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STATISTICS and ENERGETICS of the SOLID EARTH PROCESSES:
RELIEF, PLATES, EARTH PROCESSES, VOLCANOES
G.S. GolitsynA.M. Obukhov Institute
of Atmospheric Physics, RASMoscow, RF
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a)
b)
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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
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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
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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.
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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
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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
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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
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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
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Check of the convection velocities in viscous and turbulent regimes
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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
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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
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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
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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
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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
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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
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Scatter plot of observed rupture areas against the
theoretical values
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The scatterplot of the observed rupture length against the theoretical value
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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
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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
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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
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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
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