2015
3.水深分布と大地形
• 海底の大地形
• 水深はなんで決まっているか
• プレートの熱構造
• 水深との関係
• 海洋リソスフェアのたわみ ー弾性体としてのプレートがもたらす大地形
• 海山（列）周辺のたわみ
• アウターライズの形成
1
水をとりさった地球の姿
2
Heezen and Tharp(1959)
Marie Tharp, http:://www.columbia.edu
3
宇宙から見る惑星の地形
地球海底 1°=~1.8km
火星
∼数百ｍ
(Sandwell, 2004
現状では火星の地形のほうが全体としては高分解能でわかる
4
音波で測る海底地形
昔：直下水深のみ
今：広範囲を一度に調査
マルチビーム測深（スワス測深）
数10cm~数10ｍの分解能でわかるが、調査に非常に時間がかかる
5
宇宙から測る（予測する）海底地形
~800km
• 海面高度を測定して精密海底ジオイド
を求める
• ジオイドの傾斜から重力分布を算出
• 重力異常から水深を推定
(Smith et al, 2004
6
現在
ETOPO1 (1 arcmin ~1.8km)
http://www.ngdc.noaa.gov/mgg/global/global.html
7
太平洋を横断してみよう
(Press, Understanding Earth, 2003
8
lithosphere リソスフェア
• 剛体として振る舞う地球表面の固い層
（流動特性による区分） rigid
• 粘性が大きい＝荷重をかけた場合の歪み
速度が小さい層 high viscosity
地球の熱構造に起因する（成長モデル）
中央海嶺で海底にあらわれたアセノスフェアが熱を放出し
て冷えて固くなる
地球内部の水の分布に起因する
中央海嶺下でメルト生成時に脱水が起こり硬くなる（カン
ラン石は少量の水で劇的に粘性が下がる）
9
plate as a thermal boundary layer
stream line
isotherm
average temperature at depth
thermal boudary layer
thermal boudary layer
(瀬野：プレートテクトニクスの基礎：元はParsons and Richter, 1981)
• リソスフェア＝地球内部の熱構造を考えた時の上部熱境界層 地表での冷却が
• 温度勾配が大きい
• 熱は対流ではなく主に伝導で運ばれる
• 地球の場合、上部熱境界層の下限の温度は1300°C位
10
T2 # T1
"t
thermal structure of oceanic lithosphere
L
1 dQ
dT
qz = #
= #k
A dt
dz
• 概ね2次元構造を仮定してよい（プレート運動に直交する方向＝海嶺軸の走向
"Q = kA
の変化は無視できる）
• 定常状態である
!
dqz
dQz
d 2T
dz =
dz(dxdy)dt = k 2 dVdt
dz
dz
dz
• プレート運動による熱の出入り=移流と熱伝導のみを考える（対流は無視）
"Q =u:プレート速度 c p m"T
κ:熱拡散係数
advection
!
ここでtはプレートの年代とすると
!
!
c p m"T = c p #dVdT
conduction
d 2T
c p "dVdT = k 2 dVdt
dz
この式を解くにあたって
2
k ・近似
dT
dT
=
dz 2
dt c p " ・境界条件
#=
k
cp"
11
% $ 2T $ 2T $ 2T (
$T
学部・院共通講義「プレートテクトニクス」資料
= #' 2 + 2 + 2 *
$t
$y
$z )
& $x
半無限冷却モデル half space cooling model
1次元の熱伝導方程式を解く
!
"T
" 2T
=# 2
"t
"z
(1)
!
T = T1 at t = 0 z > 0
T = T0 at z = 0 t > 0
T " T1 as z " # t > 0
(２)
境界条件:
!
T # T1
- アセノスフェアは一定温度Tm
"=
T0 # T1
- アセノスフェアはある時刻（t=0)に地表
$"
（z=0)に上昇
$ 2"
=% 2
- 地表の温度は常にT0
$t
$z
- 下限の条件はない（半無限モデル）
" ( z,0) = 0
- 水平方向への熱の拡散はない
" (0,t ) = 1
" (&,t ) = 0
12
"T
"T
=# 2
"t
"z
!
T = T1 at t = 0 z > 0
学部・院共通講義「プレートテクトニクス」資料
T = T0 at z = 0 t > 0
!
T " T1 as z " # t > 0
!
"=
規格化：無次元量 θ の導入
T # T1
T0 # T1
$ 2"
$"
= % 2 (3)
$t
$z
" ( z,0) = 0
式(1)をθで書き換えると
同様に境界条件(2)式は
" (0,t ) = 1
(4)
境界条件がきわめてシンプルになる
" (&,t ) = 0
z
2 #t
z
z
d% $" d% ' 1 z
$%
"= "=
=
=
)&
無次元
similarity variable ηの導入
2 #t 2 #t
$t d" $t d" ( 4 #t
!
' 1 dz% '1* 1 d%$%
$% d$%
% $" d%d%$"
d% ' d%1 " 1*
z ')&=11*d"%*,$"
=
=
)& =
,=
= = & 2 ,#t
=
&
(3)式の左辺は
)
d"$(t 4 d#"t (t + 4 d"$#z(t t2,+dt"+ $dz" )( d"
$t d"$t$t d"
2 t+
2
2
zd% 1
$% d$%
%"$"
1
$
%
d
%
$" 1 1
= =dz % $" d% 1
(3)式の右辺は
=
=
=
t 2 =
$z d"" $=z=22 #d#t"
$z 2 2 #t d" 2 $z 4 #t
#t
$$%
$z d"' 2 #t *
z d" $"
' 2 *
$ 2%
$" =dd21%%'1) &1d12%zz 11*, =dd%d%2%' )1&11d""
12 =ddd2%%%$"
*%,
$%
&
"
=
=
=
$ $%t = d"1$t = dd%")(&$"4 1 t,1+= dd"%)&( 2 t,2+
(3),(4)はθとηを使って
$z 2 2 $t#2 t =dd""2 $$tz d4"2(#t 4d"=2##t tt + dd"" (2 22dt"+
d% $z1 4 %#(-)
t d"
$z$% d2% $"
#t d"
=0
d% $%1 =d=2d%% $" ==d% 1
(6)
$
$
z
d
"
z
d
2 #t
&"
=
2 "
(5)
2
" 2 #t
% (0)
=1
d" &$"2z2dd%"d"= $1z d d%
2
2
1 d2 %2 $" 1 1 d2 %
$2 %
% (-) = 0$ %d2 ="= 12 dd"% 2$" ==1 1 d % 2
4 #t d"
$z2
2 #t d"2 $z もっとシンプルになる
z
20 #t d" $z 4 #t d" 2 d#
% (0) = 1%$(-)
=
2
"=
d% 1 d2 %
d$
&"d% =1 d % 2 !
" d="1
= 2 d"
%&(0)
d" 2 d" 2
1 d"
d# % (-) = 0
%$" =
"=
% (-) = 0
2 d$
!
d$ % (0) = 1
% (0)d=#1
1 d"
"1 d="
%$d$ =
学部・院共通講義「プレートテクトニクス」資料
%
$"
=
2 "
!
d$
2 d$ d#
と置くと
2
= d# 1 d"
%$ = ln " % lnc1
微分方程式を解く
"1"=d
"d$
!
!
%$d$ =%$" d=$ 2 d$
2
d#
(7)
2 " 11dd""
" = c1e%$ =
(5)式は
%
$"
=
d
$
%$" = 21d$d"
%$ 2 = ln
" d%$lnc
$ %$ ' 2
%$
=21d$
# = c1 & 0 e d$'+1
1dd""
2
d# 12
" = c1e%$%%$$2=dd$$== 2 "
'
2
0 = c1 & 0 e%$ ' d$'+1
%$ 2=d$ln2""% lnc1
積分を実行すると
2
lnc1
$
%%$$ =2=lnln""2 %%lnc
d1#
# = c1 & 0 e%$ ' d%$$'+1
2
'
(
" "==cc1ee%%$$2 2 ==dd##
& 0 e%$ ' d$'= 2
'
" =%$c' 21e1 = d$
0 = c1 & 0 e d$'+1 dd$$
2
$ %$2' 2
$$ %%
$$' 2' dd
(8) c1 = %
#
=
c
e
$
'+1
#
=
c
e
$
'+1
&
#
=
c
e
d
$
'+1
1
1
積分定数c1を(6)から決める ' %$ ' 2
1 &(
0
(
& 0 e d$'= 20'0'' %%$%$'$2' 2' 2
2 $ %$ ' 2
000===cc111&&000eee dd$d$'+1
$'+1
'+1
# = 1%
& e d$'
2
( 0
c1 = %
(
22
''
(&&'0 ee%%%$$$'' '2dd$$'='= ( (
2 " $" ' 2
0e
erf (") =
2
d$'= 2
% e d" '
20 $ %$ ' 2
# 0
2
2
# = 1%
e2 d$'
cc1(1==&%%0 2
! (9)
erfc(") = 1$ erf (")
周知の定積分を用いて
c1 =2 % ("(
z
T $ T1
22 $" ' 2$$ %$%'$' 2' 2
= erfc
erf (") =##==1%
1%% 0(e && dee"
dd$$' '
T
$
T
2
&t
0
0
# (2( $
0
1
%$ ' 2
(10)
(9)を(8)式に代入すると
#
=
1%
e
d
$
'
!
erfc(") = 1$ erf ("22) 0"" $"$"' 2' 2
erf((""))== ( %%0 ee dd""' '
erf
z ## 0
T $ T1
= erfc
2 erf("("!
!
erfc(
!T0 $ T1 erf
1$
")e) $" ' 2 d"'
erfc(
t erf
(""")2))===&1$
0
#
zz
TT$$TT11
==erfc
erfc
相補誤差関数を使うと !
"T11) = 1$2erf
erfc(
TT00$$T
2 &&t(t")
"=
1 * d% ' 1 " *
,
)&
,=
t + d" ( 2 t +
d 2%
d" 2
13
&
&
&
&
%
!
!
!
z
T $ T1
= erfc
T0 $ T1
2 &t
(11)
熱構造（温度分布）が時間ｔと深さzの関数で求まる
14
!
&
' %$ ' 2
0
e
d$'=
(
2
学部・院共通講義「プレートテクトニクス」資料
2
c1 = %
!
(
誤差関数 error
function
2 $ %$ ' 2
# = 1%
e d$ '
&
( 0
誤差関数
2 " $" ' 2
erf (") =
% 0 e d"'
#
相補誤差関数
erfc(") = 1$ erf (")
z
T $ T1
=
erfc
平均０，標準偏差１とした正規分布の
T0 $ T1
2 &t
確立密度関数と同じ形で、正の部分の
みを考える
!
ｘ
erf(x)
ｘ
erf(x)
0.05
0.05637
0.6
0.60386
0.8
0.11246
0.7
0.67780
1.4
0.16800
0.8
0.74210
0.2
0.22270
0.9
0.79691
0.25
0.,7633
1.0
0.84270
0.3
0.32863
1.2
0.91031
0.35
0.37938
1.4
0.95229
0.4
0.42839
1.6
0.97635
0.45
0.47548
1.8
0.98909
0.5
0.52050
2.0
0.99532
(Fundamentals of Geophysics 2nd ed., Lowrie, 2007)
15
学部・院共通講義「プレートテクトニクス」資料
時間の代わりに片側海底拡大速度 u を使うと
t = x /u
z
T " T1
= erfc(
) (12)
(11)式は
T0 " T1
2 #x /u
T " T0
z
1"
= 1" erf (
)
T1 " T0
2 #x /u
z
T " T0
= erf (
) (13)
T1 " T0
2 #x /u
x,zの関数としても書ける
$ #T '
q0 = "k& 0°C,
)
深海では海底面付近での温度は概ね
!
% #z ( z= 0
T0=0, T1=Tm (対流するマントルの温度）とすると
''
z
#$ $
= "k(T1 " T0 ) &erf &
))
#z % % 2 *x /u (( z= 0
熱境界層の下限をT1の90%に達する等温線とすると
(T " T ) d
=k 0 1
(erf (+))+ = 0
T=0.9Tm
2 *x /u d+
erf(x)=~ 0.9 at x=1.16 (誤差関数表から読み取る）
k(T0 " T1 ) $ 2 "+ 2 '
u
=
e ) = k(T0 "T 1)
&
(+ = 0
,*x
2 *x /u % ,
!
#
zL
0
(14)
16
"dz + w" w
学部・院共通講義「プレートテクトニクス」資料
plate thickness is proportional to its square root of
age : from half-space cooling model
T0=0とした場合
• プレートの厚さは年齢の平方根に比例して増大する
t = x /u
• だんだん冷えて厚くなる（成長する）
z
T " T1
T0 " T1
2 #x /u
• 実際にどれくらいの厚さかというと... T " T
z
0
1"
= 1" erf (
)
-6
T1 " T0
2 #x /u
• 熱拡散係数 κ=0.8*10 [m2/s]として年齢1億年のプレートの厚さは
z
T " T0
• thickness =
= erf (
)
T1 " T0
2 #x5[m]=~120[km]
/u
2.32*(0.8*10-6*1*108*365*24*60*60)0.5=1.2*10
= erfc(
)
• このモデルの場合は成長率は落ちるが古ければ古いほど厚くなる
$ #T '
このモデルは適切か？？ −＞ 実際に厚さを観測することは難しい
q = "k
!
)
&
0
% #z ( z= 0
熱流量や水深（温度構造から推定可能）との比較
= "k(T1 " T0 )
=k
''
z
#$ $
erf
))
&
&
#z % % 2 *x /u (( z= 0
17
(T0 " T1 ) d
(erf (+))+ = 0
2 *x /u d+
age-depth curve:
'
k(T " T ) $ 2
=
e )
&
プレートモデルから水深を予測する
(
2 *x /u % ,
0
"+ 2
1
= k(T0 "T 1)
+= 0
u
,*x
T0=0とした場合
#
!
zL
0
"dz + w" w
1-erf(x)=erfc(x) 定義
" m (w + zL ) =
zL
#
zL
0
"dz + w" w
" $ " m = " m% (T1 $ T)
w( " m $ " w ) = " m% (T1 $ T0 ) &
w=
=
2 " m% (T1 $ T0 ) 'x
("m $ "w )
u
#
.
0
#
.
0
(z u +
erfc*
-dz
2
'
x
)
,
erfc(/)d/
2 " m% (T1 $ T0 ) 'x
0u
("m $ "w )
水深は海底の年齢の平方根に比例する
!
18
" $ " m = " m% (T1 $ T)
w( " m $ " w ) = " m% (T1 $ T0 ) &
#
.
0
(z u +
erfc*
-dz
2
'
x
)
,
2 " % (T $ T ) 'x .
実際の値を入れてみると
w= m 1 0
# erfc(/)d/
("m $ "w )
ｗ=
u
0
2 " m% (T1 $ T0 ) 'x
0u
("m $ "w )
t=x/uとして、適当と思われる以下の値（Stein and Stein, 1992) を代入
アセノスフェア密度 ρm=3300 [kg/m3]
!
海水の密度 ρw=1030 [kg/m3]
熱膨張率 α=3.1*10-5 [/K]
熱拡散係数 κ=8.04*10-7 [m2/s]
アセノスフェアの温度 T1=1450 [°C]
水の温度 T0=0 [°C]
海嶺の頂部水深 dr=2600 [m]
これは海嶺頂上を水深の基準面にしているので、実際の水深は
d=dr+w
dr=海嶺ridgeの頂部水深
d=2600+370√t 予想されるage-depth curve
19
comparison with observed depth
(Turcotte and Schubert, Geodynamics 2nd. Ed, 2002)
80Myrより古い海底では合わない（観測値が浅い）
20
年代が古くなるとプレートは半無限体モデルの予測ほどは冷えない
plate model
T=Tm (x=0)
T=T0 (z=0)
T=Tm (z=a）
•
一定の温度Tｍを持つアセノスフェアが、ある時刻t=0を
境に上面のみ温度T0に強制される（境界条件1）
•
プレートは無限には冷却しない。すなわち、ある深さa
•
x=∞ではプレート内の温度は線形に変化T=Tm*z/a
においてT=Tｍとなる(境界条件２）
21
solution of plate model
2
数学的に厳密に追いたい人はCarslaw and Jaeger(1959)
仮定などをチェックしたい人はTurcotte and Schubert “Geodynamics”
アイソスタシーの考えを取り入れて水深を予測すると
m
m
0
w
想定するプレートの厚さaによってage-depth curveが変わる
22
plate models
PSM model : Parsons and Sclater(1972)
a=125km Tm=1333°C α＝3.28×10-5
プレートGDH1 model : Stein and Stein(1992)
高温のマントルと薄いプレートプレート
a=95km Tm=1475°C α＝3.05×10-5
(Fowler, The Solid Earth, 2nd ed., 2005)
23
comparison with observed depth
(Turcotte and Schubert, Geodynamics 2nd. Ed, 2002)
24
plate model vs half-space cooling model
• 熱流量や水深の平均的な観測値は、プレート（板）モデルのほうがよりよく
説明できているように見える
• 古いプレートでは、プレート下部が不安定になってプレート物質が剥離し
て落ちていく（e.g., Parsons and McKenzie, 1978)
• マントルフローに沿ってプロットすれば、半冷却モデルで実は十分（Adam
and Vidal, 2010)
• 観測値は地域差がかなりある
• 地域やテクトニックセッティングによるグルーピングの仕方によっては値
は一定の漸近値に近づかない −＞普遍的熱モデルがあり得るか
REPORTS
25. M. T. Hurtgen, M. A. Arthur, N. S. Suits, A. J. Kaufman,
Earth Planet. Sci. Lett. 203, 413 (2002).
26. P. Gorjan, J. J. Veevers, M. R. Walter, Precambrian Res.
100, 151 (2000).
27. L. R. Kump, W. E. Seyfried Jr., Earth Planet. Sci. Lett.
235, 654 (2005).
28. K. S. Habicht, M. Gade, B. Thamdrup, P. Berg, D. E. Canfield,
Science 298, 2372 (2002).
29. D. H. Rothman, J. M. Hayes, R. E. Summons, Proc. Natl.
Acad. Sci. U.S.A. 100, 8124 (2003).
30. R. A. Berner, R. Raiswell, Geology 12, 365 (1984).
31. We are extremely grateful to R. Raiswell, S. Bates,
W. Gilhooly, B. Gill, J. Owens, A. Khong, P. Marenco,
N. Planavsky, C. Reinhard, M. Rohrssen, C. Scott,
S. Severmann, J. Huang, L. Feng, H. Chang, and
Q. Zhang for laboratory and field assistance and
helpful discussions. The NSF Earth Sciences program
プレート下面からの熱はいらない？
Mantle Flow Drives the Subsidence
of Oceanic Plates
Claudia Adam1,2* and Valérie Vidal3
The subsidence of the sea floor is generally considered a consequence of its passive cooling
and densifying since its formation at the ridge and is therefore regarded as a function of
lithospheric age only. However, the lithosphere is defined as the thermal boundary layer of mantle
convection, which should thus determine its structure. We examined the evolution of the
lithosphere structure and depth along trajectories representative of the underlying mantle flow.
We show that along these flow lines, the sea-floor depth varies as the square root of the
distance from the ridge (as given by the boundary-layer equation) along the entire plate, without
any flattening. Contrary to previous models, no additional heat supply is required at the base
of the lithosphere.
A
t mid-oceanic ridges, hot material rises
and then cools while driven away to
subduction zones, forming the tectonic
plates. The structure of the lithosphere, as the
upper thermal boundary layer, is determined by
conductive cooling after its formation at the
ridge. The lithosphere thickens away from the
mid-oceanic ridge and, as rock density increases
by cooling, slowly sinks into the underlying mantle. Therefore, the sea-floor depth is regarded as
a function of its age only and is studied along
trajectories following an age gradient (referred to
as “age trajectories”). Several models have been
proposed to describe the thermal subsidence of
the sea floor with age (1–4), but no consensus has
been reached on the origin of the flattening observed at old ages (5). These thermal subsidence
models do not directly consider the role of convection in the underlying mantle, which deforms
with a velocity on the order of a few centimeters
per year. In particular, their description of passive
lithosphere cooling ignores any change in plate
motion (in other words, in mantle convection).
Adam and Vidal ,Science, 328, 2010
1
Institute for Research on Earth Evolution, Japan Agency for
Marine-Earth Science and Technology, 2-15 Natsushima, Yokosuka, 237-0061, Japan. 2Centro de Geofísica, Universidade de
Évora, Rua Romão Ramalho 59, 7002-554 Évora, Portugal.
3
Laboratoire de Physique, Université de Lyon, Ecole Normale
Supérieure de Lyon–CNRS, 46 Allée d'Italie, 69364 Lyon cedex
07, France.
*To whom correspondence should be addressed. E-mail:
The first model that proposed to explain the
variations of sea-floor depth with age—the halfspace model (6)—considered the lithosphere as
the cold upper boundary layer of a cooling mantle, where the depth varies with the square root of
the distance from the ridge. By assuming a constant plate velocity, the sea-floor depth varies
with the square root of the age of the lithosphere.
However, subsequent studies found that the observed sea-floor depth at old ages [>70 million
years ago (Ma)] was substantially shallower than
the model prediction (1, 2). These studies
suggested that the flattening observed at old ages
could be accounted for by a model in which the
lithosphere is considered as a rigid, cooling
conductive plate with a constant basal temperature
(plate model) (2, 3, 7). However, if this constant
temperature at the base of the plate is a simple and
convenient way to introduce the additional heat
supply necessary to explain sea-floor flattening
at old ages, there is no physical reason why this
should be true for the entire plate. Different physical processes have been proposed to explain
the origin of this additional heat supply: smallscale convection (8–11), upwelling mantle plumes
(12, 13), or internal heating, including radiogenic
heating as well as the heating from secular cooling (11, 14). Nonetheless, we still do not know
which of these physical processes is truly responsible for the observed flattening at old ages.
Previous global models also do not account
(grant EAR-0720362 to G.D.L. and T.W.L. and
grant EAR-0719493 to A.L.S.), National Science
Foundation of China Fund (grant 40532012 to
X.C.), the Chinese Academy of Sciences Fund (grant
KZCX3-SW-141 to X.C.), the NASA Astrobiology Institute,
and the Agouron Institute provided funding.
25
Supporting Online Material
www.sciencemag.org/cgi/content/full/science.1182369/DC1
Materials and Methods
Figs. S1 to S7
Table S1
References
23 September 2009; accepted 27 January 2010
Published online 11 February 2010;
10.1126/science.1182369
Include this information when citing this paper.
depth, either spatial or temporal. Systematic studies
of sea-floor subsidence along the East Pacific
Rise, for instance, show that the ridge depth varies
from 2000 to 3200 m, and the associated subsidence rate from 50 to 450 m/Ma1/2 (15–19).
These variations imply spatial mantle temperature variations of about T100°C (16–18). Others
suggested that the possible change through time
of plate motion and plate-driving forces (20) [in
particular, pulsations in sea-floor spreading rates
(21), and a higher mean mantle temperature
during the Mesozoic (22)] could be responsible
for higher ridge height and subsidence rate during
this period. Estimates of a mantle ~50°C warmer
during the Mesozoic, for example, could explain
much of the observed flattening relative to a
boundary-layer model (22).
Regardless of their differences, all previous
models are based on the hypothesis that the thermal structure of the oceanic lithosphere is determined entirely by its age—that is, the time elapsed
since its creation at the mid-oceanic ridge. However, because mantle convection and plate motion evolve over time, the new thermal conditions
imposed on the base of the oceanic lithosphere
also change, thus modifying its structure. This
lithospheric structure will evolve to adapt to the
new thermal conditions imposed at its base, along
the entire plate. After a drastic change in the convective system, it will either thicken (or, alternately, become thinner) if the temperature at its
base, defined by the new convective system, is
cooler (or hotter) than it was previously. After
a time long enough (several tens of million years),
the lithosphere will tend toward the structure of the
thermal boundary layer for the new underlying
mantle flow, independently of its initial state.
To test that the structure of the oceanic lithosphere is determined by the underlying mantle
convection, we analyzed more than 770 depth
profiles, leading to a complete coverage of the
Pacific plate (23). Several kinematic models have
been tested to compute the trajectories representative of the present-day mantle convection (flow
lines) (23). The Pacific plate is an ideal candidate
to test our hypothesis for a number of reasons.
First, the Pacific plate velocity has remained constant over the last 47 to 50 million years (My) (24),
Downloaded from www.sciencemag.org on April 15, 2010
最近の論文から
16. M. Y. Zhu, J. M. Zhang, A. H. Yang, Palaeogeogr.
Palaeoclimatol. Palaeoecol. 254, 7 (2007).
17. E. Vernhet, C. Heubeck, M. Y. Zhu, J. M. Zhang,
Precambrian Res. 148, 32 (2006).
18. D. J. Des Marais, H. Strauss, R. E. Summons, J. M. Hayes,
Nature 359, 605 (1992).
19. L. Yin et al., Nature 446, 661 (2007).
20. S. Xiao, Y. Zhang, A. H. Knoll, Nature 391, 553
(1998).
21. G. D. Love et al., Nature 457, 718 (2009).
22. P. A. Cohen, A. H. Knoll, R. B. Kodner, Proc. Natl. Acad.
Sci. U.S.A. 106, 6519 (2009), and references therein.
23. D. E. Canfield, Am. J. Sci. 304, 839 (2004).
24. J. L. Kirschvink, in The Proterozoic Biosphere:
A Multidisciplinary Study, J. W. Schopf, C. Klein, D. Des
Maris, Eds. (Cambridge Univ. Press, Cambridge, 1992),
pp. 51–52.
26
bending of plate
弾性板のたわみは、一般に以下の4次微分方程式で表現できる
以降のスライドの解も含め、導出については、Turcotte and
Schubert “Geodynamics”の第3章を参照のこと
D
!
d4w
d 2w
=
V
(x)
"
H
dx 4
dx 2
Eh 3
D=
12(1" # 2 )
d4w
D 4 + ( " m # " w )gw = 0
V: 鉛直方向の荷重（単位面積あたり）
!
dx
H:水平方向の荷重（単位長さあたり）
(Fowler, The Solid Earth, 2005)
#
D:flexural
rigidity
[cos(x /# ) + sin(x /# ), x > 0
w(x)
= w 0e"x /E:ヤング率 h:プレートの厚さ σ：ポワソン比
!
w0 =
V# 3
,x = 0
8D
27
1/ 4
!
%
(
4D
" ='
*
& ( # m $ # w )g )
$ x '*
$ x'
$ x '"
moat
around
w(x) =
expseamounts
&# ),#M sin& ) + (V" + M)cos& )/, x > 0
!
2
2D
% " (+
%" (
% " (.
!
(Fowler, The Solid Earth, 2005)
関連課題あり
28
弾性板のたわみで説明すると...
2
dd44w
w
w = V (x) " H dd2 w
D
D dx 44 = V (x) " H dx22
dx
dx
3
Eh
Eh 3
2
!!
12(1"
12(1" #
# 2 ))
x=0でのみ荷重Vがあると単純化
D
D ==
!!
解は
!!
dd44w
D
)gw == 00
D 44 + ( " m # " ww )gw
dx
dx
[cos(x //##))++sin(x
sin(x//##),),xx>>00
w(x)
w(x) = w 0e"x / # [cos(x
w00 ==
w
V# 3
,x = 0
8D
ここでflexural parameter αは
!!
(1/1/44
%%
4D
"
=
*
" = ''
&&( # mm $ # ww )g ))
(Fowler, The Solid Earth, 2005)
$ xx ''**
$$ xx''
$ $xx'-'" 22
"
ふくらみ最大の点Xbとたわみのなくなる点X0は
sin
w(x)
=
exp&$&#
""++M)cos
)),,#M
&& ))++(V
& & )/),/x, x>>0 0
#M
sin
(V
w(x)
=
exp
#
M)cos
!
!
%% "
%%""((
% %""(.(.
2D
2D
" ((++
Xb=πα、 X0＝３πα/4
!!
29
海溝外縁隆起帯 アウターライズ outer rise
海溝の海側にゆるやかな高まりが存在
30
!
dx
w(x) = w 0e"x / # [cos(x /# ) + sin(x /# ), x > 0
!
w0 =
V# 3
,x = 0
8D
outer rise : plate bending model
%
(1/ 4
4D
! 片方の端x=0で垂直荷重Vがかかり、かつ曲げのモー
" ='
*
& ( # m $ # w )g )
メントMが単位長さあたりにかかっているとする
!
w(x) =
$ x '*
$ x'
$ x '"2
exp&# ),#M sin& ) + (V" + M)cos& )/, x > 0
% " (+
%" (
% " (.
2D
!
(Turcotte and Schubert, Geodynamics 2nd. Ed, 2002)
31
32
50
0
136˚
138˚
140˚
142˚
144˚
146˚
148˚
136˚
138˚
140˚
142˚
144˚
146˚
148˚
44˚
44˚
44˚
42˚
42˚
42˚
40˚
40˚
40˚
38˚
38˚
38˚
38˚
36˚
36˚
36˚
34˚
34˚
34˚
2005~2011.3.10
136˚
138˚
140˚
142˚
144˚
146˚
148˚
136˚
500
450
400
350
300
250
200
150
100
50
0
500
450
400
350
300
250
200
150
100
50
0
36˚
34˚
2011.3.11~2012.3.10
138˚
140˚
142˚
144˚
146˚
148˚
33
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