Multi-fluid MHD Study on Ion Loss from Titan’s Atmosphere

Y. J. Ma, C. T. Russell, A. F. Nagy, G. Toth, M. K. Dougherty, A. Wellbrock, A. J. Coates,

P. Garnier, J-E. Wahlund, T. E. Cravens, M. S. Richard, F. J. Crary

Magnetospheres of the Outer Planets1359, Monday, July 11, 2011

Boston, Massachusetts1

Motivation• The ion temperature is tightly coupled with the neutral temperature in the

lower ionosphere (Ti = Tn) because of frequent collisions between ions and neutrals; while the electron temperature can be significantly higher due to heating through collisions with super-thermal electrons (Te >> Ti).

• Single fluid MHD model only solves one energy (pressure) equation, and it is usually assumed that Te = Ti. Thus the electron temperature and the plasma thermal pressure are usually underestimated in the ionosphere.

• It is important to get the right temperature of the electrons since the relative strength of the thermal pressure and upstream flow pressure determine WHETHER the ionosphere is magnetized or unmagnetized.

• To calculate more accurately the thermal pressure in the ionosphere, a two fluid model is needed.

2

Continuity equation for each ion species

Momentum equation for ion fluid ( )

)(

)(

,

,

neutralsttiteiRiii

ionssssiiphiii

iiii

nknnmL

nknmS

LSt

u

n

ionsii

ionsii

neutralstit

ionsii SL

Bp

t

uuuGBBIIuuu

00

2 1

2

)(

iiions

Electron Density is calculated based onCharge neutrality:

ionsi

i ie mn /

Electron Velocity: nee /Juu 3

Two - Fluid Multi-species MHD Equations(Solved with Michigan BATSRUS Code)

Magnetic Induction Equation:

Pressure Equation for ions and electrons:

BBuuBB 2

00

1)(

t

ionsi i

iini

ionsii

ionsi

ie

ionsi neutralsniin

ni

inii

i

km

TLTSSkTTnnv

umTTkmm

PPt

P

eiie

i

2

2

2

)1(2)(

])1()(2[)()(

u

uu

neen

e

mv

ne2

0

kTnnkTTnnvEvn

kTTnnvPPt

P

ionsieeiR

neutralsn

e

neutralsn

ionsi

eieee

inennenphn

eiie

,2)()1(

2)()()(

0,

ee uu

Two - Fluid Multi-species MHD Equations

4

No. Name Components Mass(amu) Mass Range(amu)

1 L+ H+, H2

+, H3

+ 1 1-3

2 M+CH

5+, N+, CH

4+, CH

3+, CH

2+,

CH+, C+14 12-17

3 H1+ C2H

5+ 29 29

4 H2+ HCNH+ 28 28

5 MHC+ C3H+, C3H2+, C3H3

+, C3H4+,

C3H5+, C4H3

+, C4H5+, ...

44 37-53

6 HHC+ C5H3+, C5H5

+, C5H7+, C5H9

+, C6H5

+, C6H7+, C7H5

+, … 70 63-89

7 HNI+ C3H

2N+, C

5H

5N+, C

3HN+ 74 51-79

Reduced Titan Ionospheric Chemistry (7 Species)

5

Titan Simulation Details

Computational domain:-24RT ≤ X ≤ 48RT , -32RT ≤ Y, Z ≤32RT

• Spherical grid:– Radial resolution is 16 km to

800 km(~0.3RT)– Angular resolution is 2.50 to

5.00

• Inner Boundary Conditions– Inner boundary at 725 km– photochemical equilibrium

for ion densities– Zero gradient for U and B

• Ionization Sources– Solar radiation (SZA

dependent)– Magnetospheric electrons

(spherically symmetric)

X axis is along the corotation direction. Y axis is from Titan to Saturn.

Total cells: 2.0 million

A Typical run requires

2000 processor hours*

*One human work year 6

Apply the Two-Fluid MHD Model to an Ideal Case

Magnetic field B (0, 0,–5) nTPlasma flow speed u (120,0,0) km/s

Electron number density ne 0.2cm-3

L+ number density nL+ 0.1 cm-3

M+ number density nM+ 0.1 cm-3

Plasma temperature Ti=1500 eV, Te=200 eV

Mass density 1.5 amu/cm3

Total plasma pressure P 5.4×10-2 nPa=P/(B2/2µ0) 5.5

Alfven speed cA 89.4 km/s

Sonic speed cs 191 km/s

Note: Titan is located at 18 SLT in this ideal case. So the upstream co-rotational plasma would interact directly with the dayside ionosphere of Titan, as in the solar wind-planetary interactions.

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Two-Fluid MHD Model Results in the Ideal Case

Contour plots of plasma flow speed and magnetic field strength in the XY(U-E) and XZ(U-B) planes. The white arrows show projections of flow directions (upper panels) and magnetic field directions (lower panels) in the corresponding plane for the two-fluid MHD model.

U

B

U-E Plane U-B Plane

u0

E0

u0

B0

8

Two-Fluid MHD Model vs Single-Fluid MHD Model in the Ideal Case

Contour plots of plasma pressure (Pe+Pi) in the XY(U-E) and XZ(U-B) planes for the single-fluid MHD model (upper panels) and two-fluid MHD model (lower panels).

U-E Plane U-B Plane

Pi+Pe(Two-fluid)

u0

E0

u0

B0

9

Pi+Pe(Single-fluid)

Single-Fluid vs Two Fluid Model Results

Pressure profiles along the ram direction for the single fluid (a) and two-fluid (b) models. In each case, the magnetic pressure PB = B2/20, dynamic pressure PD = u2, electron thermal

pressure Pe = nekTe, ion thermal pressure Pi = nikTi, plasma thermal pressure PT = Pe + Pi, and

total pressure PTOTAL = PB + PD + PT are plotted

Significant Differences:-Electron temperature around 1500 km is much higher for two-fluid MHD case due to thermal electron heating. -magnetic field is shielded by the high conducting ionosphere for two-fluid MHD case, while it is penetrating into lower ionosphere for single fluid case.

10

T34 Flyby: CA Time: JULY 19, 2007 1:11 UTCA Altitude: 1332 kmLocation of Titan: 18.8 SLT

The only upstream, dayside pass. (Venus like)

Equatorial Pass 11

Plasma Parameters for T34 Flyby

Magnetic field B (–1.0, 2.3,–2.1) nT|B|=3.27 nT

Plasma flow speed u (150,0,0) km/sElectron number density ne 0.11cm-3

L+ number density nL+ 0.06 cm-3

M+ number density nM+ 0.05 cm-3

Plasma temperature Ti=540 eV, Te=360 eV

Mass density 0.76 amu/cm3

Total plasma pressure P 1.6×10-2 nPa=P/(B2/2µ0) 3.7

Alfven speed cA 82 km/s

Sonic speed cs 144 km/s

Note: The upstream plasma parameters are set based on CAPS and MAG observations.

12

Model and Data Comparison-T34 Flyby

Data and model comparison of the magnetic field during the T34 flyby for the single-fluid (a) and the two-fluid (b) MHD models. The T34 trajectory is shown by the black squares (10 min interval) and colored with magnetometer measurement.

Single Fluid MHD Two-Fluid MHD

B

Two fluid MHD model results match better with the magnetic observations.

13

Data and Model Comparison of Plasma Parameters During the T34 Flyby

--The single-fluid (blue) and the two-fluid (red) MHD models results are plotted with the LP and CAPS observations.

--The electron density and ion velocity of two-fluid model results are similar with the single-fluid model results

--The ion temperature of the two fluid model are the same as the single fluid model inside the ionopause. But the electron temperature of the two-fluid model is much higher than the ion temperature and matches better with the observations inside the ionopause.

The vertical purple lines is closest approach (CA). The two vertical dashed lines indicate ionopause locations during inbound and outbound passes.

14

Data and Model Comparison of the Magnetic Field During the T34 Flyby

--The single-fluid (blue) and the two-fluid (red) MHD models results are plotted with the observed (black) magnetic field.

--The vertical purple lines is closest approach (CA). The two vertical dashed lines indicate ionopause locations during inbound and outbound passes.

--Two fluid model results are different than the single fluid model results inside the ionopause and fit better with the observations.

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T34 Flyby-comparison with hybrid results

Hybrid Simulation Results [Simon et al., 2008] (Electron temperature is assumed to be adiabatic)

Two fluid MHD Model Results (Thermal electron heating is included when calculating electron temperature)

CA - MAG- Hybrid

- MAG- MHD

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Summary• The ion temperature is tightly coupled with the neutral temperature in the

lower ionosphere while the electron temperature can be significantly higher due to thermal electron heating.

• Two-fluid model solves both the electron and ion pressure equations instead of a single plasma pressure equation, and enables a more accurate evaluation of ion and electron pressures inside Titan’s ionosphere thus better reproduces the magnetic field inside the ionosphere

• Simulation results show that the dayside ionosphere thermal pressure is larger than the upstream pressure when Titan is located in the dusk region; thus the magnetic field is shielded by the highly-conducting ionosphere in this situation, similar to the interaction of Venus during solar maximum conditions

• Model results show that it is critical to include thermal electron heating in order to reproduce the key features that are observed in the ionosphere by the magnetometer and the Langmuir probe.

17