半無限冷却モデル half space cooling model -...
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半無限冷却モデル half space cooling model
!
"Q = kAT2#T
1
L"t
qz = #1
A
dQ
dt= #k
dT
dz
!
dQz
dzdz =
dqz
dzdz(dxdy)dt = k
d2T
dz2dVdt
!
"Q = cpm"T
!
cpm"T = cp#dVdT
!
cp"dVdT = kd2T
dz2dVdt
dT
dt=
k
cp"
d2T
dz2
# =k
cp"
$T
$t=#
$ 2T
$x 2+$ 2T
$y 2+$ 2T
$z2%
& '
(
) *
!
"T
"t=#
" 2T
"z2
!
T = T1at t = 0 z > 0
T = T0at z = 0 t > 0
T" T1as z"# t > 0
!
" =T #T
1
T0#T
1
$"
$t=%
$ 2"
$z2
" z,0( ) = 0
" 0,t( ) =1
" &,t( ) = 0
1次元の熱伝導方程式を解く
(1)
(2)
境界条件: - アセノスフェアは一定温度Tm - アセノスフェアはある時刻(t=0)に地表(z=0)に上昇 - 地表の温度は常にT0 - 下限の条件はない(半無限モデル) - 水平方向への熱の拡散はない
プレート冷却モデル単純に地表からの冷却だけを考慮した場合、熱境界層の厚さはどうなるか?
!
"Q = kAT2#T
1
L"t
qz = #1
A
dQ
dt= #k
dT
dz
!
dQz
dzdz =
dqz
dzdz(dxdy)dt = k
d2T
dz2dVdt
!
"Q = cpm"T
!
cpm"T = cp#dVdT
!
cp"dVdT = kd2T
dz2dVdt
dT
dt=
k
cp"
d2T
dz2
# =k
cp"
$T
$t=#
$ 2T
$x 2+$ 2T
$y 2+$ 2T
$z2%
& '
(
) *
!
"T
"t=#
" 2T
"z2
!
T = T1at t = 0 z > 0
T = T0at z = 0 t > 0
T" T1as z"# t > 0
!
" =T #T
1
T0#T
1
$"
$t=%
$ 2"
$z2
" z,0( ) = 0
" 0,t( ) =1
" &,t( ) = 0
(Fundamentals of Geophysics 2nd ed., Lowrie, 2007)
q: 熱流量
Cp: 定圧比熱
k: 熱伝導率
熱拡散係数 [length2/time]
(1)
熱伝導方程式
!
"Q = kAT2#T
1
L"t
qz = #1
A
dQ
dt= #k
dT
dz
!
dQz
dzdz =
dqz
dzdz(dxdy)dt = k
d2T
dz2dVdt
!
"Q = cpm"T
!
cpm"T = cp#dVdT
!
cp"dVdT = kd2T
dz2dVdt
dT
dt=
k
cp"
d2T
dz2
# =k
cp"
$T
$t=#
$ 2T
$x 2+$ 2T
$y 2+$ 2T
$z2%
& '
(
) *
!
"T
"t=#
" 2T
"z2
!
T = T1at t = 0 z > 0
T = T0at z = 0 t > 0
T" T1as z"# t > 0
!
" =T #T
1
T0#T
1
$"
$t=%
$ 2"
$z2
" z,0( ) = 0
" 0,t( ) =1
" &,t( ) = 0
境界条件がきわめてシンプルになる
similarity variable ηの導入
!
" =z
2 #t
$%
$t=d%
d"
$"
$t=d%
d"&1
4
z
#t
1
t
'
( )
*
+ , =
d%
d"&1
2
"
t
'
( )
*
+ ,
$%
$z=d%
d"
$"
$z=d%
d"
1
2 #t
$ 2%
$z2=
1
2 #t
d2%
d"2$"
$z=1
4
1
#t
d2%
d"2
&"d%
d"=1
2
d2%
d"2
%(-) = 0
%(0) =1
!
" =d#
d$
%$" =1
2
d"
d$
%$d$ =1
2
d"
"
%$2 = ln" % lnc1
" = c1e%$ 2
=d#
d$
# = c1
e%$ ' 2
d$'+10
$
&
0 = c1
e%$ ' 2
d$'+10
'
&
e%$ ' 2
d$'=(
20
'
&
c1
= %2
(
# =1%2
(e%$ ' 2
d$'0
$
&
!
erf (") =2
#e$" ' 2
0
"
% d"'
erfc(") =1$ erf (")
T $T1
T0$T
1
= erfcz
2 &t
!
" =z
2 #t
$%
$t=d%
d"
$"
$t=d%
d"&1
4
z
#t
1
t
'
( )
*
+ , =
d%
d"&1
2
"
t
'
( )
*
+ ,
$%
$z=d%
d"
$"
$z=d%
d"
1
2 #t
$ 2%
$z2=
1
2 #t
d2%
d"2$"
$z=1
4
1
#t
d2%
d"2
&"d%
d"=1
2
d2%
d"2
%(-) = 0
%(0) =1
!
" =d#
d$
%$" =1
2
d"
d$
%$d$ =1
2
d"
"
%$2 = ln" % lnc1
" = c1e%$ 2
=d#
d$
# = c1
e%$ ' 2
d$'+10
$
&
0 = c1
e%$ ' 2
d$'+10
'
&
e%$ ' 2
d$'=(
20
'
&
c1
= %2
(
# =1%2
(e%$ ' 2
d$'0
$
&
!
erf (") =2
#e$" ' 2
0
"
% d"'
erfc(") =1$ erf (")
T $T1
T0$T
1
= erfcz
2 &t
無次元
(3)
同様に境界条件(2)式は (4)
(3)式の左辺は
(5) (6)
規格化:無次元量 θ の導入
式(1)をθで書き換えると
(3)式の右辺は
(3),(4)はθとηを使って
もっとシンプルになる
!
" =z
2 #t
$%
$t=d%
d"
$"
$t=d%
d"&1
4
z
#t
1
t
'
( )
*
+ , =
d%
d"&1
2
"
t
'
( )
*
+ ,
$%
$z=d%
d"
$"
$z=d%
d"
1
2 #t
$ 2%
$z2=
1
2 #t
d2%
d"2$"
$z=1
4
1
#t
d2%
d"2
&"d%
d"=1
2
d2%
d"2
%(-) = 0
%(0) =1
!
" =d#
d$
%$" =1
2
d"
d$
%$d$ =1
2
d"
"
%$2 = ln" % lnc1
" = c1e%$ 2
=d#
d$
# = c1
e%$ ' 2
d$'+10
$
&
0 = c1
e%$ ' 2
d$'+10
'
&
e%$ ' 2
d$'=(
20
'
&
c1
= %2
(
# =1%2
(e%$ ' 2
d$'0
$
&
!
erf (") =2
#e$" ' 2
0
"
% d"'
erfc(") =1$ erf (")
T $T1
T0$T
1
= erfcz
2 &t
積分定数c1を(6)から決める
!
" =z
2 #t
$%
$t=d%
d"
$"
$t=d%
d"&1
4
z
#t
1
t
'
( )
*
+ , =
d%
d"&1
2
"
t
'
( )
*
+ ,
$%
$z=d%
d"
$"
$z=d%
d"
1
2 #t
$ 2%
$z2=
1
2 #t
d2%
d"2$"
$z=1
4
1
#t
d2%
d"2
&"d%
d"=1
2
d2%
d"2
%(-) = 0
%(0) =1
!
" =d#
d$
%$" =1
2
d"
d$
%$d$ =1
2
d"
"
%$2 = ln" % lnc1
" = c1e%$ 2
=d#
d$
# = c1
e%$ ' 2
d$'+10
$
&
0 = c1
e%$ ' 2
d$'+10
'
&
e%$ ' 2
d$'=(
20
'
&
c1
= %2
(
# =1%2
(e%$ ' 2
d$'0
$
&
!
erf (") =2
#e$" ' 2
0
"
% d"'
erfc(") =1$ erf (")
T $T1
T0$T
1
= erfcz
2 &t
周知の定積分を用いて
(7)
(8)
(9)
(5)式は
積分を実行すると
!
" =z
2 #t
$%
$t=d%
d"
$"
$t=d%
d"&1
4
z
#t
1
t
'
( )
*
+ , =
d%
d"&1
2
"
t
'
( )
*
+ ,
$%
$z=d%
d"
$"
$z=d%
d"
1
2 #t
$ 2%
$z2=
1
2 #t
d2%
d"2$"
$z=1
4
1
#t
d2%
d"2
&"d%
d"=1
2
d2%
d"2
%(-) = 0
%(0) =1
!
" =d#
d$
%$" =1
2
d"
d$
%$d$ =1
2
d"
"
%$2 = ln" % lnc1
" = c1e%$ 2
=d#
d$
# = c1
e%$ ' 2
d$'+10
$
&
0 = c1
e%$ ' 2
d$'+10
'
&
e%$ ' 2
d$'=(
20
'
&
c1
= %2
(
# =1%2
(e%$ ' 2
d$'0
$
&
!
erf (") =2
#e$" ' 2
0
"
% d"'
erfc(") =1$ erf (")
T $T1
T0$T
1
= erfcz
2 &t
(9)を(8)式に代入すると
相補誤差関数を使うと
熱構造(温度分布)が時間tと深さzの関数で求まる
(10)
(11)
と置くと微分方程式を解く
誤差関数 error function
平均0,標準偏差1とした正規分布の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 0.6778
1.4 0.168 0.8 0.7421
0.2 0.2227 0.9 0.79691
0.25 0.,7633 1 0.8427
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 0.5205 2 0.99532
!
" =z
2 #t
$%
$t=d%
d"
$"
$t=d%
d"&1
4
z
#t
1
t
'
( )
*
+ , =
d%
d"&1
2
"
t
'
( )
*
+ ,
$%
$z=d%
d"
$"
$z=d%
d"
1
2 #t
$ 2%
$z2=
1
2 #t
d2%
d"2$"
$z=1
4
1
#t
d2%
d"2
&"d%
d"=1
2
d2%
d"2
%(-) = 0
%(0) =1
!
" =d#
d$
%$" =1
2
d"
d$
%$d$ =1
2
d"
"
%$2 = ln" % lnc1
" = c1e%$ 2
=d#
d$
# = c1
e%$ ' 2
d$'+10
$
&
0 = c1
e%$ ' 2
d$'+10
'
&
e%$ ' 2
d$'=(
20
'
&
c1
= %2
(
# =1%2
(e%$ ' 2
d$'0
$
&
!
erf (") =2
#e$" ' 2
0
"
% d"'
erfc(") =1$ erf (")
T $T1
T0$T
1
= erfcz
2 &t
誤差関数
相補誤差関数
(Fundamentals of Geophysics 2nd ed., Lowrie, 2007)
!
t = x /u
T "T1
T0"T
1
= erfc(z
2 #x /u)
1"T "T
0
T1"T
0
=1" erf (z
2 #x /u)
T "T0
T1"T
0
= erf (z
2 #x /u)
!
q0
= "k#T
#z
$
% &
'
( ) z= 0
= "k(T1"T
0)#
#zerf
z
2 *x /u
$
% &
'
( )
$
% &
'
( ) z= 0
= k(T0"T
1)
2 *x /u
d
d+(erf (+))+= 0
=k(T
0"T
1)
2 *x /u
2
,e"+ 2$
% &
'
( ) += 0
= k(T0"T
1)
u
,*x
!
"dz + w"w0
zL#
"m (w + zL ) = "dz + w"w0
zL#" $ "m = "m%(T1 $T)
w("m $ "w ) = "m%(T1 $T0) & erfcz
2
u
'x
(
) *
+
, -
0
.
# dz
w =2"m%(T1 $T0)
("m $ "w )
'x
uerfc(/)d/
0
.
#
=2"m%(T1 $T0)
("m $ "w )
'x
0u
時間の代わりに片側海底拡大速度 u を使うと
(11)式は (12)
(14)
(13)
深海では海底面付近での温度は概ね 0°C, T0=0, T1=Tm (対流するマントルの温度)とすると
熱境界層の下限をT1の90%に達する等温線とすると T=0.9Tm erf(x)=~ 0.9 at x=1.16 (誤差関数表から読み取る)
x,zの関数としても書ける