Distributed mass balance models, which translate micrometeorological
conditions into local melt rates, have proven deficient to reflect the energy
flux variability on mountain glaciers. This deficiency is predominantly
related to shortcomings in the representation of local processes in the
forcing data. We found by means of idealized large-eddy simulations that heat
advection, associated with local wind systems, causes small-scale sensible
heat flux variations by up to 100

The complex interaction of glaciers, atmosphere and topography constitutes a
fundamental challenge in glaciological research. Countless studies aim to
identify the climatic drivers behind observed glacier changes by using
distributed mass and energy balance models

While large-scale weather shapes the environmental conditions in which
mountain glaciers exist, the mass and energy exchange on individual glaciers
is controlled by the micrometeorological conditions. Given the complex
topography around mountain glaciers with its contrasting surface
characteristics, it is not trivial to bridge the scale gap between the
large-scale conditions and the local characteristics. The micrometeorological
condition of the surface layer is directly influenced by the presence of the
earth surface and quickly responds to changes in the surface energy
budget. The radiative and turbulent heat fluxes cool and heat the
near-surface air layer and determine the temperature distribution across the
topography. Local temperature excess and deficit create buoyancy forces that
drive the thermal wind systems, including the valley circulations, slope and
glacier winds

The fluctuations of the thermal conditions are of practical interest for
distributed glacier mass balance studies. For example, winds may advect warm
air from the surroundings towards the glacier, which locally increase the
downward-directed sensible heat flux

A fundamental obstacle in studying small-scale boundary layer characteristics
is that, even on well-studied mountain glaciers, the deficiency of monitoring
activities restricts the process understanding, required for detailed
research, to a few sites and limited time periods

To overcome this difficulty, we make use of high resolution large-eddy
simulations (LES). The LES are considered as pseudo-reality – a testbed to
identify the shortcomings in the local sensible heat flux estimates when the
lack of observations restricts our micrometeorological knowledge to a few
sites. The plausibility of the temperature interpolation algorithms and the
derived sensible heat fluxes can be more strictly tested in a surrogate world
of atmospheric simulations, which offer a realization of atmospheric states
in which all target variables are known. The pseudo-reality atmosphere is not
required to be an observed real world case, but it does need to be a plausible
realization of the atmosphere in the sense that relevant processes are
realistically simulated. The advantage of such studies is that the surrogate
atmosphere provides a perfect pseudo-observation of all the variables
required to establish the skill of an interpolation method and hence the
sensible heat flux calculations. While surrogate atmospheres have been widely
used in downscaling studies, this approach is still novel in glaciological
studies

After a brief description of the LES model (Sect.

The pseudo-reality atmosphere is simulated by an OpenFOAM-based
incompressible LES solver

In the original formulation the value of the constant

Proceeding from the instantaneous internal energy equation, the conservation of
potential temperature can be derived and becomes

The filtered momentum equation is solved using the PIMPLE algorithm and a preconditioned bi-conjugate gradient solver for asymmetric matrices. To reduce numerical dissipation the convective terms are solved using a second-order central differencing scheme with a multi-dimensional limiter. The time derivative is discretized by a second-order implicit scheme with adaptive time stepping.

Map showing the surface topography of the studied glaciers and the surrounding terrain.

Even though the LES is designed as pseudo-reality, the lower boundary
condition is provided by a real topography. The designated study area is
located at the head of Martell Valley in the central Ortles-Cevedale Group,
Autonomous Province of Bozen, Northern Italy (46.28

The surface temperature,

The lateral boundaries are specified as periodic. At the top boundary a no-slip zero-stress boundary is used. The pressure gradient is set based on the Boussinesq density gradient normal to the boundary, and the potential temperature gradient is specified according to the initial profile. At the surface, the same pressure boundary condition is used as at the top boundary.

The filter and grid resolution are too coarse to resolve the near-wall motions,
including in the viscous wall region, so that their influence close to the wall
is modeled by a shear stress model. A local version of Schumann's
shear stress model is applied at the surface

The surface temperature flux is determined using Monin–Obukhov scaling laws
for velocity and potential temperature

Given the large computational costs, the analysis is confined to four
pseudo-realistic case experiments. The simulations merely differ in the
geostrophic flow direction (0, 90, 180 and 270

Adaption and relaxation of the computational domain. The blue and green lines show a east–west profile of the original and relaxed topography, respectively. The boundary grid cells have been set to the mean value of both cells (dashed line). The adjacent grid points are slowly relaxed (exponentially) to get a smooth transition from the boundaries towards the inner domain using a spline algorithm.

Besides the fluid dynamical challenges, the numerical model must be able to
cope with complex topography. The OpenFOAM solver allows for unstructured
grids, which can be adapted more easily to steep topography than commonly
used terrain following grids. The 3-D unstructured mesh is generated with the
OpenFOAM utility snappyhexmesh. The tool automatically generates hexahedra
and split-hexahedra meshes from triangulated surface geometries, i.e.,
digital elevation models (DEMs). In this study, the mesh is generated from a
high-resolution elevation model (1 m horizontal resolution) derived from
airborne laser scans conducted in September 2013

The LES resolves the large energy-containing turbulent structures, so that
the output fields are fully turbulent. A given fully turbulent variable,

The following section analyses the mean modeled flow patterns and vertical
profiles. The analysis is confined to the atmospheric boundary layer near the
glacier surface and the kinematic flow properties affecting it. For the
discussion we introduce four specific regions: (R1) ridge region, (R2) a
steep ice fall, (R3) katabatic wind region, and (R4) divergence zone of
katabatic wind. Local characteristics are discussed at four virtual sites on
the glacier (Z1–4; see Fig.

Figure

Mean velocity of the surface wind fields
(2

Mean statistics at the sites (Z1–4) at 2

On lower wider passes and gaps, the flow follows the topography and modifies
the wind systems on the lee side. This is particularly evident at the long
stretched glacier divide between Zufallferner and Fürkele Ferner (R1). The
large-scale flow enhances the katabatic wind when both wind systems are
aligned but retards it otherwise. Since the glaciers are west–east
orientated, surface wind predominantly accelerates during westerly flow (see
R2 in Fig.

Vertical profiles of the mean wind velocity
at the four sites (Z1, Z2, Z3 and Z4) for each case experiment. The dashed
line represents a neutral logarithmic wind profile with

In the central part of Zufallferner (R3), the wind velocities considerably
vary with the large-scale flow directions. For example, northerly and
easterly flows (Fig.

At the glacier tongue (R4), the large-scale flow hardly affects the surface
winds. The katabatic winds gently drain down the glaciers (see
Table

The temperature deficit increases towards the glacier tongue (from Z4 to Z1)
and implies a larger forcing to the glacier wind (see Table

Vertical temperature profiles at
location Z2 for each experiment. The dashed line indicates the altitude of
the Cevedale ridge. The grey solid line represents a lapse rate of

Standard deviation of the vertical velocity fluctuation at 2 m above ground for each of the four case experiments.

Example of a rectified wavelet power spectrum of the temperature signal at location Z2 for southerly flow (upper left column), the time average-wavelet power spectra (right column), and the scale-averaged time series (lower left column). Red and blue indicate high and low scaled powers (in base 2 logarithm), respectively. Black lines outline the wavelet spectrum at a 95 % confidence level. The cross-hatched region marks the cone of influence, where edge effects become important.

Mean sensible heat flux from the LES runs for each of the four case experiments.

Figure

At some distance away from the mountain ridges the boundary layer is less
turbulent (

The vertical mixing of momentum and heat is a non-stationary process with
changing frequency and intensity across time

Potential temperature at 2

According to the principle of energy conservation the local change in the
potential temperature tendency of dry air at any given point is related to
the advective, turbulent and the radiative heat fluxes. The latter one is not
explicitly modeled in this study but indirectly given by the prescribed
surface temperature. In this case, the heating and cooling of the
near-surface layer is only a result of the advective and turbulent transport.
Local advection is usually negligible over flat terrain and during weak wind
conditions, but it is considered a relevant process on mountain glaciers with
consequences on the spatial variations of the surface heat flux

Comparison of the surface heat fluxes. Shown are the
glacier-wide averaged surface heat fluxes from the LES and the estimated
fluxes calculated with the bulk approach using different temperature fields
as predictor (linearly interpolated and a thermodynamic glacier wind model
based on

Figure

Between the individual experiments the spatial variability shows striking
differences (see Fig.

While advection is essential for local estimates the question remains whether
the impact of recurrent mixing events is of the same order of magnitude (see
Sect.

Even though the pseudo-reality atmosphere seems to describe realistically the physical processes and patterns, the simulations must be interpreted with care. The patterns depend on the model assumptions, which include parameterizations and idealized boundary conditions.

A crucial assumption is the surface roughness length. To obtain more general
results, uniform values of

The roughness lengths of snow and ice are relatively small compared to non-uniform roughness elements at a scale of tens of meters such as deep seracs or ice falls. The scales of these elements are approximately of the same order as the horizontal model resolution. Enhanced mixing due to the sudden roughness changes are therefore not resolved by the model, and it is very likely that the model underestimates the overall variability.

In general, the model resolution is very decisive for the overall quality of
the LES simulations. LES require that

Differences in the mean surface sensible heat fluxes between the LES and the bulk method for different wind direction. Positive differences correspond to an overestimation of the surface heat flux by the bulk approach.

The Lagrangian averaged dynamic Smagorinsky model assumes that the energy
transfer from the resolved-scale eddies to the residual motions is entirely
balanced by the dissipation of kinetic energy. However, dissipation is not
necessarily in balance with the energy production in stably stratified
boundary layers. As a consequence, the SGS model is likely to dissipate too
much energy. We have shown that the calculated statistics, such as the
integral turbulence scales, skewness and vertical velocity variance, are in
the same order of magnitude as those obtained from observations

We like to note that the current version of the solver ignores differential surface heating by radiation and is therefore only suitable for idealized simulations. Differences in insolation on slopes due to exposure, aspect or shadow cause upslope flows to be inhomogeneous. The different onsets of the slope winds then lead to more asymmetric cross-valley circulations.

Physically based distributed mass balance models are often applied to
translate the local-scale weather conditions into net mass gain and loss at
the glacier surface. The ablation process, which removes ice and snow, is
controlled by the net energy balance at the ice–atmosphere interface. Direct
measurements of energy balance components exist in most cases only for
radiation, while surface heat and moisture fluxes are rarely measured
directly on glaciers. The simplest and most widely used method to
parameterize the turbulent energy exchange from available meteorological
observations is the bulk approach. The approach is based on the
Monin–Obukhov theory and assumes constant fluxes within the surface layer.
This is not necessarily true in the presence of a LLJ, but the method is
found to give good results when measurements are below the wind velocity
maximum

It is straightforward to apply Eq. (

To illustrate how the sensible heat flux estimates depend on the local flow
conditions, we define two pseudo-observation sites at Zufallferner, with
preferable great vertical differences between the sites (S1 and S2, see
Fig.

Figure

In contrast, the fluxes are largely overestimated along the glacier
centerlines and tongues. In these regions the well-developed katabatic flow
prevents warm air advection from the surroundings (see Sect.

Mean glacier-wide sensible heat fluxes using
the bulk approach with linearly extrapolated temperature and wind fields. The
table shows extrapolation scenarios based on different pseudo-observations
(S1, S2–S5). The exact location of the pseudo-observations is given in
Fig.

On a glacier scale, the bulk approach in concert with linearly interpolated
temperature fields underestimates the average heat flux between 5.2
(

The choice and number of observation sites on glaciers is always a compromise between logistic feasibility, financial expenditure and scientific issue. These factors usually restrict the monitoring activities to a few sites along the glacier centerlines. Even from a purely scientific perspective the choice of observation sites that meet all requirements is challenging.

To explore how the choice of observation sites influences the spatial
variation of the sensible heat flux estimates, we define a set of
pseudo-observation on Zufallferner (S1–S5; see Fig.

Differences in the surface sensible heat
fluxes between the LES and the bulk method. The cases

Figure

The results confirm that the phenomenological understanding at few locations and weather situation is not valid beyond the case and insufficient to infer on the micrometeorological conditions on mountain glaciers.

We have shown how complex topography influences the micrometeorological
conditions on three midlatitude mountain glaciers in the Italian
Ortles-Cevedale Group. The idealized LES experiments demonstrate that heat
advection associated with the wind systems shapes the thermal conditions on
the glaciers during the course of a summer day with clear sky conditions. In
particular, the cross-valley circulations, and bluff body formations behind
sharp ridges, transport warm air from the surroundings to the peripheral
zones of the glaciers and locally increase the sensible heat fluxes by
50–100

Our pseudo-reality experiments demonstrate that it is challenging to fully
characterize the micrometeorological conditions over glacier surfaces from a
limited number of observations. Linearly extrapolated forcing fields fail to
reflect the temperature variability that originates from insufficient
characterization of advection. The shortcomings in the forcing fields have
direct consequences on estimated sensible heat fluxes (e.g., by the bulk
approach). Local differences in the sensible heat fluxes of up to
60

The choice of observations sites, and thus the derived temperature gradients,
determines the magnitude of the local sensible heat flux errors. Calculated
temperature lapse rates are shallower (

As a glacier-wide average, the choice of observation sites causes differences
of about

We can conclude that a profound knowledge of the heat advection process is
needed when small-scale variations of surface energy balance are required for
distributed mass balance studies. Current thermodynamic and statistical
centerline models describe temperature variations along the flow line of
glaciers, but they do not resolve the cross-glacier variability

The LES data are available
upon request from Tobias Sauter. The open-source CFD software OpenFOAM can be
downloaded from

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG), no. SA 2339/4-1. This work was also supported and partly financed by the Autonome Provinz Bozen – Südtirol, Abteilung Bildungsförderung, Universität und Forschung. We thank both reviewers for their detailed comments and constructive criticism of the original manuscript. Edited by: V. Radic Reviewed by: two anonymous referees