Inductive-Dynamic Magnetosphere-Ionosphere

Coupling via MHD Waves

Jiannan Tu Center for Atmospheric Research

University of Massachusetts

Collaborators: Paul Song, Vytenis VasyliunasBoston College Seminar 11/5/2013

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Outline

• Background

• Governing Equations

• Numerical Method

• Simulation Results

• Conclusion

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Background• Coupling must be through waves in addition to flow• Transient responses of ionosphere/thermosphere to

changes in solar wind/IMF variations• Conventional approach: electrostatic, quasi-stead state

0B

= Et

Because 0

0 0, , 0BB

B B B Bt t

(In fact ) 0(

0 0B B)B B

= Et t t

0=

t

in plasma momentum equations and

Implication: time averaging or quasi-steady state. Missing wave or oscillation information

(Detailed discussion in Vasyliunas, Angeo., 30, 357, 2012)

( )J σ E u Bn

Governing Equations

Plasma and neutral continuity, momentum equations

( ) ( ) ( )

( ) ( ) ( )n

u

u

uuu J B u u J

g u u

uu u u u J

nn n

ein n en in

n

n en n n n in n en in

S Lt

L St

mp

t e

S L

mp

t e

g u un nL S

(1)

(2)

(3)

(4)

4

2 2 2

2 2 2

3 1( ) ( ) ( )

2 2

3 1( ) ( ) ( )

2 2

u u q J E u u

u u q u u

in n n

nn n n n n in n n

pp p w w

t

Q

pp p w w

t

n nQ C

Plasma and neutral dissipation equations [Vasyliunas and Song, JGR, 110, A02301, 2005]

(5)

(6)

(7)

Generalized Ohm’s law

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( )1( )E= u B J B+ Je en ei

e e

m

en e n

Faraday’s law and Ampere’s law

, 0

B= E B= J

t

(8)

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Eliminate J with Ampere’s law and E with generalized Ohm’s law, and write eq. (1)-(6) and Faraday’s law in a compact form

2

02,0

u

u

BBuu I

Fu u I

uB Bu

u

u

n n

n n n n

n n

Bp

p

p

p

W F+ R

t z

0

0

2

0 0

2 220

( ) ( )

( ) ( )

1 1( ) ( ) ( )

1( ) ( )

2

u u B g u u

u u B g u u

RB B B B B B B B

V B u u

ei in n en in n

ei in n en in n n n

e

e e

nin n

n

S L

L S

mS L

e

mS L

e

n

en n

p pp Q

21( ) ( )

2

q

u u u qnn n in n n n n

n

p pp Q C

( , , , , , , , , , , , )W Tn x y z n nx n ny n nz x y nu u u u u u B B p p

where

(9)

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Numerical Method• One-dimensional geometry for the central polar cap

(northern hemisphere) with a uniform background magnetic field 0 ˆB zB

• Eq. (9) is a system of stiff partial differential equations because of very large (up to 106 s-1) ion-neutral collision frequency vin at low altitudes

y

z - vertical

x – antisunwardz = 80 km

z = 1000 km

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A scaled M-IT system

Altitude distribution of ion-neutral, electron-neutral, and electron-ion collision frequencies

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• Fully implicit difference scheme1 1 1

1/ 2 1/ 2 1F FR

n n n nj j j j n

j

W W

t z

(10)

• Implicit difference overcomes the stiffness so that 5-6 order of magnitude larger time step (up to tens of sec), compared to explicit difference scheme, can be used while still obtaining stable solutions

• Eq. (10) is a set of nonlinear algebraic equations, which is solved by an iterative Newton-Krylov subspace method

• An implicit framework has been developed based on a portable, extensible toolkit for scientific computation (PETSc) package (http://www.mcs.anl.gov/petsc), which includes various kinds of Newton-Krylov solvers

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Initialization:background & runtime param

PhotoionizationChemistry

Collision-frequencies

Main Routine

PETSc Nonlinear Solver (SNES)

PETSc Krylov Linear Solver

Functions Evaluation

JacobianMatrix

PostprocessingFlexible: 1-D, 2-D or 3-D,easy to add variables

Time Stepping

Optional

Large time step (up to tens of seconds) can be used with implicit solver – quite efficient

Components of implicit framework for the inductive-dynamic ionosphere/thermosphere model

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• At the top boundary (1000 km), the density, velocity, pressure are linearly extrapolated, the free boundary condition (zero first order derivative) is applied to the perturbation magnetic field.

• At the bottom boundary (80 km) the plasma and neutral velocities, and perturbation magnetic field is set to zero. The plasma and neutral mass densities are determined by photo-chemical equilibrium. The plasma and neutral pressures are assumed fixed in time.

• The initial ionosphere/thermosphere is specified by IRI 2011 ionospheric model and NRLMSISE00 thermospheric model for the solar minimum night time polar cap.

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• Spatial cell size Δz = 5 km and time step Δt = 0.01s. Choose Δt = 0.01 s so that Δt is less than the transition time (0.1 s) of the convection velocity imposed at the top boundary, and also less than the time for the Alfven wave to propagate from the top to bottom boundary. If the simulation domain is extended to the magnetopause the Alfven travel time will be about 1 min and we may use a larger time step, say 10 sec.

• The system is driven by an antisunward convection velocity (Vx) at the top boundary, changing from 0 to 600 m/s in 0.1 s, keeping at 600 m/s for 2 min, and then to 2000 m/s in 0.1 s, keeping at 2000 m/s for 1 min. Simulation results for the last 1 min are shown.

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Time variation of an antisunward convection (Vx) imposed at the top boundary

Transient time 0.1 s

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Simulation Results

Oscillations & overshoots. Also Vy & By

The transient to quais-steady state takes ~18 sec (~40 Alfven travel times) for scaled M-IT system but will be proportionally longer with higher altitude of the top boundary so that the Alfven travel time tA is used to scale the time

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Bristow et al., 2003

80 km

1000 km

Alfven travel time tA ~ 0.46 s. t = 0 equilibrium state. Topside Vx increases from 600 m/s to 2000 m/s in 0.1 s, velocity perturbation propagates downward along the field line. Localized enhancement and all-altitude overshoot.

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Superposition of incident and reflected waves causes oscillations and overshoots

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High Alfven speed (~10,000 km/s) at high altitude, decreases to ~3000 km/s in the F region. Topside Vx increases from 600 m/s to 2000 m/s in 0.1 s. Prompt perturbation in Vx at 600 km, about one tA to reach 120 km. Perturbations in Vy begins at 120 km when Vx perturbation arrives and propagates upward.

Dynamic Hall effect not present in electrostatic models

0

ˆ

ˆ

( ) /

B= z E

J z B

E= u B J B + Je

t

en

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Ionosphere behaves as a damped resonator in response to magnetospheric perturbations

The oscillation period depends on the ionospheric mass or inertia.

,( ) ( )

uu u J B

u u Jein n en in

t

m

e

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Compressional waves similar to acoustic-gravity waves but with much faster propagation speed because of the different parameter range.

Pressure oscillations with progressive phase delay with altitudes. The phase velocity is ~ 200 km/s at lower altitudes and ~400 km/s at higher altitudes, and correlate with the plasma temperature variations

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Variation of total heating rate q with time and height. Strong heating below about 250 km. Heating rate included both frictional and true Joule heating rates, with dominant contribution from the frictional heating rate.

2 2 2

2 2 2

2

3 1( ) ( ) ( )

2 2

3 1( ) ( ) ( )

2 2

( )

u u q J E u u

u u q u u

J E u u

in n n

nn n n n n in n n

in n

pp p w w

t

pp p w w

t

q

• The M-I coupling is through MHD waves and produces strong variation of the ionospheric state variables during the transient stage, which persists for 20-40 Alfven travel times (20-30 minutes).

• Inductive-dynamic (non-electrostatic and retaining time derivative terms in the momentum equations) approach is necessary to correctly to describe transient variations

• The overshoot and oscillation is caused by the superposition of the incident and reflected Alfven waves with the period dependent on IT inertial.

• The dynamical Hall effect is an inherent aspect of the M-I coupling.

• The ionosphere-thermosphere responds to magnetospheric driving forces as a damped oscillator.

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Conclusions