PUBLICATIONS
Journal of Geophysical Research: Earth Surface
RESEARCH ARTICLE
10.1002/2013JF002930
Key Points:
• Coupled moisture-heat process of
permafrost near thermokarst pond
is analyzed
• The specific time when talik beneath
pond penetrates permafrost
is determined
• A formula on the penetrative time
versus rate of temperature rise
is obtained
Correspondence to:
S. Li and H. Zhan,
[email protected];
[email protected]
Citation:
Li, S., H. Zhan, Y. Lai, Z. Sun, and W. Pei
(2014), The coupled moisture-heat process of permafrost around a thermokarst
pond in Qinghai-Tibet Plateau under
global warming, J. Geophys. Res. Earth
Surf., 119, 836–853, doi:10.1002/
2013JF002930.
Received 22 JUL 2013
Accepted 4 MAR 2014
Accepted article online 26 MAR 2014
Published online 9 APR 2014
The coupled moisture-heat process of permafrost
around a thermokarst pond in Qinghai-Tibet
Plateau under global warming
Shuangyang Li1,2, Hongbin Zhan2, Yuanming Lai1, Zhizhong Sun1,3, and Wansheng Pei1
1
State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute,
Chinese Academy of Sciences, Lanzhou, Gansu, China, 2Department of Geology and Geophysics, Texas A&M University,
College Station, Texas, USA, 3Beiluhe Observation and Research Station on Frozen Soil Engineering and Environment in
Qinghai-Tibet Plateau, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of
Sciences, Lanzhou, Gansu, China
Abstract Due to environmental disturbances such as local human activity and global warming, melting of
massive ground ice has resulted in thermokarst ponds, which are extensively distributed in the Qinghai-Tibet
Plateau (QTP). Besides the global warming, the thermokarst pond, as a major heat source, speeds up the moisture
change and degradation of its surrounding permafrost. To analyze the long-term coupled moisture-heat
process near a representative nonpenetrative thermokarst pond in a permafrost region, abundant temperature
data over multiple years at different depths and horizontal distances from the center of the thermokarst pond
have been collected at a field experimental station in QTP. A numerical model is built to analyze this
thermokarst pond. The temperature and moisture processes of surrounding permafrost are simulated by this
model and compared with measured temperature data. Our results show that if the rate of air temperature rise
is 0.048°C/yr, which refers to a 2.4°C temperature rise over 50 years, the thawing fronts underneath the
thermokarst pond move downward at a linear rate of 0.18 m/yr and the permafrost beneath the pond center
would disappear after the year of 2281. Beyond that time, the impact range of the pond on the natural
ground increases to about 50 m in horizontal direction. So a dish-shape thawing zone occurs around the
thermokarst pond. Simultaneously, the moisture state is greatly changed in 2281 and becomes completely
different from that in 2013. All of these would inevitably deteriorate the ecological and environmental system
in QTP.
1. Introduction
Permafrost is widespread around the world, particularly in places such as Eurasia, North America, and
Antarctica continents [Zhou et al., 2000], and it is very sensitive to global temperature changes that are
likely to occur in the coming millenniums [Klein et al., 2007; Zhang et al., 2001, 2005]. Understanding the
dynamics of permafrost and its response to environmental perturbation is of great international concern
because of its obvious impact to the ecological community and socioeconomic stability [Qiu, 2012; Jones
et al., 2011; Ling and Zhang, 2003]. This study uses a well-monitored permafrost field site in Qinghai-Tibet
Plateau (QTP) to illustrate the fundamental hydrological and thermal processes controlling the permafrost
dynamics. Particularly, we will investigate the response of permafrost dynamics to the potential
environmental perturbation factors such as global warming.
Permafrost is widely distributed in QTP which has an average altitude of over 4000 m above mean sea level
(msl) (Figure 1) [Zhou et al., 2000; Cheng and Wu, 2007; Ge et al., 2011]. Such permafrost typically has the
widest area, the greatest thickness, and the lowest temperature among the middle-low latitudinal zones
in the northern hemisphere [Zhou et al., 2000]. Due to its high altitude and large area, the permafrost in
QTP plays a significant role in regional environmental changes and even affects the global climate
system [Yao et al., 2012; Jin et al., 2000], so QTP is often regarded as the “third pole” of the earth by
the public.
There are more than 1500 lakes/ponds in QTP at present [Ling et al., 2012]. Such an abundant surface feature
is a consequence of local permafrost degradation following post environmental disturbances, such as human
activities and climate warming [Jones et al., 2011; Lin et al., 2010]. Normally, the mean annual temperatures of
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
836
Journal of Geophysical Research: Earth Surface
10.1002/2013JF002930
Figure 1. Permafrost distribution in QTP [Cheng and Wu, 2007; Ge et al., 2011].
water in these lakes are greater than 0°C, whereas the temperature of the neighboring land surface may
fluctuate near 1.0°C or lower [Cheng and Wu, 2007; Ling and Zhang, 2003; Harris, 2002]. Therefore, these
ponds are major heat sources responsible for accelerating their surrounding permafrost degradations. After
several decades or longer, a perennial thaw layer called talik will form, and the talik can even penetrate the
entire permafrost eventually, with simultaneous and gradual lateral permafrost degradation as well. Up to
now, some thermokarst ponds previously regarded as nonpenetrative have become penetrative and the
lateral thaw of permafrost has also speeded up in QTP [Sun et al., 2012; Cui et al., 2010]. The permafrost
degradation not only results in ground subsidence but also changes both surface and subsurface water
storage, thus affecting river discharge as well [Qiu, 2012; Karlsson et al., 2012; Yang et al., 2002]. Even worse,
the thawing permafrost can release great amounts of trapped carbon into atmosphere, which further
exacerbates global warming [Wu and Zhang, 2008; Ling and Zhang, 2003]. Therefore, the permafrost
degradation triggered by the thermokarst pond/lake has significant influence on the regional physical,
geomorphological, environmental, and climate processes.
During the past decades, the subject on permafrost degradation induced by thermokarst ponds/lakes has
attracted attention on several main permafrost regions. For instance, in Siberia region, Agafonov et al. [2004]
estimated the rate of thermokarst expansion in Western Siberia by dendrochronological analysis. Katamura
et al. [2009] reconstructed the initiation and development of a thermokarst lake in Central Yakutia from
macroscopic charcoal records. Kirpotin et al. [2009] and Karlsson et al. [2012] mapped thermokarst lake
changes using remote sensing analyses. In North America, Harris [2002] compared water temperature of Fox
Lake in Yukon Territory with soil temperature at 10 cm depth in the adjacent ground and found that water
could absorb between 5 and 7 times as much solar energy as the soil on an annual basis. Through
monitoring borehole temperatures over a period of 20 years, Yoshikawa and Hinzman [2003] observed that
some thermokarst ponds grew larger and initiated large taliks that had completely penetrated permafrost
near Council, Alaska. Jones et al. [2011] employed high spatial resolution remotely sensed imagery to detect
long-term mean expansion rate of thermokarst lakes with 0.35–0.39 m/yr on the northern Seward Peninsula,
Alaska. In QTP region, Cui et al. [2010] observed significant temperature changes in the soil around Honglianghe
thermokarst lake compared with those in natural ground. Through monitoring thermal regimes of a
thermokarst lake and adjacent natural ground in QTP, Lin et al. [2010, 2011], Niu et al. [2011], and Luo et al.
[2012] discovered that the mean annual ground temperature beneath the thermokarst lake was more than
5°C higher than that in the surrounding terrain at the same depth, and the talik had completely penetrated
the permafrost. Sun et al. [2012] analyzed the thermal characteristics of a nonpenetrative permafrost near a
thermokarst lake in Beiluhe Basin in QTP.
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
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Journal of Geophysical Research: Earth Surface
10.1002/2013JF002930
However, the thermokarst is a long-term physical geographical process, whose influence on permafrost
degradation in the future could not be predicted by short-term monitoring. In addition, the scattered
measurement data cannot give an adequate description of the spatial temperature distribution. A practical
alternative to the in situ monitoring is to simulate the sequential spatial and temporal temperature evolution
of the ground underneath and around the thermokarst lake/pond by adequate numerical methods based on
a physically sound conceptual model. To our knowledge, Ling and Zhang [2003, 2004] and Brouchkov et al.
[2004] estimated the permafrost thermal regime and talik development beneath thermokarst lakes in Alaska
and Yakutia, respectively. Until now, there is only one numerical analysis on open-talik formation and
permafrost lateral thaw under a thermokarst lake in QTP [Ling et al., 2012]. These theoretical studies may help
us estimate thermal development of the permafrost around a thermokarst lake/pond in the future.
Three important physical processes existing in the permafrost around a thermokarst lake/pond, including
moisture migration and formation of ice lenses at the freezing fringe and water replenishment from the lake/
pond, are not taken into account in previous studies. In fact, when the ground is subjected to continuous
freezing-thawing changes, the ground temperature gradients induce soil moisture migration from warm
to cold regions, and simultaneously, some ice lenses form at the freezing front. Moreover, the moisture
migration may be an important process to consider due to external water supply from the thermokarst
lake/pond. These moisture movement and freezing-thawing process are coupled and will influence the
thermal regime and the rate of heat transfer [Jame and Norum, 1980; Frampton et al., 2013; Painter et al.,
2013]. The existing theoretical studies could not reveal the actual temperature changes without
consideration of these three processes. In particular, as the so-called “third pole” of the earth and the
head water areas of the Yangtze River, the Yellow River, and the Lantsang River watersheds, QTP plays an
important role in regional environmental system. Besides, there are abundant evidences showing
accelerating permafrost degradation in QTP owing to global warming in recent decades [Wu and Zhang,
2008; Cheng and Wu, 2007; Jin et al., 2000; Zhang et al., 2007; Pang et al., 2009]. Therefore, it is urgent
and necessary to carry out experimental as well as theoretical studies on the influence of thermokarst lake/
pond upon permafrost in QTP under global warming.
To achieve this objective, we have taken into account the following procedures in our study. First, we built a
numerical moisture-heat coupled model for permafrost based on theories of soil moisture dynamics, heat
transfer, and physics of frozen soil. Second, we took a representative nonpenetrative thermokarst pond in
QTP as an example to simulate the temperature and moisture processes of permafrost around the
thermokarst pond by this model. Third, the simulated results were compared with field measured
temperature data to test the robustness of the model and to gain further insights into the coupled processes.
Through this investigation, we can delineate the characteristics of permafrost degradation and further
determine the times needed for a talik to penetrate the entire permafrost under different rates of air
temperature rise. This study is helpful to better understand the interaction between the thermokarst pond
and the permafrost, and it can serve as a reference for further investigation. Some limitations of the present
study are also summarized in the discussion.
2. Mathematical Model and Governing Equations
The shape of a nonpenetrative thermokarst pond is nearly circular (Figure 2), and it may be assumed as
spatially axial-symmetrical, so the numerical model is built under a cylindrical coordinate system.
During the process of heat transfer in soil, the heat convection is very small and could be neglected
compared with heat conduction [Nixon, 1975; Fuchs et al., 1978; Jame and Norum, 1980; Li et al., 2001].
However, moisture migration and ice-water phase change must be included in heat transport equation,
which is expressed as [Li et al., 2012]
∂T
∂
∂T
λ ∂T
∂
∂T
∂θi
cρ ¼
λ
þ
þ
λ
þ L ρi
;
∂t ∂r
∂r
r ∂r ∂z
∂z
∂t
(1)
where r and z are radial and vertical coordinates, respectively (m), T is temperature (°C), c represents specific
heat capacity of soil (J kg 1 °C 1), λ denotes thermal conductivity of soil (W m 1 °C 1), ρ and ρi are soil and
ice densities, respectively (kg m 3), t represents time (s), L is the latent heat of water (J kg 1), and θi denotes
i
volumetric ice content. In the unfrozen zone, the last term, L ρi ∂θ
∂t , is equal to 0.
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
838
Journal of Geophysical Research: Earth Surface
10.1002/2013JF002930
Figure 2. The thermokarst pond field site and the layout of measurement points [Sun et al., 2012]. (a) Thermokarst pond in
summer and (b) Thermokarst pond in winter.
The boundary conditions are as follows:
1. The temperature on boundary of S1 is known, that is,
T ¼ T;
(2)
where T is a known function of space and time, T ¼ T ðr; z; tÞ, and its most simple and common form is constant.
But it is a sinusoidal function to reflect the annual temperature change at the ground surface in this study.
2. The heat flux on boundary of S2 is fixed, which is
λ
∂T
∂T
nr þ λ nz ¼ qT ;
∂r
∂z
(3)
where nr and nz are the radial and axial components of the outward unit vector normal to the surface of S2,
respectively; qT is heat flux (W m 2), and it is also a known function of space and time, qT ¼ qT ðr; z; tÞ. Its
most particular and common form is constant, which is used in this study.
3. There is heat convection on another boundary of S3, and this boundary condition is
λ
∂T
∂T
nr þ λ nz ¼ hðT a T Þ;
∂r
∂z
(4)
where h is the convection coefficient (W m 2 °C 1) and Ta denotes ambient temperature (°C). Although there
is no heat convection in this study, we still list three boundary conditions above to provide a reference to
other problems.
Obviously, the above three boundary conditions are all linear and cannot be used to deal with the nonlinear
boundary conditions such as heat radiation and convection radiation.
The generalized moisture transport equation in unsaturated soil during freezing could be written as [Lei et al.,
1988; Shang et al., 1997]
∂θu
∂
∂θu
Dθ ∂θu ∂
∂θu
∂k θu ρi ∂θi
Dθ u
Dθu
¼
þ u
þ
þ
;
(5)
∂r
∂z
∂t
∂r
r ∂r
∂z
∂z
ρw ∂t
where Dθu is moisture diffusion coefficient of soil (m2 s 1); k θu denotes unsaturated hydraulic conductivity of
soil (m s 1); ρw is water density (kg m 3); θu is volumetric content of unfrozen water, and it is water content
θw in the unfrozen zone but is expressed as a temperature function in the frozen soil [Andersland and
Landanyi, 2004; Xu et al., 2010]:
b1
T θu ¼ a1 (6)
T0
where a1 and b1 are experimental constants; T0 is a reference temperature, and its value is 20°C in this study.
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
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Journal of Geophysical Research: Earth Surface
10.1002/2013JF002930
The moisture boundary conditions include three types [Lei et al., 1988]:
1. The moisture content on boundary of S1 is fixed, that is,
θu ¼ θu ;
(7)
where θu is a known volumetric water content, and it can be expressed as a known function of space and
time, θu ¼ θu ðr; z; tÞ. The moisture content is known at the bottom of the thermokarst pond, so this kind of
boundary condition is used in this study.
2. The moisture flux on boundary of S2 is known, which is
Dθu
∂θu
∂θu
nr þ Dθu
nz þ k θu ¼ qθu ;
∂r
∂z
(8)
where qθu is moisture flux (m s 1), qθu ¼ qθu ðr; z; tÞ. This kind of boundary condition can express the known
infiltration or evaporation rate at the ground surface [Lei et al., 1988].
3. The moisture flux on boundary of S3 changes linearly with θu, which could be expressed as
Dθu
∂θu
∂θu
nr þ Dθu
nz þ k θu ¼ cθu þ d;
∂r
∂z
(9)
where c and d are constants. This boundary condition is mainly used to reflect the second stage of
evaporation, in which the evaporation intensity is linear to water content [Lei et al., 1988].
Similar to the treatment of temperature boundary conditions, although this study utilizes the first and second
moisture boundary conditions, we prefer to list three boundary conditions above to provide a general
theoretical framework that can be used for a broad range of problems in the future.
In fact, the moisture and heat exchanges between the ground surface and atmosphere is a very complex land
surface processes, which is impossible to be disclosed completely by the above several moisture and heat
boundary conditions. Up to now, lots of studies are focus on this subject.
Substituting equation (5) into equation (1), introducing boundary conditions equations (2)–(4), and then
discretizing equation (1) by the Galerkin finite element method (FEM) will yield
ð½A þ Δt χ ½BÞ fT gtþΔt ¼ ð½A Δtð1 χ Þ½BÞ fT gt þ Δt χ ½C fk θu gtþΔt
þΔt ð1 χ Þ ½C fk θu gt þ Δt χ ½DtþΔt þ Δt ð1 χ Þ ½Dt ;
(10)
in which
½A ¼ ∫ ½NT c ½N rd Ω;
Ω
½ B ¼
∫
Ω
!
∂½N T ∂½N ∂½N T ∂½N λ
þ
rd Ω þ ∫ ½N T h½N rdS;
λ
λ
∂r
∂z
∂r
∂z
λ
s3
½C ¼ ∫ ½N T Lρw
Ω
½D ¼
∂ ½N rd Ω;
∂z
λ
∫ ½N T λ qT rdS þ ∫ ½N T
s2
s3
λ
hT a rdS;
λ
(11)
(12)
(13)
(14)
and
c ¼ c ρ þ L ρw
∂θu
;
∂T
λ ¼ λ þ Lρw Dθu
∂θu
;
∂T
(15)
(16)
where Δt represents time step, χ is integral weight factor between 0 and 1, and [N] denotes shape functions
matrix. In addition, there is no term of fk θu g in the unfrozen zone.
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
840
Journal of Geophysical Research: Earth Surface
10.1002/2013JF002930
Thus, the ice content in the frozen zone can be obtained from equation (1) as
½E fθi gtþΔt ¼ ½E fθi gt þ ð½F þ Δt χ ½G Þ fT gtþΔt þ ð½F þ Δt ð1 χ Þ ½G Þ fT gt
(17)
þΔt χ ½H tþΔt þ Δt ð1 χ Þ ½H t ;
in which
T
½ E ¼ ∫½N L ρi ½N rd Ω;
(18)
Ω
T
½ F ¼ ∫½N c ρ½Nrd Ω;
(19)
Ω
½G ¼
∫
Ω
!
∂½N T ∂½N ∂½N T ∂½N þ
rd Ω þ ∫ ½N T h½N rdS;
λ
λ
∂r
∂z
∂r
∂z
s3
½H ¼ ∫ ½NT qT rdS ∫ ½N T hT a rdS:
s2
(20)
(21)
s3
Introducing moisture boundary conditions equations (7)–(9) and discretizing equation (5) by FEM result in
ð½ I þ Δt χ ½J Þ fθu gtþΔt ¼ ð½ I Δtð1 χ Þ½ J Þ fθu gt þ ½K fθi gtþΔt ½K fθi gt þ Δt χ ½L fk θu gtþΔt
þΔt ð1 χ Þ ½L fk θu gt þ Δt χ ½M tþΔt þ Δt ð1 χ Þ ½Mt ;
(22)
in which
T
½ I ¼ ∫½N ½N rd Ω;
(23)
Ω
½J ¼
∫
Ω
∂ ½N T
∂½N ∂½N T
∂ ½N þ
Dθu
Dθu
∂r
∂z
∂r
∂z
!
rd Ω ∫ ½N T a½N rdS;
ρi
½Nrd Ω;
ρw
(25)
∂ ½N T
½N rd Ω;
Ω ∂z
(26)
½K ¼ ∫ ½N Ω
T
½L ¼ ∫
½M ¼
∫ ½N T qθ
s2
(24)
s3
rdS þ ∫ ½N T b rdS:
(27)
s3
Equations (6) and (10)–(27) make up the numerical moisture-temperature coupled model for frozen soil, and
corresponding FEM program is developed by FORTRAN.
During the numerical simulation, there is dramatic ice-water phase change in freezing-thawing fronts, and
even small latent heat of phase change will cause much temperature change in one time step, so the
computational program sometimes does not converge in this time step despite many iterations. To resolve
this numerical issue, a sequential series of decreasing time steps, determined by the bisection method, are
adopted and continued until the criterion for convergence is met.
3. Field Site Description and Numerical Analyses
3.1. Field Site Description
The thermokarst pond field site selected for this study is the Beiluhe test site in QTP with an elevation of
4666 m above msl (Figure 1). Beiluhe region is located in hinterland of QTP and belongs to alluvial and aeolian
high plain in geomorphology. In this site, the terrain is gently rolling and low mounds alternate with
depressions. The surface ground is mainly quaternary alluvial and diluvial silty sand, and the underlying
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
841
Journal of Geophysical Research: Earth Surface
10.1002/2013JF002930
stratum is tertiary mudstone. The type
of frozen soil is ice-rich permafrost with
the thickness of 50–80 m, and its mean
annual temperature ranges from 1.8°C
to 0.3°C and natural permafrost table
is 1.8 to 2.2 m, where negative sign
implies distance below ground surface
hereinafter [Zhang et al., 2013; Lin et al.,
2010]. The recorded meteorological data
from 2001 to 2010 show that the mean
annual air temperature is 3.6°C and
the mean annual precipitation is about
300 mm [Luo et al., 2012].
In order to monitor thermal regime of
the thermokarst pond, four boreholes
shown in Figure 2 were drilled in the
pond and surrounding ground, and
temperature measurements indicate
that the permafrost table was at a depth
of 6.0 m underneath the pond center
whereas it was 2.0 m underneath the
Figure 3. Numerical model of the thermokarst pond.
natural ground [Sun et al., 2012]. In
addition, the mean annual water
temperature at the pond bottom was about 5.5°C. Obviously, the thermokarst pond, acting as a heat resource,
has been triggering the thawing of surrounding permafrost.
3.2. Numerical Model
According to in situ geophysical conditions and geometric shape of the thermokarst pond, the detailed
computational model is shown in Figure 3. There are plenty of experimental and theoretical researches on
various kinds of testing embankments of Qinghai-Tibet Railway near the thermokarst pond [Liu et al., 2001; Xu
et al., 2010; Li et al., 2009], so some fundamental thermal parameters used in this paper are directly borrowed
from those studies and given in Table 1.
When a soil is freezing, the formation of ice can disrupt the paths of moisture transfer and the coefficients of
hydraulic conductivity and diffusivity function for the unfrozen soil cannot be applied to the frozen soil. To
account for the reduced transfer in the frozen soil, an impedance factor is introduced as the following [Jame
and Norum, 1980; Newman and Wilson, 1997].
k θu ¼
a 2 θ u b2
1010θi
(28)
Dθ u ¼
a 3 θ u b3
1010θi
(29)
where a2, b2, a3, and b3 are constants and their values are listed in Table 1.
According to climate predictions of QTP and long-term in situ monitoring of temperature on Beiluhe test site
[Qin, 2002; Sun et al., 2012; Lin et al., 2010, 2011], the boundary conditions can be expressed as follows.
a
Table 1. Thermal Parameters of Ground
1
Physical
Variable
ρ
3
(kg/m )
Silty clay
Mudstone
1.9 × 10
3
2.0 × 10
3
2
λf
(J/(yr m °C))
7
4.26 × 10
7
5.74 × 10
2
cf
(J/(kg °C))
2
9.44 × 10
2
7.42 × 10
2
λu
(J/(yr m °C))
7
3.56 × 10
7
4.64 × 10
2
cu
(J/(kg °C))
3
1.26 × 10
3
1.05 × 10
3
1,2
a1
1,2
b1
0.09
0.01
-0.22
-0.56
3
a2
(m/yr)
2
4.73 × 10
2
4.19 × 10
3
b2
a3
(m2/yr)
b3
11.27
10.13
47.9
38.5
4.45
4.38
3
a
The parameters with subscript f and u are the corresponding physical components in the frozen and unfrozen zones, respectively. The parameters with
subscripts 1, 2, and 3 come from Liu et al. [2001], Li et al. [2009], and Xu et al. [2010], respectively.
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
842
Journal of Geophysical Research: Earth Surface
a)
-6
-2
0
0
2
6
-4
-2
-60
-100
-100
d)
2
4
6
-6
4
6
Simulated data
Measured data
Temperature /
-4
-2
0
-20
2
-60
-80
0
0
-40
-80
-2
0
2
4
6
0
Simulated data
Measured data
-20
-40
Simulated data
Measured data
-40
Depth /m
Depth /m
0
-20
-40
Temperature /
-4
-6
Depth /m
Depth /m
c)
4
Temperature /
Simulated data
Measured data
-20
-6
b)
Temperature /
-4
10.1002/2013JF002930
-60
-60
-80
-80
-100
-100
Figure 4. Geotemperature curves of T1–T4 on 1 December 2009. (a) T1, (b) T2, (c) T3, and (d) T4.
The water and air temperature boundary changes according to the following function:
T b ¼ A þ B sinð2π t þ α0 Þ þ Rtr t;
(30)
where A is mean annual temperature (°C), B denotes temperature amplitude (°C), t represents time (year), α0 is
phase angle and is determined by initial computational time, and Rtr is the rate of temperature rise (°C/yr).
The temperature at the pond bottom AB (Figure 3) changes according to the following function:
π
T b ¼ 5:5 þ 6:5 sin 2π t þ
þ 0:048t;
(31)
2
where Rtr is set to be 0.048°C/yr, which reflects a 2.4°C temperature increase over a 50 year period, as commonly
used by other investigators in QTP [Cheng et al., 2008; Lin et al., 2011; Jin et al., 2012]. The temperature at the
natural surface BC (Figure 3) varies as follows [Zhang et al., 2013; Lin et al., 2012]:
π
T b ¼ 0:5 þ 12:0 sin 2π t þ
þ 0:048t:
(32)
2
In horizontal direction, there is no heat exchange at the lateral boundaries AE and CD; hence, they are assumed
to be adiabatic. In vertical direction, the geothermal heat flux at boundary DE is qT = 0.047W/m2 according to
the monitored data [Sun et al., 2012].
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
843
Journal of Geophysical Research: Earth Surface
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0
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Measured data
Temperature /
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-2
Simulated data
Measured data
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2
2
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0
0
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-20
Depth /m
Temperature /
-4
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Temperature /
-4
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Depth /m
Depth /m
(c)
2
Simulated data
Measured data
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-6
(b)
Temperature /
-4
0
-20
Depth /m
(a)
10.1002/2013JF002930
0
2
4
6
Simulated data
Measured data
-40
-60
-80
-80
-100
-100
Figure 5. Geotemperature curves of T1–T4 on 1 December 2010. (a) T1, (b) T2, (c) T3, and (d) T4.
The boundary AB is pond bottom, so it is saturated and its volumetric water content is 0.43. Li et al.
[2011] and Xiao et al. [2013] monitored the moisture-heat process of active layer of permafrost in
QTP where the active layer refers to the top layer of ground in which temperature fluctuates above
and below 0°C during the year and found the water content was seasonal variation and its
interannual difference was very small; hence, the BC is regarded as closed boundary. In addition, the
other boundaries are assumed to have no moisture exchanges with their adjacent zones and are
water resisting.
Because the thermokarst pond may form many decades ago and it has been causing the underlying
permafrost thaw, the initial moisture-heat conditions are no longer uniform and complex, which
should be determined by trial computation. At first, the initial moisture-heat conditions are assumed
to be the mean values, and the above boundary conditions with no global warming are used to carry
out a series of trial computations. If the consecutive 2 years simulated temperatures on 1 December
agree with measured data in four boreholes of T1–T4 to maximum extent (see Figures 4 and 5),
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
844
Journal of Geophysical Research: Earth Surface
(b)
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z /m
z /m
(a)
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10.1002/2013JF002930
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0
10
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30
40
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60
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90 100
0
10
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30
40
r /m
(c)
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90 100
r /m
0
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z /m
-40
-50
-60
-70
-80
-90
-100
0
10
20
30
40
50
60
70
80
90 100
r /m
Figure 6. Initial moisture-heat conditions. (a) Temperature (unit: °C) (b). Volumetric unfrozen water content. (c) Volumetric
ice content.
the consecutive times are assigned to 1 December 2009 and 2010, respectively. Consequently, the moistureheat states on 1 December 2010 are assumed to be initial conditions in the ultimate simulation (Figure 6).
3.3. Numerical Results and Analyses
3.3.1. Temperature Analysis
In order to calibrate the numerical model, the simulated results are compared with measured temperatures
at the locations of T1–T4 (Figures 7 and 8). As can be seen from Figures 7a, 7b, 8a and 8b the computed
temperatures are in good agreement with the measured ones in most parts underneath the thermokarst
pond. For example, as far as the temperature profile of T1 is concerned, the simulated and measured
permafrost tables are 6.30 m and 6.55 m on 1 December 2012, respectively, and their difference is merely
0.25 m. On 1 December 2013, the difference is 0.36 m. Moreover, below the permafrost table, the maximum
temperature discrepancy is no more than 0.15°C at 14.74 m. But there are obvious differences between the
simulated and measured temperature values in the range of 2.2 to 4.5 m in Figures 7c, 7d, 8c and 8d and
1.6 to 4.0 m in Figures 8a and 8b, and these differences are probably caused by the simplified boundary
conditions adopted in the numerical model instead of the complex and variable weather in QTP. In spite of
that, the numerical results agree well with the monitored temperatures in large parts of boreholes. Generally
speaking, the theoretical model and the numerical program developed in this study are found to be reliable
and can be applied to solve the problem of heat and mass transfer in frozen soils in QTP.
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
845
Journal of Geophysical Research: Earth Surface
(a)
10.1002/2013JF002930
(b)
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0
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4
6
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(c)
2
Simulated data
Measured data
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Depth /m
Depth /m
4
(d)
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0
Depth /m
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0
2
4
6
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-4
-2
Simulated data
Measured data
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0
0
-20
Depth /m
-6
2
4
6
Simulated data
Measured data
-40
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-80
-80
-100
-100
Figure 7. Geotemperature curves of T1–T4 on 1 December 2012. (a) T1, (b) T2, (c) T3, and (d) T4.
Since the dimension of the thermokarst pond is relatively small, it has limited influence on the deep permafrost
whose base is at 56.86 m at the boreholes of T1–T4 (Figures 6a, 7, and 8). However, the thermokarst pond has
great influence on its surrounding ground; thus, the freezing-thawing states at different locations are somewhat
different. On 1 December 2013, the permafrost table has dropped to 6.7 m underneath the thermokarst pond
center, whereas the depth of the thawing front is no more than 2.0 m deep under the natural ground (Figure 8).
Moreover, from Figures 9a and 9b, the thawing fronts (the 0°C isotherm) underneath the pond move downward
at the same linear rate of 0.18 m/yr on the profiles of T0 and T2. If the permafrost degrades at this rate, the
permafrost underneath the pond center (T0) would eventually be penetrated in the year of 2281 and the
permafrost may completely thaw at the profile of T2 in 2291. It is found that the farther the natural ground is
from the thermokarst pond center, the later its underlying permafrost thaws completely, as expected.
Compared to the evolution of the thawing front underneath the thermokarst pond, the developing processes
of the 0°C isotherm under natural ground exhibit bilinearity and their thawing rates are slower at early stage
than at later stage. As for the profile of T3, the thawing rate is 0.03 m/yr in 2010–2050 and it increases to
0.18 m/yr after 2050 (Figure 9c). Because T4 is far from the thermokarst pond, the thawing rate of its profile
will begin to accelerate in 2080, 30 years later than T3. That is to say, the effect of the pond on natural ground
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
846
Journal of Geophysical Research: Earth Surface
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-4
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0
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Measured data
Temperature /
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-2
Simulated data
Measured data
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4
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0
0
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-20
Depth /m
Temperature /
-4
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Temperature /
-4
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Depth /m
Depth /m
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6
Simulated data
Measured data
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(c)
4
0
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Depth /m
(a)
10.1002/2013JF002930
0
2
4
6
Simulated data
Measured data
-40
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-80
-80
-100
-100
Figure 8. Geotemperature curves of T1–T4 on 1 December 2013. (a) T1, (b) T2, (c) T3, and (d) T4.
expands at a rate of about 0.40 m/yr in the horizontal direction. Evidently, the expansion rates of influence of
the thermokarst pond on the underlying ground are large in both horizontal and vertical directions, so the
surrounding permafrost degrades rapidly under combined impacts of the thermokarst pond (a heat source)
and climate change, and the nonpenetrative thermokarst pond becomes penetrative in 2281.
To fully understand the spatial temperature changes, the mean annual temperature distributions in 2013 and
2281 are given in Figure 10. As indicated in this figure, the mean temperatures exhibit great difference in
2013 and 2281 because of the effects of the thermokarst pond and global warming. In 2013, the mean 0°C
isotherm is almost a straight line underneath the pond and its depth is 6.6 m. This isotherm gradually rises
and eventually intersects the natural ground surface in a transitional zone between the thermokarst pond
and natural ground. And the 1°C isotherm does not exist in this zone. In contrast, all of the isotherms including
the 1°C isotherm are horizontal lines under the natural ground, and two parallel isotherms of 1°C deviate
from their original directions and merge into one line in the transitional zone (see Figure 10(a)). Approximately,
the impact range of the thermokarst pond on the natural ground is no more than 20 m at present. In the
following years, the thawing layer would penetrate the entire permafrost in 2281 and the impact range of the
thermokarst pond on natural ground increases to somewhat around 50 m in horizontal direction. At the same
time, the thawing front will move down to 36.1 m under natural ground. Consequently, a thawing layer with a
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
847
Journal of Geophysical Research: Earth Surface
(b)
Depth /m
Depth /m
(a)
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2010
2050
2090
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2170
2210
2250
2290
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2010
2050
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2130
Time /year
2170
2210
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2290
2210
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2290
Time /year
(c)
(d)
0
Depth /m
Depth /m
10.1002/2013JF002930
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2010
2050
2090
2130
2170
2210
2250
2290
0
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-60
2010
2050
2090
2130
Time /year
2170
Time /year
Figure 9. Freezing-thawing processes at four different locations (unit: °C): (a) T0, (b) T2, (c) T3, and (d) T4.
dish shape emerges around the thermokarst pond. Meanwhile, the thickness of the permafrost would get thinner
and its temperature may rise to 0.2–0°C. In summary, most of the permafrost degrades due to simultaneous
effects of the thermokarst pond (a heat source) and global warming, which inevitably leads to deterioration
of ecological and environmental system in QTP. This change could be a global environmental concern
because of the importance of QTP in regulating the global climate as the so-called “third pole” of the earth.
3.3.2. Total Water Content Analysis
Under the forcing of temperature gradient, the unfrozen water content and ice content in ground change
with temperature; hence, the total water content shows significant difference as time goes on. For instance,
as there are periodic freezing-thawing alternations underneath the natural ground, which would cause ice
lenses formation and growth in the freezing-thawing fronts, the total water content would increase
correspondingly in this area. Therefore, the mean annual water contents are higher in active layer than in
other zones (the permafrost and talik zones) in 2013 and the maximum volumetric content reaches to 74.2%.
Because the moisture migrates toward the freezing-thawing fringes, the volumetric content of mean annual
water is only 4%–30% in most of permafrost zone. In addition, although there is direct and enough water
recharge from the thermokarst pond, the mean annual water content of ground underneath the pond is
much less compared to those in the active layer (Figure 11a). So the main factor that determines the distribution
and amount of water content is the ground temperature with seasonal alternations in permafrost regions. After
long-term combined effects of the thermokarst pond (a heat source) and global warming, a large portion of the
permafrost thaws near the thermokarst pond in 2281 and the freezing-thawing cycle also turns weaker; thus,
the distribution of the mean annual water content is relatively uniform in most of thawed ground except for
(b)
0
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40
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60
70
80
90 100
0
10
20
30
r /m
40
50
60
70
80
90 100
r /m
Figure 10. Mean annual temperature distributions in (a) 2013 and (b) 2281 (unit: °C).
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
848
Journal of Geophysical Research: Earth Surface
(b)
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z /m
(a)
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90 100
0
10
r /m
20
30
40
50
60
70
80
90 100
r /m
Figure 11. Mean annual water content distributions in (a) 2013 and (b) 2281.
the ground directly underneath the thermokarst pond. At the moment, the mean annual water contents are
much less in the permafrost at the deep ground and its volumetric content is 10%–25% (Figure 11b). However,
with the continuing degradation of the permafrost, the distribution of total water content is expected to
become increasingly uniform. At that time, the moisture states in the ground around the thermokarst pond is
thoroughly changed. As a result, the groundwater environment would also be entirely altered in QTP.
4. Discussions
One notable point deserving further discussion is about the initial condition used in the numerical models. It
is well known that the initial conditions are very important, but such conditions in most theoretical studies
are often artificially chosen because of lacking data to formulate the actual initial states. As a result, it is
impossible to accurately predict the moisture-heat states of adjacent permafrost around the thermokarst
pond, especially the specific time at which the talik penetrates the permafrost layer. By contrast, this problem
is well solved in this study by tens of trial computations, and the initial conditions are determined at several
times when the computational results are in good agreement with monitoring temperatures in the four
boreholes (Figures 4 and 5). Therefore, the numerical results discussed in this study may reflect actual
moisture and temperature variations in the surround permafrost around the thermokarst pond.
The magnitude of temperature rise in the following decades is affected by a number of factors, such as
human activities, solar radiation, atmospheric compositions, sea water temperature and surface reflectance,
and the variations of these factors are often uncertain. Therefore, it is difficult and sometime impossible to
accurately predict the rate of air temperature rise. To resolve this issue, there are many studies on the rate of
air temperature rise in QTP by different methods [Tong and Wu, 1996; Wei et al., 2003; Wu and Zhang, 2008; Liu
et al., 2010; Yang et al., 2010; D. Wu et al., 2010; Q. Wu et al., 2010]. To sum up, the range of the rate of air
temperature rise is 0.02–0.07°C/yr in QTP. It has been shown that the air temperature in QTP will increase
about 2.2–2.6°C during the next 50 years [Qin, 2002; Lin et al., 2011; Li et al., 2009; D. Wu et al., 2010; Q. Wu
et al., 2010]. We adopted an average temperature rising rate of 0.048 °C/yr in our simulation.
In this study, in order to fully investigate the influence of global warming on the permafrost around the
thermokarst pond, a sensitivity experiment for the rate of air temperature rate is conducted. A series of rates
of air temperature rise, such as 0.0, 0.02, 0.03, 0.04, 0.044, 0.052, 0.06, and 0.07 °C/yr, are selected among the
prediction range to carry out corresponding computations, and their results are given in Figure 12.
As shown in Figure 12, the penetrative time of the thermokarst pond becomes shorter when the rate of air
temperature rise is higher, and their relationship could be fitted by the following quadratic polynomial.
T y ¼ 1322575R3tr þ 211385:5R2tr 12517:45Rtr þ 2542:094 R2 ¼ 0:9990
(33)
where Ty denotes the time (year) in which the thermokarst pond becomes penetrative, Rtr is the rate of air
temperature rise (°C/yr), and R2 is the coefficient of determination. The R2 is very close to 1, suggesting a
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
849
Journal of Geophysical Research: Earth Surface
2550
nearly perfect matching between the
simulated and fitted curves. By using
this fitted equation of (33), the time at
which the talik penetrates the permafrost
layer could be easily estimated at a given
rate of air temperature rise. For example,
if the rate of air temperature rise is
0.01°C/yr, Rtr = 0.01, the thermokarst
pond would become penetrative in 2436.
Simulated curve
2500
Fitted curve
Time /year
10.1002/2013JF002930
2450
2400
2350
2300
In order to show the evolution of talik
layer solely due to presence of the
thermokarst pond, the freezing-thawing
process underneath the thermokarst
Figure 12. Relationship between the rate of air temperature rise and the
pond center (T0) is given in Figure 13.
penetrative time of thermokarst pond.
At this situation, the thawing front
underneath the pond moves downward
at a rate of 0.083 m/yr approximately, which is far smaller than 0.18 m/yr with the air temperature rise rate of
0.048°C/yr. So the permafrost underneath the pond center would be penetrated until the year of 2542. At that
time, the moisture-heat states in 2542 (Figure 14) are quite different from those in Figures 10 and 11.
2250
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
As shown in Figure 14a, there still exists plenty of permafrost under the natural ground though the
nonpenetrative thermokarst pond becomes penetrative. The maximum radius of the talik is no more than
39 m. If the permafrost underneath the natural ground, whose horizontal distance to the pond side, is
beyond 50 m, its geotemperature will not be affected by the thermokarst pond at all. As the distribution
and amount of water content is mainly determined by the geotemperature with seasonal freezing-thawing
cycles in permafrost regions, the water content distribution in Figure 14b correspondingly exhibits
obvious difference compared with that in Figure 11b. The volumetric contents of the mean annual water
are relatively high in or near the active layer in 2542, and their values range from 30% to 75% while they
are merely 4%–30% in the permafrost and 20%–50% in the talik, respectively. Consequently, the
hydrological balance in the ground near the thermokarst pond will be broken, which would result in some
natural hazards in QTP.
Depth /m
Although some important characteristics of frozen soil during freezing-thawing process are taken into
account in this study and there is much progress compared with previous studies, some limitations still exist.
For example, the simulation results of moisture content are not calibrated fully by measurement data, so a
field measurement on moisture content should be carried out in the future to support the theoretical result.
In addition, the geometry of the thermokarst pond is assumed to be unchanged. In fact, there is high thawed
water content along the edge of the thermokarst pond during the thawing process, which worsens
mechanical stability of soil, so the slump often happens along the edge and the thermokarst pond expands
gradually. Therefore, a coupled moisture-temperature-mechanics model should be established, and the
expansion process of the thermokarst pond needs to be simulated by the discrete element method (DEM)
[Utili and Crosta, 2011; Wang and Xin, 1991] instead of the finite element method (FEM). The DEM is proven to
be effective in handling the slump development process, which is often not well simulated via FEM [Utili and
Crosta, 2011]. By doing so, a better description of the moisture-heat process of permafrost around a thermokarst
pond may be obtained. Notwithstanding such limitations, this study is expected to provide a theoretical basis
and reference for further research on the coupled moisture-heat process of permafrost around a thermokarst
lake/pond in permafrost regions.
-20
-40
-60
2050
2100
2150
2200
2250
2300
2350
2400
2450
2500
2550
Time /year
Figure 13. Freezing-thawing process underneath the thermokarst pond center (T0) (unit: °C).
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
850
Journal of Geophysical Research: Earth Surface
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0
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-40
z /m
z /m
(a)
-50
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-80
-80
-90
-90
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0
10
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40
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80
90 100
r /m
10.1002/2013JF002930
0
10
20
30
40
50
60
70
80
90 100
r /m
Figure 14. Moisture-heat states in 2542 with no global warming. (a) Mean annual temperature (unit: °C). (b) Mean annual
water content.
5. Conclusions
The long-term interaction of the thermokarst pond and surrounding permafrost is very complex, which
involves heat conduction, moisture migration, ice-water phase change, and other factors. It is impossible to
disclose its physical variation process through short-term in situ monitoring. So a numerical moisture-heat
coupled model for the permafrost is provided and the corresponding computer program is written in this
study. The temperature and moisture processes of the permafrost around the thermokarst pond is simulated
and analyzed under global warming for a carefully documented field site (Beiluhe test site) in QTP. The
following conclusions can be drawn.
1. The comparison between computed and measured temperatures shows a satisfactory agreement, which
implies that the theoretical model as well as the numerical program is valid. The numerical model could
be used to simulate moisture-heat interaction process of the thermokarst pond and surrounding permafrost.
2. Like a heat source, the thermokarst pond has great influence on the surrounding ground, especially the
underlying permafrost. At present, a talik has been formed under the thermokarst pond and the permafrost table has dropped to 6.7 m underneath the thermokarst pond center, while all of the isotherms are
horizontal lines and the depth of the thawing front is no more than 2.0 m under the natural ground.
3. In vertical direction, the thawing fronts underneath the thermokarst pond move downward at a linear rate
of 0.18 m/yr and the permafrost under the pond center would be penetrated in 2281. However, the
developing processes of the 0°C isotherm under the natural ground exhibit bilinearity and their thawing
rates are slower at early stage than at later stage. In horizontal direction, the impact range of the pond on
the natural ground increases to about 50 m in 2281. So a thawing layer with a dish shape would emerge
around the thermokarst pond.
4. According to the penetrative times of the thermokarst pond at a series of rates of air temperature rise, a cubic
on the penetrative time versus rate of temperature rise is obtained, by which the time at which the talik
penetrates the permafrost layer could be easily determined at any given rates of air temperature rise.
5. The main factor that determines the distribution and amount of water content is the ground temperature
with seasonal alternations in permafrost regions. Since there are seasonal freezing-thawing alternations
under the natural ground at present, the mean annual water contents are higher in the active layer than
in other zones in 2013. However, after long-term combined effects of the thermokarst pond (a heat
source) and global warming, the freezing-thawing alternation becomes weaker in the deep ground with
the talik expanding, so the distribution of the mean annual water content is relatively uniform in most of
thawed ground in 2281and the volumetric contents are higher than those in 2013.
6. If there is no global warming, the nonpenetrative thermokarst pond would become penetrative in 2542. At this
situation, the maximum radius of the talik underneath the pond is no more than 39 m, and the permafrost
area existing under the natural ground is far greater than that with the air temperature rise rate of 0.048°C/yr.
LI ET AL.
©2014. American Geophysical Union. All Rights Reserved.
851
Journal of Geophysical Research: Earth Surface
10.1002/2013JF002930
This study indicates that a majority of the permafrost around the thermokarst pond will degrade rapidly under
the simultaneous effects of the thermokarst pond (a heat source) and global warming, and the nonpenetrative
thermokarst pond would become penetrative in later centuries. At that time, the moisture states in the
thawed ground would also be changed entirely. All of these would inevitably deteriorate the ecological and
environmental system in QTP and further worsen environment on the earth. To avoid or delay occurrence of
this environment disaster in QTP, some treatment methods, such as artificial drainage of water and vegetation
recovery, should be adopted to prevent the permafrost degradation caused by the thermokarst pond/lake.
Acknowledgments
The authors would like to express their
gratitude to the editor and three
anonymous for their constructive and
helpful comments. This research was
supported by the National Natural
Science Foundation of China (41101068,
41230630, 40801024, and 41172281),
the National Basic Research Program of
China (973) (2011CB710602), the CAS
Action-Plan for West Development
(KZCX2-XB3-19), the Knowledge
Innovation Program of the Chinese
Academy of Sciences (KZCX2-EW-QN301),
and the Foundation of State Key
Laboratory of Frozen Soil Engineering
(SKLFSE-ZQ-03).
LI ET AL.
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