1. Electric field by pick-up ions and electrons
M. Yamauchi, E. Behar, H. Nilsson and M. Holmstrom
Swedish Institute of Space Physics, Kiruna
Observations by the Rosetta Plasma Consortium (RPC) ICA instrument indicates strong azimuthal deflection of the solar wind. We tried to understand this by modeling the E-field caused by the pickup ions, by assuming quasi-2D (ignoring the X component, where X is the Sun direction)
(1) The start location and the guiding center location of a newly born cold ion are different by a gyroradius (rB) in the solar wind E-field (ESW) direction. This causes secondary E-field (∆Epickup) in the opposite direction to ESW.
(2) The solar wind electrons are also moved by ∆Epickup, and these electrons tries to cancel ∆Epickup in the same way as the plasma oscillation.
poster (EGU ), Thursday ( ) 1

2. Pickup ion motion by Efinal = ESW + ∆E
for T<1/ΩB: Ex + Ey
for T>>1/ΩB: Ey only 
consider only ion
consider only UV case y x
Dpickup: azimuthal charge separation 2

3. What is the difference from most simulations?
Charge determines E (e0·E = rc instead of the Ohms law)
E determines u (udrift = E/B instead of E = - uxB)
J determines B (xB = m0Jpickup instead of J = m0-1xB) 
To solve from upstream to downstream, we need some assumptions,
e.g., Only d/dx ≠ 0 (1D variation)
& Directions of B or E is constant
 ∆Epickup = – spickup/e0 instead of full solution of e0·E = rc
The result must be the same as simulations, but we can follow the causality-chain in this way. 3

4. Estimating ∆E by the pickup ions
Quasi-2D assumption:
⇒ ∆E = – spickup/e0
(s: surface charge density)
Estimate spickup/e0 ⇒ estimate ∆Epickup
Ion production/sec: dnpickup = pUVnNdx
where pUV is the ionization rate
Density change in the x direction = dnpickup/uix
Average separation = Rgyro = miEfinal/qiBfinal2
⇒ Average charge density at dx length:
dspickup = qi·(dnpickup/uix)·Rgyro
= (pUVminN/Bfinal)·(dx/uix)·(Efinal/Bfinal)
⇒ Change in the E-field over dx:
d(∆Epickup)/dx = - (pUVminN/e0Bfinal2)·(Efinal/uix) (1)
If we can assume uix = E/B
d(∆Epickup)/dx = - (pUVminN/e0Bfinal) (1)’ 4

5. Effect of the solar wind: reduce ∆E
Quasi-2D assumption:
⇒ ∆E = + ssw/e0 (note + sign!)
Estimate sSW/e0 ⇒ ∆ESW
Additional charge separation
x  x+dx:
where duey = -(qe/m)∫x[(Efinal-ESW)·dx/uSW]
⇒ Corresponding charge density:
dsSW = qe·nSW·(uey·dx/uSW)
= e0wp2·(dx/uSW)·∫x[(ESW-Efinal)·dx/uSW]
where wp2 = qe2nSW/e0me
⇒ Change in the E-field over dx:
d(∆ESW)/dx
≈ (wp2/uSW)·∫x[(ESW-Efinal)·dx/uSW] (2)
Action items:
Neither (1) or: (2) includes rSW
Should use distribution function for SW e- 5

6. Add both contributions (1) and (2)
dEfinal/dx = d(∆E)/dx
= - (pUVminN/e0Bfinal2)·(Efinal/uix) - (wp2/uSW)·∫x(∆E·dx/uSW) or
df(x)/dx = - g(x)·[1-f(x)] - (wp2/uSW)·∫x(f(x)·dx/uSW) (3)
where f(x) = ∆Efinal/ESW
g(x) = pUVminN/e0Bfinal2uix ≈ pUVminN/e0BfinalEfinal
wp2 = qe2nSW/e0me
Extreme case (a): no pickup
Extreme case (b): thin solar wind
nN = 0 (i.e., g(x)=0)  d2f(x)/dx2 = f(x)/lp2
where lp = uSW/wp ≈ 10 km oscillation
 The same as B=constant solution
nSW ~ 0 (i.e., wp2 = 0) 
dEfinal/Efinal = - g(x)dx
ln(Efinal) = -∫xg(x)dx
where∫xg(x)dx is the ratio of total picked-up ions mass upstream to the SW pointing flux 6

7. How about xB ≈ m0Jpickup?
Extreme case (c): E/B=const
dE2final/dx2 - (pUVminNuSW/e0Efinal2)·dEfinal/dx + Efinal/lp2 = ESW/lp2
How about xB ≈ m0Jpickup?
dspickup/dx also means motion of positive charge per unit length. Multiplying uix, we can get the current density
Jpickup = (pUVminN/Bfinal)·(Efinal/Bfinal) (not dependent on qe) , and
Bpickup/x = m0pUVminNuix/Bfinal where uix=Efinal/Bfinal
If we ignore the electron current (oscillating with wp), we can obtain
[Bfinal2-BSW2]/2m0 = pUVminN∫xuix(x)2 (4)
The same as “pressure balance”, and therefore, we probably need to include the change in the solar wind dynamic pressure in (4) for completeness. 7

8. Summary and future work
We tried a new approach to understand the physics of azimuthal momentum transfer by estimating E-field from the displacement of pickup ions and solar wind electrons.
With proper assumptions, we can derive governing equations (3) and (4) for E and B, respectively.
The solutions for extreme cases converges to ideal cases.
In the future, we have to include the solar wind density in some way in equation (3).
We also have to consider thermal electron effect (Debye shielding) in equation (3).
Finally we will aim numerical integration of (3) and (4). 8