Characteristic time of thermal and diffusional relaxation
It is shown that frequently used dimensional estimate of a layer thermal relaxation time, tr, as L²/k based on the layer thickness, L, and thermal diffusivity, к, strongly overestimates tr. The correct estimate tr=L²/π²k should contain a factor of 1/π² which is of the order of 0.1. Показано, что час...
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Aryasova, O.V. Khazan, Ya.M. 2016-06-23T13:31:17Z 2016-06-23T13:31:17Z 2015 Characteristic time of thermal and diffusional relaxation / O.V. Aryasova, Ya.M. Khazan // Геофизический журнал. — 2015. — Т. 37, № 1. — С. 99-104. — Бібліогр.: 3 назв. — англ. 0203-3100 https://nasplib.isofts.kiev.ua/handle/123456789/103741 It is shown that frequently used dimensional estimate of a layer thermal relaxation time, tr, as L²/k based on the layer thickness, L, and thermal diffusivity, к, strongly overestimates tr. The correct estimate tr=L²/π²k should contain a factor of 1/π² which is of the order of 0.1. Показано, что часто используемая размерная оценка времени тепловой релаксации tr=L²/k, где L - мощность слоя, а к - температуропроводность, дает сильно завышенное значение tr. Правильная оценка tr=L²/π²k, должна содержать множитель 1/π² порядка 0,1. ru Інститут геофізики ім. С.I. Субботіна НАН України Геофизический журнал Научные сообщения Characteristic time of thermal and diffusional relaxation Характерное время тепловой и диффузионной релаксации Article published earlier |
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Characteristic time of thermal and diffusional relaxation |
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Characteristic time of thermal and diffusional relaxation Aryasova, O.V. Khazan, Ya.M. Научные сообщения |
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Characteristic time of thermal and diffusional relaxation |
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Characteristic time of thermal and diffusional relaxation |
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Characteristic time of thermal and diffusional relaxation |
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Characteristic time of thermal and diffusional relaxation |
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characteristic time of thermal and diffusional relaxation |
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Aryasova, O.V. Khazan, Ya.M. |
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Aryasova, O.V. Khazan, Ya.M. |
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Научные сообщения |
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Научные сообщения |
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2015 |
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Геофизический журнал |
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Інститут геофізики ім. С.I. Субботіна НАН України |
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Article |
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Характерное время тепловой и диффузионной релаксации |
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It is shown that frequently used dimensional estimate of a layer thermal relaxation time, tr, as L²/k based on the layer thickness, L, and thermal diffusivity, к, strongly overestimates tr. The correct estimate tr=L²/π²k should contain a factor of 1/π² which is of the order of 0.1.
Показано, что часто используемая размерная оценка времени тепловой релаксации tr=L²/k, где L - мощность слоя, а к - температуропроводность, дает сильно завышенное значение tr. Правильная оценка tr=L²/π²k, должна содержать множитель 1/π² порядка 0,1.
|
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0203-3100 |
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https://nasplib.isofts.kiev.ua/handle/123456789/103741 |
| citation_txt |
Characteristic time of thermal and diffusional relaxation / O.V. Aryasova, Ya.M. Khazan // Геофизический журнал. — 2015. — Т. 37, № 1. — С. 99-104. — Бібліогр.: 3 назв. — англ. |
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2025-11-25T22:33:15Z |
| last_indexed |
2025-11-25T22:33:15Z |
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| fulltext |
CHARACTERISTIC TIME OF THERMAL AND DIFFUSIONAL RELAXATION
Геофизический журнал № 6, Т. 37, 2015 99
Introduction. Notwithstanding the correct
way to estimate thermal and diffusional relaxation
time is well known and straightforward, in some
publications estimates of thermal relaxation time
contain major errors. Most typically, when esti-
mating the thermal relaxation time based upon
dimensional analysis, tr=CL2 , where L is a layer
thickness and κ is heat diffusivity, one assumes by
default that the numerical factor, , is of the order
of unity while for a 1D geometry it is, in fact, 1/π2.
As a consequence, a statement that a character-
istic thermal relaxation time for a lithosphere of,
say, L=200 km thickness is of the order of a billion
years is incorrect. The same relates to an estimate
of time necessary for a thermal perturbation at a
depth L to manifest itself in the surface heat flow.
Actually, in both cases the characteristic time is
of L2/π2κ, i. e. as short as ca. 140 m.y. for L=200 km
and κ=10–6 m2s–1.
In the present letter we remind at the begin-
ning what is meant by the term «the relaxation
time», and then discuss a few most common prob-
lems where this value is useful. To be specific, in
what follows we speak of the thermal relaxation
although all results up to notation are valid for the
problem of the diffusional relaxation.
For those who are not interested in the mathe-
matical details, Table 1 summarizes the relaxation
time expressions for a number of boundary condi-
tions and the geometry of the system (one-dimen-
sional or spherically symmetric). Also, Table 2 lists
the thermal relaxation time estimates for the conti-
nental (Precambrian) and oceanic lithosphere. The
estimates assume κ=0.8·10–6 m2s–1. The values of
Archaean and Proterozoic lithosphere thickness-
es are from [Artemieva, 2009]. In the suboceanic
mantle, a thickness of the conductive lithosphere
as well as a thickness of the thermal boundary lay-
er (the lithosphere together with a layer accom-
Characteristic time of thermal and diffusional relaxation
© O. V. Aryasova, Ya. M. Khazan, 2015
Institute of Geophysics, National Academy of Sciences of Ukraine, Kiev, Ukraine
Received October 12, 2015
Presented by the Editorial Board Member V. N. Shuman
Показано, что часто используемая размерная оценка времени тепловой релаксации tr=L2 ,
где L — мощность слоя, а κ — температуропроводность, дает сильно завышенное значение tr.
Правильная оценка tr=L2 2 должна содержать множитель 1/π2 порядка 0,1.
Ключевые слова: характерное время, тепловая релаксация.
modating a transition to the convecting mantle)
depends on rock rheology and potential tempera-
ture, Tp, of the mantle convection [Khazan, Ary-
asova, 2014]. The values in Table 2 are based upon
the laboratory data by Hirth and Kohlstedt [2003]
and Tp range of 1350 °C to 1300 °C (see details in
[Khazan, Aryasova, 2014]).
It is instructive to observe in Figs. 1 and 2 how
a layer thermally perturbed at its base approaches
a new steady state.
The definition of relaxation time. Let a depen-
dence of a system on time, t, be characterized by
a function A(t) with the value A=A0 of the function
corresponding to the stable equilibrium. The latter
means that if the system is brought out of the equi-
librium then the sign of the rate dA/dt is opposite
to that of the deviation A–A0 and dA/dt vanishes
at A=A0. Therefore, the series expansion of dA/dt
contains only odd powers of A–A0. In the simplest
case only linear term remains:
0( )dA A A
dt
, (1)
where γ is a positive factor.
The solution to Eq. (1) is as follows
0 1 0( ) ( ) tA t A A A e , (2)
where A1 is the initial value of A, i. e. (0)=A1.
One can see from Eq. (2) that the absolute value
of the system deviation from the equilibrium, |A(t)–
A0|, decreases exponentially, and the time required
for |A(t)–A0| to decay from its initial value |A1–A0|
to 1/e of that value is tr=1/γ. This time interval is
commonly referred to as the relaxation time. Note
that the relaxation time is independent of the ini-
tial deviation.
Thermal relaxation time for an infinite ho-
mogeneous layer with fixed temperature at the
НАУЧНЫЕ СООБЩЕНИЯ
O. V. ARYASOVA, Y. M. KHAZAN
100 Геофизический журнал № 6, Т. 37, 2015
Fig. 1. The temperature increment (Eq. (14); the term in square brackets) vs. time in units of the characteristic relaxation
time (Eq. (13)).
Ta b l e 1. Thermal/diffusional relaxation time, tr , depending on the geometry and the boundary
conditions of the problem (κ is thermal diffusivity, D is diffusion coefficient)
Problem Boundary conditions Relaxation time
Relaxation of the temperature in
a layer of thickness L after an in-
stantaneous change of basal tem-
perature.
Temperature boundary condition on both
boundaries
2
2r
Lt =
Time necessary for a thermal
perturbation at a depth L to mani-
fest itself in the surface heat flow
The same
2
2r
Lt =
Relaxation of the temperature in a
layer after an instantaneous change
in heat flow through its base
Temperature and heat flow boundary con-
ditions on top and bottom boundary, res-
pectively
2
2
4
r
Lt =
Relaxation of the temperature (mi-
nor element abundance) in a sphe-
rical body
Zero temperature (zero abundance) boun-
dary condition. In the center of the body
the temperature (abundance) is finite.
2
2r
Rt = ,
2
2r
Rt
D
=
boundaries. To be specific, we assume that the
layer is horizontal, the axis z is directed downward,
and the temperature at the surface is zero.
The heat propagation in the layer is described
by the 1D heat equation:
2
2 ( )T T Q z
t z
, (3)
where T(t,z) is temperature, t is time, z is depth, κ
is the heat diffusivity, and Q (z) is to account for
the heat generation.
CHARACTERISTIC TIME OF THERMAL AND DIFFUSIONAL RELAXATION
Геофизический журнал № 6, Т. 37, 2015 101
Assume that at t<0 the layer was in its steady
state T01(z), satisfying the steady state version of
Eq. (3)
2
2 ( ) 0T Q z
z
, (4)
with the basal temperature T(L)= 1.
At t=0 the basal temperature changes stepwise
from 1 to 2= 1 and the layer relaxes to the
new steady state
02 01( ) ( ) zT z T z T
L
, (5)
satisfying Eq. (4) with the basal temperature
T02(L)= 2.
The solution T(t,z) to Eq. (3) with the boundary
conditions T(t,0)=0, T(t,L)=T2 and the initial condi-
tion T(0,z)=T01(z) may be expressed as
02( , ) ( ) ( , )T t z T z u t z , (6)
where u(t,z)=T02(z)–T(t,z) is the difference between
the steady state temperature, T02(z), and the cur-
rent one, T(t,z). |u(t,z)| is maximum at t=0 and tends
to zero with time. u(t,z) is the solution to Eq. (3)
without sources
2
2
u u
t z
, (7)
with zero boundary conditions u(t,0)=u(t,L)=0 and
the initial condition u(0,z)=T02(z)–T01(z Tz/L. In
terms of dimensionless coordinate z/L, ,
and time t /L2 the boundary value problem Eq.
(7) for function w T T may be rewritten as
2
2
w w
t
= , w w w . (8)
To apply the standard method of separation
of variables we make the substitution of the form
( , ) ( ) ( )w and for the space part obtain
an eigenvalue problem:
" 0, (0) (1) 0 , (9)
where λ is the separation parameter.
A general solution of Eq. (9) satisfying the
boundary conditions is
( ) sinn n nC , 2 2
n n , 1,2,...n = . (10)
Solving also the equation for the time part
one may write a general solution to problem Eq.
(8) as an exponentially converging series:
2 2
1
( , ) sinn
n
n
w C e n
=
. (11)
Here Cn=2(–1)n+1 n is Fourier coefficients of
sine expansion of the initial condition w .
Since the series (11) converges exponentially,
the dimensionless relaxation time may be esti-
mated based upon its first term:
2
1
1 1 0,101r . (12)
The dimensional relaxation time is as follows:
2
2r
Lt = . (13)
Now, one may estimate from Eq. (13) that the
characteristic relaxation time of Archaean litho-
sphere of L=200 km thick is tr=168 m. y. while for
a 40 km crust tr=6.8 m. y. (in both estimates the
value κ=0.8·10–6 м2с–1 is used).
The closed dimensional solution to Eq. (3) may
be written as:
01( , ) ( )T t z T z= +
2 2
2
1
( 1)1 2 sin
tn n
L
n
z L zT n e
L z n L=
. (14)
The second term in the latter equation char-
Ta b l e 2. Some useful estimates of the relaxation time
Thickness, km Age, Ma tr, Ma
Continental lithosphere
Archaean
Proterozoic
180—250
150—180
>2500
500—2500
137—264
95—137
Suboceanic mantle
Lithosphere
Thermal boundary layer
50—70*
75—110*
<200 11—21
25—49
* The range of thicknesses corresponds to the range of values of the mantle convection potential temperature of 1350 °C to
1300 °C (see text).
O. V. ARYASOVA, Y. M. KHAZAN
102 Геофизический журнал № 6, Т. 37, 2015
acterizes an increment of the temperature due to
a thermal perturbation at the bottom of the layer.
This term is zero initially and tends to ΔTz/L af-
terwards. The most slowly the temperature ap-
proaches to its steady state value in the vicinity of
the surface, i.e. the farthest from the perturbation
source. However, even in the close vicinity of the
surface, e. g. at z/L=0.01, this term is of 0.3ΔTz/L at
t=tr, 0.73ΔTz/L at t=2tr, and 0.9ΔTz/L at t=3tr. The
approach of the temperature to the new steady
state is illustrated in Fig. 1.
It is also instructive to calculate the variation
of the heat flow q(t, T(t due to the ther-
mal perturbation operating at the depth L. It is
straightforward to see from Eq. (14) that
01( , ) ( )q t z q z= +
2 2
2
1
1 2 ( 1) cos
tn
n L
n
zq n e
L=
, (15)
where q01 T01 is the initial value of the
heat flow at the depth z, and q T/L is the ad-
ditional steady state heat flow due to the pertur-
bation. The relaxation of the heat flow at differ-
ent depths is illustrated in Fig. 2. In particular, it
follows from Eq. (15) that the characteristic time
necessary for the perturbation operating at the
depth L to affect the surface heat flow is tr. Actu-
ally, the surface heat flow increment (the second
term in the right hand side of Eq. (15) taken at z=0)
is of 0.3Δq at t=tr, 0.7Δq at t=2tr, and 0.9Δq at t=3tr.
Thermal relaxation time of a homogeneous
layer in the case of the heat flow bottom bound-
ary condition. From the geophysical viewpoint,
this problem does not make much sense because
under natural conditions in the Earth a system in
which one part supplies a constant heat flux into
another part cannot occur. On the other hand,
one can imagine a body irradiated by a beam of
particles absorbed by the body surface and dif-
fusing inward. This situation is well described by
constant particle flux boundary condition.
The evolution of the system with constant heat
flow boundary condition is described by Eq. (3)
with different boundary conditions:
2
2 ( )T T Q z
t z
; T(t,0)=0; 0
( , )T t L q
z
. (16)
Let T01(z) be the steady state solution to Eq.
(16), i. e. the solution satisfying Eq. (16) with zero
Fig. 2. Increment of the heat flux at different depths (Eq. 15; the term in the square brackets) vs. time in units of the characteristic
relaxation time (Eq. (13)).
CHARACTERISTIC TIME OF THERMAL AND DIFFUSIONAL RELAXATION
Геофизический журнал № 6, Т. 37, 2015 103
left hand side. If at t=0 the heat flow through the
bottom boundary changes by Δq, the bottom
boundary condition becomes as follows:
( )
0
,u t L
q q
z
. (17)
After the relaxation is over, the new steady
state is
02 01( , ) ( ) qzT t z T z . (18)
It is convenient to look for the solution to Eq.
(16) with boundary condition Eq. (17) in the form
02( , ) ( ) ( , )T t z T z w t z= + , (19)
where w(t,z) is the solution to Eq. (16) without
sources, with zero surface temperature, zero bot-
tom boundary heat flow, and initial condition
w(0,z)=T(0,z)–T02(z)=z q .
Introducing now dimensionless coordinate
z/L, time t/L2, and temperature u= w/L q, we
arrive at the problem
2
2
u u
= , u
( ,1) 0u
= , u (20)
Similarly to the preceding problem Eq. (8), this
one may be solved using the separation of vari-
ables method. The final solution may be written as
01( , ) ( ) qL zT t z T z
L
(21)
2
2
2
1
2
2 2
0
8 ( 1) 11 sin
2(2 1)
tn n
L
n
L zn e
z Ln=
.
Note that if the heat flows upward then Δq is
negative. The characteristic relaxation time for
this problem is
2
2
4
r
Lt = , (22)
i. e. in a factor of 4 longer than that for the problem
Eq. (3) with temperature boundary condition at the
bottom boundary. At t=tr the temperature incre-
ment (the second term in the right hand side of Eq.
(21)) is about 0.6 of the steady state value q|z ,
at t=2tr is of 0.85 q|z , and at t=3tr is of 0.95 q|z .
Relaxation time of impurity abundance/tem-
perature in a sphere. Consider a problem of diffu-
sion of an absolutely incompatible impurity from
a spherical «crystal» into melt. Since the distribu-
tion coefficient for the impurity equals to zero, the
boundary condition for the problem is zero impu-
rity abundance. The second boundary condition
is the finiteness of the abundance in the sphere
center. Up to the notation, the problem is the same
as the problem of a sphere cooling.
The governing equation for the diffusion prob-
lem is as follows:
2
2
1C CD r
t r rr
= ,
C(t,R)=0, C(t, C(0,r)=C0 r<R, (23)
where C(t,r) is abundance, r is the distance from
the sphere center, R is the sphere radius, D is dif-
fusivity.
Similarly to preceding problems it is useful to
introduce dimensionless radius r/R and time
Dt/R2 and use the separation of variables meth-
od, i. e. look for the solution in the form C
. As usually, for the space part one obtains the
eingenvalue problem
2" ' 0P P P ,
P(1)=0, , (24)
where λ is the separation parameter.
A general solution to Eq.(24), satisfying the
condition of the abundance finiteness in the cen-
ter, is B -1/2J1/2
1/2 , where В is a constant,
J1/2( ) is the first kind Bessel function of real ar-
gument and of order 1/2: J1/2( )1/2sin . The
eigenvalue of the problem is 2 2, and a general
solution to Eq. (23) is:
2 2
21
10
sin2 ( 1)
Dtn
n R
n
C nr Re
C nr R
+
=
. (25)
The complete content of the impurity in the
«crystal» decreases with time in agreement with
the next equation
2 2
2
2
0
3 2 2
10
4 ( , )
16
4 / 3
R
Dtn
R
n
r C t r dr
e
R C n=
= . (26)
The convergence of the abundance to the
steady state zero value is controlled by the slow-
est varying first term of the series (26). Therefore,
the characteristic time of the diffusional relaxation
in the sphere of radius R is tr=R2 2D.
O. V. ARYASOVA, Y. M. KHAZAN
104 Геофизический журнал № 6, Т. 37, 2015
References
Artemieva I. M., 2009. The continental lithosphere:
Reconciling thermal, seismic, and petrologic data.
Lithos 109(12), 23—46.
Hirth G., Kohlstedt D. L., 2003. Rheology of the upper
mantle and the mantle wedge: A view from the ex-
perimentalists. In: Inside the Subduction Factory.
Vol. 138. Geophys. Monogr. Ser., AGU, Washington,
D.C. P. 83—105.
Khazan Ya. M., Aryasova O. V., 2014. Stability of the boun-
dary layer between the lithosphere and convecting
mantle and the steady-state lithospheric geotherm.
Izvestiya. Physics of the Solid Earth 50(4), 543—561.
Characteristic time of thermal and diffusion relaxation
© O. V. Aryasova, Y. M. Khazan, 2015
It is shown that frequently used dimensional estimate of a layer thermal relaxation time, tr,
as L2 based on the layer thickness, L, and thermal diffusivity, κ, strongly overestimates tr. The
correct estimate tr=L2 2 should contain a factor of 1/π2 which is of the order of 0.1.
Key words: characteristic time, thermal relaxation.
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