Modeling core fluid motions and the drift of magnetic field patterns at the CMB by use of topography obtained by seismic inversion - grl4962.pdf

GEOPHYSICAL

RESEARCH

LETTERS,

VOL.

17,

NO.

I0,

PAGES

1473-1476,

SEPTEMBER

1990

MODELING

CORE

FLUID

MOTIONS

AND

THE

DRIFT

OF

MAGNETIC

FIELD

PATTERNS

AT

THE

CMB

BY

USE

OF

TOPOGRAPHY

OBTAINED

BY

SEISMIC

INVERSION

Monica

D.

KoMer

Seismological

Lab,

California

Institute

of Technology

David.

J.

Stevenson

Division

of Geological

and

Planetary

Sciences,

California

Institute

of Technology

Abstract.

The

thermal

wind

equations,

in which

the

Coriolis

force

is balanced

by

pressure

gradients

and

horizontal

density

gradients

rather

than

by

Lorentz

forces,

are

used

to describe

patterns

of

magnetic

field

drift

associated

with

core

fluid

motions

near

the

core-mantle

boundary

(CMB).

The

advection

of

magnetic

field

may

be

due

in

part

to the

flow

driven

by

such

horizontal

temperature

gradients,

just

as East-

West

air

flow

is driven

by

North-South

temperature

gradients

in the

Earth's

atmosphere.

It is argued

that

this

flow

may

be

concentrated

in

a shell

near

the

CMB,

and

the

horizontal

temperature

gradients

are

expected

to be

directly

proportional

to

horizontal

gradients

in

CMB

topography,

the

lowest

harmonics

of

which

are

approximately

constrained

in

seisinology.

Part

of the

zonal

drift

is then

associated

with

the

1=2,

m---0

harmonic

of

CMB

topography,

and

anticyclones

are

attached

to topographic

Mghs

(thermal

highs).

Comparison

of

our

derived

flow

pattern

with

those

determined

purely

by

magnetic

field

observations

provides

tentative

support

for

our

model.

Introduction

The

relationship

between

the

Earth's

mantle

and

core

is the

focus

of

numerous

studies

which

consider

the

thermal

and

morphological

nature

of the

core-mantle

boundary

(CMB)

and

its

influence

on

core

fluid

motions.

Various

types

of

core-

mantle

interactions

have

been

proposed

by

which

the

morphology

of

the

CMB,

and

the

dynamics

and

temperature

variations

in

the

lower

mantle

affect

motions

of

outer

core

fluid.

Hide

(1969)

proposed

a coupling

mechanism

which

involved

a hydrodynami_cal

interaction

between

core

fluid

motions

and

undulations

in CMB

topography.

Jones

(1977)

proposed

that

thermal

interaction

influenced

field

geometry

and

caused

some

geomagnetic

properties

(e.g.,

reversal

frequency)

to vary

on

a manfie

convection

timescale.

In their

secular

variation

study,

Bloxham

&

Gubbins

(1985)

suggested

thermal,

electromagnetic,

and

topographic

core-

mfie

intexactions

to explain

the

existence

of static

features

in

the

Earth's

magnetic

field.

The

relationship

was

explored

further

by

Bloxham

& Gubbins

(1987)

who

proposed

thermal

interaction

between

the

core

and

lower

manfie,

where

large

lateral

temperature

variations

just

above

the

CMB

influenced

convection

in

the

core.

Com-•tle

interaction

has

also

been

invoked

to explain

certain

features

of secalax

variation

in the

Earth's

magnetic

field

such

as

westward

drift.

Such

Copyright

1990

by

the

American

Geophysical

Union.

Paper

number

90GL0!479

0094-8276/90

/ 90GL-O!479503.00

nondipole

features

are

believed

by

some

to

be

the

result

of

differential

rotation

of outer

core

fluid

relative

to the

deep

outer

core.

In

addition

to

using

core-mantle

interaction

to

explain

stationary

features

of

sectflax

variation

(Gubbins

& Richards,

1986),

Olson

(1989)

maintained

that

azimuthal

drift

of

core

fluid

could

be

partially

accounted

for

by

thermal

wind

flow

and

that

lateral

temperature

differences

as small

as 10 -3 K•rn

could

significantly

excite

thermal

winds.

The

Model

We

develop

a simple

model

for

part

of the

core

fluid

motion

due

to core-mantle

boundary

topography.

The

basic

idea

is as

follows:

relatively

cold

mantle

just

above

the

CMB

is more

dense

and

will

sink

causing

a depression

in the

CMB,

whereas

relatively

hot

manfie

is less

dense

and

will

rise,

resulting

in

positive

CMB

topographic

relief

or

"bumps".

The

relationship

between

the

deformation

of

an

interface

and

the

pattern

of

convective

thermal

anomalies

is not

necessarily

so

simple

[cf.

Hager

and

Clayton,

1989]

especially

if the

lowermost

manfie

(D")

is a compositional

layer

[Lay,

1989]

but

we

have

chosen

the

most

commonly

assumed

correlation.

Cold,

topographic

lows

cool

the

core

from

above

and

promote

more

vigorous

vertical

convection

in

the

locally

cold

core

fluid.

Warm,

topographic

highs

axe

associated

with

lower

vertical

heat

flow

and

warmer

core

fluid.

This

is illustrated

in Figure

1.

Simple

mixing

length

recipes

for

turbulent

convection

suggest

temperature

deviations

away

from

an adiabat

of order

10-3

K

but

with

a large

uncertainty

[Stevenson,

1987].

These

will

also

be

the

horizontal

temperature

variations

on

an

equipotential.

Horizontal

as well

as vertical

heat

flows

can

be

expected

[Flasar

and

Gierasch,

1978]

and

the

relationship

between

the

magnitude

of the

mantle

tempera•

fluctuations

and

those

in the

core

may

well

be

non-linear,

but

the

sense

of

the

correlation

is clear:

topographic

highs

of

the

CMB

are

associated

with

thermal

highs

in the

outermost

core.

In

our

consideration

of

the

dynamics,

we

adopt

the

stan•

approach

in which

fluid

acceleration

relative

to the

rotating

frame

is ignored

and

viscous

effects

are

ignored.

We

also

neglect

the

Lorentz

force

in

the

outermost

core

because

the

toroidal

component

of the

field

should

decrease

to a low

value

in

the

outermost

few

hundred

kilometers

of

the

core

due

to

the

low

conductivity

of the

manfie

['Merrill

and

McElhinny,

!983].

The

Lorentz

force

is dominated

by

a term

that

is proportional

to the

cross

product

of

this

toroidal

field

.with

the

poloidal

current.

Here,

we

exclude

the

artificial/ty

of

current

sheets

that

arise

in

some

dynamo

models.

We

also

adopt

the

Boussinesq

approximation

in

which

the

only

dynamically

significant

density

variations

axe

those

caused

by

therural

1473

1474

Kohler

and

Stevenson:

Modeling

Core

Fluid

Motions

mantle

cold /

mantle

Equipotential

Depth

Fig.

1.

Schematic

representation

of temperature

profries

near

CMB.

Temperature

and

depth

scales

are

shown

grossly

distorted

to

clarify

the

important

features.

Profile

A

corresponds

to a low

heat

flow

region

and

topographic

high.

Profile

B is

a high

heat

flow

region

and

topographic

low.

Notice

that

at a given

depth

within

the

core,

the

temperature

on

profile

A is high

relative

to proffie

B.

This

correlation

of

topographic

and

thermal

highs

is

central

to

the

model

described

in

the

text.

Note

a/so

that

the

absolute

temperature

at the

CMB

is lower

for

the

high

heat

flow

case

(A)

than

for

the

low

heat

flow

case

lB),

because

the

CMB

on

profile

A is at

a lower

pressure.

This

effect

is actually

much

bigger

than

the

temperature

difference

at

a fixed

equipotential

depth,

but

dynamically

irrelevant.

expansion.

We

then

arrive

at

the

standard

thermal

wind

equation,

well

known

to

atmospheric

dynamicists

[e.g.,

Pedlovsky,

1979]

and

occasionally

discussed

for

the

core

[e.g.,

Olson,

1989]'

2 fix:•

= - VP.•' _ •,c• T

(1)

Do

where

fl

is the

mean

Earth

angular

velocity,

-7 is the

fluid

velocity

relative

to this

fratrie,

P'

is

the

non-hydrostatic

component

of the

fluid

pressure,

Po

is the

mean

fluid

density,

a is the

coefficient

of thermal

expansion,

T is the

deviation

of

the

temperature

away

from

some

mean

core

adiabat

and

•, is

the

gravitational

acceleration.

The

solution

to the

equation

of

motion

can

be

found

by

taking

the

curl

of

both

sides;

however,

by

doing

so,

information

concerning

any

flow

that

is

purely

geostrophic

(i.e.,

Coriolis

force

balanced

by

a

pressure

gradient)

is lost.

We

get

(2 fl. •)•.=

• x (•otT)

(2)

We

now

make

the

further

assumption

that

the

thermal

wind

is primarily

confined

to

a layer

that

is thin

compared

to

the

core

radius.

This

is not

a boundary

layer

(which

is much

thinner

yet)

but

a layer

in

which

the

horizontal

temperature

gradients

and

the

associated

wind

are

dynamically

dominant.

This

layer

could

be

of

order

one

hundred

kilometers

thick

(but

with

a large

uncertainty)

and

arise

in two

ways.

First,

it could

be

the

region

within

which

the

boundary-generated

buoyancy

driving

thermal

convection

is confined.

This

is in the

spirit

of

local

prescriptions

for

turbulent

convection

[Monin

and

Yaglom,

1971],

if one

assumes

that

the

mixing

length

is equal

to

the

distance

from

the

CMB.

According

to

this

view,

the

temperature

anomaly

T decays

in

amplitude

over

some

characteristic

distance

D <<

R c, the

core

radius,

as

illustrated

in Figure

1.

The

second

possible

reason

for

this

thin

layer

approximation

is that

it may

correspond

to the

reg•'on

in which

the

Lorentz

force

is small.

At

deeper

levels,

there

is a large

enough

toroidat

field

that

flows

of

the

type

we

consider

are

effectively

damped.

In

either

case

we

must

seek

solutions

to

equation

(2)

in which

the

velocity

decays

over

a characteristic

•listance

D

as

one

goes

downward

into

the

core.

These

solutions

have

the

property

that

the

shear

is

parallel

to

the

flow,

i.e.,

Since

D <<

R c, it follows

that

0 and

• derivatives

of the

velocity

(but

not

the

temperature)

can

be

neglected

relative

to

radial

derivatives.

(This

is

exactly

analogous

to

the

meteorological

approach

to thermal

winds.)

We

then

have

g•e

•r

v0 = - 2fl sin0

cos0

•q•

(3)

glxe

BT

for

the

values

of

these

velocity

components

at

the

CMB,

where

e =--•.

As usual,

0 is the

colatitude,

• is longitude,

and

g=l•l.

Continuity

dictates

that

Vr is smaller

than

v 0 or

v•__by

a

factor

of

• <<

1.

Notice,

however,

that

the

predicted

tlow

is

not

purely

toroidal

in

general,

although

toroidal

components

tend

to dominate.

In the

limit

of fairly

rapid

O-variation

of T

not

near

the

equator

(i.e.,

-• >> I-I

>> T; 0 • •) it is clear

that

the

vorticity

of the

towns

rach*'•l'

and

proportff)nal

to V2

T

with

a positive

(negative)

constant

of

proportiona!ity

in

the

Northern

(Southern)

hemisphere.

This

means

that

anticyclones

are

associated

with

temperature

highs,

the

same

as meteorological

experience.

Quantitatively,

the

fundamental

assumption

of our

model

is

the

expression

that

core

temperature

variations

are

proportional

to CMB

topography

variations,

or

T(0,,)

=

f(0,,)

(5)

where

13 is the

constant

of proportionality,

and

T and

f are

assumed

to have

zero

means.

if CMB

topography

is given

in

terms

of

a spherical

harmonic

expansion,

the

temperature

is

expressed

in

terms

of

the

same

expansion

coefficients

multiplied

by

1•:

L

T(0,(•)

= [• Z

Z P•n(cøs0)[g•

cosm(•

+ h• a sinm(•]

(6)

1=0

m=O

L

l

f(0,.)

= Z

I P•n(cøs0)[g•

cosm(•

+ h• sinm•]

(7)

œ=0

m=O

We

cannot

justify

rigorously

this

choice.

However,

the

numerical

values

are

plausible.

For

I• = 10 '3

K•m,

and

topographic

relief---

kin,

we

have

temperature

fluctuations

not

Kohler

and

Stevenson:

Modeling

Core

Fluid

Motions

1475

90N

..................

60N

30N

0

"•

0

60E

120E

180

120W

60W

0

a

-5

km

l

[! i:liilli:._.iilili'-iiilli!iiiiiil+5

km

not

the

magnitude

of

that

flow.

For

the

same

reason,

uncertainties

in the

magnitude

of

CMB

topography

even

by

factors

of ten

are

less

important

to assessing

the

model

than

the

pattern

of

CMB

topography.

Discussion

and

Results

Maps

of the

velocity

field

are

given

in Figures

2 and

3 with

corresponding

CMB

topography

maps.

Figure

2b

shows

the

velocity

flow

field

using

the

CMB

results

of

Morelli

&

Dziewonski

(1987)

and

Figure

3b

shows

the

velocity

flow

field

using

the

results

of

Gudmundsson

(1989).

Both

maps

illustrate

that

closed

loops

of fluid

flow

are

closely

related

to

topography

highs

and

lows,

as

expected.

Although

.....

=====================

topography

maps

vary

wide!y

from

worker

to worker,

there

are

features

common

to the

maps

studied

here.

For

example,

a0 2// -

--.--•rT/ti!

• :'•. }}}i'' :• •

'•......."

'"•'•;'"';'•

ao

all maps

show

an anticyclone

(due to the CMB

topography

high)

underneath

the

Indian

Ocean.

Note

that

this

flow

is

0

:: "":::

:•

!'

:•••••..a

I mostly but not purely toro idal. The con tinents h ave been

-a

0

drawn

only

for

establ/shing

the

frame

of reference.

b -80

"'"'"'"'"'-"•-60

•-----

Fig.

2 a)

Map

of CMB

topography

obtained

by

Morelli

and

Dziewonski

(1987).

(Results

are

from

inversion

of

.

compressional

core phase data with spherical

harmonic

expansion

up

to degree

and

order

4.)

b)

Map

of core

fluid

• •

",

patterns

based

on the

thermal

wind

model

described

in the

text

, -

,.

,_

•-•,

'+2.0

krn

-2.0

km

60

60

30•'..

30

0 '•

t3

-3

-30

Fig

3.

a)

Map

of

CMB

topography

map

obtained

by

Gudmundssen

(1989).

(Results

are

from

inversion

of

compressional

core

phases

with

spherical

harmonic

expansion

up

to

degree

and

order

5.)

b)

Map

of

core

fluid

patterns

based

on

the

thermal

wind

model

described

in

the

text

using

results

from

part

a).

enormously

different

from

values

suggested

by

turbulent

convection

models

[Stevenson,

!979,

1987],

and

we

find

I•l

- 10 -2 crrgsec

comparable

to westward

drift.

Clearly

our

model

is most

suited

to estimating

the

pattern

of core

fluid

and

-

20

km/yr

Fig

4.

Core

fluid

velocity

map

of Bloxham

(1989a)

obtained

from

magnetic

field

data

for

time

interval

1935-1940.

Comparisons

with

magnetic

field

data

results

of

Bloxham

(1989a)

given

in

Figure

4,

indicate

that

thermal

winds

may

indeed

be

largely

responsible

for

core

fluid

motions.

The

fluid

flow

map

in Figure

4 is a result

of

magnetic

field

data

during

the

time

interval

1935-1940

but

it displays

similar

general

characteristics

to

maps

of

flow

during

other

time

intervals.

For

example,

as

in the

velocity

map

given

by

thermal

winds,

fluid

flow

in

Figure

4 is anticyclonic

underneath

the

Indian

Ocean

and

over

south-central

Asia.

Our

model

is probably

only

relevant

to field

components

that

are

steady

although

it is

possible

for

dynamos

to

have

steady

velocity

fields

yet

unsteady

magnetic

fields.

The

expression

for

velocity

containing

the

1=2,

m=0

component

of CMB

topography

expansion

is given

by

v0=0

3 gc•e

sin 0 g20

v•=

2

fl

This

expression

describes

the

pattern

of

westward

drift

if the

1=2,

m=0

topography

expansion

coefficient

is positive;

fluid

velocity

is eastward

if the

coefficient

is negative.

It is the

only

term

which

globally

describes

this

pattern.

Figure

2b

shows

strong

westward

drift

in regions

south

of New

Zealand

and

in

the

Indian

Ocean.

Figure

3b

shows

less

westward

drift

and

regions

of

strong

eastward

drift,

possibly

due

to

the

large

1 -[76

Kohler

and

Stevenson:

Modeling

Core

Fluid

Motions

negative

1=2,

m=0

CMB

topography

coefficient.

This

model

does

not

work

near

the

equator

since

certain

velocity

terms

diverge.

This

arises

through

our

failure

to

retain

0 and

{ derivatives

in the

derivation

of v 0 and

v•

(equations

3 and

4).

More

fundamental/y,

the

Coriolis

forc•

cannot

be

expected

to

dominate

near

the

equator

and

other

effects,

neglected

here,

will

be

important.

However,

the

model

behaves

very

well

to within

ten

degrees

of

the

equator

and

it can

be

used

to predict

patterns

in core

fluid

velocity.

In

several

respects,

it

is

surprising

that

this

model

shows

tentative

evidence

of

validity.

There

are

surely

other

sources

of

near-CMB

flow

and

we

have

only

determined

one

component.

Moreover,

the

validity

of

the

thin

layer

approximation

(e

<<

1) is debatable.

Models

that

do

not

use

this

approximation

(e.g.,

Bloxham,

1989b)

show

less

correlation

between

flow

and

topography.

As

seismic

and

other

models

improve,

it may

become

possible

to

decide

whether

a relatively

thin

layer

thermal

wind

component

contributes

to core

flow;

we

can

thereby

learn

more

about

the

core.

Conclusions

If outer

core

horizontal

fluid

flow

is driven

by

horizontal

temperature

gradients

due

to

CMB

topography,

then

the

thermal

wind

equations,

expressed

as

a function

of

CMB

topography,

describe

the

advection

of magnetic

field

resulting

from

core

fluid

motions.

This

model

describes

zonal

drift

by

the

velocity

expression

containing

the

1=2,

m=0

topography

coefficient

and

predicts

that

westward

drift

should

not

be

a

dominant

feature

in

core

fluid

motions

if CMB

topography

harmonic

terms

other

than

1=2,

m=0

contribute

much

power.

The

thin

layer

version

of our

model

predicts

that

anticyclones

are

associated

with

thermal

and

topographic

highs,

and

a

comparison

of magnetically

derived

flow

maps

with

our

maps

provides

tentative

support

for

this

version.

Acknowledgments.

We

thank

Jeremy

Bloxham

for

discussions

and

Rob

Clayton

for

the

use

of

his

computer

graphics

programs.

This

work

is supported

in part

by

NSF

grant

EAR-8816268.

Contribution

number

4872

of

the

Division

of Geological

and

Planetary

Sciences.

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