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Melt rate of Palcaraju glacier, Cordillera Blanca, Peru: attribution of anthropogenic influence and
proposed methodology for calculating adaptation cost
FHS in Geography
Rupert Stuart-Smith
January 2019
Word count: 11,491
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Acknowledgements
The author would like to thank Professor Myles Allen for supervising this dissertation, for advice on
conceiving the study and comments on the text; Dr Sihan Li for co-supervision, providing Regional
Climate Model data, assistance with Python, and comments on the text; Dr Karsten Haustein for his
help with climate data operators; and Felix Heilmann for his translation of a court document from
German.
Contents
Acknowledgements 2
List of figures 3
1 Abstract 4
2 Summary 4
3 Introduction 5
4 Data 8
4.1 Regional Climate Model 8
4.2 Observations 9
4.3 Digital Elevation Model 10
5 Methods 11
5.1 Temperature bias correction 11
5.2 Precipitation bias correction 11
5.3 Distributed energy and mass balance (DEMB) glacier model 13
6 Results 21
6.1 Model evaluation 21
6.2 Energy fluxes 24
6.3 Meteorological data 26
6.4 Attribution of anthropogenic influence on glacier melt rate 28
7 Further modelling work 30
7.1 RCM outputs 30
7.2 Glacier model 31
8 Discussion of legal context 33
8.1 Implications for climate litigation 37
9 Conclusion 38
10 References 39
Appendix 47
3
List of figures
Figure 1: Satellite image of the southern Parque Nacional Huascarán. 8
Figure 2: Map of HadRM3P domain used for this study. 9
Figure 3: Palcaraju glacier outline. 10
Figure 4: Ablation time series for Shallap and Palcaraju glaciers (2007). 22
Figure 5: Mass balance time series for Shallap and Palcaraju glaciers (2007). 23
Figure 6: Energy fluxes for Shallap glacier. 25
Figure 7: Modelled energy fluxes for Palcaraju glacier. 25
Figure 8: Precipitation time series for the raw and bias corrected RCM data. 26
Figure 9: Bias corrected daily mean RCM temperature data. 27
Figure 10: Difference between actual and natural RCM temperature. 27
Figure 11: Full time series ablation rate for Palcaraju glacier. 30
Figure 12: Influence of the discount rate on costs attributable to climate change. 35
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1 Abstract
Litigation concerning the influence of anthropogenic greenhouse gas (GHG) emissions on the melt rate
of Palcaraju glacier in the Peruvian Andes, and the resultant change in risk of a glacier lake outburst
flood (GLOF) from Lake Palcacocha, has entered an evidentiary phase. While previous attribution
studies on anthropogenic influence on glacier mass balance have been largely restricted to the
regional or global scale, evidence of the influence of emissions on the scale of the individual glacier is
needed to establish causation in the specific case before the courts. This paper assesses the
contribution of climate change to the melt rate of Palcaraju glacier. A vertically distributed surface
energy and mass balance model was developed and run at daily resolution for 29 years (1988-2016).
Boundary conditions were provided by a coupled global/regional climate model simulating the climate
both with and without anthropogenic influence, to quantify impacts attributable to past emissions.
Anthropogenic climate forcing is found to increase melt rate for all years by 20-80% and advances the
timing of Lake Palcacocha reaching its current depth by 15-37 years. Additionally, an original
methodology is proposed for calculating the costs of damages attributable to GHG emissions that is
applicable to ‘slow-onset’ impacts, such as glacial melt, in which climate change can be shown to
materially affect the timing of the occurrence of a natural disaster that would, eventually, have
occurred naturally.
2 Summary
The impact of anthropogenic climate forcing on the height-resolved surface energy balance and melt
rate of Palcaraju glacier (9°23’49”S 77°22’47”W, 4625-6175 m above sea level (a.s.l.)) was calculated
as follows. A vertically distributed surface energy and mass balance model was developed (henceforth
referred to as the ‘glacier model’), following the methodology of Gurgiser, Marzeion, et al. (2013) as
closely as the available data permitted, and where necessary, using parameterisations from further
sources, most notably Arnold et al. (1996). The glacier model was driven by surface temperature and
fluxes from the Hadley Centre regional climate model (RCM) version 3P (HadRM3P) coupled to the
global atmosphere model HadAM3P from the weather@home distributed climate modelling system.
RCM precipitation was bias-corrected using satellite-based measurements from the Tropical Rainfall
Measuring Mission (TRMM), in the absence of complete local surface observations, while temperature
was bias-corrected using elevation-adjusted observations from a meteorological station in the nearby
village of Anta. Two ensembles of simulations were compared, one in which the global atmosphere is
driven with observed sea-surface temperatures (SSTs), sea ice fractions (SIFs), and atmospheric
composition (denoted ‘actual’), and the other with SSTs, SIFs and atmospheric composition adjusted
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to simulate a counterfactual ‘world that might have been’ in the absence of human influence on
climate (denoted ‘natural’).
An increase in melt rate as a result of anthropogenic GHG emissions is detected for all 29 years studied.
The exact magnitude of this increase is subject to considerable uncertainty. Nevertheless a 20-50%
increase in melt rate can be attributed to anthropogenic influence on the climate, across the modelled
period. In the natural scenario (without human influence), meltwater production does occur in a
comparatively small ablation zone near the glacier snout. The primary glacier response to climate
change is a rise in the equilibrium line altitude, expanding the area of the ablation zone, supplemented
by an acceleration in the melt rate of the ablation zone present in the natural run. The increased rate
of glacier meltwater production suggests that the rate of expansion of Lake Palcacocha has increased
as a result of human GHG emissions, and that the lake has reached a critical level, 15-37 years sooner
than it would have done under a natural climate, necessitating the implementation of costly risk
reduction measures.
Existing literature on climate change-induced loss and damage focuses primarily on the impacts of
meteorological events. The results presented here indicate that the application of frameworks based
solely on meteorological events to slow-onset events such as increasing GLOF risk is inappropriate. For
many extreme weather events, GHG emissions may be thought of as increasing the likelihood of
occurrence of a meteorological event of a given intensity whose likelihood may otherwise be assumed
to be constant over time. For GLOFs, which occur naturally, the consequence of emissions is to bring
the next flood forward in time. In this paper, I propose a methodology for calculating the costs
attributable to the GHG emissions of individual emitters for the impacts of glacial melt, and other
instances in which the impact of climate change is to advance the timing of an event’s occurrence.
3 Introduction
The risk of a GLOF from Lake Palcacocha, approximately 20km upstream of the city of Huaraz (see
figure 1), has been widely documented (e.g. Somos-Valenzuela et al., 2016). Reduced to 514,800 m3
by a GLOF in 1941 (Vilímek et al., 2005), the lake volume has since increased 34-fold to 17,325,207 m3,
primarily due to meltwater from the Palcaraju glacier, and, as of 2014, had a maximum depth of 73 m
(Autoridad Nacional del Agua, 2014; Portocarrero Rodríguez and Engility Corporation, 2014). A small
landslide-induced GLOF occurred in 2003 but caused little damage to Huaraz. The 1941 GLOF was the
result of a large breach of the lake’s moraine dam, resulting in a debris flow which destroyed one third
of the city of Huaraz, and caused between 1800 and 6000 deaths (Evans and Clague, 1994; Vilímek et
al., 2005; Somos-Valenzuela et al., 2016). Recent research on Lake Palcacocha has indicated that the
6
lake has very high outburst potential, and the occurrence of a GLOF would most likely be due to
avalanche-induced waves overtopping the moraine dam (Emmer and Vilímek, 2013).
The main controls on a GLOF of this type are the lake depth and avalanche size; a 30m reduction in
lake depth would reduce overtopping volume by 90% following an avalanche (Somos-Valenzuela et
al., 2016). In light of this, siphons were installed in 2011 to lower the water surface by 3-5m and leave
a freeboard of 12m, slightly reducing the GLOF risk to Huaraz. Any anthropogenic influence on the rate
of meltwater production above the lake, and therefore the rate of rise of lake level can therefore be
linked to altering the GLOF hazard (Somos-Valenzuela et al., 2016).
Ongoing litigation in the German courts (Lliuya v RWE) seeks to recover, from RWE (a German power
utility company), costs incurred by Huaraz in protecting the city from this threat, proportional to RWE’s
contribution to historical anthropogenic GHG emissions (Frank, Bals and Grimm, 2019). Now in an
evidentiary phase, question 2c of the court’s call for evidence concerns the role of GHG emissions in
altering meltwater production from Palcaraju Glacier: the question that this paper seeks to answer.
While there is no doubt that GHG emissions cause harm, tracing specific damages to specific emitters
is more complex. For most meteorological events, climate science can provide only probabilistic
attribution (linking emissions to an increased likelihood of an event occurring). It has been suggested
that more than doubling of risk may be sufficient to attribute damages to emissions (Marjanac, Patton
and Thornton, 2017; Simlinger and Mayer, 2019). For GLOFs, however, it may be possible to
demonstrate a robust link between emissions and glacial melt rates; the question for the courts
concerns the liability of emitters for advancing the timing of events: a matter hitherto not discussed
in literature concerning climate change attribution, loss and damage, and litigation.
Tropical glaciers are particularly sensitive to climatic changes, due in part to the thermal homogeneity
of the tropical troposphere (Kaser, 1999; Rabatel et al., 2013). Palcaraju (see Figure 3) is a high-
altitude, outer tropical glacier for which subtropical conditions prevail with a dry season in the austral
winter of May to August, a transitional season from September to December, and a wet season from
January to April. Notable accumulation occurs only in the wet season, predominantly due to an
easterly flow of moisture from the Amazon Basin, when increased snowfall raises albedo and reduces
the melt rate (Rabatel et al., 2013).
Few studies have sought to attribute glacier mass loss to anthropogenic forcing, particularly due to
challenges with downscaling modelled atmospheric circulation to the scale of glaciers, complexities in
time lag of glacier response to climate change (10-40 years, depending on size of ice mass, its dynamic
and thermodynamic regime, slope hypsography and the climatic environment) and poor long term
7
precipitation and humidity records (Marshall, 2007; Rabatel et al., 2013; Schauwecker et al., 2014). All
Cordillera Blanca glaciers are documented to be retreating (Thompson et al., 2017) and the
Intergovernmental Panel on Climate Change states that there is ‘high confidence that a substantial
part of the mass loss of glaciers is likely due to human influence’ (Bindoff, et al., 2013). The global
synchronicity of recent mountain glacier fluctuations also indicate that their causes are global in scale
(Kaser, 1999). 25% ± 35% of global glacier mass loss over the period 1851-2010 has been attributed to
anthropogenic causes, rising to 69% ± 24% for the period 1991-2010. Anthropogenic influence on
glacier mass is, however, spatially heterogeneous, underlining the importance of local attribution of
climate change impacts on melt rates (Marzeion et al., 2014).
The glacier coverage in the Cordillera Blanca has shrunk from its Little Ice Age peak of 900 km2 to 482
km2 in 2010 (Somos-Valenzuela et al., 2016). This rate of retreat is above the global average and has
been highest in the past 50 years, with a 40% loss in glacier-coverage in Peru observed between 1970
and 2003-2010 (Rabatel et al., 2013; Thompson et al., 2017). Significantly, the freezing line altitude
rose by 160 m on precipitation days between 1964/72 and 1983/92 (Schauwecker et al., 2014), and
some Cordillera Blanca glaciers show a reasonable correlation of Equilibrium Line Altitude (ELA) with
temperature (Kaser, Ames and Zamora, 1990). However, regional variations in anthropogenic
influence on the climate system preclude an assumption that enhanced glacier meltwater production
can be attributed to human influence alone. Individual glacier topography and aspect mediate its
response to changing local climate conditions and in the Cordillera Blanca, glaciers with eastern aspect
have seen a 71% reduction in volume, compared with 39% for southwestern aspect glaciers (such as
Palcaraju), hypothesised as being the result of the presence of late afternoon clouds protecting
western-facing glaciers from direct solar irradiation (Vilímek et al., 2005). The largest source of
variation glacier melt rates in the Cordillera Blanca is precipitation, whereas temperature remains
fairly constant (Gurgiser, Marzeion, et al., 2013). Whilst widely documented in history, GLOFs are
becoming increasingly common in the Andes, with outbursts occurring from other proglacial lakes in
2008 and 2009 (McGuire, 2012).
Glacier melt is controlled predominantly by energy balance at the interface between the atmosphere
and the glacier (Hock, 2005). This energy balance is determined by the meteorological conditions
above the glacier and the physical properties of the glacier itself. The interactions between the glacier
and the atmosphere are complex and atmospheric supply of energy for melt is mediated by the
presence of snow and ice.
8
Figure 1: Satellite image of the southern Parque Nacional Huascarán, showing the city of Huaraz (blue
outline), Palcaraju glacier (red outline) and the location of the meteorological station at Anta (yellow
outline).
4 Data
4.1 Regional Climate Model
Meteorological inputs to the glacier model were taken from regional climate model outputs from the
weather@home simulations. The weather@home project is run on climateprediction.net, a
distributed volunteer computing climate modelling framework. Weather@home (Massey et al., 2015;
Guillod et al., 2017) consists of the Met Office Hadley Centre Atmosphere-only model, with HadAM3P
running globally at 1.25x1.875 degrees (latitude by longitude) to drive the Met Office Hadley Centre
regional model (HadRM3P). The HadRM3P simulations used in this study were run at 0.22° resolution
over South America (figure 2). This is a one-way nesting where HadAM3P provides boundary
conditions for HadRM3P, but HadRM3P does not feed back into HadAM3P. The HadAM3P model
generates ensembles of climate conditions for a counterfactual world with the climate conditions
without anthropogenic influence on the climate (‘natural’) and present day conditions (‘actual’),
driven by prescribed sea surface temperatures and radiative forcing (Massey et al., 2015; Guillod et
al., 2017). The glacier model was driven by an ensemble of 22 simulations of daily precipitation,
temperature, downwelling shortwave radiation, and wind speed under both actual and natural
conditions, over 29 years from 1988-2016. Each ensemble members differs in the initial conditions of
the driving HadAM3P.
9
Figure 2: Map of the domain modelled by HadRM3P that was used in this study. Width of the yellow
band denotes the ‘sponge’ layer over which RCM fields are relaxed to the driving AGCM. Figure used
courtesy of Dr Sihan Li.
4.2 Observations
Daily temperature observations were taken from the nearby Comandante Fap German Arias
meteorological station at Anta (figure 1; -9.347°,-77.598°, 2749.9m), sourced from the Global
Historical Climatology Network Daily dataset (Menne et al., 2012). The temperature dataset was
incomplete, with only 1792 days of data available over a 23-year period (1988-2010). For the bias
correction detailed below, months with greater than five days of observations available were used to
form the reference period (for which both observations and RCM data was available).
Precipitation observations in the Pacific-Andean region, including the area surrounding Palcaraju
glacier, were scarce (Ochoa et al., 2014). It was not possible to use surface observations since only few
days of data were available for most months and the data that was available was recorded at an
elevation approximately 2000m lower than the glacier snout. The topographic effect on precipitation
in the Cordillera Blanca is such that a strong elevational precipitation gradient is expected. It was also
necessary to have complete observational data for the correction of dry day frequency. Therefore,
following Malone et al. (2015), daily precipitation data was derived from Version 7 of the TRMM Real-
Time Multi-Satellite Precipitation Analysis (TMPA-RT). TRMM combines microwave, microwave-
10
calibrated infrared, and combined microwave-IR estimates of precipitation (Goddard Earth Sciences
Data and Information Services Center, 2016) and is one of the most widely used satellite-based
methods for producing global precipitation estimates. TRMM data is only available from March 2000,
limiting the bias correction reference period for precipitation to the latter half of the RCM time series.
Research in the Peruvian Andes suggests that TRMM tends to underestimate precipitation intensity
(Malone et al., 2015), but represents precipitation seasonality well (Scheel et al., 2011; Ochoa et al.,
2014).
4.3 Digital Elevation Model
Palcaraju glacier (shown in figure 3) was divided into 32 grid boxes with 50m vertical resolution,
covering the elevations from 4625 to 6175m above sea level, with the glacier model solving the daily
melt rate for each vertical level on the glacier surface, following the method of Caidong and Sorteberg
(2010). Glacier hypsometry was taken from the Randolph Glacier Inventory and was valid as of
12/08/2008, the most recent data available (glacier ID: RIGI60-16.02263;
http://www.glims.org/maps/info.html?anlys_id=312068). The average slope of Palcaraju glacier is
32.4° (Pfeffer et al., 2014; RGI Consortium, 2017).
Figure 3: Palcaraju glacier outline. Data from the Randolph Glacier Inventory (Pfeffer et al., 2014; RGI
Consortium, 2017).
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5 Methods
5.1 Temperature bias correction
Since RCM bias may be of sufficient magnitude to introduce significant errors if fed directly into the
glacier model, temperature data was first bias corrected against observations from the closest
meteorological station. Daily temperature data from the climate model was corrected using a variance
scaling methodology described by Chen et al. (2011) and detailed below. The model temperatures
were lapse rate adjusted from the elevation of the nearest grid box to the weather station elevation
using a lapse rate of 8 °C/km, which is the lapse rate calculated over a similar elevational range for the
Cordillera Blanca by Carey et al. (2012) and Schauwecker et al. (2014).
Model temperatures (𝑇𝑅𝐶𝑀) were first adjusted by adding the difference between monthly mean
observational and model temperatures, sampled identically over the same reference period (the
period for which there is observational and modelled data), �̅�𝑜𝑏𝑠 and �̅�𝑅𝐶𝑀,𝑟𝑒𝑓, respectively (equation
1). Variations about this mean temperature were then adjusted to the correct standard deviation by
subtracting the monthly mean corrected temperature, �̅�𝑐𝑜𝑟, from the daily corrected temperature,
𝑇𝑐𝑜𝑟, and multiplying this by the ratio of the observed (𝑆𝑜𝑏𝑠) to model (𝑆𝑅𝐶𝑀,𝑟𝑒𝑓) standard deviation
over the same period (equation 2). This approximately doubled the RCM standard deviation.
Temperatures were then returned to the correct magnitude by multiplying by �̅�𝑐𝑜𝑟 (equation 3).
𝑇𝑐𝑜𝑟 = 𝑇𝑅𝐶𝑀 + (�̅�𝑜𝑏𝑠 − �̅�𝑅𝐶𝑀,𝑟𝑒𝑓) (1)
𝑆𝑐𝑜𝑟 = 𝑆𝑜𝑏𝑠 / 𝑆𝑅𝐶𝑀,𝑟𝑒𝑓 (2)
𝑇𝑐𝑜𝑟,𝑎𝑙𝑙 = (𝑇𝑐𝑜𝑟 − �̅�𝑐𝑜𝑟) × 𝑆𝑐𝑜𝑟 + �̅�𝑐𝑜𝑟 (3)
5.2 Precipitation bias correction
For precipitation data, a similar bias correction approach as is used for temperature is inappropriate
as precipitation data only has non-negative values and variance scaling could give rise to impossible
negative precipitation values (Hempel et al., 2013). A number of methods of statistical bias correction
were considered for precipitation, including local intensity scaling (LOCI) and power transformation
(Schmidli, Frei and Vidale, 2006; Teutschbein and Seibert, 2012). Linear scaling methods account for
bias in the mean but cannot correct for biases in the wet-day frequency and intensity. Evaluation of
TRMM in the La Plata Basin of South America found good agreement between TRMM and gauge
precipitation data on monthly time scales and on the occurrence of daily precipitation events, but
comparatively poor agreement on precipitation intensity estimates at daily scales (Su, Hong and
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Lettenmaier, 2008). Since daily precipitation data was required by the glacier model and biases in both
wet-day frequency and intensity appeared to be present in the RCM, the Inter-Sectoral Impact Model
Intercomparison Project (ISI-MIP) approach described by Hempel et al. (2013) was selected and
adapted to correct only the monthly mean precipitation intensity and wet-day frequency. Monthly
mean precipitation was first scaled, using a month-specific correction factor, and wet-day frequency
was then corrected.
Bias-corrected data is inevitably limited by the availability of the observational dataset and the
representation of physical processes in the climate model. Stationarity in the bias in the actual data
(March 2000-2016 was available for TRMM, and 2001-2016 was used for the correction, which is a
shorter period than the RCM simulation years) with respect to natural data is assumed when applying
the bias correction for the natural run and years outside of the reference period for which there are
observations. Given that non-stationarities in the bias may be present, this introduces additional
uncertainty (Maraun, 2012; Hempel et al., 2013).
A correction factor, 𝑐, is derived from monthly mean precipitation, with 𝑐 limited to 10 to avoid
unrealistically high precipitation values, shown by equation 4. Month specific values of 𝑐 ranged from
0.95 to 4.67.
𝑐 = ∑ 𝑃𝑖𝑜𝑏𝑠
𝑚=16
𝑖=1
∑ 𝑃𝑖𝑅𝐶𝑀
𝑚=16
𝑖=1
⁄ (4)
where 𝑃𝑖𝑜𝑏𝑠 and 𝑃𝑖
𝑅𝐶𝑀 are the observational and modelled monthly mean precipitation for each month
in the reference time series, and 𝑚 is the number of occurrences of each month in the reference
period. Daily precipitation (𝑃𝑖𝑗𝑅𝐶𝑀) is then multiplied by the month-specific correction factor, 𝑐, to give
the mean-corrected daily precipitation, �̃�𝑖𝑗𝑅𝐶𝑀, accounting for the fact that the magnitude of bias in
RCM precipitation may vary seasonally, as shown by equation 5 (Maraun et al., 2010).
�̃�𝑖𝑗𝑅𝐶𝑀 = 𝑐 ⋅ 𝑃𝑖𝑗
𝑅𝐶𝑀 (5)
A wet day threshold was calculated to match the dry day frequency of the RCM to the dry day
frequency in the observational time series (𝑃1𝑅𝐶𝑀[𝑁𝑑𝑟𝑦]). The number of dry days in a month in the
observational dataset was multiplied by the number of ensemble members (22) to determine the dry
day threshold of the ensemble aggregate such that the average number of dry days in an RCM month
across all ensemble members was equal to the number of dry days in the same months of the
observations (equation 6). The frequency of dry days was then adjusted by setting all values below the
dry day threshold 𝜖𝑑 to zero (equation 7).
13
𝜖𝑑 = 0.5 ⋅ 𝑃𝑖𝑗𝑅𝐶𝑀|( 𝑃𝑖𝑗
𝑅𝐶𝑀 ≤ 𝑃1𝑅𝐶𝑀[𝑁𝑑𝑟𝑦]) + 0.5 ⋅ 𝑃𝑖𝑗
𝑅𝐶𝑀|(𝑃𝑖𝑗𝑅𝐶𝑀 > 𝑃1
𝑅𝐶𝑀[𝑁𝑑𝑟𝑦]) (6)
�̂�𝑖𝑗𝑅𝐶𝑀 = 0 if (𝑃𝑖𝑗
𝑅𝐶𝑀 ≤ 𝜖𝑑) (7)
This accounts for the fact that there are artificially large numbers of ‘drizzle days’ in RCMs (Maraun et
al., 2010; Rauscher et al., 2010; Hempel et al., 2013). Therefore, the number of observed dry days,
𝑁𝑑𝑟𝑦, are determined during the reference period by counting the number of days with precipitation
of less than 1 mm (to measurement noise in the TRMM data). Since precipitation values smaller than
𝜖𝑑 are set to zero, frequency of dry days can be increased in the model data. If there are more days
with zero precipitation in the RCM than observational dataset, 𝑁𝑑𝑟𝑦 is selected to equal the number
of days in order to calculate the threshold. Days with rainfall less than or equal to 𝜖𝑑 are adjusted to
0mm. Additional wet days cannot be introduced as this would introduce physical inconsistencies to
the glacier model, such as days with rainfall but no clouds.
This exclusion of drizzle days would alter monthly means and therefore could affect the long-term
trend, so each month is normalised and the sum of precipitation in days adjusted to dry-days is
redistributed uniformly among wet days by the additive constant 𝑚𝑖𝑑𝑎𝑡𝑎 as per equation 8:
�̂�𝑖𝑗𝑑𝑎𝑡𝑎 = {
𝑃𝑖𝑗𝑑𝑎𝑡𝑎 + 𝑚𝑖
𝑑𝑎𝑡𝑎 if wet
0 if dry(8)
In the application period (that not covered by observations), monthly mean values were scaled using
the month-specific correction factor, 𝑐, and then dry day frequency was corrected with the monthly
mean of the ensemble aggregate dry days set to the monthly mean number of dry days in the TRMM
observations. The eliminated drizzle precipitation amount was then redistributed among the wet days
of the month, as noted above.
Due to lack of observations, insolation and wind outputs from the RCM were not bias corrected and
were used directly.
5.3 Distributed energy and mass balance (DEMB) glacier model
The distributed surface energy and mass balance model followed the methodology of Gurgiser,
Marzeion, et al. (2013), which was used for the nearby Shallap glacier, as closely as possible. Due to
the absence of surface measurements on Palcaraju glacier, alternative parameterisations for some
energy fluxes were required to be taken from other sources (notably Arnold et al. (1996) and Hock
(2005)) and it was therefore necessary to code up the model from scratch (the full script is presented
in the appendix). Glacier modelling studies without in situ measurements of all or most of snow depth,
14
relative humidity, global radiation, cloud cover, surface albedo and glacier temperature, are highly
unusual. As far as possible, all parameterisations used in the model have been previously
demonstrated to be effective on other high-altitude tropical glaciers. Similar process-based mass
balance models have been used in various climatic conditions for alpine glaciers (e.g. Mölg and Hardy,
2004; Mölg et al., 2012; Gurgiser, Marzeion, et al., 2013; Gurgiser, Molg, et al., 2013; Prinz et al., 2016).
Energy balance melt models are a physically based approach for melt computation which assesses
energy fluxes to and from the glacier surface. A central assumption in such models is that at a surface
temperature of 0 °C, any surplus energy at the surface-air interface is assumed to be used immediately
for melting (Hock, 2005). Similar studies have run model simulations at hourly time-steps (e.g.
Gurgiser, Molg, et al., 2013) but the resolution of available input climatological data rendered this
impossible. For this study, the DEMB was run at daily resolution and forced with daily bias-corrected
model precipitation and bias corrected daily mean temperature data. Observations of the diurnal cycle
of temperature and precipitation in this region were not available: if they had been, then adding a
realistic diurnal cycle to daily bias-corrected modelled data would have been a preferable approach
given the non-linearities. That said, other approximations, such as the omission of shadowing effects
from surrounding steep topography, and the relatively coarse resolution of the RCM inputs, might
render more detailed modelling less than ideal.
The DEMB model calculates available melt energy, 𝑄𝑀, for each grid cell, using the following
relationship:
𝑄𝑀 = 𝑄𝑁 + 𝑄𝑆 + 𝑄𝐿 (9)
where 𝑄𝑁 is net radiation to the glacier surface, 𝑄𝑆 is the sensible heat flux, and 𝑄𝐿 is the latent heat
release due to sublimation (Gurgiser, Marzeion, et al., 2013). Variations in 𝑄𝑁 dominate 𝑄𝑀, although
increases in 𝑄𝐿 in the austral winter can also reduce the energy available for melt. An additional term,
𝑄𝑅, the heat flux due to rain on the ice surface, is sometimes included in melt models, but due to its
small magnitude, is often excluded (e.g. Arnold et al., 1996; Malone et al., 2015), as is done here,
although refreezing of meltwater was accounted for in calculating ablation. The effects of subsurface
melting and sublimation are not considered (as is the case in Hock and Holmgren, 2005), with all
ablation presumed to be in liquid form. Rupper and Roe (2008) note that in regions with high
precipitation (as is the case for the tropical Andes), ablation is dominated by melt and surface runoff,
and sublimation is comparatively small in magnitude. Subsurface melt is more sensitive to refreezing
than surface melt, and is of much smaller magnitude than surface melt, justifying its exclusion (Mölg
et al., 2012). The melt rate, 𝑀, is therefore calculated as follows:
15
𝑀 =𝑄𝑀
𝜌𝑤𝐿𝑓− 𝑅𝑀 (10)
where 𝜌𝑤 denotes the density of water (1000 kg m-3), 𝐿𝑓 the latent heat of fusion (3.34 × 105 J kg-1),
and 𝑅𝑀 is refrozen meltwater (Hock, 2005).
The vertical air temperature gradient was assumed to be -0.55 °C / 100m, as used by Gurgiser, Molg,
et al. (2013), for the nearby Shallap Glacier, based on measurements at Zongo Glacier, Bolivia (Sicart
et al., 2011). Following Gurgiser, Marzeion, et al. (2013), a vertical gradient in precipitation was not
included, since the vertical precipitation gradient on high tropical mountains is very low.
5.3.1 Net radiation
Evidence from Antisana Glacier, Ecuador (0°28’ S) demonstrates that there is a close relationship
between net shortwave radiation and surface ablation throughout the year (Favier et al., 2004). Net
radiation balance was calculated as follows:
𝑄𝑁 = (𝐼 + 𝐷𝑡)(1 − 𝛼) + 𝑅↓ + 𝑅↑ (11)
where 𝐼 is direct and diffuse solar radiation, 𝐷𝑡 is reflected radiation from surrounding terrain. 𝛼 is
albedo, 𝑅↓ is longwave sky radiation, and 𝑅↑ is emitted longwave radiation (Hock, 2005). As in Gurgiser,
Marzeion, et al. (2013), longwave radiation from the surrounding terrain was not included.
5.3.2 Shortwave radiation
Distributed modelling generally requires global radiation to be split into diffuse and direct components
as shaded grid boxes receive only diffuse shortwave radiation (Hock, 2005). However, since this model
does not attempt to account for obstruction of direct shortwave radiation by surrounding topography
(and Palcaraju glacier’s orientation and low-latitude location is such that this is likely of minor
influence), there is no need to model direct and diffuse shortwave radiation separately. RCM
shortwave radiation outputs could therefore be used directly.
The final component of global radiation, radiation reflected from surrounding terrain, 𝐷𝑡 is
parameterised as:
𝐷𝑡 = 𝛼𝑚 ⋅ 𝐼 ⋅ sin2 (𝛽
2) (12)
where 𝛽 is the gradient of the slope, 𝛼𝑚 is the mean albedo of surroundings, and 𝐼 is downwelling
shortwave radiation from the RCM (Arnold et al., 1996). Arnold et al. (1996) use a value of 0.4 for 𝛼𝑚.
The value of 𝐷𝑡 was found to be very small and could have been excluded from the model.
16
5.3.3 Longwave radiation
Emitted predominantly by atmospheric water vapour, CO2 and ozone, variations in longwave radiation
incident on the glacier surface are primarily controlled by cloud cover, and the amount and
temperature of water vapour. Melt models usually estimate longwave radiation from its correlation
with air temperature and vapour pressure at the surface, parameterised as:
𝑅↓ = 𝜖𝑐𝜎𝑇𝑎4𝐹(𝑛) (13)
where 𝜖𝑐 is the full-spectrum clear-sky emissivity, 𝜎 is the Stefan Boltzmann constant,
5.67 × 10−8Wm-2K-4, 𝑇𝑎 is air temperature (K) and 𝐹(𝑛) is a cloud factor which describes the
increase in radiation due to clouds, as a function of cloud amount, 𝑛. Effective emissivity, 𝜖𝑒 is defined
as the product of 𝜖𝑐 and 𝐹(𝑛) and ranges from 0.7 under clear-sky conditions to close to 1 under
overcast conditions (Hock, 2005). A parameterisation proposed by Arnold et al. (1996) is employed
here for 𝜖𝑒:
𝜖𝑐 ⋅ 𝐹(𝑛) = 𝜖𝑒 = 8.733 × 10−3 ⋅ 𝑇𝑎0.788(1 + 𝑘𝑛) (14)
where 𝑘 is a constant depending on cloud type, and 𝑛, from 0 to 1, is the fraction of the sky covered
by cloud. A constant value of 𝑘 = 0.26, the mean values for altostratus, altocumulus, stratocumulus,
stratus and cumulus cloud types, is used (Arnold et al., 1996). In the absence of model output data on
cloud coverage, 𝑛 was set at 1 for precipitation days and 0.3 for dry days, although these
approximations must be used cautiously. The choice of 𝑛 = 0.3 for dry days was based on the
following relationship, and an approximation of 10K for the daily temperature range (equation 15),
where 𝑥 is the daily temperature range (Arnold et al., 1996).
𝑛 = −0.098𝑥 + 1.285 (15)
Longwave outgoing radiation is controlled by the glacier temperature, and approximated by the
relationship in equation 16:
𝑅↑ = 𝜖𝜎𝑇𝑠4 (16)
where 𝜖 is the surface infra-red emissivity, assumed to be 1 (Gurgiser, Marzeion, et al., 2013), 𝜎 is the
Stefan Boltzmann constant and 𝑇𝑠 is the absolute temperature of the emitting surface (K). If the energy
available for melt is positive, the surface temperature is assumed to be 0 °C (Hock, 2005; Hock and
Holmgren, 2005). Outgoing longwave radiation is often prescribed as 316 Wm-2 (Oerlemans, 1993), as
infra-red emissivity of snow is fairly insensitive to snowpack parameters (Warren, 1982). Snow
17
emissivity has been measured as 99% (Griggs, 1968), but was here assumed to be 100% for both snow
and ice (Gurgiser, Molg, et al., 2013).
5.3.4 Albedo
Glacier albedo varies considerably, ranging from 0.1 for dirty ice to greater than 0.9 for fresh snow
(Hock, 2005). This is a significant control on the spatial and temporal distribution of meltwater
production. Summer snowfall, and resultant increased albedo reduces melt and runoff. Snow aging is
also an important consideration: fresh snow albedo may fall by 0.3 within a few days (Hock, 2005).
The performance of an energy-balance model is highly sensitive to the treatment of albedo
(Oerlemans, 1992).
Glacier surface albedo increases with cloudiness, as clouds preferentially absorb near-infrared
radiation, increasing the fraction of visible light incident on the glacier surface, for which albedo is
higher, and atmospheric water content. Snow albedo has been found to increase 3-15% on transition
from clear-sky to overcast conditions. Albedo modelling is complicated by difficulties in quantitatively
relating variations in albedo to their causes. Some studies have kept albedo constant, such as Malone
et al. (2015) which uses a value of 0.6. This was deemed insufficient given the high temporal resolution
of this model, and albedo was calculated here at daily, and 50m vertical, resolution.
Internal generation of albedo by the mass balance model is considered necessary due to considerable
variations of albedo in space and time. The dependence of albedo on crystal structure, ice and snow
morphology, dust and soot concentrations, morainic material, liquid water in veins, surface water
runoff, solar elevation and cloudiness complicates albedo parameterisation considerably, but these
are rarely accounted for in glacier modelling, and this paper also does not account for these factors.
Gurgiser, Marzeion, et al. (2013) use a parameterisation proposed by Oerlemans and Knap (1998). This
parameterises the albedo of a snow-covered glacier site at day 𝑖 as dependent on the age of the snow
at the surface:
𝛼𝑠𝑛𝑜𝑤𝑖 = 𝛼𝑓𝑖𝑟𝑛 + (𝛼𝑓𝑟𝑠𝑛𝑜𝑤 − 𝛼𝑓𝑖𝑟𝑛) ⋅ exp (
𝑠 − 𝑖
𝑡∗) (17)
where 𝛼𝑓𝑖𝑟𝑛 is 0.53, 𝛼𝑖𝑐𝑒 is 0.34, 𝛼𝑓𝑟𝑠𝑛𝑜𝑤 is 0.75. 𝛼𝑓𝑖𝑟𝑛 is the characteristic albedo for firn (old,
compacted snow), 𝛼𝑓𝑟𝑠𝑛𝑜𝑤 is that of fresh snow, and 𝑡∗ controls the rate at which the snow albedo
approaches the firn albedo following a snowfall event. 𝑠 is the number of the day on which the last
snowfall occurred. When the snow depth, 𝑑, decreases, albedo gradually decreases from that of snow
to that of ice (𝛼𝑖𝑐𝑒). Therefore, a final equation for albedo is as follows:
18
𝛼𝑖 = 𝛼𝑠𝑛𝑜𝑤𝑖 + 𝛼𝑖𝑐𝑒 − 𝛼𝑠𝑛𝑜𝑤
𝑖 ⋅ exp (−𝑑
𝑑∗) (18)
where 𝑑∗ is a characteristic scale for snow depth, with a value of 32 (Oerlemans and Knap, 1998).
When snow depth equals 𝑑∗, snow cover contributes 1/𝑒 to the albedo, and the underlying surface,
(1 −1
𝑒). If snow depth equals 3𝑑∗, the underlying surface contributes about 5% to the albedo.
Snowfall events are defined as days with ≥ 20mm of snow accumulation (Oerlemans and Knap, 1998),
and snowfall was calculated from precipitation according to the method used by Caidong and
Sorteberg (2010) and Tarboton et al. (1995):
𝑃𝑟 = 𝑃 | 𝑇𝑎 ≥ 3℃ (19)
𝑃𝑟 = 𝑃 (𝑇𝑎 − 𝑇𝑏
𝑇𝑟 − 𝑇𝑏) | − 1℃ < 𝑇𝑎 < 3℃ (20)
𝑃𝑟 = 0 | 𝑇𝑎 < −1℃ (21)
𝑃𝑠 = (𝑃 − 𝑃𝑟) (22)
where 𝑃 is the daily precipitation, which is partitioned into rain, 𝑃𝑟, and snow, 𝑃𝑠 (mm water equivalent
(w.e.)) based on air temperature, 𝑇𝑎. As daily mean temperature data is used, the threshold air
temperatures for rain and snow (3°C and -1°C, respectively) deviate significantly from zero to take into
account diurnal temperature variations, as suggested by Caidong and Sorteberg (2010). Due to
unavailable data, no wind-blown drifting snow or accumulation due to avalanches is accounted for,
and following the methodology of Caidong and Sorteberg (2010), snow thickness, 𝐻 (mm w.e.) is
calculated as follows:
𝐻 = 𝐻1 + 𝑃𝑠 −𝑄𝑚
𝜌𝑤𝐿𝑓
(23)
where 𝜌𝑤 is water density, 𝐿𝑓 is the specific latent heat of fusion, 𝑃𝑠 is daily snow accumulation and
𝐻1 is the snow thickness of the day before. The model was spun up for one year to equilibrate snow
depth over a full seasonal cycle (Caidong and Sorteberg, 2010).
5.3.5 Turbulent heat fluxes
Turbulent heat fluxes are driven by the temperature and moisture gradients between the air and the
surface, and by lower atmosphere turbulence. Although these are generally small when averaged over
extended periods of time, turbulent heat fluxes may exceed radiation flux on certain days, and the
19
highest melt rates frequently coincide with high turbulent flux values, notably on temperate glaciers
(Hock, 2005).
Arnold et al. (1996) provide a parameterisation for distributed energy and mass balance modelling, in
which turbulent heat fluxes are calculated for each cell of a melting glacier surface with a temperature
of 0 °C and vapour pressure equal to the saturated vapour pressure at 0 °C. This method requires an
assumption of adiabatic stratification in a Prandtl-type boundary layer, for which vertical fluxes of
energy and momentum constant with height (Kraus, 1973). Relationships for 𝑄𝑆 and 𝑄𝐿, the sensible
and latent heat fluxes, respectively, are therefore:
𝑄𝑆 = 𝐾𝑠𝑃𝑇𝑉 (24)
𝑄𝐿 = 𝐾1𝛿𝑒𝑉 (25)
where 𝐾𝑠 (4.42 × 10−6m kg-1 K-1 s2) and 𝐾1 (7.77 × 10−3m kg-1 s2) are coefficients, 𝑃 is atmospheric
pressure (Pa), 𝑇 is air temperature (°C), 𝑉 is wind speed (ms-1, taken from the RCM) and 𝛿𝑒 is the
difference between the vapour pressure of air and the saturation vapour pressure at the glacier
surface (Pa, Arnold et al., 1996). The vapour pressure of a melting surface was taken as 6.11 hPa, and
saturation vapour pressure calculated using an exponential approximation to the Clausius-Clapeyron
relation for moist air over ice and water (Hock, 2005). Numerical values of 𝐾𝑠 and 𝐾1 include an
adjustment for latent heat of fusion and water density, and vary with snow and ice surfaces and, for
𝐾1, whether the surface is experiencing condensation or evaporation.
Atmospheric pressure was calculated at the grid box elevation according to the barometric formula:
𝑃 = 𝑝0 ⋅ (1 −𝐿𝑅 ⋅ ℎ
𝑇0)
𝑔⋅𝑀𝑅0⋅𝐿
(26)
where 𝑝0 is the standard sea level atmospheric pressure (101325 Pa), 𝐿R is the lapse rate for dry air
(0.0065 K/m), ℎ is the grid box altitude, 𝑇0 is the sea level standard temperature (288.15K), 𝑔 is the
Earth-surface gravitational acceleration (9.80665 m s-2), 𝑀 is the molar mass of dry air (0.0289644 kg
mol-1) and 𝑅0 the universal gas constant (8.31447 J mol-1 K-1).
5.3.6 Other heat fluxes
The sensible heat flux of rain is generally unimportant in the overall surface energy balance of a glacier
and often neglected in models (e.g. Hock and Holmgren, 2005). As its magnitude, compared to other
fluxes, is small, heat flux from precipitation is not considered in the model, nor is mass redistribution
due to wind and gravity (Gurgiser, Marzeion, et al., 2013).
20
5.3.7 Refreezing of meltwater
Accurate parameterisation of meltwater refreezing is challenging even in complex snowpack models
(Etchevers et al., 2004), and some modelling efforts do not take this into account (e.g. Jiang et al.,
2010; Malone et al., 2015; Sagredo et al., 2014). However, this is still a variable widely included in
glacier modelling (e.g. Banwell et al., 2016, 2012) and its exclusion risks overestimating ablation losses
from the glacier and underestimating glacier mass balance. Banwell et al. (2012) find that 6% of surface
meltwater and rainwater refreezes in the snowpack (rather than becoming runoff) in West Greenland;
for a Tibetan glacier, Mölg et al. (2012) found an average of 13% of surface melt refreezes; and for
another Tibetan glacier, Fujita and Ageta (2000) as much as 20% of infiltrated water is refrozen and
does not run off from the glacier. Refrozen meltwater, 𝑀𝑅 for each time-step was calculated as follows,
such that the fraction of meltwater refreezing decreases as the glacier temperature approaches the
melting point (Oerlemans, 1991; Wright et al., 2007).
𝑀𝑅 = 𝑄𝜃 (27)
where 𝑄 is the surface energy budget, 𝜃 is the temperature of the thermally active layer, taken as
equivalent to an upper 2m thickness of solid ice, set to the mean annual air temperature, at the start
of the year (Oerlemans, 1991). Due to the negativity constraint on glacier temperature, temperature
was not permitted to rise above 273.15 K.
𝑥𝑖 =𝑄𝑖{1 − exp(𝜃𝑖)}
𝐿𝑟
(28)
Where 𝑥𝑖 is the mass of refrozen meltwater, 𝐿𝑟 is the latent heat generated by refreezing. Through
the year, glacier temperature is adjusted only by the addition of latent heat due to refreezing of
meltwater, calculated as:
𝐻𝑖𝑐𝑒 = 𝑄𝑖 − 𝑀𝑅 (29)
5.3.8 Initial conditions
Separate model simulations were run for each precipitation and shortwave radiation ensemble
member (to avoid physically improbable sunny days with large amounts of precipitation), and the
ensemble mean of wind velocity and bias corrected ensemble mean of temperature, for each year.
Using the methodology of Rye et al. (2010) each cell of the digital elevation model is initialised as bare
ice and spun-up for one year to generate initial conditions of snow depth on the glacier. Rye et al.
(2010) found that 20 annual cycles were required for mass balance and firn to reach a steady state (in
the high Arctic), Banwell et al. (2012) used 5 years of spin up for the West Greenland Ice Sheet,
21
Bougamont et al. (2005) used a spin up period of 6 months, and Gurgiser, Molg, et al. (2013) use a
three-month spin-up period for the nearby Shallap Glacier, also in the Cordillera Blanca of Peru. These
variations are attributed to varying initial conditions, which influence the time taken for the iterative
process to reach a stable solution, rather than the steady state itself (Banwell et al., 2012). For
Palcaraju glacier, a 1-year spin-up period was found to produce a stable melt rate, and therefore used
for this paper.
6 Results
6.1 Model evaluation
In the absence of measurements of ablation rate at Palcaraju glacier, the glacier model was evaluated
through a comparison with the results of Gurgiser, Marzeion, et al. (2013) from the nearby Shallap
glacier. The comparison was undertaken for 2007, the only complete year modelled for the Shallap
glacier. This was based on the expectation of a similar ablation rate over the same time period on
Shallap and Palcaraju glaciers and the success of Gurgiser, Marzeion, et al. (2013) in reproducing
measured ablation. Data from Shallap glacier indicates a strong semi-annual cycle, whereas the model
of Palcaraju glacier appears to produce only an annual cycle in melt rates. This is most likely the result
of reductions in albedo and therefore increasing net shortwave radiation to the glacier in April and
October, for Shallap glacier, which does not appear in the model use here, due to differences in the
meteorological inputs to the model, the comparatively slow response of albedo to dry periods in the
model, and differences in the glacier site. Nonetheless, across the whole year, ablation rate is found
to be of similar magnitude (20-40 mm/day in the ablation zone), although Palcaraju glacier (as
modelled here) tends to have a slightly higher melt rate for much of April – December 2007, in the
actual scenario. The altitudinal extent of the ablation zone is similar for both glaciers, peaking at
around 5000m (shown in figure 4). Melt rates are lowest in the wet season (austral summer) and the
elevation of the ablation zone increases through the austral winter, peaking at around 5100m in
September. The actual scenario shows a higher melt rate in the ablation zone for much of 2007, and
the equilibrium line altitude is raised from around 4800m (in the natural scenario) to over 5000m in
the actual scenario. Attribution of changed melt rate to anthropogenic climate forcing will be
discussed further in the in section 6.4.
22
Figure 4: Ablation time series for Shallap glacier calculated as daily means for 10m steps in altitude
(top; Gurgiser, Marzeion, et al., 2013) and ensemble mean modelled ablation rate of Palcaraju glacier
under actual (middle) and natural (bottom) climate conditions (50m vertical resolution), all for 2007.
The use of the ensemble mean in the middle and bottom panels smooths out short term variations in
melt rate.
23
A comparison of modelled mass balance values was also conducted with Shallap glacier (Gurgiser,
Marzeion, et al., 2013; figure 5). In line with the ablation data, above, an annual cycle in mass balance
is found for Palcaraju glacier, as compared to the semi-annual cycle present on Shallap glacier, where
mass balance peaks at the glacier snout in April and October. However, the equilibrium line altitude
and mass balance magnitude are similar for the two glaciers in the austral winter of June to September
(with the exception of a brief increase in Shallap mass balance in late July, following a snowfall event).
Glacier flow modelling would be required to evaluate whether the negative mass balance found below
5000m would lead to glacier retreat.
Figure 5: Mass balance for Shallap glacier (top, Gurgiser, Marzeion, et al., 2013) and Palcaraju
glacier, averaged across all ensemble members of the actual run, for 2007. The use of the ensemble
mean in the middle and bottom panels smooths out short term variations in melt rate.
24
6.2 Energy fluxes
Melt rate is most sensitive to variations in net shortwave radiation. This is controlled by seasonal
variations in radiation intensity (which are comparatively moderate at such low latitudes), cloud cover,
and albedo. Cloud cover alters the downwelling shortwave radiation produced by the RCM on a daily
scale. Albedo drives more gradual, high magnitude changes in net shortwave radiation, and is
modelled as being dependent on snow depth and the time elapsed since the last snowfall event. Net
shortwave radiation falls to its lowest levels in the wet season when cloud cover is high, and albedo at
the level of fresh snow due to frequent snowfall and thick snow depth. In the dry season, albedo is
reduced to that of ice (0.34) and net shortwave energy flux increases.
As shown in figure 7, similar to the results of Gurgiser, Marzeion, et al. (2013; figure 6), net longwave
radiation peaks in the months of the austral winter, although the modelled values are, in the dry
months, significantly lower than those found by Gurgiser, Marzeion, et al. (2013). The most significant
differences between the model results on Palcaraju and those from Shallap glacier is in the magnitude
of latent heat flux, and the near-absence of sensible heat flux, partly explained by the low or sub-
freezing surface temperatures for much of the glacier. In the model parameterisation, latent heat flux
was calculated as the product of wind velocity, the difference between the vapour pressure of air and
the saturation vapour pressure at the glacier surface, and a constant. In the absence of surface
measurements of saturation vapour pressure at the glacier surface, this was calculated using the
Arden Buck equation and therefore estimated based on daily ensemble mean air temperature at the
grid box elevation. The use of ensemble mean wind speeds as model inputs might also contributed to
the comparatively small variability in latent heat flux, particularly in the austral winter months.
Sensible heat flux was generally low on Shallap glacier, but could be high in June and July, due to high
surface air temperatures. The use of ensemble mean air temperatures to drive this glacier model
appears to obscure any short-term increases in air temperature and surface air temperature rarely
rises much above 0 °C, even at the lowest elevations of the glacier.
This attempt at modelling the energy fluxes on Palcaraju glacier is hindered by the lack of surface
observations, but the modelled energy available for melt is still similar for Palcaraju and Shallap
glaciers. In light of this, the model was considered to be sufficient for an assessment of the influence
of climate change on melt rate of Palacaraju glacier, although the exact magnitude of this change
predicted by this model (given the constraints in data availability) should be treated with caution.
25
Figure 6: Stacked chart of three-day running means of energy fluxes (orange = net shortwave, blue =
net longwave, red = sensible flux, grey = latent flux, QG = ground heat flux, black = energy available for
melt) spatially averaged over the glacier area, for 2007. Melt energy in the subsurface is indicated as
a dotted line. Grey and blue bars at top denote the dry and wet season, respectively (adapted from
Gurgiser, Marzeion, et al., 2013).
Figure 7: Stacked chart of modelled energy fluxes (net shortwave, net longwave, sensible and latent
fluxes) for the ablation zone (< 5000m) of Palcaraju glacier (actual run), averaged across all ensemble
members, 2007. Energy flux values are in W m-2. Note change in vertical axis scale relative to figure 6.
26
6.3 Meteorological data
6.3.1 Precipitation
One randomly selected ensemble member of the precipitation time series used for the glacier model
runs is presented below, in figure 8, for both the actual and natural scenarios alongside the TRMM
data used for the bias correction. For 20/29 years, the annual mean precipitation values are higher in
the actual runs, and in some years, by as much as 1 mm/day. Corrected model biases are noted in
section 5.2.
Figure 8: Precipitation time series for the raw and bias corrected RCM data (1 randomly chosen
ensemble member, 1987-2016) with the bias corrected data shown in blue, and raw RCM daily
precipitation values in green. The TRMM data used for bias correction is plotted for 2001-2016, in red.
6.3.2 Temperature
As explained above, RCM temperature data was bias corrected against lapse rate-adjusted
observations from the nearest meteorological station to the glacier. RCM data was corrected to
increase its monthly mean and standard deviation to that of the observational dataset, the results of
which are presented in figure 9 for actual and natural runs. Figure 10 shows the strong warming trend
in the temperature anomaly between 1987 and 2016. Over this 30-year period the temperature
anomaly rises by around 1 °C, with the strongest warming trend between 1992 and 2016. Periods of
27
cooling are observed in both the natural and actual runs in 1988-1989, 2007-2008, and 2010-2011, all
during La Niña events.
Figure 9: Bias corrected daily mean RCM temperature data from the factual (green) and natural (blue)
runs, at the bottom of Palcaraju glacier, Peru.
Figure 10: Bias corrected difference between actual and natural RCM temperature, Comandante
German Arias Graziani meteorological station, 2749.9m elevation, 1987-2016.
28
6.4 Attribution of anthropogenic influence on glacier melt rate
The model results for the area of the glacier below 5000m averaged across all ensemble members and
each year show an increase in melt rate in the actual scenario, as compared to the natural scenario,
for all years (figure 11, bottom panel). Melt rate anomalies are generally greatest in the late austral
winter (the dry season, with highest melt rates) and for most years, the greatest anomalies occur at
the top of ablation zone, due to the temperature driven increase in the equilibrium line altitude.
Changes in melt rate as a result of anthropogenic forcing are attributed here on century timescales,
and melt rate is elevated throughout the timescale studied. However, on the decadal time scales, the
contribution of rising temperatures between 1988 and 2016 to melt rate is dampened by other factors,
most likely increases in precipitation and cloud cover, and therefore a reduction in net shortwave
radiation flux at the glacier surface, due to increased albedo and decreased downwelling shortwave
radiation. The attributable increase in melt rate of the Palcaraju glacier ablation zone ranges from 5
mm/day to 35 mm/day, and natural melt rates in the ablation zone range from 25 to 45 mm/day.
While the exact magnitude of this increase is subject to considerable uncertainty, an increase in
ablation zone melt rate of 20-80% is seen for most of the time series. Since the increase in ablation
zone melt rate is between 20 and 50% for most of the time series, this would be equivalent to
advancing the timing of Lake Palcacocha reaching its current levels by approximately 15-37 years (since
lake levels were reduced significantly by the 1943 GLOF). Further research would be required to
explain why the dampening effect on increased melt rate described above appears to be
comparatively small prior to 1988 but prevents further increases in melt rate over the period studied,
despite the significant rise in temperature (ap proximately 1°C) over the same period, as shown in
figure 10.
29
30
Figure 11: Ablation rate, Palcaraju glacier, Cordillera Blanca, Peru, 1988-2016 under actual (top) and
natural (middle) climate conditions, and the attributable change in melt rate across the glacier, over
the same period (bottom). Note the episodic nature of the attributable change in melt rate, with strong
inter annual and decadal variability.
7 Further modelling work
7.1 RCM outputs
Due to the steep orography of the Cordillera Blanca, individual RCM grid boxes in this region cover a
large elevational range and may not accurately represent the values of meteorological variables at the
glacier site. Additionally, errors in the RCM itself, some of which may be inherited from the driving
GCM contributes to the uncertainty in the data and statistically adjusted results of these bias
corrections (Maraun, 2016). The unavailability of usable surface observations of wind speed,
precipitation and shortwave radiation is an obstacle to effective bias correction of the RCM outputs.
Wind speed and shortwave radiation were therefore taken directly from the RCM without bias
correction, precipitation was bias corrected for its dry day frequency and precipitation intensity using
data from the TRMM. These data have been found to accurately capture seasonality in the
precipitation rate of the Peruvian Andes (Malone et al., 2015) while underestimating total annual
precipitation rates observed by Thompson et al. (2013). Evaluation of TRMM in the La Plata Basin of
South America found good agreement between TRMM and gauge precipitation data on monthly time
scales and on the occurrence of daily precipitation events, but comparatively poor agreement on
precipitation intensity estimates at daily scales (Su, Hong and Lettenmaier, 2008). As a result, an
adapted version of the ISI-MIP bias correction approach (Hempel et al., 2013) was applied for
precipitation, in which precipitation was corrected for monthly mean intensity and dry day frequency,
whereas the fitting of a transfer function to the precipitation intensity of wet days was left out. This
could lead to biases remaining in the precipitation data driving the glacier model. Bias correction of
TRMM data with surface precipitation observations would allow the full ISI-MIP bias correction
methodology to be applied to RCM precipitation.
Surface temperature values from the RCM were bias corrected using a variance-scaling methodology
for lapse rate-adjusted observational data. This dataset was incomplete for parts of the time series.
Justified by the assumption of normally distributed daily mean temperature values within individual
months, months with greater than five days of temperature observations were included within the
reference period for bias correction.
31
The bias correction methods assume that the temperature precipitation bias within and outside of the
reference period (for which there are observations) of the actual run is constant, and that the
correction function remains valid for the natural RCM run. Accordingly it is unable to correct for any
errors in the natural RCM run which are not present in the actual data (Maraun et al., 2010).
Furthermore, the effectiveness of statistical bias correction in aligning the RCM values with
observations is dependent on the quality of the observational data set, deficiencies of which have
been noted previously.
7.2 Glacier model
The distributed surface energy and mass balance model devised for this paper, as described above,
provides parameterisations for all significant processes contributing to glacier melt rate. Following
protocols which have been demonstrated to be valid for high alpine glaciers, certain factors which
might contribute to surface energy balance and melt rate were excluded from the model. A high sky
view factor was assumed for the model, and so topographic shading was excluded (e.g. Oerlemans,
1992). The glacier’s orientation (facing a valley) and low-latitude location (i.e. the sun is high in the sky
throughout the year) suggests that little topographic shading of the ablation zone is likely. As a result
of this, diffuse and direct shortwave radiation incident on the glacier surface were taken together
directly from the RCM. Separating diffuse and direct radiation, as would have been required if shaded
grid cells received only diffuse and shortwave radiation (and light reflected from surrounding terrain)
would have introduced additional uncertainties, likely of greater magnitude than the exclusion of this
factor from the melt model.
Additionally, as is the case for the model used by Sicart et al. (2011) for modelling the Zongo glacier in
Bolivia, the melt model is not able to account for subsurface meltwater production. The exclusion of
sub-surface energy fluxes increase the melt rate sensitivity to albedo (Sicart et al., 2011). Although
subsurface meltwater has a very high likelihood of refreezing within the glacier, studies on a Tibetan
glacier have suggested that its exclusion can increase mass balance by as much as 21% (although as
mass balance is comprised of both accumulation and ablation, this represents a smaller proportion of
total melt from the glacier) (Mölg et al., 2012). The lower mean temperatures in the natural scenario
suggests that a greater proportion of subsurface melt would be refrozen. Including subsurface
meltwater production would, therefore, most likely increase the difference in melt rate between the
natural and actual runs. Furthermore, a widely acknowledged source of uncertainty in glacier
modelling is in the determination of turbulent energy fluxes (thought to be much larger than often
modelled) and surface albedo (Hock, 2005). Parameterisations for latent and sensible heat fluxes used
32
ensemble mean RCM output wind speed and air temperature data. More accurate parameterisations
require observations of humidity for calculating latent heat flux. This would reduce latent heat flux in
the wet season and significantly increase its magnitude in the dry season, a variation not captured by
this model (which uses an exponential approximation of the Clausius-Clapeyron relation to calculate
saturation vapour pressure with air temperature).
Confidence in results is also reduced by the lack of surface measurements of humidity, albedo, glacier
temperature, cloud cover, solar radiation, and melt rates, which are a significant obstacle to model
development and validation. For this reason, and as noted previously, remote glacier modelling
(without any surface observations) is highly unusual, and it was therefore necessary to source
parameterisations from several sources for the glacier model. The lack of measurements at the glacier
surface reduced the model sensitivity to short term variations in latent and sensible heat fluxes (in
particular), which can make large contributions to melt rate over short time scales, particularly in the
dry season of May to September.
However, given the motivation to undertake this work, in the context of ongoing legal proceedings
concerning the melt rate of this glacier, this assessment of the sign and approximate magnitude of the
change in melt rate in response to anthropogenic climate forcing was considered appropriate. Current
evidence provided to the court by plaintiffs notes the IPCC’s observation of glacial retreat on the global
and regional scale. This work aligns the impact of climate change on Palcaraju glacier with the global
trend of increasing glacial melt rates, and despite the uncertainties, a clear anthropogenic signal in
melt rate is detected. The assumptions made in constructing this model are consistent with those used
in previously validated glacier models (e.g. Arnold et al., 1996; Hock and Holmgren, 2005; Gurgiser,
Marzeion, et al., 2013), in almost all cases, for tropical glaciers. Nonetheless, there would be
considerable benefit to validating these results and improving the parameterisations using surface
observations. Further research to quantify the rate of increase of depth of Lake Palcacocha
attributable to anthropogenic climate forcing would need to include hydrological modelling of the lake
(including water outflow and evaporation from the lake surface). This would better constrain the
advance in the time at which point it becomes necessary to implement adaptation measures for
Huaraz. This is required if the cost attributable to climate change is to be calculated, as explained
below.
33
8 Discussion of legal context
Many natural disasters occur independently of human alteration of atmospheric GHG concentrations,
but growing numbers of events are demonstrably not ‘acts of god’ and can be attributed to
anthropogenic climate change (Stone and Allen, 2005; Thompson and Otto, 2015). Furthermore, there
is growing acknowledgement in legal and scientific circles that it may be possible to hold major
emitters liable for the proportion of the costs of events attributable to climate change for which they
are responsible (Allen, 2003; Allen and Lord, 2004; Simlinger and Mayer, 2019). As a result, growing
numbers of legal cases seeking damages for climate change impacts are being brought against fossil
fuel companies.
In circumstances in which the likelihood of an event happening increases with cumulative GHG
emissions and has at least doubled, it has been argued that increasing the risk of the event occurring
and causing the damage can be considered synonymous (Marjanac, Patton and Thornton, 2017). The
risk of such events occurring is determined by the atmospheric composition and global temperature
at the time of the event’s occurrence. Therefore, risk increases only as a result of anthropogenic
emissions and does not increase with time, even in the absence of emissions. In these cases, litigation
has sought to hold firms liable for the portion of the total cost of damages caused by the given event,
equal to the proportion of historical anthropogenic emissions for which the firms are responsible
(Thornton and Covington, 2016). However, this argument is valid only for events for which risk
increases only as a result of emissions, typically meteorological events such as floods and heatwaves.
Non-meteorological slow-onset events for which risk also increases with time even in the absence of
human influence, such as the accumulation of meltwater in a proglacial lake, need to be considered
differently.
Climate change may increase the likelihood of a meteorological event occurring in a given year,
whereas increased glacier melt accelerates the rate at which a proglacial lake is filled and brings
forward the next occurrence of a GLOF. If a GLOF would have occurred in the absence of human
influence on the climate, and if the return period of the event is long or the total number of
occurrences of a GLOF from a lake is limited, it is not accurate to attribute damages for the full cost of
the event to climate change. In the case of GLOFs, the return period is typically of the order of decades-
to-centuries, and erosion of terminal moraine may limit the total number of occurrences. Even if the
time to recharge the lake following a flood is reduced as a result of anthropogenic GHG emissions, it
is unlikely that the number of occurrences of the event will have increased over socially or
economically relevant time periods (for example, an additional occurrence of a 1/100-year GLOF
34
reduced to 1/80 by increased melt rates appears only once every 400 years). Rather, the period of
time before an upcoming event may have been reduced.
Further, unlike meteorological events, deterministic attribution to climate change and therefore the
collective action of emitters of long-lived GHGs of an increased rate of glacier meltwater production is
possible. This avoids a common obstacle to the success of litigation: whether an increase in the risk of
an event occurring can be considered sufficient to attribute damages to emitters. In the case of glacier
melt, attribution of changed melt rate may be demonstrable; but how to calculate the costs associated
with this is hitherto undiscussed in climate litigation literature and appears to have few parallels in
legal history (Simlinger and Mayer, 2019).
It can therefore be argued that the harm of a GLOF attributable to climate change is the additional
costs arising from the event occurring sooner than it otherwise would have done. One option,
consistent with mainstream economic practice in the climate damage literature (e.g. Markandya and
González-Eguino, 2019) is for the cost attributable to climate change to be calculated as the difference
between the present-day cost of, in the case of Huaraz, risk reduction (adaptation), and the discounted
present value of the natural event, that is, the discounted present value of the future costs that would
be incurred in the absence of human GHG emissions to obtain the same level of risk reduction. The
discounted present value is calculated as the cost divided by one plus the discount rate, raised to
power of the number of years forward in time the event has been moved as a result of anthropogenic
emissions, as shown in equation 30:
𝐶𝑎𝑛𝑡ℎ = 𝐶𝑎𝑙𝑙 − 𝐶𝑛𝑎𝑡 = 𝐶𝑎𝑙𝑙 −𝐶𝑎𝑙𝑙
(1 + 𝑑)𝑦= 𝐶𝑎𝑙𝑙 (1 −
1
(1 + 𝑑)𝑦) (30)
Equation 30: The cost of slow-onset damages attributable to climate change for events for which risk
is cumulative with both time and emissions. 𝐶𝑎𝑛𝑡ℎ is the cost attributable to anthropogenic GHG
emissions, 𝐶𝑎𝑙𝑙 is the total current cost of the event as a result of climate change, 𝐶𝑛𝑎𝑡 is the discounted
cost of the event occurring when it would have done in the absence of climate change, 𝑑 is the discount
rate, and 𝑦 is the number of years forward in time the event has been moved.
35
Figure 12: Schematic representation of the change in discounted present value of adaptation costs (left
axis) and costs attributable to anthropogenic GHG emissions (right axis) over a 30-year period for
Huaraz. Horizontal lines show the estimated cost attributable to anthropogenic climate forcing based
on the glacier model results under the three discount rates used in policy or proposed by prominent
authors: 1.4% (the average discount rate over the 200-year modelling period in the Stern Review on
the Economics of Climate Change (there is no single discount rate used in the Review since a separate
social discount rate is applied to possible future states of the world due to the uncertainties around
climate change impacts; Dietz, 2008; Stern et al., 2006)), 3.8% (UK Government, 2018) , and 6%
(Nordhaus, 2007, p. 690). This demonstrates how, for a slow-onset event whose timing is brought
forward by anthropogenic influence on climate, the use of a higher discount rate increases the present
value attributable to human influence, in contrast to the social cost of carbon.
Figure 12 demonstrates the importance of the discount rate (the rate of fall of ‘the value of an increase
in consumption at a time in the future relative to now’ (Stern et al., 2006)) for the calculation of the
costs of non-meteorological slow-onset events to climate change. In calculations of the social cost of
carbon (the marginal harm done by emitting one tonne of carbon dioxide) higher discount rates reduce
the costs attributable to emitters (Allen, 2016). By contrast, since higher discount rates ascribe a lower
discounted present value to future costs, it is under higher discount rates that the costs attributable
to climate change are larger according to this methodology, for a given damage. Therefore, as shown
by figure 12, if anthropogenic GHG emissions are responsible for advancing by 30 years the time at
which point adaptation costs of $4 million are required of the city of Huaraz, the cost attributable to
36
climate change is $3.304 million under a 6% discount rate, but only $1.364 million given a 1.4%
discount rate (compared to a present cost of $4 million).
For GLOFs, it is also inappropriate to divide the cost attributable to climate change among emitters in
proportion to their historic contribution to cumulative GHG emissions. Rather, the cost of damages
attributable to any one emitter is dependent on the proportion of the cost of moving the event
forward in time (𝐶𝑎𝑛𝑡ℎ) for which their emissions are responsible, and this depends on the length of
time over which an emitter’s historical emissions have contributed to GLOF risk. Carbon dioxide
emissions released in 1950, for example, will have contributed to the accumulation of meltwater in
Lake Palcacocha (and therefore the increase in GLOF risk) for 50 years longer than emissions produced
in 2000. Consequently, the age of a firm’s emissions is a significant factor in calculating their
contribution to the cost attributable to anthropogenic GHG emissions, as shown by equation 31.
Further research is necessary to calculate the accumulation of meltwater attributable to a pulse
emission of carbon dioxide over time, and therefore the time-dependent contribution of emissions of
different ages to GLOF risk. It is noted that the increase in melt rate in response to a pulse CO2 emission
is unlikely to be constant with time as melt rate increases non-linearly with emissions (as
demonstrated in this paper, and as is the case for numerous impacts of climate change (Harrington
and Otto, in press.)). Therefore, the contribution of the pulse emission to melt rate will change as
atmospheric CO2 levels increase.
𝐶𝑥 = 𝐶𝑎𝑙𝑙 (1 −1
(1 + 𝑑)𝑦) ×
𝐸𝑥
𝐸𝑎𝑙𝑙× 𝑤𝑥 (31)
Equation 31: The cost of damages attributable to an individual emitter for events for which risk of
occurrence is cumulative with time and emissions. 𝐶𝑥 is the cost attributable to emitter 𝑥, 𝐶𝑎𝑙𝑙 is the
total cost of damages, 𝑑 is the discount rate, 𝑦 is the number of years forward in time the event has
moved, 𝐸𝑥 is the historical emissions of firm 𝑥, 𝐸𝑎𝑙𝑙 is cumulative anthropogenic GHG emissions, and
𝑤𝑥 is a weighting factor which accounts for the time at which the emissions of the firm were produced
and therefore their cumulative contribution to the risk of the event occurring.
GLOF risk is dependent, at least in part, on lake volume and therefore cumulative meltwater
production. Elevated meltwater production in response to climate change may increase the rate of
proglacial lake expansion, bringing forward the timing of an outburst flood (or the time at which
adaptation measures are required to be implemented). If meltwater influx to a proglacial lake exceeds
lake drainage and evaporation in a natural climate, then damages sought from emitters should be
calculated according to this equation. If, however, it can be demonstrated that a GLOF would not have
37
happened in the absence of human enhancement of meltwater production through GHG emissions,
litigation could reasonably seek the full cost of the damages of the GLOF or of the cost of adaptation
to heightened GLOF risk (in effect, 𝑦 would be infinite in equations 30 and 31).
8.1 Implications for climate litigation
In Lliuya v. RWE, the claimant seeks to recover damages from RWE AG ‘proportionate to [RWE’s] level
of impairment (share of global GHG emissions) to cover the expenses for appropriate safety
precautions in favour of the claimant’s property from a glacial lake outburst flood from Lake
Palcacocha’ (Rechtsanwälte Günther, 2015). The validity of this claim is dependent on meltwater
production in the absence of anthropogenic emissions being insufficient to trigger a GLOF from Lake
Palcacocha. If this is not the case, the influence of climate change on the timing of the event provides
the justification for claiming of damages. The proposed formula (equation 31) using the difference
between realised costs and their discounted present value (in the absence of climate change) may
provide a useful means to help the courts accurately assess the quantum of damages. Furthermore,
the formula draws attention to the time dependency of the contribution of emissions to GLOF risk:
older emissions will have a greater cumulative contribution to meltwater, and the value of damages
attributed to individual emitters must take this into account.
It is important to stress that equations 30 and 31 are new expressions, developed for this study, with
a much broader application than GLOF risk alone: it would also apply, for example, to liability for the
costs of mitigating the risks of sea level rise in a location in which sea level is rising naturally due to
isostatic rebound. While it is consistent with conventional economic thinking regarding the calculation
of the present value of climate damages, it clearly merits further discussion with stakeholders and
alternative methods of attributing costs to impacts of this type might emerge.
The need to assess discounted future damages adds an additional layer of complexity to climate
litigation. In the case of GLOFs, scientific questions on attribution to climate change, economic analysis
of costs and ethical considerations (including time preferences and concerns about intergenerational
equity (e.g. Stern et al., 2006; Beckerman and Hepburn, 2007)) to determine the discount rate may all
need to be accepted by the courts, if litigation is to succeed. As this paper demonstrates, the impacts
of climate change are increasingly quantifiable by the scientific community. The question remains,
however, is whether in practice the courts will find emitters liable for damages in the face of this multi-
disciplinary complexity.
38
9 Conclusion
A strong anthropogenic signal is identified in the melt rate of Palcaraju glacier based on the output of
a regional climate model and an idealised glacier model. Below 5000m, the modelled melt rate for the
current climate exceeds the natural rate throughout. As discussed in section 6, there are a number of
uncertainties in modelling of a glacier in the absence of surface observations, but all assumptions used
in the glacier model developed for this paper have been used by other authors for alpine glaciers and
demonstrated to be valid.
Although the modelled melt rates under natural conditions are comparatively low, these are still likely
to be sufficient to fill Lake Palcacocha over time. It is clear, however, that the elevated melt rate in
response to anthropogenic GHG emissions will fill the lake much more rapidly. Uncertainties act to
both increase and decrease the meltwater production attributable to human influence on the climate
system: higher mean temperatures in the actual scenario would, for instance, reduce subsurface
refreezing of meltwater, but likely increase the evaporated fraction of ablation. Consequently, the lake
will reach a critical volume, at which point costly adaptation measures will be required to mitigate
GLOF risk, much sooner than would have been the case in the absence of human influence on the
climate system. While the findings of the present study are too uncertain (particularly in the absence
of direct in situ measurements) to determine precisely the proportion of adaptation costs attributable
to an individual emitter, this study does provide the basis for further research which, if undertaken,
would allow this question more accurately. Further research should be focussed in the following four
areas:
i. Further development of the glacier model to include a glacier flow model, and validation and
tuning of model parameterisations with surface observations, particularly of key surface fluxes
over this glacier.
i. Improvements to RCM bias correction with observations of surface downwelling shortwave
solar radiation, wind speed, temperature, and precipitation at locations close to the glacier.
Alternatively, if resources permit, a much higher-resolution non-hydrostatic model would
provide much more reliable information about attributable changes in precipitation, in
particular.
ii. Hydrological modelling of Lake Palcacocha to quantify outflow and evaporation and therefore
the rate at which the lake fills under the actual and natural climate scenarios.
iii. Age of the defendant company’s emissions to calculate cumulative contribution to meltwater
production.
39
With improvements to the glacier model and modelling of lake hydrology it would be possible to
calculate how much sooner the lake will reach a sufficient depth for a GLOF to occur than would have
been the case in a natural climate. For slow-onset climate change impacts, conventional approaches
to climate change litigation may make incorrect assumptions about the responsibility of emitters for
damages. Rather than attributing the entire cost of an event to human GHG emissions, an alternative
methodology for calculating emitters’ liability is proposed. In addition to how much sooner an event
occurs as a result of human influence on the climate system, the discount rate used is also a significant
control on the costs that can be recovered by plaintiffs. Higher discount rates decrease the discounted
present value of events under counterfactual conditions to a greater extent, and therefore attribute a
greater proportion of the current costs to GHG emissions.
Although devastating when they occur, GLOFs are rare and represent only a minor component of the
impacts of climate change on human societies. However, it is possible that this work could have
implications for climate litigation concerning other climate impacts, most notably sea level rise.
Further research is required to assess the time-dependency of the contribution of emitters to sea level
rise, but it is possible that a similar methodology may be required if sea level rise litigation is to align
itself with the best available science and if the contribution of emitters to these impacts is to be
correctly quantified.
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Appendix
#--- Vertically Distributed Surface Energy and Mass Balance Model ---#
#--- November 2018 ---#
#--- Import required packages ---#
import numpy as np
import math
import statistics
import matplotlib.pyplot as plt
import matplotlib.cbook as cbook
import matplotlib.cm as cm
from matplotlib.path import Path
from matplotlib.patches import PathPatch
from netCDF4 import Dataset
import scipy.special as special
import glob
from datetime import datetime
from netCDF4 import date2num,num2date
import matplotlib.patches as mpatches
from math import *
from tempfile import TemporaryFile
import datetime as dt
import matplotlib.dates as mdates
#--- Subroutines that input data from excel spreadsheet as matlab arrays ---#
import Topo_Import
import THREE_VALUE_GRAB
#--- Conversion Factors and Constants ---#
secs_per_day = 3600. * 24.
pi = 3.14159
deg_rad = pi/180.
days_per_year = 360.
#--- Parameters ---#
# Lapse Rate
LR = 0.0055 # degrees per km Gurgiser et al 2013
Lap = 0.0065
Tfreeze = 273.15 # freezing temp of H20 (K)
rhow = 1000. # Liquid at 0C
Lf = 334000.0 # Latent heat of fusion of water - Arnold et al 1996
psia = 0.84 # Atmospheric clear-sky transmissivity (Cuffey & Paterson, 2010)
atm0 = 101325.0 # Atmospheric pressure at sea level (??)
g = 9.80665 # earth-surface gravitational acceleration
M = 0.0300644 # Molar mass of dry air
gc = 8.31447 # Ideal gas constant
Kelvin = 273.15
L = 1367.0 # Solar constant (W/m^2) - Varley et al 1996.
e = 0.017 # eccentricity
precession = 0. # precessional angle Rel. to solstaice (degrees)
obliquity = 23.44 # obliquity angle (degrees)
48
latitude = -9.375 # latitude of site (degrees)
precess = precession * deg_rad # precessional angle (radians)
obl = obliquity * deg_rad # obliquity angle (radians)
lat = latitude * deg_rad # latitude (radians)
psi = lat
sigma = 0.0000000567 # Stefan Boltzman constant (Hock, 2005)
alpham = 0.4 # Arnold et al (1996) - albedo of surroundings
# Define constants for longwave terrain radiation:
Lta = 0.77 # Wm**-2sr**-1
Ltb = 0.54 # Wm**-2sr**-1 from Pluss & Ohmura (1997)
Lb = 100.2 # Wm**2sr**-1 Emitted radiance of a 0C black body (Hock & Holmgren, 2005)
etas = 1.0 # Emissivity of snow cover = 1, Hock & Holmgren (2005)
cp = 1004.0 # Gas phase specific heat (J/(kg K)), at 298K and 1 bar
Ks = 0.00000442 # Sensible heat scalar (snow)
K1 = 0.00777 # for evaporating snow.
#K1 = 0.01114 # for evaporating ice
#--- Model definitions ---#
def xTa(nat_Tgl,LR,b, T_step):
yTa = nat_Tgl[T_step] - (LR * (b - 4625.0))
return yTa
def xpd(nat_ppt, T_step):
ypd = nat_ppt[T_step]
return ypd
def xELA(nat_Tgl, LR, T_step):
yELA = 4625.0 + ((nat_Tgl[T_step] - 273.15) / LR)
return yELA
def xs(Ta, pd, T_step, lev, j, k):
if Ta[lev,T_step] < 276.15 and pd[lev,T_step,k,j] > 10.0:
ys = T_step
elif Ta[lev,T_step-1] < 276.15 and pd[lev,T_step-1,k,j] > 20.0 and T_step > 0:
ys = T_step - 1
elif Ta[lev,T_step-2] < 276.15 and pd[lev,T_step-2,k,j] > 20.0 and T_step > 1:
ys = T_step - 2
elif Ta[lev,T_step-3] < 276.15 and pd[lev,T_step-3,k,j] > 20.0 and T_step > 2:
ys = T_step - 3
elif Ta[lev, T_step-4] < 276.15 and pd[lev,T_step-4,k,j] > 20.0 and T_step > 3:
ys = T_step - 4
elif Ta[lev, T_step-5] < 276.15 and pd[lev,T_step-5,k,j] > 20.0 and T_step > 4:
ys = T_step - 5
elif Ta[lev, T_step-6] < 276.15 and pd[lev,T_step-6,k,j] > 20.0 and T_step > 5:
ys = T_step - 6
elif Ta[lev, T_step-7] < 276.15 and pd[lev,T_step-7,k,j] > 20.0 and T_step > 6:
ys = T_step - 7
elif Ta[lev, T_step-8] < 276.15 and pd[lev,T_step-8,k,j] > 20.0 and T_step > 7:
ys = T_step - 8
elif Ta[lev, T_step-9] < 276.15 and pd[lev,T_step-9,k,j] > 20.0 and T_step > 8:
ys = T_step - 9
elif Ta[lev, T_step-10] < 276.15 and pd[lev,T_step-10,k,j] > 20.0 and T_step > 9:
ys = T_step - 10
elif Ta[lev, T_step-11] < 276.15 and pd[lev,T_step-11,k,j] > 20.0 and T_step > 10:
ys = T_step - 11
elif Ta[lev, T_step-12] < 276.15 and pd[lev,T_step-12,k,j] > 20.0 and T_step > 11:
ys = T_step - 12
elif Ta[lev, T_step-13] < 276.15 and pd[lev,T_step-13,k,j] > 20.0 and T_step > 12:
ys = T_step - 13
elif Ta[lev, T_step-14] < 276.15 and pd[lev,T_step-14,k,j] > 20.0 and T_step > 13:
49
ys = T_step - 14
elif Ta[lev, T_step-15] < 276.15 and pd[lev,T_step-15,k,j] > 20.0 and T_step > 14:
ys = T_step - 15
elif Ta[lev, T_step-16] < 276.15 and pd[lev,T_step-16,k,j] > 20.0 and T_step > 15:
ys = T_step - 16
elif Ta[lev, T_step-17] < 276.15 and pd[lev,T_step-17,k,j] > 20.0 and T_step > 16:
ys = T_step - 17
elif Ta[lev, T_step-18] < 276.15 and pd[lev,T_step-18,k,j] > 20.0 and T_step > 17:
ys = T_step - 18
elif Ta[lev, T_step-19] < 276.15 and pd[lev,T_step-19,k,j] > 20.0 and T_step > 18:
ys = T_step - 19
elif Ta[lev, T_step-20] < 276.15 and pd[lev,T_step-20,k,j] > 20.0 and T_step > 19:
ys = T_step - 20
elif Ta[lev, T_step-21] < 276.15 and pd[lev,T_step-21,k,j] > 20.0 and T_step > 20:
ys = T_step - 21
elif Ta[lev, T_step-22] < 276.15 and pd[lev,T_step-22,k,j] > 20.0 and T_step > 21:
ys = T_step - 22
elif Ta[lev, T_step-23] < 276.15 and pd[lev,T_step-23,k,j] > 20.0 and T_step > 22:
ys = T_step - 23
elif Ta[lev, T_step-24] < 276.15 and pd[lev,T_step-24,k,j] > 20.0 and T_step > 23:
ys = T_step - 24
elif Ta[lev, T_step-25] < 276.15 and pd[lev,T_step-25,k,j] > 20.0 and T_step > 24:
ys = T_step - 25
elif Ta[lev, T_step-26] < 276.15 and pd[lev,T_step-26,k,j] > 20.0 and T_step > 25:
ys = T_step - 26
elif Ta[lev, T_step-27] < 276.15 and pd[lev,T_step-27,k,j] > 20.0 and T_step > 26:
ys = T_step - 27
elif Ta[lev, T_step-28] < 276.15 and pd[lev,T_step-28,k,j] > 20.0 and T_step > 27:
ys = T_step - 28
elif Ta[lev, T_step-29] < 276.15 and pd[lev,T_step-29,k,j] > 20.0 and T_step > 28:
ys = T_step - 29
elif Ta[lev, T_step-30] < 276.15 and pd[lev,T_step-30,k,j] > 20.0 and T_step > 29:
ys = T_step - 30
elif Ta[lev, T_step-31] < 276.15 and pd[lev,T_step-31,k,j] > 20.0 and T_step > 30:
ys = T_step - 31
elif Ta[lev, T_step-32] < 276.15 and pd[lev,T_step-32,k,j] > 20.0 and T_step > 31:
ys = T_step - 32
elif Ta[lev, T_step-33] < 276.15 and pd[lev,T_step-33,k,j] > 20.0 and T_step > 32:
ys = T_step - 33
elif Ta[lev, T_step-34] < 276.15 and pd[lev,T_step-34,k,j] > 20.0 and T_step > 33:
ys = T_step - 34
elif Ta[lev, T_step-35] < 276.15 and pd[lev,T_step-35,k,j] > 20.0 and T_step > 34:
ys = T_step - 35
elif Ta[lev, T_step-36] < 276.15 and pd[lev,T_step-36,k,j] > 20.0 and T_step > 35:
ys = T_step - 36
elif Ta[lev, T_step-37] < 276.15 and pd[lev,T_step-37,k,j] > 20.0 and T_step > 36:
ys = T_step - 37
elif Ta[lev, T_step-38] < 276.15 and pd[lev,T_step-38,k,j] > 20.0 and T_step > 37:
ys = T_step - 38
elif Ta[lev, T_step-39] < 276.15 and pd[lev,T_step-39,k,j] > 20.0 and T_step > 38:
ys = T_step - 39
elif Ta[lev, T_step-40] < 276.15 and pd[lev,T_step-40,k,j] > 20.0 and T_step > 39:
ys = T_step - 40
elif Ta[lev, T_step-41] < 276.15 and pd[lev,T_step-41,k,j] > 20.0 and T_step > 40:
ys = T_step - 41
elif Ta[lev, T_step-42] < 276.15 and pd[lev,T_step-42,k,j] > 20.0 and T_step > 41:
ys = T_step - 42
elif Ta[lev, T_step-43] < 276.15 and pd[lev,T_step-43,k,j] > 20.0 and T_step > 42:
ys = T_step - 43
elif Ta[lev, T_step-44] < 276.15 and pd[lev,T_step-44,k,j] > 20.0 and T_step > 43:
ys = T_step - 44
elif Ta[lev, T_step-45] < 276.15 and pd[lev,T_step-45,k,j] > 20.0 and T_step > 44:
50
ys = T_step - 45
elif Ta[lev, T_step-46] < 276.15 and pd[lev,T_step-46,k,j] > 20.0 and T_step > 45:
ys = T_step - 46
elif Ta[lev, T_step-47] < 276.15 and pd[lev,T_step-47,k,j] > 20.0 and T_step > 46:
ys = T_step - 47
elif Ta[lev, T_step-48] < 276.15 and pd[lev,T_step-48,k,j] > 20.0 and T_step > 47:
ys = T_step - 48
elif Ta[lev, T_step-49] < 276.15 and pd[lev,T_step-49,k,j] > 20.0 and T_step > 48:
ys = T_step - 49
elif Ta[lev, T_step-50] < 276.15 and pd[lev,T_step-50,k,j] > 20.0 and T_step > 49:
ys = T_step - 50
elif Ta[lev, T_step-51] < 276.15 and pd[lev,T_step-51,k,j] > 20.0 and T_step > 50:
ys = T_step - 51
elif Ta[lev, T_step-52] < 276.15 and pd[lev,T_step-52,k,j] > 20.0 and T_step > 51:
ys = T_step - 52
elif Ta[lev, T_step-53] < 276.15 and pd[lev,T_step-53,k,j] > 20.0 and T_step > 52:
ys = T_step - 53
elif Ta[lev, T_step-54] < 276.15 and pd[lev,T_step-54,k,j] > 20.0 and T_step > 53:
ys = T_step - 54
elif Ta[lev, T_step-55] < 276.15 and pd[lev,T_step-55,k,j] > 20.0 and T_step > 54:
ys = T_step - 55
elif Ta[lev, T_step-56] < 276.15 and pd[lev,T_step-56,k,j] > 20.0 and T_step > 55:
ys = T_step - 56
elif Ta[lev, T_step-57] < 276.15 and pd[lev,T_step-57,k,j] > 20.0 and T_step > 56:
ys = T_step - 57
elif Ta[lev, T_step-58] < 276.15 and pd[lev,T_step-58,k,j] > 20.0 and T_step > 57:
ys = T_step - 58
elif Ta[lev, T_step-59] < 276.15 and pd[lev,T_step-59,k,j] > 20.0 and T_step > 58:
ys = T_step - 59
elif Ta[lev, T_step-60] < 276.15 and pd[lev,T_step-60,k,j] > 20.0 and T_step > 59:
ys = T_step - 60
elif Ta[lev, T_step-61] < 276.15 and pd[lev,T_step-61,k,j] > 20.0 and T_step > 60:
ys = T_step - 61
elif Ta[lev, T_step-62] < 276.15 and pd[lev,T_step-62,k,j] > 20.0 and T_step > 61:
ys = T_step - 62
elif Ta[lev, T_step-63] < 276.15 and pd[lev,T_step-63,k,j] > 20.0 and T_step > 62:
ys = T_step - 63
elif Ta[lev, T_step-64] < 276.15 and pd[lev,T_step-64,k,j] > 20.0 and T_step > 63:
ys = T_step - 64
elif Ta[lev, T_step-65] < 276.15 and pd[lev,T_step-65,k,j] > 20.0 and T_step > 64:
ys = T_step - 65
elif Ta[lev, T_step-66] < 276.15 and pd[lev,T_step-66,k,j] > 20.0 and T_step > 65:
ys = T_step - 66
elif Ta[lev, T_step-67] < 276.15 and pd[lev,T_step-67,k,j] > 20.0 and T_step > 66:
ys = T_step - 67
elif Ta[lev, T_step-68] < 276.15 and pd[lev,T_step-68,k,j] > 20.0 and T_step > 67:
ys = T_step - 68
elif Ta[lev, T_step-69] < 276.15 and pd[lev,T_step-69,k,j] > 20.0 and T_step > 68:
ys = T_step - 69
elif Ta[lev, T_step-70] < 276.15 and pd[lev,T_step-70,k,j] > 20.0 and T_step > 69:
ys = T_step - 70
elif Ta[lev, T_step-71] < 276.15 and pd[lev,T_step-71,k,j] > 20.0 and T_step > 70:
ys = T_step - 71
elif Ta[lev, T_step-72] < 276.15 and pd[lev,T_step-72,k,j] > 20.0 and T_step > 71:
ys = T_step - 72
elif Ta[lev, T_step-73] < 276.15 and pd[lev,T_step-73,k,j] > 20.0 and T_step > 72:
ys = T_step - 73
elif Ta[lev, T_step-74] < 276.15 and pd[lev,T_step-74,k,j] > 20.0 and T_step > 73:
ys = T_step - 74
elif Ta[lev, T_step-75] < 276.15 and pd[lev,T_step-75,k,j] > 20.0 and T_step > 74:
ys = T_step - 75
elif Ta[lev, T_step-76] < 276.15 and pd[lev,T_step-76,k,j] > 20.0 and T_step > 75:
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ys = T_step - 76
elif Ta[lev, T_step-77] < 276.15 and pd[lev,T_step-77,k,j] > 20.0 and T_step > 76:
ys = T_step - 77
elif Ta[lev, T_step-78] < 276.15 and pd[lev,T_step-78,k,j] > 20.0 and T_step > 77:
ys = T_step - 78
elif Ta[lev, T_step-79] < 276.15 and pd[lev,T_step-79,k,j] > 20.0 and T_step > 78:
ys = T_step - 79
elif Ta[lev, T_step-80] < 276.15 and pd[lev,T_step-80,k,j] > 20.0 and T_step > 79:
ys = T_step - 80
elif Ta[lev, T_step-81] < 276.15 and pd[lev,T_step-81,k,j] > 20.0 and T_step > 80:
ys = T_step - 81
elif Ta[lev, T_step-82] < 276.15 and pd[lev,T_step-82,k,j] > 20.0 and T_step > 81:
ys = T_step - 82
elif Ta[lev, T_step-83] < 276.15 and pd[lev,T_step-83,k,j] > 20.0 and T_step > 82:
ys = T_step - 83
elif Ta[lev, T_step-84] < 276.15 and pd[lev,T_step-84,k,j] > 20.0 and T_step > 83:
ys = T_step - 84
elif Ta[lev, T_step-85] < 276.15 and pd[lev,T_step-85,k,j] > 20.0 and T_step > 84:
ys = T_step - 85
else:
ys = 0.
return ys
def xPr(Ta, pd, T_step, lev, j, k): # parameterisation from Caidong and Sorteberg 2010
if Ta[lev,T_step] >= 276.15:
yPr = pd[lev,T_step,k,j]
elif 276.15 > Ta[lev,T_step] >= 272.15:
yPr = pd[lev,T_step,k,j] * ((Ta[lev,T_step] - 272.15)/4.0)
else:
yPr = 0.0
return yPr
def xPs(Pr, pd, T_step, lev, k,j):
yPs = pd[lev,T_step,k,j] - Pr
return yPs
def xSNDEPTH(snow_depth, T_step, lev, Ps, model_elev, rhow, Lf,k,j):
if k == 0:
if T_step == 0:
ySNDEPTH = 0.
else:
ySNDEPTH = snow_depth[lev,T_step-1,k,j] + Ps - model_elev[lev,T_step-1,k,j]
else:
if T_step == 0:
ySNDEPTH = snow_depth[lev,359,0,j] + Ps - model_elev[lev,359,0,j]
else:
ySNDEPTH = ySNDEPTH = snow_depth[lev,T_step-1,k,j] + Ps - model_elev[lev,T_step-1,k,j]
return ySNDEPTH
def zSNDEPTH(aSNDEPTH):
if aSNDEPTH >= 0.:
bSNDEPTH = aSNDEPTH
else:
bSNDEPTH = 0.
return bSNDEPTH
def xSNALBEDO(T_step, s):
ySNALBEDO = 0.53 + (0.22 * math.exp((s - T_step) / 21.9))
return ySNALBEDO # Oerlemans & Knap 1998
def xALBEDO(SNALBEDO, SNDEPTH):
52
yALBEDO = SNALBEDO + 0.34 - (SNALBEDO * math.exp((-1. * SNDEPTH)/32.))
return yALBEDO # Oerlemans & Knap 1998
def aALBEDO(zALBEDO):
if zALBEDO > 0.75:
bALBEDO = 0.75
elif zALBEDO < 0.34:
bALBEDO = 0.34
else:
bALBEDO = zALBEDO
return bALBEDO
def xn(pd, T_step, lev, k,j):
if pd[lev,T_step,k,j] < 0.1:
yn = 0.3
else:
yn = 1.0
return yn
# 2. Incoming shortwave radiation calculation
def xKt(alpham, nat_glob_rad, T_step): # Shortwave reflected from surrounding terrain
yKt = alpham * nat_glob_rad[T_step] * 0.07783604
return yKt
def xSHORTIN(nat_glob_rad, T_step, Kt):
ySHORTIN = nat_glob_rad[T_step] + Kt
return ySHORTIN
def xSHORTNET(SHORTIN, ALBEDO):
ySHORTNET = SHORTIN * (1. - ALBEDO)
return ySHORTNET
#--- 3. Longwave radiation ---#
# General equation for incoming L: Hock, 2005; using Arnold et al (1996) for ec*F(n)
def xLONGIN(sigma, Ta, T_step, n, lev, k,j):
yLONGIN = sigma * ((Ta[lev,T_step])**4.) * 0.008733 * ((Ta[lev,T_step])**0.788) * (1. + (0.26*n)) # Incoming solar radiation (l/w)
return yLONGIN
#--- Glacier surface temperature (Oerlemans, 1991) ---#
def xmean_temp(Ta):
ymean_temp = np.array(np.mean(Ta, axis=1)) # deg C
return ymean_temp
#def xlatent_minus(melt_en, glac_temp, lev, T_step, k, j):
# if T_step == 0:
# zlatent_release = 0.
# else:
# zlatent_release = melt_en[lev,T_step-1,k,j] - (melt_en[lev,T_step-1,k,j]**glac_temp[lev,T_step-1,k,j])
# return zlatent_release
#def xglacier_temp(mean_temp, lev, T_step, glac_temp, latent_min, k,j):
def xglacier_temp(mean_temp, lev):
yglacier_temp = mean_temp[lev] - 273.15
return yglacier_temp
def zglacier_temp(glacier_tem):
if glacier_tem > 0.:
aglacier_temp = 0.
else:
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aglacier_temp = glacier_tem
return aglacier_temp
# Longwave radiation from surrounding terrain
#def xLONGTERR(glacier_temp, Lb, Lta, Ltb, T_step, Ta, lev, k,j): # Pluss and Ohmura, 1997. Does not include obstruction by
surrounding terrain.
# yLONGTERR = Lb + (Lta * (Ta[lev,T_step,k,j] - 273.15)) + (Ltb * glacier_temp)
# return yLONGTERR
# Longwave outgoing Radiation
def xLONGOUT(sigma, glacier_temp):
yLONGOUT = sigma * ((glacier_temp+273.15)**4.)
return yLONGOUT
def xLONGNET(LONGIN, LONGOUT):
yLONGNET = LONGIN - LONGOUT
return yLONGNET
#--- 4. Turbulent heat fluxes ---#
#--- a) Sensible heat ---#
def xAIR_PRESS(atm0, Lap, g, M, gc, b):
#yAIR_PRESS = atm0 * ((1.0 - (Lap * b)/288.15)**((g*M)/(gc*Lap))) # Atmospheric pressure at grid-box level
yAIR_PRESS = atm0 *((288.15/(288.15+(Lap * b)))**((g*M)/(gc*Lap)))
return yAIR_PRESS
# Based on Arnold et al (1996)
def xSENSIBLE(Ks, AIR_PRESS, Ta, T_step, nat_vw, lev, k,j):
ySENSIBLE = Ks * AIR_PRESS * (Ta[lev,T_step] - 273.15) * nat_vw[T_step]
return ySENSIBLE
#--- b) Latent heat ---#
# Based on Arnold et al (1996)
#--- Arden Buck equation for saturation vapour pressure over ice ---#
def xsatvpi(Ta, lev, T_step):
if Ta[lev,T_step] >= 273.15:
esice = 100.0 * 0.61121 * math.exp((18.678 - ((Ta[lev,T_step]-273.15)/234.5)) * ((Ta[lev,T_step]-273.15)/(257.14 + Ta[lev,T_step]
- 273.15)))
else:
esice = 100.0 * 0.61115 * math.exp((23.036-((Ta[lev,T_step]-273.15)/333.7))*((Ta[lev,T_step]-273.15)/(279.82+Ta[lev,T_step]-
273.15)))
return esice
#def vapour_press():
# yvappress = ?????
# return yvappress
def xdeltae(satvpi): # Difference between vapour pressure of air and saturation vapour pressure at the glacier surface
ydeltae = satvpi - 611.
return ydeltae
def xLATENT(K1, deltae, nat_vw, T_step):
yLATENT = K1 * deltae * nat_vw[T_step]
return yLATENT
#--- Calculate energy available for melt ---#
def xMELT_ENERGY(SHORTIN, LONGNET, ALBEDO, SENSIBLE, LATENT):
yMELT_ENERGY = (SHORTIN * (1.0 - ALBEDO) + LONGNET + SENSIBLE + LATENT)
if yMELT_ENERGY > 0.:
zMELT_ENERGY = yMELT_ENERGY
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else:
zMELT_ENERGY = 0.
return zMELT_ENERGY
#--- Calculate refreezing (Oerlemans, 1991) ---#
def xmeng(MELT_ENERGY, glacier_temp):
ymeng = MELT_ENERGY**glacier_temp
return ymeng
def xlatent_release(MELT_ENERGY, meng):
ylatent_release = MELT_ENERGY - meng
return ylatent_release
def xrefrozen(latent_released, Lf): # Oerlemans, 1991
yrefrozen = latent_released / Lf
return yrefrozen
def zrefrozen(arefrozen, rhow, Lf, MELT_ENERGY, secs_per_day):
if arefrozen < (MELT_ENERGY * secs_per_day * 1000. / (rhow * Lf)):
brefrozen = arefrozen
else:
brefrozen = MELT_ENERGY * secs_per_day * 1000. / (rhow * Lf)
return brefrozen
#--- Calculate net melt ---#
def xTOTAL_MELT_ELEV(rhow, Lf, MELT_ENERGY, secs_per_day, refrozen):
yTOTAL_MELT_ELEV = MELT_ENERGY * secs_per_day * 1000. / (rhow * Lf) - refrozen # Melt
if yTOTAL_MELT_ELEV < 100.:
zTOTAL_MELT_ELEV = yTOTAL_MELT_ELEV
else:
zTOTAL_MELT_ELEV = 1.
return zTOTAL_MELT_ELEV
def xDAY_TOTAL_MELT(TOTAL_MELT_ELEV):
yDAY_TOTAL_MELT = np.array(np.mean(TOTAL_MELT_ELEV, axis=0)) # Is this correct?
return yDAY_TOTAL_MELT
def xensemble_mean(model_elev):
yensemble_mean = np.array(np.mean(model_elev, axis=3))
return yensemble_mean
def xensemble_mean_sw(short_net):
yensemble_mean_sw = np.array(np.mean(short_net, axis=3))
return yensemble_mean_sw
def xensemble_mean_lw(longnet_arr):
yensemble_mean_lw = np.array(np.mean(longnet_arr, axis=3))
return yensemble_mean_lw
def xensemble_mean_sens(sensflux):
yensemble_mean_sens = np.array(np.mean(sensflux, axis=3))
return yensemble_mean_sens
def xensemble_mean_latent(latentflux):
yensemble_mean_latent = np.array(np.mean(latentflux, axis=3))
return yensemble_mean_latent
def xlevmean_sw(sw_mean):
ylevmean_sw = np.array(np.mean(sw_mean,axis=0))
return ylevmean_sw
def xlevmean_lw(lw_mean):
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ylevmean_lw = np.array(np.mean(lw_mean,axis=0))
return ylevmean_lw
def xlevmean_sens(sens_mean):
ylevmean_sens = np.array(np.mean(sens_mean,axis=0))
return ylevmean_sens
def xlevmean_latent(latent_mean):
ylevmean_latent = np.array(np.mean(latent_mean,axis=0))
return ylevmean_latent
def main():
list_of_ensmem = glob.glob('/soge-home/users/shil4723/dissertation/rcm/precip/factual/raw/model/2007/x_sr_precip_*.nc')
n_ens = len(list_of_ensmem)
nat_list_of_ensmem = glob.glob('/soge-
home/users/shil4723/dissertation/rcm/precip/counterfactual/raw/model/2007/x_sr_precip_*.nc')
nat_ens = len(nat_list_of_ensmem)
melt_en = np.zeros((8,360,2,22))
snow_depth = np.zeros((8,360,2,22))
glac_temp = np.zeros((8,360,2,22))
latent_release = np.zeros((8,360,2,22))
pd = np.zeros((8,360,2,22))
mean_temp = np.zeros((8))
ensemble_mean = np.zeros((8,360,2))
albedo_array = np.zeros((8,360,2,22))
model_elev = np.zeros((8,360,2,22))
Ta=np.empty((8,360))
Ta[:] = np.nan
short = np.zeros((8,360,2,22))
short_terr = np.zeros((8,360,2,22))
short_net = np.zeros((8,360,2,22))
long = np.zeros((8,360,2,22))
longterrain = np.zeros((8,360,2,22))
longoutgoing = np.zeros((8,360,2,22))
longnet_arr = np.zeros((8,360,2,22))
sensflux = np.zeros((8,360,2,22))
latentflux = np.zeros((8,360,2,22))
refreeze = np.zeros((8,360,2,22))
snow = np.zeros((8,360,2,22))
sw_in = np.zeros((8,360,2,22))
sw_mean = np.zeros((8,360,2))
lw_mean = np.zeros((8,360,2))
sens_mean = np.zeros((8,360,2))
latent_mean = np.zeros((8,360,2))
sw = np.zeros((360,2))
lw = np.zeros((360,2))
sens = np.zeros((360,2))
latent = np.zeros((360,2))
#--- Set up mountain topography ---#
# topo = 'Palcaraju_topo.xlsx'
# x, y, w = THREE_VALUE_GRAB.GetData(topo)
y = [4625., 4675., 4725., 4775., 4825., 4875., 4925., 4975.]
az = 215. # slope azimuth
grad = 32.4 # slope gradient
# Temperature
file_temp = '/soge-home/users/shil4723/dissertation/rcm/temp/counterfactual/biascorr/model/glacier_2007.nc'
temp = Dataset(file_temp, mode='r')
Tgl = temp.variables['temp'][:] # Daily temp at bottom of glacier
Tgl_units = temp.variables['temp'].units
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temp.close()
nat_file_temp = '/soge-home/users/shil4723/dissertation/rcm/temp/factual/biascorr/model/glacier_2007.nc'
nat_temp = Dataset(nat_file_temp, mode='r')
nat_Tgl = nat_temp.variables['temp'][:]
nat_Tgl_units = nat_temp.variables['temp'].units
nat_temp.close()
for T_step in np.arange(360):
for lev in np.arange(8):
b = y[lev]
tmp = xTa(Tgl, LR, b, T_step)
print tmp
Ta[lev,T_step] = tmp
lev += 1
T_step += 1
for j in np.arange(6):
precipitation_file = '/soge-home/users/shil4723/dissertation/rcm/precip/factual/raw/model/21/x_sr_precip_' + str(j+1) + '.nc'
precip = Dataset(precipitation_file, mode='r')
time = precip.variables['time0'][:]
time_units = precip.variables['time0'].units
t = num2date(time[:],time_units)
ppt = precip.variables['precip'][:] # Daily precipitation
ppt_units = precip.variables['precip'].units
glob_rad = precip.variables['sr_flux'][:]
glob_rad_units = precip.variables['sr_flux'].units
precip.close()
nat_precipitation_file = '/soge-home/users/shil4723/dissertation/rcm/precip/counterfactual/raw/model/21/x_sr_precip_' +
str(j+1) + '.nc'
nat_precip = Dataset(nat_precipitation_file, mode='r')
nat_ppt = nat_precip.variables['precip'][:]
nat_glob_rad = nat_precip.variables['sr_flux'][:]
nat_precip.close()
# Temperature
# file_temp = '/soge-home/users/shil4723/dissertation/rcm/temp/factual/biascorr/model/glacier_' + str(q+1) + '.nc'
file_temp = '/soge-home/users/shil4723/dissertation/rcm/temp/factual/biascorr/model/glacier_21.nc'
temp = Dataset(file_temp, mode='r')
Tgl = temp.variables['temp'][:] # Daily temp at bottom of glacier
Tgl_units = temp.variables['temp'].units
temp.close()
nat_file_temp = '/soge-home/users/shil4723/dissertation/rcm/temp/counterfactual/biascorr/model/glacier_21.nc'
nat_temp = Dataset(nat_file_temp, mode='r')
nat_Tgl = nat_temp.variables['temp'][:]
nat_Tgl_units = nat_temp.variables['temp'].units
nat_temp.close()
# Wind Speed
wind = '/soge-home/users/shil4723/dissertation/rcm/wind/factual/wind_21.nc'
vel = Dataset(wind, mode='r')
vw = vel.variables['v_wind'][:]
vel.close()
nat_wind = '/soge-home/users/shil4723/dissertation/rcm/wind/counterfactual/wind_21.nc'
nat_vel = Dataset(wind, mode='r')
nat_vw = nat_vel.variables['v_wind'][:]
nat_vel.close()
n_days = len(t)
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for k in np.arange(2):
# Calculate the glacier dynamics and surface energy balance for the number of steps per year
for T_step in np.arange(360):
#--- Defining grid-cell elevations ---#
for lev in np.arange(8):
b = y[lev]
ypd = xpd(nat_ppt, T_step)
pd[lev,T_step,k,j] = ypd
ELA = xELA(nat_Tgl, LR, T_step)
s = xs(Ta, pd, T_step, lev, j, k)
Pr = xPr(Ta, pd, T_step, lev, j, k)
Ps = xPs(Pr, pd, T_step, lev, k,j)
snow[lev,T_step,k,j] = Ps
aSNDEPTH = xSNDEPTH(snow_depth, T_step, lev, Ps, model_elev, rhow, Lf,k,j)
SNDEPTH = zSNDEPTH(aSNDEPTH)
snow_depth[lev,T_step,k,j] = SNDEPTH
SNALBEDO = xSNALBEDO(T_step, s)
zALBEDO = xALBEDO(SNALBEDO, SNDEPTH)
ALBEDO = aALBEDO(zALBEDO)
albedo_array[lev,T_step,k,j] = ALBEDO
n = xn(pd, T_step, lev,k,j)
Kt = xKt(alpham, nat_glob_rad, T_step)
short_terr[lev,T_step,k,j] = Kt
SHORTIN = xSHORTIN(nat_glob_rad, T_step, Kt)
sw_in[lev,T_step,k,j] = SHORTIN
SHORTNET = xSHORTNET(SHORTIN, ALBEDO)
short_net[lev,T_step,k,j] = SHORTNET
LONGIN = xLONGIN(sigma, Ta, T_step, n, lev, k,j)
long[lev,T_step,k,j] = LONGIN
mean_temp[:] = np.squeeze(xmean_temp(Ta))[:]
glacier_tem = xglacier_temp(mean_temp, lev)
glacier_temp = zglacier_temp(glacier_tem)
glac_temp[lev,T_step,k,j] = glacier_temp
LONGOUT = xLONGOUT(sigma, glacier_temp)
longoutgoing[lev,T_step,k,j] = LONGOUT
LONGNET = xLONGNET(LONGIN, LONGOUT)
longnet_arr[lev,T_step,k,j] = LONGNET
AIR_PRESS = xAIR_PRESS(atm0, LR, g, M, gc, b)
SENSIBLE = xSENSIBLE(Ks, AIR_PRESS, Ta, T_step, nat_vw, lev,k,j)
sensflux[lev,T_step,k,j] = SENSIBLE
satvpi = xsatvpi(Ta, lev, T_step)
deltae = xdeltae(satvpi)
LATENT = xLATENT(K1, deltae, nat_vw, T_step)
latentflux[lev,T_step,k,j] = LATENT
MELT_ENERGY = xMELT_ENERGY(SHORTIN, LONGNET, ALBEDO, SENSIBLE, LATENT)
melt_en[lev,T_step,k,j] = MELT_ENERGY
meng = xmeng(MELT_ENERGY, glacier_temp)
latent_released = xlatent_release(MELT_ENERGY, meng)
latent_release[lev,T_step,k,j] = latent_released
arefrozen = xrefrozen(latent_released, Lf)
refrozen = zrefrozen(arefrozen, rhow, Lf, MELT_ENERGY, secs_per_day)
refreeze[lev,T_step,k,j] = refrozen
TOTAL_MELT_ELEV = xTOTAL_MELT_ELEV(rhow, Lf, MELT_ENERGY, secs_per_day, refrozen)
model_elev[lev,T_step,k,j] = TOTAL_MELT_ELEV
lev += 1
T_step += 1
k += 1
j += 1
ensemble_mean[:] = np.squeeze(xensemble_mean(model_elev))[:]
#--- Plotting script inserted here ---#
main()
