This article is an extension of Don Scott's work on modelling Birkeland currents. Search for his articles here. These Birkeland currents (BCs) are cosmic low beta plasma discharges delivering electric current along filaments in space. We recall that $\beta$ is the ratio of the plasma thermal pressure $p=Nk_B T$ with the so-called "magnetic pressure" $B^2/2\mu_0$. So low beta means that so-called magnetic pressure is the domainant force in the interaction.
Professor Scott's model is derived from Maxwell's equations and Lorentz' force law. Our goal is to reconsider his model in light of Ampere's electrodynamics. Thus the subject of this article is ABC, or Ampereian Birkeland Currents.
Ampere's Force Law
Ampere's investigations in 1820-1826 led to his proposing a force law between two current elements $I_1 ds_1$ and $I_2 ds_2$. The current element has a scalar $I$ which represents the net current intensity, and $ds$ is a differential vector element representing arc length along the circuit. The magnitude of $I$ typically represents the current intensity, while the magnitude of $ds$ represents the current velocity.
Alternatively we could follow Weber and write $I ds = q dv$. That is, an electric current element is essentially equivalent to charge in motion, i.e. a charge $q$ in velocity $v$.
Ampere determined that the force between two current elements was radial and proportional to the current intensity and to current velocities. Thus the force $F$ was proportional to $I_1 I_2 ds_1 ds_2$. Ampere discovered the inverse square distance proportionality. And moreover determined the precise coefficient of proportionality. In cgs units Ampere's force law is expressed:
$$ F_{~I_2 ds_2 ~~ \text{on} ~~ I_1 ds_1 } =F_{21}= - I_1 I_2\frac{\hat{r}_{12}}{r_{12}^2} [ 2(ds_1 \cdot ds_2) - 3 (\hat{r}_{12}\cdot ds_1) ( \hat{r}_{12}\cdot ds_2) ] =-F_{12} .$$
The following code uses Ampere's formula:
def dot(v1, v2):
vx1, vy1, vz1 = v1
vx2, vy2, vz2 = v2
return vx1*vx2+vy1*vy2+vz1*vz2
def scalar(a, v1):
vx1, vy1, vz1 = v1
return [a*vx1, a*vy1, a*vz1]
## Ampere's Force
def F(state1, state2):
x1, y1, z1, vx1, vy1, vz1 = state1
x2, y2, z2, vx2, vy2, vz2 = state2
v1=[vx1,vy1,vz1]
v2=[vx2,vy2,vz2]
I1=dot(v1,v1)**0.5
I2=dot(v2,v2)**0.5
rho_12 = ((x2-x1)**2+(y2-y1)**2+(z2-z1)**2)**0.5
rhat12 = scalar(rho_12**-1, [x1-x2, y1-y2, z1-z2])
coefficient = -I1*I2*(2*dot(v1,v2)-3*dot(rhat12, v1)*dot(rhat12, v2))*(rho_12)**-1
return [x1,y1,z1]+scalar(coefficient, rhat12)
s0=[0,0,0,.1,.7,0]
s1=[1,0,0,0.9,0,0]
print(F(s0,s1))
print(F(s1, s0))
Ampere's force law has several predictions.
- Parallel collinear current elements repel;
- Reversing a parallel colinear current attracts by 2/3rd of the repelling force;
- Parallel adjacent current elements attract;
- Parallel oppositely oriented current elements repel.
These can all be tested via various samples.
s0=[ 0,0,0, 0,0,1 ]
s1=[ 0,0,2, 0,0,1 ]
s3= [ 0,0,3, 0,0,-1 ]
print(F(s0,s1))
print(F(s1,s0))
print(F(s0,s3))
print(F(s3,s0))
# Parallel Adjacent Current Elements Attract
s2=[ 1,0,0, 0,0,1]
print(F(s0, s2))
What is a Birkeland current?
Here we imagine a direct current through space which is being conducted in a plasma cylinder. We cannot assume that the cylinder is rigid and strictly right angled throughout, i.e. the Birkeland currents will possibly be radially contracting. We do not wish to assume rotational symmetry around the central axis, although it is convenient in some settings. For example, two doubly infinite parallel filaments will attract/repel depending on their relative orientations. This interaction contributes to the internal potential energy, and the filaments will attract/repel towards a lower energy state.
Note: in the above right hand image, the force between the Birkeland currents is probably computed according to Coulomb's formula, and not Ampere's. So we would expect the force to differ from that graph.
We are trying to find the mathematical equations for the Birkeland currents. We model the currents as a cylindrical configuration of plasma ions in motion, and deliverying a sustained direct current. Therefore we want to know something about the distributions $j$ of electrical charges in motion satisfying:
$$\int_{area} \mathbf{j} \cdot \mathbf{n} dS=I=constant,$$ where the integral is defined over a two-dimensional cross-section of the cylinder.
Now we follow Scott's analysis, namely that the Birkeland currents are ideally suited to satisfy the minimum total potential energy principle. In otherwords, the stable ground state of a Birkeland current is that with minimal internal energy. In the Ampere perspective, we have charges in motion and the primary dominant force is Ampere's. Now we must proceed cautiously: if we follow Scott, then the primary force is $F=q(E+v \times B)$ and the internal potential energy is measured by the force $j\times B$. Thus Scott's interpretation of "force free field aligned currents" leads to the equation $$j\times B =0$$ throughout the current. Moreover Maxwell's equation says $\nabla \times B = \mu j$, and therefore $$(\nabla \times B) \times B =0.$$ This leads to the hypothesis ("ansatz") that $$\mu j = \alpha B.$$ Moreover Scott discovered that if we assume $B_r=0$, then we find $B_z$ satisfies a zero order Bessel equation in the variable $r$.
The above gives an interpretation of the Birkeland current from the standard Lorentz force and Maxwell equation perspective. But we are proceeding with strictly Ampere's force law as the only force acting on the current. Therefore we must carefully think about the meaning of minimal total potential energy in the current.
[To be continued]