11.1 Magnetic Dipole Field

11.1 Magnetic Dipole Field

To first order, Earth's magnetic field can be described as the field of an

earth-centered magnetic dipole whose axis is tilted by 11.5 re the rotational axis.

Since 0 because m curl grad

B

B B

2

3

0.

The magnetic scalar potential is given by

cos, , 11.1

where , , are spherical coordinates (Fig.11.2).

1 1Note, for spherical coordinates:

sin

2 cos sin

m

m

m

mr

rr

r r r

m mr

r r

r θ φ

B

3

2 2 2 2 23 3

0 11.2

4cos sin 1 3cos 11.3

r

r

B B

m mB B B

r r

r θ

11.1 Magnetic Dipole Field (2)

3 3

2 2 2 2 23 3

3 3 5

2 cos sin0 11.2

, 4cos sin 1 3cos 11.3

,90 where 3 10

:

The dip angle is the inclination of the magnetic fie

m r

r

E E E E E E

m mB B

r rm m

B r B Br r

B B R m R m B R B T

B r θ r θ

Magnetic dip angle I

2

2

ld vector with the horizontal.

Notice that is a tangential vector to the sphere, i.e., in the horizontal plane.

sincos cos

1 3cos

2cossin sin

1 3cos4sin cos

2sin cos1 3cos

I IB

I IB

I I

θ

Bθ

Bθ

2 2

12

4sin cossin 2

1 3cos1 4sin cos

sin2 1 3cos

I

I

B

11.1 Magnetic Dipole Field (3)

3 3

3

3

2

0 0 0 0

0 o

:

2 cos sin

sinrd tan

2 cosdr 2

d

tan 2

sin 1 sinln ln ln ln

sin 2 sin

For =90 , R is the Earth's radius at the equator. Then

r

r

m mB B

r rm

B rmBr

dr

r

r r

R R

Equation of a fieldline

B r θ r θ

2 20 0sin or sin where r R L L r R

B

B

Br

dr

rd

r

r

11.2 Geomagnetic Field The most accurate description of Earth's geomagnetic field is given by the

international geomagnetic reference field (IGRF) using gegraphic

spherical coordinaates r', ', '. The magnetic potential func

1

m1

22

2

0

tion is:

1', ', ' cos ' sin ' (cos ')

'

are the associated Legendre polynomials:

1 1

2 211 Legendre polynomials

2

nn

m m mn n n

n m o E

mn

mmm nm

n m

nm n m

n nm

r g m h m Pr R

P

d P xP x x

dxn n m

P x xm n

P

2 31 2 3

11, , 3 1 , 5 3 , ....

2P x P x P x

11.2 Geomagnetic Field (2)

1

m1

22

2

0

1', ', ' cos ' sin ' (cos ')

'


1 1 ; cos '


2

nn

m m mn n n

n m o E

mn

mmm nm

n m

nm n m

n nm

r g m h m Pr R

P

d P xP x x x

dxn n m

P x xm n

P

2 31 2 3

2

0m 1

2

0 1 1m 1 1 1

11, , 3 1 , 5 3 , ....

21. Axial-centered dipole - n=1, m=0:

1', ', ' cos '

'

2. Tilted centered dipole - n=1, m=0,1:

1', ', ' cos ' cos ' sin ' sin '

'

E

E

P x P x P x

r gr R

r g g hr R

11.3 Geomagnetic variations

1

m1

22

2

0

1', ', ' cos ' sin ' (cos ')

'


1 1 ; cos '


2

nn

m m mn n n

n m o E

mn

mmm nm

n m

nm n m

n nm

r g m h m Pr R

P

d P xP x x x

dxn n m

P x xm n

P

2 31 2 3

2

0m 1

2

0 1 1m 1 1 1

11, , 3 1 , 5 3 , ....

21. Axial-centered dipole - n=1, m=0:

1', ', ' cos '

'

2. Tilted centered dipole - n=1, m=0,1:

1', ', ' cos ' cos ' sin ' sin '

'

E

E

P x P x P x

r gr R

r g g hr R

11.3 Geomagnetic variations (1)

•The geomagnetic field shows significant variability during quiet times and magnetically disturbed periods.•Quiet time variability results from tidal waves caused by solar heating and lunar gravitational forces. Tidal motion moves ions across field lines generating dynamo electric fields.

–Solar-quiet current system – Sq currents

–Lunar current system – L currents

11

11 0 1 0 0

Faraday's Lawxt

xt

BE

EB J

11.3 Geomagnetic variations (2)

•Disturbance field D=B-BSq-BL

–D=Dst-Ds

•Dst is the storm-time variation caused by the ring current disturbances

•Ds is the disturbance caused by auroral particle precipitation

•Kp index–Local K index: 12 stations (in the northern mid-latitudes) measure the average magnetic variation for every 3 hours with log scale

–00, 0+, 1-, 10, 1+, 2-, … 9-, 90, 9+

–Planetary Kp index: average of the 12 stations

11.4 Ionospheric Layers

Earth's ionosphere is usually stratified in D, E, F1, and F2 layers during

the daytime. At night, only the F2 layer survives. When transport

processes can be neglected, the ion continuity equation bec

e

e e

omes

dn11.49

dtIn equilibrium, i.e., during most of the daytime .

The loss is mainly caused by recombination of electrons with ions.

If we assume one dominant ion, then

n , and the loss L

e e

e e

i

P L

P L

n

2

2

. Therefore

, 11.51

where is ion/electron recombination rate.

e i e

e d e

d

n n n

P k n

k

11.4 Ionospheric Layers (2)

0

e

a

00

1 00

1 a00 0

Using the Chapman production function (9.21) for P :

11.52

with

sec 11.53

exp 11.54

exp 1

exp 1 sec e

ae

ae

z za H

P I n z e

z n z H

z zn z n z

H

z zP I n z e

H

z zI n z e Hn z

H


01 a0

0 0

0 max

amax

exp 1 sec e

For convenience (and convention) we select the height of

maximum production as our reference height z . But

sec 1. Therefore the production

z za H

e

z zP I n z e Hn z

H

z

Hn z

01 0

0

2

rate becomes

exp 1 e

Using 11.51 :

z za H

e

e d e ed

z zP I n z e

H

PP k n n

k


0

0

1 1 00

00 0 0

1exp 1 e

2

1exp 1 e Chapman profile; , are the

2

electron density and altitude of the ionization maximum, i.e., the layer peak.

An alternate fo

z za H

e d

z z

He

z zn k I n z e

H

z zn n n z

H

0

a a0 0,0

1 00

rm is obtained if we choose as reference height the

altitude of maximum production for overhead sun, i.e., for =0.

At this height z , 0 1 and

z , 0 sec0 0,0 1 1

exp 1ae

n H n H

z zP I n z e

H

0

0

00,0

sec e

1exp 1 sec e

2

z z

H

z z

He

z zn n

H

11.4 Dominant Processes in the Ionosphere

• E and F1 region: photochemistry is dominant

• F2 region: photochemistry and diffusion

• Topside F2 region: diffusion is dominant

• Plasma motion along B• Ambipolar diffusive forces are vertical: Fa~g, dT/dz, dn/dz

Fa||=FasinI

uaz=ua||sinI

See Fig.11.7 for effects

of neutral wind and E.

I

Fa

uaz

Fa||

B

ua||I


0

00

2

0 0 00

2 2

0 00 0

2

00

:

1exp 1 e

2

1 1exp 1 1

2 2

1exp 1

4 2

1 Parabolic di2

z z

He

e

z zn n

H

z z z z z zn

H H H

z z z zn n

H H

z zn n

H

Profile shape near the peak

0 0

stribution around the layer peak.

, are the electron density and altitude of the layer peak.n z

Documents

11.1 Magnetic Dipole Field