STATISTICS and ENERGETICS of the SOLID EARTH PROCESSES:

RELIEF, PLATES, EARTH PROCESSES, VOLCANOES

G.S. GolitsynA.M. Obukhov Institute

of Atmospheric Physics, RASMoscow, RF

a)

b)

ReliefDecomposition on spherical harmonics up to number

Earth

Venus

Mars (200 km)

Legendre polinomials

planetary radius

2S n n4 180n

4 50n

75n

01 0

, 1 cos sin sinl

clm slm lml m

r a c m c m P

sinlmP

0a

n

Spectral dispersion

Earth km, km

Venus km , km km

Relief spectrum for regional transsects

, , km

Turcotte for 60 trajectories

Synthetic spectrum

1

2 2 2 20l clm slm hm v

V k a c c

2 6.1h 2

2.5h

2 0.9h 2 0.95h 1

nkS k k 2

kl

0.3 30l

2.03 0.04n

2.01n

Dependence of the spectral power density S on wave number k for the synthetic depositional topography. The data points are the average of

50 simulations.

Slope Spectrum

white noise in the space

m for large scale harmonics of Earth

erosion:

water flows on slopes

and wind decelerates over slopes

0.3k

2kk S k

Structure function

If , then

m, km, then m

2

hh x r h x D r

2 1 cosD r kr S k dk

2kS k k kD r r

0.3k 1000r 1/ 2 550D r

In vertical motions the whole crust with km is involved

The relative dispersion of the potential energy

- mean geothermal flux density

Myr

30H

hU Hg

2 29 10F Wm

3 3 4 2 3

min 2 2 7

3 10 3 10 10 2.5 101

9 10 3 10 /

U kgm m ms m

U Wm s yr

Convection

In the viscous fluid

the rate of energy dissipation/generation of kinetic

energy at convection

thermal expansion of mantle

ms , kgm , m /s K

km, m s m s

cm yr , stress at core?

1/ 20.1 /u v d/ pgF c

5 12 10 K

10g 2 33 10 3 700pc 2 2

3000d 172 10 2 1 1110 2 3

6u 1

Check of the convection velocities in viscous and turbulent regimes

Earthquakes

Gutenberg-Richter, 1942, Ishima, 1939 (see Kasahara, 1981)

Correlation function of random forces

where is the stress minimum correlation time

(the free energy of deformation)

0f f fB f t f t

0

f

d

dt

Frequency spectrum

white noise

motion equation, momentum

red noise

0

cosf f fE B d const

dpf

dt

22p fD p t p t

2p fE

p

Flow of events

, cumulative frequency

mean waiting time (Feller)

energy

maxE

E

N E N E dE

1

EN E

2 2p fE p D

/ ffE N E N E

E

Examples

Tsunami:

land slides upon the soil mass

Number of lakes

,

Turcotte, Ryanzhin

nN h h 1.01n

nN M M 0.95 1.1n

nN S S 0.93 0.95n

M

Earthquakes again

magnitude m = 1 ar

Gutenberg-Richter

for , for

S.S. Grigorian 1988, Rundle 1989

Young modulus, mean slip

Seismic moment

0lg /m S S 0 100S 2

N m a bm

1b 8m 1.5b 8m

M Su

2lg 63

m M

u

Energy of seismic waves (Kanamory 1994)

total energy of an event

stress drop Bar = 4 MPa Aki, 1967

length scale

Similarity criterium

at , the process 2 D

1 D

Chile , Sumatra

56 10 ,sE M 00.07sE E

40

1/ 3/L M

/L H

1 7.5,m 7.5m 7.5m

9.5m 9.2m

Scatter plot of observed rupture areas against the

theoretical values

The scatterplot of the observed rupture length against the theoretical value

Dimensional analysis: energy dimension, E, time, T

seek , by geothermal flux

seismic moment

similarity criterium

at …

from global catalogs

1N M 1,F ET

M E

,FN M f

M

1/ 3/MH

L H

1: 1f a

1/ 3

2 / 31/ 3

/ 0.4M FN M M

H H

at

for large

2lg 63

m M

lg ,N m a bm 1/ 3lg 12 / ;a F H 1b

1 0.4f

0.4F

N MM

m 1.5,b lg 0.4a F

time in years.

globally: one quake per year with

global geothermal flux

effective action coefficientaveraged over millenia

7.6m

2lg 63

m M 202.5 10M N m E310E M

3 3 2010

7

10 10 2.5 1010

1 3 10 /

MW

yr s yr

134.5 10F W 10

413

102 10 0.02%

4.5 10F

!!fEN E

lg 7.6 ,N m m

volume energy

/ 31 ,n nN V aV a r 2.14;n / 3 0.71n

V E

2lg lg ,

3N V a V 1lg ;a a 510