, C., Scott, W. M., Keefer, D., Moeller, T., and Rhodes, R., Hyperbolic Algorithm for Coupled Plasma/Electromagnetic Fields

Including Conduction and Displacement Currents,AIAA Journal, Vol. 49, No. 5, 2011, pp. 909920.7Yoshida, Z., Mahajan, S. M., and Ohsaki, S., Scale hierarchy created in plasma flow, Physics of Plasmas, Vol. 11, No. 7, 2004, pp. 3660

3664.8Jackson, J. D.,Classical Electrodynamics, John Wiley & Sons, Inc., 3rd ed., 1999.9Towne, D. H.,Wave Phenomena, Addison-Wesley Publishing Company, 1967.

10Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, Journal of Computational

Physics, Vol. 27, No. 1, 1978, pp. 131.11Brio, M. and Wu, C. C., An Upwind Differencing Scheme for the Equations of Ideal Magnetohydrodynamics, Journal of Computational

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a novel set of unified maxwell equations describing both fluid and electromagnetic behavior

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