In recent years, observations have highlighted seasonal and interannual variability in rock glacier flow. Temperature forcing, through heat conduction, has been proposed as one of the key processes to explain these variations in kinematics. However, this mechanism has not yet been quantitatively assessed against real-world data.
We present a 1-D numerical modelling approach that couples heat conduction to an empirically derived creep model for ice-rich frozen soils. We use this model to investigate the effect of thermal heat conduction on seasonal and interannual variability in rock glacier flow velocity. We compare the model results with borehole temperature data and surface velocity measurements from the PERMOS and PermaSense monitoring network available for the Swiss Alps. We further conduct a model sensitivity analysis in order to resolve the importance of the different model parameters. Using the prescribed empirically derived rheology and observed near-surface temperatures, we are able to model the correct order of magnitude of creep. However, both interannual and seasonal variability are underestimated by an order of magnitude, implying that heat conduction alone cannot explain the observed variations. Therefore, we conclude that non-conductive processes, likely linked to water availability, must dominate the short-term velocity signal.
For several rock glaciers, and especially in Switzerland, surface
displacements have been calculated over long time periods
Overview of the four case study sites. Overview site map (middle)
and aerial view of the four studied rock glaciers
In order to explain the above introduced observations, classical concepts
from related disciplines – geotechnical engineering and glaciology – have
been applied to rock glacier research. Interannual velocities have been compared against climatic
variables and external temperature forcing has been proposed as one of the
key factors controlling the observed long-term flow variations
Summary of field sites. The borehole locations are given in the
CH1903+ coordinate system. The values of the geometrical and physical
properties of the rock glaciers are given (thickness, slope, volumetric ice
content and thermal diffusivity) as discussed in Sect.
Even though great improvements have been achieved in this field, our
understanding of the processes governing rock glacier dynamics and their
relation with external forcings and controlling factors remains at a
qualitative level, and many questions remain unanswered. However, it is clear
that, in order to understand rock glacier dynamics, the complex
thermo-hydro-mechanical behaviour of the ice-rich frozen soil and its
coupling with the climate have to be considered. In particular, when aiming
to understand the influence of temperature forcing on permafrost creep and
its relative importance on rock glacier dynamics, we have to consider two
aspects. On one hand, the thermal regime of a rock glacier is mainly
controlled by heat conduction, driven by external temperature forcing
In this study, we quantify the relative importance of the conductive thermal
influence on flow and extend previous research
For constraining the numerical modelling investigations, we use observational
data from four rock glaciers in the Swiss Alps, namely from the rock glaciers
Ritigraben located in the Valais, and Murtèl-Corvatsch (hereafter called
Murtèl), Schafberg, and Muragl all located in the Engadine
(Fig.
These four rock glaciers cover a wide range of geometric settings and dynamic
states: thickness, slope and flow velocity from decimetres to several metres
per year; for an overview see Table
The Ritigraben rock glacier is located above the village of Grächen (VS) and
originates from the northern slope of the Gabelhorn (
The Murtèl rock glacier originates from the north wall of Piz Murtèl
(
Contour plots of ground temperatures time series (colour coded) for
rock glaciers
The Schafberg rock glacier originates in a cirque south of the Piz Muragl ridge,
has an extent of less than 300 m and an average surface slope of
The Muragl rock glacier is located on the western side of the ridge of Piz Muragl
(
We designed a suite of 1-D numerical models, based on
finite differences, to simulate the response of viscous and plastic flow to
external near-surface temperature forcing using the software
Here, we provide a detailed description of the data used for model input and for comparison with the model results.
The surface slope values are calculated on the basis of the swisstopo digital
cartography
Several types of velocity data are available. For Murtèl and Muragl
rock glacier mean annual surface velocities are available between 2009 and
2015 from terrestrial surveys with total station from the PERMOS network. For
all study cases but Murtèl, daily velocities from continuous single
frequency DGPS measurements are available from 2012 from the PermaSense
network. For a summary of the study case sites see Table
We model vertical heat conduction throughout the rock glacier unit by solving
the diffusion equation for temperature evolution with depth
For modelling ice creep we use the empirically derived creep relation
proposed by
Assuming an infinitely wide surface parallel slab, the shear stress
Given the temperature field and assuming an inclined infinite parallel slab,
the velocities are solved from Eq. (
The model is calibrated to best fit the average observed surface velocities
by varying the volumetric ice content parameter within the range of
literature values. Because of the mathematical formulation of the applied
rheology, calibrating the model by varying the volumetric ice content gives a
concave curve for the mean velocities, with a maximum corresponding to around
Normalised surface velocities for the four case studies with
volumetric ice content values. For each rock glacier the velocities are
normalised with the velocity value corresponding to
Within the range of possible volumetric ice content values proposed in the
literature
Summary of modelling inputs and results. Values of geometrical and physical input parameters for the modelling are listed (thickness, slope, volumetric ice content and thermal diffusivity). The mean modelled surface velocity and its relative interannual and seasonal variations are reported in the last three columns.
Parameters and results of the sensitivity experiments.
The first column indicates the scenario with a label in the form of
Scn
We consider further uncertainties in input parameters: slope
The creep model strongly depends on the temperature input. In order to assess uncertainties related to the heat conduction model and to take into account all possible heat transfer process, we perform additional numerical experiments forcing the ice-creep relation directly with the observed temperature fields with depth.
The above described approach for creep does not consider enhanced deformation
in the shear horizon, where most of the displacement takes place. In order to
investigate the sensitivity of the model to such a phenomenon, we perform
additional numerical experiments. We approximate the behaviour of the shear
horizon with a pseudo-plastic creep relation, by increasing the flow law
exponent of Eq. (
We perform additional synthetic sensitivity experiments in order to explore
the influence of the different input parameters on our model results. For
these experiments we simulate seasonal temperature forcing by prescribing the
temperature below the active layer as a sinusoidal function with a mean of
Additionally, the thickness sensitivity experiments have been repeated using
the pseudo-plastic rheology (Eq.
The modelled and measured temperatures are shown in
Fig.
The observed and modelled surface flow velocities with time are shown in
Fig.
Observed and modelled surface flow velocities for
For the chosen volumetric ice content values within the proposed range, we obtain the correct order of magnitude of the observed surface velocities for all four case study rock glaciers. For Ritigraben and Muragl rock glaciers the modelled velocities are smaller (by a factor 2), for Murtèl the average velocity matches the observations and for Schafberg the modelled velocities are overestimated (by a factor 3) in comparison to the observed ones.
The modelled amplitudes in seasonal and multi-annual velocity variations
(values in Table
For the three rock glaciers with continuous DGPS measurements, we
are also able to compare the phase of the seasonal variations. The modelled
velocity maxima occur in late winter and are substantially delayed in
comparison to the observed velocity peaks in autumn. The above findings
(amplitude underestimation and phase shift) do not change when the observed
temperature fields are used as input for the creep model. The exception is
the case of Ritigraben, where, as discussed above, substantial discrepancies
between velocities obtained using modelled (blue solid line) and observed
(red solid line) temperatures occur. When using the observed temperature
field, the modelled and the observed seasonal velocity variations are phase
synchronous (see red and green line in Fig.
The discrepancies found between observed and modelled velocity variations do
not improve when using the pseudo-plastic creep model for all rock glaciers.
On the contrary, the seasonal velocity amplitude further reduces and the
phase shift increases further (see yellow line in Fig.
The results of the sensitivity experiments are plotted in
Figs.
In Fig.
In Fig.
Plots of the time-averaged values of surface velocities for the
sensitivity experiments with the different parameter scenarios. The mean
velocities are normalised with the mean values of the reference scenario
Scn1.0. Panel
Results of the modelling sensitivity experiments of the model to
In this study, we developed a simple numerical model approach to investigate
the dynamical behaviour of rock glaciers with the aim of resolving the
influence of external temperature forcing through heat conduction on
rock glacier surface velocities. When choosing volumetric ice contents within
a physical range of values proposed in the literature (see
Sects.
We model rock glacier temperature evolution based on near-surface temperatures
as measured below the active layer (see Sect.
The assumption of constant bottom temperature agrees well with the observed borehole temperatures. This is further supported by the good agreement between the modelled velocities from prescribed observed and modelled temperatures. We assume the physical properties of the rock glacier (density, ice content and thermal diffusivity) to be constant in time and homogeneous in space, which seems justified at the considered short (seasonal to multi-annual) timescales and is supported by the good performance of the temperature evolution model.
For Schafberg (Fig.
Using the modelled and observed temperature fields respectively, we force the empirical creep relation for rock glacier material. Additionally, we run a separate experiment with the pseudo-plastic rheology to investigate the impact on the model from including enhanced deformation within the shear horizon.
When applying the creep rheology of
For Murtèl, the mean surface velocities (averaged over the whole time
series) match the observations. For Ritigraben and Muragl the modelled
average velocities are between
For all rock glaciers, we find that both seasonal and in particular
interannual variations are strongly underestimated. This result is also
coherent when considering relative velocity variations (see
Fig.
In general, our modelled seasonal variations for the four rock glaciers as
well as for the sensitivity experiments are consistent with the obtained
3
Consistent with the results for similar thicknesses of
A phase lag of about 2–3 months between the seasonal summer peak in the observed
ground surface temperatures and measured surface velocity has been detected
on several rock glaciers including Ritigraben, Schafberg and Muragl (see
Fig.
Temperature and velocity time series in the
In contrast, the seasonal winter minima in measured temperatures below the
active layer (used as model input forcing) only have a lag of 2 months from the
surface temperatures and seem in phase with the observed velocity minima
(Fig.
Despite the highly asymmetric seasonal temperature pattern, the resulting
modelled surface flow variations are almost symmetric
(Fig.
The clear overestimation of the time lag in the modelled surface variations
is a further sign that the process of heat conduction alone cannot explain
the observed variations in deformation. Infiltration of surface meltwater
into the permafrost in the summer season could reduce this time lag and
through advection of water affect the flow in two ways. Firstly, the
infiltrating water can effectively advect heat and warm up the rock glacier
body at depth as observed in the case of Ritigraben, which in our modelling
removed the phase lag when using the observed temperature field that includes
the talik formation in summer. For the other three rock glaciers, water
infiltration may also occur but it does not seem to significantly warm the
temperatures at depth, as confirmed by the good agreement between observed
and modelled temperatures, and we can therefore exclude this heat advection
process. Secondly, with increasing water infiltration the water content and
pore-water pressure within the rock glacier material are expected to increase
which in turn may reduce the shear strength and thereby enhance deformation
and flow. This process has been suggested in other studies
Regarding the multi-annual variations, which are well documented and
synchronous for many rock glaciers in Switzerland
Including a shear horizon with a pseudo-plastic rheology (with the same
temperature dependency as for the main rock glacier body and enforcing the
same mean flow velocity) does not improve our results. On the contrary,
interannual and seasonal variations are even more underestimated, because at
the shear horizon depth, where the main deformation occurs, the signal of
seasonal temperature variations from the surface is too small, being close to
one-tenth of a degree
In summary, the strong underestimations in both amplitude in seasonal and multi-annual variations, as well as the overestimation in time lag of seasonal peaks in our modelling, suggest that heat conductive processes alone cannot explain the observed variation in flow velocity, suggesting the need for other processes, such as the interaction of rock glacier rheology with surface water advecting into the rock glacier body.
Sensitivity experiments were carried out to explore the influence of
different input geometries and parameters on the simulated surface velocity
in a systematic way. In their set-up and results, the experiments are an
extension of the earlier modelling study by
Absolute mean velocities are strongly affected by variations in geometry due
to changing stress conditions (Eq.
The model used shows a dependency of the surface velocities on the volumetric
ice content value (Figs.
Decreasing thicknesses lead to very different absolute mean velocities, but
more interestingly, they also lead to stronger seasonal variations.
Considering realistic thicknesses of
For all other remaining parameters, except the rock glacier thickness, the
modelled seasonal velocity variations do not change much and stay again below
By considering the pseudo-plastic relation, the seasonal variations are
coherently decreasing for all the scenarios (even for shallow rock glaciers)
and the velocity peaks are considerably shifted in time, with a delay of up
to 6 months (see Table
We quantitatively investigated the contribution of heat conduction to
seasonal and multi-annual variations in rock glacier flow velocity on the
basis of numerical modelling and a multi-year time series of observed surface
velocities and borehole temperatures from four different rock glaciers. The
numerical model couples heat conduction to an empirically derived rheology of
rock glacier creep that accounts for temperature and ice content. We find
that, using standard parameters from the literature, our modelling reproduces the
correct order of magnitude of mean surface velocities for all chosen
rock glaciers. However, the magnitudes of seasonal and multi annual variations
are strongly underestimated by our modelling, and the phase-lags of the
seasonal peaks are too long. This suggests that the effect of heat conduction on
the observed variations in surface flow is very limited and cannot explain
more than about
Additional sensitivity experiments underpin the robustness of these conclusions within expected parameter uncertainties, also when including a shear horizon at the bottom of the rock glacier. Our idealised sensitivity experiments further indicate that, when the temperature changes over the full depth of the rock glacier (changing bottom temperature), the mean deformation maybe affected substantially, but this requires changes in climate over periods of several decades or centuries.
From our quantitative process modelling approach we can therefore exclude
heat conduction as the governing process for seasonal to multi-annual
variations in rock glacier flow. Considering the phase-lag information of the
summer peak (e.g. the case of case of Ritigraben) and indications from
earlier qualitative and statistical analysis of rock glacier velocities
Data on rock glacier kinematics and temperature are
available from the PERMOS office upon request. The reference web link is
Andreas Vieli is an editor of
The work presented in this paper is part of the project X-Sense 2 funded by Nano-Tera.ch (ref. no. 530659). This work could not have been possible without the pioneering results obtained from the DGPS pilot project funded by the Swiss Federal Office for the Environment BAFU. We thank Johann Müller for the coherent fieldwork in Engadine, and Vanessa Wirz for her preliminary study on this topic. We further acknowledge the editor, Christian Hauck, and the referees, Martin Hoelzle and Lukas U. Arenson, for their constructive and thorough comments, which helped to improve the manuscript. Edited by: Christian Hauck Reviewed by: Lukas U. Arenson and Martin Hoelzle