PUBLICATIONS
Journal of Advances in Modeling Earth Systems
RESEARCH ARTICLE
10.1002/2014MS000404
Key Points:
Entropy production is calculated with
a glacier-atmosphere model
In situ measurements support a state
of maximum entropy production
This state is most important for
internal system parameters
Correspondence to:
T. M€
olg,
[email protected]
Citation:
M€
olg, T. (2015), Exploring the concept
of maximum entropy production for
the local atmosphere-glacier system,
J. Adv. Model. Earth Syst., 7, 412–422,
doi:10.1002/2014MS000404.
Received 9 NOV 2014
Accepted 9 MAR 2015
Accepted article online 16 MAR 2015
Published online 10 APR 2015
Exploring the concept of maximum entropy production for the
local atmosphere-glacier system
€ lg1
Thomas Mo
1
Climate System Research Group, Institute of Geography, Friedrich-Alexander-University Erlangen-N€
urnberg (FAU),
Erlangen, Germany
Abstract The concept of maximum entropy production (MEP) is closely linked to the second law of thermodynamics, which explains spontaneous processes in the universe. In geophysics, studies have argued
that planetary atmospheres and various subsystems of Earth also operate at maximum dissipation through
MEP. One of the debates, however, has concerned the degree of empirical support. This article extends the
topic by considering measurements from a high-altitude, cold glacier in the tropical atmosphere and a
numerical model, which represents the open and nonequilibrium system of glacier-air exchanges. Results
reveal that several sensitive system parameters, which are mainly tied to the shortwave radiation budget,
cause MEP states at values that coincide closely with the in situ observations. Parameters that set up the
forcing of the whole system, however, do not show this pattern. Empirical support for the detection of MEP
states, therefore, is limited to parameters that regulate the internal efficiency of energy flow in the glacier.
System constraints are shown to affect the solutions, yet not critically in the case of the two most sensitive
parameters. In terms of MEP and geophysical fluids, the results suggest that the local atmosphere-glacier
system might be of relevance in the further discussion. For practical purposes, the results hold promise for
using MEP in single or multiparameter optimization for process-based mass balance models of glaciers.
1. Introduction
The second law of thermodynamics states that the entropy of the universe increases in the course of any
spontaneous change. The universe, in technical parlance, means the system of interest together with its surroundings. The term spontaneous, furthermore, is used as a synonym of ‘‘natural’’ and implies that no work
is required to drive the change in the universe. While the first law of thermodynamics identifies a feasible
change among all conceivable changes (by imposing energy conservation), the second law provides the
basis for explaining spontaneous processes among the feasible changes [Atkins, 2007]. Yet, the second law
of thermodynamics does not say which path, out of the available paths, a system will take to implement the
spontaneous process.
C 2015. The Authors.
V
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€
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This is the point where the Maximum Entropy Production (MEP) principle connects [e.g., Dewar, 2003], as it
aims to deliver information about the trajectory a system in nonequilibrium state will take over time [Dyke
and Kleidon, 2010]. At this trajectory, entropy is maximized at the fastest rate given the system constraints.
Likewise, the thermodynamic force is minimized at the fastest rate. Many systems of Earth and of other
planets are indeed in nonequilibrium states, and paths are the central objects of interest in nonequilibrium
systems from a microscopic (Boltzmann’s) viewpoint [Dewar, 2003], which suggests a role of MEP for modeling complex, nonlinear fluids. The classical study in this context was conducted by Paltridge [1975] with a
simple multibox model. He demonstrated that the mean state of the global climate can be reproduced well
if horizontal heat transport in the atmosphere and in the oceans is constrained by MEP. Subsequent studies
for different applications obtained essentially the same result, namely that a MEP state exists for geophysical
systems. Examples include atmospheres on other planets [Lorenz et al., 2001]; processes in subsystems of
the global climate such as atmospheric convection [Ozawa and Ohmura, 1997; Pujol and Fort, 2002] or the
thermohaline circulation [Shimokawa and Ozawa, 2002]; mantle convection in the Earth’s interior [Vanyo
and Paltridge, 1981]; and crystal growth morphology [Hill, 1990].
The review articles by Ozawa et al. [2003] and Dyke and Kleidon [2010] present further examples and, in general, support MEP as a possible principle of a variety of nonlinear fluid systems. These papers also discuss
GLACIER-ATMOSPHERE ENTROPY PRODUCTION
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different mechanisms that may explain MEP states, but basically argue that nonequilibrium states as well as
conservation of energy and mass are central to Earth systems, which justifies their description by thermodynamics and delivers the foundation for MEP. Dyke and Kleidon [2010], however, also emphasize that the
MEP principle is not necessarily a natural law, but rather a method for increasing information about realworld systems. Therefore, explanations for MEP are also based on the application of information theory in
statistical mechanics [Jaynes, 1957; Dewar, 2003]. Problems of the MEP principle are highlighted by Goody
[2007] concerning a particular MEP study of planetary atmospheres and, more generally, with regard to the
meaning of constraints in the employed models. However, Goody [2007] states that MEP may be useful for
application to individual climate processes, and that agreement with observed data is vital for strengthening the foundation of MEP as a ‘‘principle.’’
The aim of the present paper is not to contribute to the theoretical discussion of MEP. Rather, the present
study targets the empirical aspect of the problem. The major goal is to test with the help of in situ measurements whether a particular climate process, which is the local interaction between atmosphere and glaciers,
behaves like a MEP system. To the best of my knowledge, the local atmosphere-glacier system has not
been considered in MEP analyses so far. A secondary goal is to discuss the results in light of model parameter choice, since the MEP concept was examined more recently in terms of parameter tuning for climate
modeling [Kleidon et al., 2006; Kunz et al., 2008]. This suggests potential value for glacier-atmosphere models as well.
2. Methods, Model, and In Situ Measurements
2.1. Mass Balance Model
The specific climatic mass balance of a glacier (MB; kg m22) over a specified unit of time is the result of
accumulation (mass gain processes) and ablation (mass loss processes) in response to the in situ atmos€lg et al. [2008, 2009a, 2012] and has
pheric conditions. The model of this linkage is described in detail in Mo
been applied to various questions and climatic environments around the world [e.g., Conway and Cullen,
€lg et al., 2014].
2013; Gurgiser et al., 2013; MacDonell et al., 2013; Nicholson et al., 2013; Cullen et al., 2014; Mo
Therefore, only a brief model description is given here.
For each time step, the model receives meteorological input data, from which it calculates the MB as the
sum of three mass gain processes (solid precipitation, refreezing of liquid water in the snow, surface deposition of water), three mass loss processes (surface sublimation, surface and subsurface melt), and the change
in liquid water storage in the snow. All these processes are linked directly or indirectly to the energy balance
at the glacier surface. This balance considers the two energy sources of incoming solar radiation and atmospheric longwave radiation, the two energy sinks of reflected solar radiation (depending on the surface
albedo) and longwave radiation emitted by the glacier surface, as well as four terms that can be either
energy sources or sinks (depending on the surface-air or surface-subsurface gradients of the respective
potentials): the turbulent sensible and latent heat fluxes, a heat flux from precipitation, and the ground
heat flux. If the surface temperature of the glacier in a time step reaches 273.15 K and the energy balance
yields excess energy, this energy is expended by surface melt. In the case of no surface melt, energy conservation is obtained by solving for the surface temperature value that balances the fluxes [e.g., Van den Broeke
€lg et al., 2009a].
et al., 2006; Mo
Energy flow within the glacier is calculated from the thermodynamic energy equation on a numerical grid
between glacier surface and a specified depth at several meters (section 2.2). The subsurface part not only
treats heat conduction, but also energy release due to penetrating solar radiation and refreezing of liquid
€lg et al., 2008], as well as energy consumption by subsurface melting [Mo
€lg et al.,
water in the snow [Mo
€lg et al. [2009a] for the governing MB equation and Mo
€lg et al. [2012] for the surface
2012]. Refer to Mo
energy balance equation. The specific form of the thermodynamic energy equation can be found in Greuell
and Konzelmann [1994].
To summarize the model in a thermodynamic framework, we have two distinct thermal reservoirs. The
upper reservoir is determined by the meteorological conditions in the atmospheric surface-layer directly
above the glacier, and is therefore highly variable at various time scales (diurnal and annual cycles, climate
variability). The lower reservoir is represented by the model bottom, where energy fluxes are assumed to be
negligibly small and temperature is constant (section 2.2). Hence, our glacier acts as the connecting system
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4
10.1002/2014MS000404
between these two thermal reservoirs and experiences a continuous
flow of energy. This flow implies a
certain efficiency, which aims to dissipate the existing gradients and is
driven by the nonequilibrium state
of the whole (atmosphere-glacier)
system. As in many previous applications of the present model, our interest
concerns
a
high-altitude
mountain glacier (Figure 1), where
extensive field research has provided
the basis for the data requirement of
this study.
2.2. System Forcing and Model
Parameters
At the upper system boundary, as
outlined above, the energy and mass
fluxes are driven by the atmospheric
variations in the air above the glacier. The full system forcing consists of local air temperature and relative
humidity, air pressure, wind speed, cloud cover fraction, and precipitation amount. These data are provided
at hourly steps from measurements with an automatic weather station (AWS) on Kersten Glacier on Kilimanjaro at 5873 m above sea level, equatorial East Africa (Figure 1). Here this station’s established 3 year record
€lg et al. [2009a] subjected these data to a rigorous
from 9 February 2005 to 23 January 2008 is used, since Mo
€lg et al. [2009a] also summarize (in their
check and found them to be of high quality for MB modeling. Mo
Table 1) how the six forcing variables are tied to the energy and mass fluxes. The AWS also records short
and longwave radiation components (from which cloud cover was derived), which are used below in the
discussion of results. Note that solid precipitation is recorded as actual height with unknown density.
Figure 1. Kersten Glacier on the southern slope of Kibo, Kilimanjaro’s main massif
(3.1 S, 37.4 E). Automatic weather stations 3 (5873 m a.s.l.) and 4 (5603 m a.s.l.) on
the glacier are indicated by the numbers. The image was taken in July 2005 during
an aerial photography campaign (photo courtesy of Nicolas J. Cullen).
The lower thermal boundary is specified at several meters depth in the glacier, where the effects of diurnal
and seasonal temperature waves penetrating from the surface vanish [Klok and Oerlemans, 2002]. On
Kersten Glacier, this depth was found to be 3 m, where the temperature is held constant at 268.2 K, and
hence dT/dt 5 0. This parameter, as well as all the other model parameters, initial conditions, and structural
€lg et al. [2009a]. Their study
settings (parameterizations and numerical grid spacing), are taken from Mo
demonstrated for the mentioned period (February 2005 to January 2008) that the MB model reproduced
the measured surface temperature at the AWS site, and surface height change and snow depth at different
locations, with good accuracy (i.e., within the measurement uncertainty). Hence, similarly to other MEP studies [Kunz et al., 2008], the present paper relies on a standard model parameter set. In light of the evaluation
€lg et al., 2009a], this set can also be understood as an ‘‘ideal’’ parameter combination. This does
results [Mo
not rule out that there could be other ‘‘ideal’’ parameter spaces as well [Rye et al., 2012].
The total number of model parameters in the current code (version 2.4) for glacier-wide runs is around 30
€lg et al., 2012, Table A1]. This number includes elevational gradients of the two forcing
[see, for example, Mo
variables of air temperature and precipitation, while relative humidity, cloud cover, and wind speed variations
with altitude are typically neglected for MB models of small mountain glaciers (and air pressure is known from
the barometric equation). The necessity for considering gradients exists since input data stem from one point
on the glacier, and thus the distributed forcing for glacier-wide model application is obtained by extrapolation
with these gradients. Hence, the lower model boundary also consists of a topographic boundary condition,
€lg et al., 2009a]. This means the glacier (Figure 1) is diswhich is a digital terrain model of Kersten Glacier [Mo
cretized by 100 m grid cells, and every cell has its own altitude, slope, aspect, and sky view factor.
The mentioned AWS on Kersten Glacier, termed AWS3 and installed in 2005, is only one part in a network of
€lg and Kaser [2011]). All of them have provided records withpresently four stations (e.g., see the map in Mo
out significant data gaps so far, which has made the field program on Kilimanjaro overly successful in view
of the difficult measurement circumstances at high altitude. Moreover, our past research put the local
€
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Reflected radiation (W m−2)
1400
AWS3
1200
α=
1000
0.9
.85
0
α=
800
600
400
200
0
0
200
400
600
800
1000
1200
1400
Incoming radiation (W m−2)
10.1002/2014MS000404
measurements in a multiscale modeling
context of climate dynamics (summarized
€lg et al. [2009a]). The resultant data
in Mo
basis is the main reason for choosing Kilimanjaro in the present paper. In particular, both (i) data from AWS4 in a lower
section of Kersten Glacier (Figure 1),
which recorded air temperature/humidity
and precipitation since 2009 for more
insights into the elevational gradients
mentioned above, and (ii) the results
from high-resolution atmospheric model€lg and Kaser, 2011] provide
ing [e.g., Mo
additional opportunities for discussing
the MEP results in section 3.
Figure 2. Hourly measurements of incoming versus reflected shortwave radiation at AWS3 between 9 February 2005 and 23 January 2008. The lines represent albedos of 0.9 and 0.85.
2.3. Entropy Production
Formally, the change in entropy of a system (dS) is defined as heat supplied
reversibly (dQ) divided by the temperature (T), hence dS 5 dQ/T with units [J K21] [Atkins, 2007]. On the
€lg et al., 2009a], the entropy production
numerical grid of the model glacier, which comprises 17 layers [Mo
(g) is calculated directly from the temperature tendencies @T/@t [e.g., Fraedrich and Lunkeit, 2008; Kunz
et al., 2008]. As a glacier-wide and time-mean value between the surface and the depth of the model bottom (zbot), this is written in units of [W m22 K21] as
(ð
)
0
q @T
g5cp
dz
(1)
zbot T @t
where {X} and X signify the horizontal and time mean, respectively, z is the thickness of the layer, q is the
density, and cp is the specific heat capacity of snow and ice.
The thermodynamic energy equation solved on the model grid [Greuell and Konzelmann, 1994] includes
four terms that govern @T/@t and thus the entropy production. (i) The incoming energy from the atmosphere (from the full surface energy balance in the uppermost model layer, and from penetrating solar radiation in the subsurface); (ii) heat conduction, which is controlled by the effective thermal diffusivity of snow
and ice; and (iii) and (iv) the water phase changes by refreezing or melting in the subsurface. Note that
material entropy exchange at the surface by water fluxes (sublimation/deposition, precipitation, melt) also
affects g through the respective energy sources or sinks in the surface energy balance.
2.4. Parameter Focus
Two central criteria determine the parameter focus of this study. First, the parameter must be known as a
sensitive (a so-called active) parameter in MB modeling. Second, field observations from Kersten Glacier, or
from a similar tropical glacier environment, must be available for the parameter. This second criterion is vital
in order to achieve the main goal of the present paper, namely to test whether MEP states occur for a highaltitude glacier on the basis of empirical evidence.
Combined experimental and MB modeling studies for climate zones around the world basically agree in the
result that solar radiation absorbed by the glacier, also called net shortwave radiation, is by far the domi€lg and Hardy, 2004;
nant energy source [e.g., Bintanja and van den Broeke, 1995; Wagnon et al., 1999; Mo
Andreassen et al., 2008; MacDougall and Flowers, 2011; Sicart et al., 2011; Gurgiser et al., 2013]. This condition
makes MB models most sensitive to parameters related to the shortwave radiation budget, in particular to
those of the surface albedo scheme. Albedo parameters, therefore, always participate in classical parameter
sensitivity tests [e.g., Klok and Oerlemans, 2004; Hock and Holmgren, 2005] and received attention in recent,
more computationally expensive approaches that targeted model uncertainty from parameter uncertainty
€lg et al., 2012; Rye et al., 2012]. In the group of parameters for the spatial distribu[Fitzgerald et al., 2012; Mo
tion of the atmospheric forcing, the elevational gradients in air temperature and precipitation, as well as the
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a
fresh snow albedo
η (mW m−2 K−1)
0.4
b
fresh snow density
0.35
0.3
0.3
0.25
0.25
diffuse radiation
0.4
0.4
0.35
c
10.1002/2014MS000404
0.38
0.36
0.34
0.32
0.2
0.5
0.65
0.8
Albedo
0.95
0.2
100
150
200
250
300
Density (kg m−3)
2
6
10
14
18
Fraction (%)
Figure 3. Mean entropy production g as a function of the model parameters (a) fresh snow albedo, (b) fresh snow density, and (c) fraction
of diffuse radiation in clear sky. Measured values (see text) from Kersten Glacier and from other tropical glacier sites are indicated in grey.
Solid lines (or the rectangle for the range) indicate multiannual mean values, broken lines measured events.
temperature threshold separating solid from liquid precipitation, are known to influence MB model solutions. Thus, these three parameters are often considered in sensitivity and uncertainty analyses as well [e.g.,
€lg et al., 2012; Rye et al., 2012].
Klok and Oerlemans, 2002; Machguth et al., 2008; Mo
With regard to this background and the demand of observations, the present study examines variations in
entropy production as a function of the following six parameters: the aforementioned three parameters for
the spatial distribution of the atmospheric forcing; as well as fresh snow albedo, fraction of diffuse solar
radiation in clear sky, and fresh snow density. While the solar radiation and albedo parameters are obviously
related to the shortwave radiation budget, fresh snow density also exerts a major effect on the albedo
€lg et al.,
(lighter snow induces less mass input). This effect has only been highlighted more recently [Mo
2009a; Rye et al., 2012].
The extended discussion (section 3.2) will consider further active model parameters, which were not measured on Kilimanjaro directly but are fairly well known from other sources. These include more parameters
for the shortwave radiation budget (albedo, penetration of solar radiation in the ice), and a roughness
parameter for the turbulence scheme. The latter is also known to be a sensitive part of process-based MB
€lg and Hardy, 2004; Machguth et al., 2008; Conway and Cullen, 2013]. The total number of
models [e.g., Mo
examined parameters is therefore higher than in typical previous MEP studies. This should allow us to make
a first assessment of the hypothesis that a MEP state exists for a high-altitude atmosphere-glacier system.
3. Results and Discussion
3.1. MEP for Parameters Observed by Corresponding In Situ Measurements
Fresh snow has a huge impact on solar radiation absorption by glaciers and, therefore, on glacier melt [e.g.,
€lg and Hardy, 2004]. The measurements on Kersten Glacier show that the albedo of fresh snow typically
Mo
lies in the range 0.85–0.90 (Figure 2). This range coincides closely with a MEP state, which exists near 0.83 in
the MB model (Figure 3a). Note that we are not looking at ice albedo in this study, because the albedo of
€lg et al. [2009a].
ice was formulated as a time-varying function of dew point temperature in the runs of Mo
The fact that the lower envelope in Figure 2 tends to show two ‘‘notches’’ (at 700 and 900 W m22 incoming radiation) partly explains why incorporating ice albedo as a constant parameter is less suited at this site.
The fresh snow density on Kilimanjaro can also, and more generally, be called the density of solid precipita€lg and Kaser, 2011]. At high and midlatitudes, this density can be
tion, since graupel may occur as well [Mo
lower than 100 kg m23 [Judson and Doesken, 2000]. In the tropics, it is usually higher. Sicart et al. [2002]
report a mean value of 250 kg m23 from 2 year measurements on Zongo Glacier, while single events vary
in the 150–300 kg m23 range. On Kilimanjaro, we could rarely measure the density of fresh solid precipitation, as field trips have to be conducted in the dry season when precipitation events are infrequent. One of
the biggest surveys of surface snow in January 2006 yielded a sample mean of 241 kg m23 from a snowfall
event during the previous night (Figure 3b). A distinct peak in entropy production indeed occurs in the
€lg and Scherer [2012],
range of measured values (Figure 3b). Additional support are the results (i) from Mo
€
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a
air temperature gradient
b
c
precipitation gradient
10.1002/2014MS000404
phase threshold
η (mW m−2 K−1)
0.5
0.4
0.4
0.4
0.35
0.3
0.3
0.2
0.38
0.25
0.36
0.1
0.2
0
−1
−0.5
0
Gradient (K (100 m)−1)
0.15
−0.2
−0.1
0
Gradient (% m−1)
0.1
0.34
−5
−2.5
0
2.5
5
Temperature (°C)
Figure 4. Mean entropy production g as a function of the model parameters (a) vertical air temperature gradient, (b) vertical precipitation
gradient, and (c) temperature threshold for the precipitation phase. Measured values (see text) from Kersten Glacier and the slope of Kilimanjaro are indicated by grey lines.
who obtained a mean value of 269 6 23 kg m23 from a joint analysis of high-resolution atmospheric model
€lg and Kaser, 2011] and the measured actual height of precipitation at AWS3, and (ii) from Mo
€lg
output [Mo
et al. [2009a] who found an effective value of 285 kg m23 in their model runs.
€lg et al. [2009b] from 2 year
The fraction of diffuse radiation in clear-sky conditions was determined by Mo
records of shortwave radiation at AWS3. They obtained 4.6%, and on the nearby glacierized Mount Kenya,
Hastenrath [1984] made a direct measurement of this fraction during field work. His result at somewhat
lower altitude (590 hPa compared to 502 hPa at AWS3) was 5–10%. A MEP state also exists for this parameter and occurs close to the measured values around 10% (Figure 3c).
Results for the parameters that distribute the meteorological forcing variables over the glacier surface (Figure 4) differ markedly from the above results. In Figure 4a, entropy production shows a monotonic decrease
as the decrease of air temperature with elevation gets weaker. However, the measured gradient in air temperature between AWS3 and AWS4 (for the overlapping period 9 October 2009 to 23 September 2012)
shows a decrease of 0.75 K per 100 m altitude. Likewise, the measured precipitation at the two stations
reveals a decrease of 16% per 100 m, and Røhr and Killingtveit [2003] also measured a decrease (8% per
100 m) on the southern mountain slopes below Kersten Glacier. Entropy production, on the other hand,
shows a minimum in the region of observed values (Figure 4b). Another monotonic pattern emerges for the
entropy production as a function of the air temperature threshold separating solid from liquid precipitation
(Figure 4c). With the help of AWS3 data, and using the physical background of wet bulb temperature, this
€lg et al., 2008]. High-resolution atmospheric modeling agrees
threshold was found to be around 2.5 C [Mo
€lg
with the measurements and shows similar air temperature and precipitation decreases with altitude [Mo
and Kaser, 2011], as well as increasing probability of solid precipitation below air temperatures of 3 C
€lg and Scherer, 2012].
[Mo
The above findings resemble the results of Kunz et al. [2008] remarkably. In their model study of entropy
production in the global atmosphere, they found a MEP state for ‘‘internal’’ parameters that regulate the
thermodynamic efficiency of the atmosphere. However, they could not detect a MEP behavior for ‘‘external’’
parameters, which set up the forcing field for the global circulation. The present results also suggest that
external parameters, which shape the forcing (upper boundary condition) of the whole atmosphere-glacier
system, are not connected to a MEP state (Figure 4). By contrast, a MEP system is suggested by the other
parameters (Figure 3), which are a direct part of the energy and mass balance formulations and can therefore be considered ‘‘internal’’ to the glacier.
From a general model perspective, it should be noted that a model parameter must sometimes differ from
the respective observation due to structural shortcomings in models [Fitzgerald et al., 2012; Rye et al., 2012].
For example, if a MB model in the case of very low fresh snow densities (<100 kg m23) would not consider
rapid initial snow settling, fresh snow density must be higher in the model than in reality [Rye et al., 2012].
Hence, even if it were proven that a certain system is a MEP system, a perfect agreement between the
parameter values during MEP and the observations cannot be expected. However, the close coincidence
between the two is evident in Figure 3 for our system of interest.
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a
b
albedo time scale
η (mW m−2 K−1)
0.33
c
albedo depth scale
d
subsurface radiation
10.1002/2014MS000404
fresh snow roughness
0.45
0.35
0.4
0.328
0.445
0.3
0.326
0.3
0.44
0.324
0.25
0.2
0.322
0.435
0.32
2
4
6
8
Time (days)
10
12
0.2
10
0.1
20
30
40
Depth (cm)
50
60
0
3
6
9
12
Extinction (m−1)
15
0
0.5
1
1.5
2
Length (mm)
Figure 5. Mean entropy production g as a function of the model parameters (a) albedo time scale, (b) albedo depth scale, (c) extinction coefficient for penetrating shortwave radiation in
ice, and (d) aerodynamic roughness length of fresh snow. Estimates from various sources (see text) are indicated by the grey lines (or the grey rectangle in Figure 5d for a measured
range).
3.2. MEP for Parameters Constrained by Other Measurements
To elucidate the possibility of a MEP state for internal parameters, this section examines further active
model parameters. Unlike in section 3.1, there are no comparable direct observations from tropical glaciers
for these parameters, but there is information from other measurement efforts or from indirect estimates
about reasonable values.
The albedo scheme of the model [Oerlemans and Knap, 1998] incorporates two further sensitive parameters.
First, the albedo time scale is a parameter for how fast the albedo of new snow decreases toward the one
of old snow or firn. This scale cannot be measured directly but only be estimated, by comparing the albedo
€lg et al.
record with the record of surface height change (a proxy of snow depth). This was done by Mo
[2009a], and accordingly they adopted 5.4 days. A well-defined MEP state occurs close to this value around
4 days (Figure 5a). Second, the albedo depth scale parameterizes the impact of the underlying ice surface
€lg et al. [2009a] found an effective
on snow albedo. It is also unknown from direct measurements, but Mo
value of 36 cm using again the albedo/surface height change records. The entropy production (Figure 5b)
rises sharply up to the maximum rates in the range 25–30 cm, and then falls slowly, which supports a depth
scale of at least 25 cm from a MEP perspective.
Another important parameter in the glacier’s shortwave radiation budget is the extinction of penetrating
€lg et al. [2009a]
solar radiation with depth in the ice. It is unmeasured in tropical environments, so Mo
accepted a standard value of 2.5 m21 from other studies. Highest entropy production also occurs for rather
small values of 1–2 m21 (Figure 5c), which supports that the extinction in ice is about an order of magnitude smaller than in snow [Bintanja and van den Broeke, 1995]. Note that solar radiation penetration in
€lg et al. [2009a], since subsurface temperature measurements at AWS3 only
snow was neglected by Mo
€lg et al., 2008].
allowed us to detect a role of radiation penetration in ice [Mo
Finally, MB model sensitivity and uncertainty runs often target the parameterization of turbulence and the
involved aerodynamic roughness length. This length scale varies strongly in the case of ice and snow surfaces [Brock et al., 2006], so any determinations from short-term measurements like on Kilimanjaro’s Northern
Icefield [Cullen et al., 2007] may reflect particular circumstances. The least spread can be expected for the
roughness length of fresh snow. The MB model implementation with the time-varying roughness length
€lg et al., 2009a] shows MEP for roughness lengths of 0.05–0.1 mm in the case of fresh snow (Figscheme [Mo
ure 5d). A recent laboratory experiment by Gromke et al. [2011] suggested a value of 0.24 6 0.05 mm.
Hence, MEP supports that the roughness length of fresh snow can be 1–2 orders of magnitude smaller than
the one for old, ablating snow [Brock et al., 2006].
3.3. System Constraints and Mutual Parameter Dependency
One of the major discussion points in the geophysical MEP literature has concerned the role of constraints
for the system of interest [e.g., Kleidon et al., 2003; Goody, 2007; Fraedrich and Lunkeit, 2008]. Dyke and Kleidon [2010], for example, interpret the increasing number of constraints as enhanced information. Goody
[2007], on the other hand, expresses that the true significance of the imposed constraints is hard to evaluate. The discussion is tied to the background that geophysical MEP studies typically reduced the problem to
€
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one constraint, for example, the
meridional heat transport [Paltridge,
1975] or the surface temperature discontinuity [Pujol and Fort, 2002]. The
influence of mutual parameter
dependency on MEP solutions, however, has hardly been considered so
far. Yet this deserves attention since
the application of unconstrained MEP
(i.e., for a system only constrained by
energy and mass conservation) is generally not warranted for climatic problems [Goody, 2007], as climate
processes are subject to many constraints, e.g., input of solar radiation
from outer-space, gravitation, planetary rotation, or surface friction.
fresh snow albedo and density
M
0.9
0.35
albedo constraint
0.3
mW m–2K–1
0.7
density constraint
Albedo
0.8
0.6
0.5
100
150
200
250
0.25
0.2
0.15
300
350
400
10.1002/2014MS000404
0.1
Density (kg m–3)
Figure 6. Mean entropy production as a function of the two model parameters,
fresh snow albedo and fresh snow density. ‘‘M’’ locates the maximum rate. The
cross symbols show the MEP states from the single-parameter variations (Figures
3a and 3b), where the constraining value (dotted line) was prescribed by the
‘‘ideal’’ parameter combination of M€
olg et al. [2009a], as explained in the methods
section.
The empirical evidence in this study
can thus be investigated further by
looking at the mutual dependency of
the two most sensitive model parameters, fresh snow albedo and density
€lg et al., 2009a], observations of which coincide closely with the MEP state of the system (Figure 3). Such
[Mo
multiple parameter analyses are particularly helpful with regard to parameter optimization [Kunz et al.,
2008]. The result in Figure 6 demonstrates indeed the mutual dependency and shows a ‘‘ridge’’ in the
topography, which runs from very high albedo and densities of 225 kg m23 to an albedo of 0.75 and a
density of 350 kg m23. The ridge location makes sense, as less fresh snow input to the system (lower density) complicates the development of a high albedo due to sufficiently deep snow, which can be compensated for by setting fresh snow albedo rather high. The opposite relation also holds. The MEP states from
the single-parameter variations (Figure 3) are contained in the zone of maximum entropy production and
approximate the MEP state from the two-parameter variation (Figure 6). Hence, the mutual parameter
dependency does not impact the system’s MEP state critically, although a small effect is evident.
To develop the discussion of MEP and system constraints further in the context of process-resolving MB
modeling, a possibility exists in connection with recent Monte-Carlo-type approaches [Machguth et al.,
€lg et al., 2012, 2014; Rye et al., 2012; Cullen et al., 2014]. These studies varied the whole parameter
2008; Mo
space of the model in thousands of runs and accepted several parameter combinations as ‘‘optimal’’ sets,
which enhances the possibility of quantifying model uncertainty considerably. Typically, in these and in
more traditional MB modeling studies [e.g., Klok and Oerlemans, 2002], the measured MB on the glacier provided the basis for the objective functions (performance criteria) employed in the parameter optimization
fresh snow albedo
fresh snow density
18
14
3
ΔT
12
2.2
F
MEP
3.4
18
3
2.6
14
ΔT
F (W m−2)
ΔT (K)
16
ΔT (K)
3.8
3.4
2.6
22
MEP
F
F (W m−2)
3.8
2.2
10
0.5
0.65
0.8
0.95
Albedo
100
150
200
250
300
10
350
Density (kg m−3)
Figure 7. Mean temperature difference DT (model bottom minus surface temperature) and mean heat flux F (conductive heat flux at surface) as a function of two model parameters (fresh snow albedo and density). Grey lines indicate at which parameter value the MEP state
occurs.
€
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GLACIER-ATMOSPHERE ENTROPY PRODUCTION
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Journal of Advances in Modeling Earth Systems
fresh snow density
fresh snow albedo
2
2mo
2mo
1
0
)
3a
l(
ful
−1
−2
0.5
1yr
3b
)
.
Fig
ig.
−1
1yr
l (F
0
1
ful
η (standardized)
2
0.65
0.8
0.95
−2
100
150
200
250
10.1002/2014MS000404
and/or model evaluation. In future, the
MEP criterion could be included as
additional, more theoretical objective
function. If high-altitude glaciers
indeed strive for MEP states, the
‘‘good’’ parameter combinations should
show higher entropy production than
the ‘‘bad’’ combinations.
300
3.4. Physical Interpretation
The above results of well-defined MEP
states require a physical consistency
Figure 8. Mean entropy production g as a function of two model parameters
(fresh snow albedo and density) and the length of the modeling period. ‘‘2mo’’
check at this stage. In climatic applicameans the first two months of the record (February and March 2005), ‘‘1yr’’ is the
tions, a MEP state is typically underfirst year (2005), and ‘‘full’’ is the multiannual period 2005–2008. Note standarstood as trade-off between the
dized units on the y axes for easier comparability.
temperature difference (DT) in, and the
associated heat flux (F) through, a system. This can be derived from considering two extreme conditions
[e.g., Ozawa et al., 2003]: in a static state, where F 0, DT will be largest; in an extreme diffusion case, where
F is very large, DT will become negligible. Since MEP implies the maximum generation of available potential
energy, the MEP state must occur as the mentioned trade-off between the two extreme conditions [Ozawa
et al., 2003; Kunz et al., 2008].
Albedo
Density (kg m−3)
Figure 7 exemplifies for the important parameters, fresh snow albedo and density, that rather small DT in
the atmosphere-glacier system coincides with a strong heat flux (left regions of the graphs), while the large
DT cases are accompanied by weak heat diffusion (right regions of graphs). The MEP states, in turn, exist for
parameter values that induce intermediate DT and F values, which conform to the trade-off situation. The
same pattern exists for the internal parameters of Figure 5 (not shown). It is therefore unlikely that the MEP
states in the model are a coincidence or a numerical artifact of the model.
3.5. System Stationarity and Modeling Period
A final noteworthy point is that the nonequilibrium systems considered in MEP analyses should be in steady
state. In terms of the Earth’s climate, this stationarity should be encapsulated in a long-term mean state of a
particular system [e.g., Kunz et al., 2008]. This most probably implies consideration of a climatic normal
period, but such long records are not available from high-altitude glaciers. While the 3 year record of observations used as the basis here can certainly not represent a long-term mean state, it can at least be considered as a ‘‘fortunate’’ approximation. This is because the observation period included a rather dry year
(2005), a rather wet one (2006), and an average year (2007) in terms of the moisture climate, 1978–2007, in
€lg et al., 2009a]. Moisture variability, in turn, characterizes the climate of East Africa and Kilithe region [Mo
manjaro on various time scales.
While it is not possible to resolve the stationarity issue for the present case due to the available data basis,
the basic influence of the steady state feature in the system can be investigated. Figure 8 illustrates, again
for the two sensitive model parameters fresh snow albedo and density, that a few months of MB modeling
do not induce a MEP state at intermediate values at all. Considering one full annual cycle seems at least necessary to approximate the multiannual results and, thus, to achieve empirical support. However, the results
from the 3 year record show the best agreement with observations (Figures 3a and 3b). This means that as
the energy-balance related observations on mountain glaciers become longer, in the best case the ones
that started in the late 1990s [Oerlemans and Knap, 1998] or shortly after 2000 [Hardy, 2011], the assessment
of the role of glaciers in the MEP problem will improve.
4. Concluding Remarks
The present study aims to contribute a concise paper to the discussion of MEP states in geophysical fluid
systems. In this regard the atmosphere-glacier linkage at high altitude, which represents a nonequilibrium
open system, was analyzed for MEP with the help of a numerical model and in situ measurements. A robust
feature of the results emerges for parameters of the glacier’s shortwave radiation budget, for example, the
€
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10.1002/2014MS000404
fresh snow albedo or fresh snow density. This parameter group shows MEP states at values that are in close
agreement with the in situ observations or other empirical evidence. By contrast, variations of parameters
that shape the forcing of the whole system (e.g., elevation gradients in meteorological variables) do not
evolve into empirically supported MEP states. This is in line with a previous study on entropy production in
the atmosphere [Kunz et al., 2008]. Therefore, the results underline that MEP states may be particularly
important for processes that affect the internal efficiency of energy flow in a climatic system.
A critical condition for the detection of a MEP system concerns the system constraints [Goody, 2007]. The
results demonstrate that such constraints (i.e. fixed parameters) do have an effect on the calculations. Yet in
the present case, multiparameter variations for two key parameters (fresh snow density and albedo) do not
yield a MEP state that differs significantly from the solution where the problem is reduced to one parameter.
The role of stationarity of nonequilibrium systems for MEP is hard to evaluate here, since meteorological observations from high-altitude glaciers are not yet available for the length of climate normal periods. A preliminary
assessment, however, indicates that at least multiannual records are required for the analysis of MEP states.
The present results provide a first support for the hypothesis that cold high-altitude glaciers operate at maximum entropy production. Emphasis must be put on the adjective cold, i.e., a glacier at subfreezing temperatures outside the near-surface region that is subjected to the seasonal atmospheric variability [Cuffey and
Paterson, 2010]. Temperate glaciers (which are constantly at the melting point below the seasonal layer) or
polythermal glaciers (the hybrid between cold and temperate) could behave differently.
If the requirement of multiannual observations is met, a practical aspect of the results concerns parameter
optimization in physical mass balance models. If a certain internal parameter is unknown but the other system constraints are reasonably well-defined, MEP could be the selection criterion for the parameter value.
In a more extended framework, MEP could be included as additional objective function in studies that allow
€lg et al., 2012; Rye
the whole parameter space to vary in thousands of different model realizations [e.g., Mo
et al., 2012]. Such a procedure would also shed light on the role of system constraints for MEP from a cryosphere angle.
Acknowledgments
All data are available from the author
upon request ([email protected]).
Field campaigns on Kilimanjaro since
2005 were led by N. J. Cullen, D. R.
Hardy, G. Kaser, and M. Winkler.
Research permits were granted by
several Tanzanian authorities
(COSTECH, KINAPA, TAWIRI, and
TANAPA). Doug Hardy provided the
snow density data for January 2006.
The comments of two anonymous
reviewers helped to clarify the
manuscript. Discussion of individual
aspects with K. Fraedrich and A.
Jarosch were also appreciated.
€
MOLG
References
Andreassen, L. M., M. R. van den Broeke, R. H. Giesen, and J. Oerlemans (2008), A 5 year record of surface energy and mass balance from
the ablation zone of Storbreen, Norway, J. Glaciol., 54, 245–258.
Atkins, P. (2007), Four Laws That Drive the Universe, Oxford Univ. Press, Oxford, U. K.
Bintanja, R., and M. van den Broeke (1995), The surface energy balance of Antarctic snow and blue ice, J. Appl. Meteorol., 34, 902–926.
Brock, B. W., I. C. Willis, and M. J. Martin (2006), Measurement and parameterization of aerodynamic roughness length variations at Haut
Glacier d’Arolla, Switzerland, J. Glaciol., 52, 281–297.
Conway, J. P., and N. J. Cullen (2013), Constraining turbulent heat flux parameterisation over a temperate maritime glacier in New Zealand,
Ann. Glaciol., 54, 41–51.
Cuffey, K. M., and W. S. B. Paterson (2010), The Physics of Glaciers, Academic, Burlington, Vermont.
Cullen, N. J., T. M€
olg, G. Kaser, K. Steffen, and D. R. Hardy (2007), Energy-balance model validation on the top of Kilimanjaro, Tanzania, using
eddy covariance data, Ann. Glaciol., 46, 227–233.
Cullen, N. J., T. M€
olg, J. Conway, and K. Steffen (2014), Assessing the role of sublimation in the dry snow zone of the Greenland ice sheet in
a warming world, J. Geophys. Res., 119, 6563–6577, doi:10.1002/2014JD021557.
Dewar, R. (2003), Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in
non-equilibrium stationary states, J. Phys. A Math. Gen., 36, 631–641.
Dyke, J., and A. Kleidon (2010), The maximum entropy production principle: Its theoretical foundations and applications to the earth system, Entropy, 12, 613–630.
Fitzgerald, P. W., J. L. Bamber, J. K. Ridley, and J. C. Rougier (2012), Exploration of parametric uncertainty in a surface mass balance model
applied to the Greenland ice sheet, J. Geophys. Res., 117, F01021, doi:10.1029/2011JF002067.
Fraedrich, K., and F. Lunkeit (2008), Diagnosing the entropy budget of a climate model, Tellus, Ser. A, 60, 921–931.
Goody, R. (2007), Maximum entropy production in climate theory, J. Atmos. Sci., 64, 2735–2739.
Greuell, W., and T. Konzelmann (1994), Numerical modeling of the energy balance and the englacial temperature of the Greenland Ice
Sheet, calculation for the ETH-Camp location (West Greenland, 1155 m asl), Global Planet. Change, 9, 91–114.
Gromke, C., C. Manes, B. Walter, M. Lehning, and M. Guala (2011), Aerodynamic roughness length of fresh snow, Boundary Layer Meteorol.,
141, 21–34.
Gurgiser, W., B. Marzeion, L. Nicholson, M. Ortner, and G. Kaser (2013), Modeling energy and mass balance of Shallap Glacier, Peru, Cryosphere, 7, 1787–1802.
Hardy, D. R. (2011), Kilimanjaro, in Encyclopedia of Snow, Ice and Glaciers, edited by V. P. Singh et al., pp. 672–678, Springer, Dordrecht,
Netherlands.
Hastenrath, S. (1984), The Glaciers of Equatorial East Africa, D. Reidel Publishing Company, Dordrecht, Netherlands.
Hill, A. (1990), Entropy production as the selection rule between different growth morphologies, Nature, 348, 426–428.
Hock, R., and B. Holmgren (2005), A distributed surface energy-balance model for complex topography and its application to Storglaci€aren,
Sweden, J. Glaciol., 51, 25–36.
Jaynes, E. T. (1957), Information theory and statistical mechanics, Phys. Rev., 106, 620.
GLACIER-ATMOSPHERE ENTROPY PRODUCTION
421
Journal of Advances in Modeling Earth Systems
10.1002/2014MS000404
Judson, A., and N. Doesken (2000), Density of freshly fallen snow in the central Rocky Mountains, Bull. Am. Meteorol. Soc., 81, 1577–1587.
Kleidon, A., K. Fraedrich, T. Kunz, and F. Lunkeit (2003), The atmospheric circulation and states of maximum entropy production, Geophys.
Res. Lett., 30(23), 2223, doi:10.1029/2003GL018363.
Kleidon, A., K. Fraedrich, E. Kirk, and F. Lunkeit (2006), Maximum entropy production and the strength of boundary layer exchange in an
atmospheric general circulation model, Geophys. Res. Lett., 33, L06706, doi:10.1029/2005GL025373.
Klok, E. J., and J. Oerlemans (2002), Model study of the spatial distribution of the energy and mass balance of Morteratschgletscher, Switzerland, J. Glaciol., 48, 505–518.
Klok, E. J., and J. Oerlemans (2004), Modeled climate sensitivity of the mass balance of Morteratschgletscher and its dependence on albedo
parameterization, Int. J. Climatol., 24, 231–245.
Kunz, T., K. Fraedrich, and E. Kirk (2008), Optimisation of simplified GCMs using circulation indices and maximum entropy production, Clim.
Dyn., 30, 803–813.
Lorenz, R. D., J. I. Lunine, P. G. Withers, and C. P. McKay (2001), Titan, Mars and Earth: Entropy production by latitudinal heat transport, Geophys. Res. Lett., 28, 415–418.
MacDonell, S., C. Kinnard, T. M€
olg, L. Nicholson, and J. Abermann (2013), Meteorological drivers of ablation processes on a cold glacier in
the semi-arid Andes of Chile, Cryosphere, 7, 1513–1526.
MacDougall, A. H., and G. E. Flowers (2011), Spatial and temporal transferability of a distributed energy-balance glacier melt model, J. Clim.,
24, 1480–1498.
Machguth, H., R. S. Purves, J. Oerlemans, M. Hoelzle, and F. Paul (2008), Exploring uncertainty in glacier mass balance modelling with
Monte Carlo simulation, Cryosphere, 2, 191–204.
M€
olg, T., and D. R. Hardy (2004), Ablation and associated energy balance of a horizontal glacier surface on Kilimanjaro, J. Geophys. Res., 109,
D16104, doi:10.1029/2003JD003546.
M€
olg, T., and G. Kaser (2011), A new approach to resolving climate-cryosphere relations: Downscaling climate dynamics to glacier-scale
mass and energy balance without statistical scale linking, J. Geophys. Res., 116, D16101, doi:10.1029/2011JD015669.
M€
olg, T., and D. Scherer (2012), Retrieving important mass-balance model parameters from AWS measurements and high-resolution mesoscale atmospheric modeling, J. Glaciol., 58, 625–628.
M€
olg, T., N. J. Cullen, D. R. Hardy, G. Kaser, and L. Klok (2008), Mass balance of a slope glacier on Kilimanjaro and its sensitivity to climate,
Int. J. Climatol., 28, 881–892.
M€
olg, T., N. J. Cullen, D. R. Hardy, M. Winkler, and G. Kaser (2009a), Quantifying climate change in the tropical midtroposphere over East
Africa from glacier shrinkage on Kilimanjaro, J. Clim., 22, 4162–4181.
olg, T., N. J. Cullen, and G. Kaser (2009b), Solar radiation, cloudiness and longwave radiation over low-latitude glaciers: Implications for
M€
mass balance modeling, J. Glaciol., 55, 292–302.
M€
olg, T., F. Maussion, W. Yang, and D. Scherer (2012), The footprint of Asian monsoon dynamics in the mass and energy balance of a
Tibetan glacier, Cryosphere, 6, 1445–1461.
M€
olg, T., F. Maussion, and D. Scherer (2014), Mid-latitude westerlies as a driver of glacier variability in monsoonal High Asia, Nat. Clim.
Change, 4, 68–73.
Nicholson, L., R. Prinz, T. M€
olg, and G. Kaser (2013), Micrometeorological conditions and surface mass and energy fluxes on Lewis Glacier,
Mount Kenya, in relation to other tropical glaciers, Cryosphere, 7, 1205–1225.
Oerlemans, J., and W. H. Knap (1998), A 1 year record of global radiation and albedo in the ablation zone of Morteratschgletscher, Switzerland, J. Glaciol., 44, 231–238.
Ozawa, H., and A. Ohmura (1997), Thermodynamics of a global-mean state of the atmosphere—A state of maximum entropy increase, J.
Clim., 10, 441–445.
Ozawa, H., A. Ohmura, R. Lorenz, and T. Pujol (2003), The second law of thermodynamics and the global climate system: A review of the
maximum entropy production principle, Rev. Geophys., 41(4), 1018, doi:10.1029/2002RG000113.
Paltridge, G. W. (1975), Global dynamics and climate—A system of minimum entropy exchange, Q. J. R. Meteorol. Soc., 101, 475–484.
Pujol, T., and J. Fort (2002), States of maximum entropy production in a one-dimensional vertical model with convective adjustment, Tellus,
Ser. A, 54, 363–369.
Røhr, P. C., and Å. Killingtveit (2003), Rainfall distribution on the slopes of Mt. Kilimanjaro, Hydrol. Sci. J., 48, 65–78.
Rye, C. J., I. C. Willis, N. S. Arnold, and J. Kohler (2012), On the need for automated multiobjective optimization and uncertainty estimation
of glacier mass balance models, J. Geophys. Res., 117, F02005, doi:10.1029/2011JF002184.
Shimokawa, S., and H. Ozawa (2002), On the thermodynamics of the oceanic general circulation: Irreversible transition to a state with
higher rate of entropy production, Q. J. R. Meteorol. Soc., 128, 2115–2128.
Sicart, J. E., P. Ribstein, J. P. Chazarin, and E. Berthier (2002), Solid precipitation on a tropical glacier in Bolivia measured with an ultrasonic
depth gauge, Water Res. Res., 38(10), 1189, doi:10.1029/2002WR001402.
Sicart, J. E., R. Hock, P. Ribstein, M. Litt, and E. Ramirez (2011), Analysis of seasonal variations in mass balance and meltwater discharge of
the tropical Zongo Glacier by application of a distributed energy balance model, J. Geophys. Res., 116, D13105, doi:10.1029/
2010JD015105.
Vanyo, J. P., and G. W. Paltridge (1981), A model for energy dissipation at the mantle—core boundary, Geophys. J. Int., 66, 677–690.
Van den Broeke, M. R., C. Reijmer, D. van As, and W. Boot (2006), Daily cycle of the energy balance in Antarctica and the influence of clouds,
Int. J. Climatol., 26, 1587–1605.
Wagnon, P., P. Ribstein, B. Francou, and B. Pouyaud (1999), Annual cycle of the energy balance of Zongo Glacier, Cordillera Real, J. Geophys.
Res., 104, 3907–3923.
€
MOLG
GLACIER-ATMOSPHERE ENTROPY PRODUCTION
422