Some remarks on seismic wave
attenuation and tidal dissipation
Shun-ichiro Karato
Yale University
Department of Geology & Geophysics
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Why Q?
orbital evolution
tidal heatingQ -> internal state
T, water, grain-size-----
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• What is the relation between seismological Q
and tidal energy dissipation?
– frequency, T-dependence of microscopic Q and tidal
energy dissipation (phenomenology)
• Q and internal structure of a planet
– What controls Q?
• T, water, strain, grain-size, ??
– Why is tidal dissipation of the Moon so large ?
– What controls the Q of a giant planet (what controls the
tidal evolution of extra-solar planets)?
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Conditions of deformation
(tele-)seismic wave propagation
tidal deformation
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Depth variation of tidal dissipation
Energy dissipation occurs in
most part in the deep interior
of a planet.
High-temperature non-elastic
properties control tidal Q
(similar to seismic waves but
at lower frequencies and
higher strain amplitude).
(Peale and Cassen, 1978)
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Phenomenology
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models of anelasticity
Absorption band model
log t
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Maxwell model
Q -1 (w ) =
1
wt
q (w ) = mw ×
wt
1+w 2t 2
× e2
and q (w ) =
m
× wt
w 1+ w 2 t 2
× e2
Voigt model
Q -1 (w ) = wt
q (w ) =
mw
2
Zener model
Q -1 = D
× wt × e 2 and q (w ) =
m
w
× wt × e 2
wt
1+w 2t 2
q (w ) = w M R × D
wt
1+w 2t 2
× e2
and q (w ) =
MR
w
×D
wt
1+ w 2t 2
× e2
absorption-band model
-a
Q -1 µ (wt )
(wt )a
2
q (w ) µ mw ×
2a × e
1+ (wt )
» mw × (wt )
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-a
and
q (w ) µ
× e 2 (for small Q-1) »
MR22B-01
(wt )
m
×
w 1+ ( wt ) 2 a
m
w
a
× (wt )
-a
× e2
× e2
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Experimental observations on Q
olivine
MgO
(Jackson et al., 2002)
Fe
(Getting et al. 1997)
(Jackson et al., 2000))
• Most of actual results for minerals, oxides and metals at high-T
and low frequencies show weak frequency dependence of Q.
(absorption band model)
Q -1 ~ w -a
à
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Q
-1
a » 0.3 .
~ (wt )
-a
µ exp
(
a H*
- RT
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)
( t = t 0 exp
( ))
H*
RT
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Other effects
Q -1 ~ (wt )
tµ
tµ
à
-a
µ t -a
1
(for small grain-size), d : grain-size
d
1
(for most cases), CW : water content
CWr
Q -1 µ d -a , Q -1 µ CW a r
“wet”
“dry”
Aizawa et al. (2008)
Tan et al. (2001)
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Non-linear anelasticity
(Lakki et al. (1998))
Amplitude of anelasticity increases with stress at high T
(above a critical stress (strain)). This tendency is
stronger at lower frequencies --> enhanced anelasticity
for tidal dissipation?
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Non-linear anelasticity?
e > ec
• For
strain (stress).
e c » 10 -1
s conv
m
, energy dissipation increases with
» 10 -6
• Linear anelasticity for seismic wave propagation, but
non-linear anelasticity for tidal dissipation?
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Frequency dependence of Q from
geophysical/astronomical observations
tide (Goldreich and Soter, 1966)
seismic waves
(+ Chandler wobble, free oscil.)
(Karato and Spetzler, 1990)
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lunar T-z (selenotherm) model
(Hood, 1986)
Lunar Q model
ar
Q-1 / Q0-1 = ( CW / CW 0 ) , a r = 0.3
Water-rich (Earth-like) deep mantle ?
(Saal et al., 2008)
Due to non-linear anelasticity ?
Williams et al. (2001)
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conclusions
• Tidal energy dissipation and seismic Q are related but
follow different frequency and temperature dependence
(for some models).
• Tidal Q is likely smaller than seismic Q because of low
frequency and high strain (no data on strain-dependent
Q for Earth materials).
• Solid part of a planet can have large energy dissipation
(low Q) at high temperatures.
• Influence of grain-size is modest, but the influence of
water is likely very large (not confirmed yet).
• Low tidal Q of the Moon is likely due to high water content
(+ high strain amplitude).
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Tidal Q
• lower Q than seismological Q
• low frequency, high strain
• non-linear anelasticity, distantdependent Q ( Q ( r ) )?
• time-dependent Q (t) (due to cooling of
planets)?
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MgO (Getting et al., 1997)
(
Q -1 µ exp - HRT*
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Deformation (generation of dislocations)
enhances anelasticity
¬ deformed
¬ undeformed
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Q in terrestrial planets
• Liquid portion
– Small dissipation (Q~105)
• Liquid-solid mixture
– Not large because a mixture is not stable under the
gravitational field (liquid and solid tend to be separated)
• Solid portion
– Large dissipation (Q~10-103)
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olivine
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Laboratory studies of Q
(on mantle minerals, olivine)
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Conclusions
• Significant energy dissipation (Q-1) occurs in the solid part of terrestrial
planets (due to thermally activated motion of crystalline defects).
• The degree of energy dissipation depends on temperature (pressure),
water content (and grain-size) and frequency.
• Seismological observations on the distribution of Q can be interpreted
by the distribution of temperature (pressure) and water content.
• Energy dissipation for tidal deformation is larger (smaller Q) than that
for seismic waves. The degree of tidal dissipation depends on
temperature (T/Tm) and water content of a terrestrial planet.
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Jackson et al. (2002)
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Orbital evolution and Q
(Jeffreys, 1976)
dx
dt
xº
» A ×x
-12
c
co
A µ Q -1
*
t µQ
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Macroscopic processes
causing Q
• Giant planets
– Dynamic, wave-like mode of deformation
– Very small energy dissipation (Q~105)
• Terrestrial planets
– Quasi-static deformation
– Elastic deformation, plastic flow, anelasticity
– Large energy dissipation (Q~10-103)
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Depth variation of Q in Earth’s mantle
-8
strain ~10
® soft
¬ hard
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