

gl031637 1..5


On the evolution of the snow surface during snowfall

H. Löwe,
1
L. Egli,

1
S. Bartlett,

1
M. Guala,

1
and C. Manes

1

Received 9 August 2007; revised 28 September 2007; accepted 12 October 2007; published 14 November 2007.

[1] The deposition and attachment mechanism of settling
snow crystals during snowfall dictates the very initial
structure of ice within a natural snowpack. In this letter we
apply ballistic deposition as a simple model to study the
structural evolution of the growing surface of a snowpack
during its formation. The roughness of the snow surface is
predicted from the behaviour of the time dependent height
correlation function. The predictions are verified by simple
measurements of the growing snow surface based on digital
photography during snowfall. The measurements are in
agreement with the theoretical predictions within the
limitations of the model which are discussed. The
application of ballistic deposition type growth models
illuminates structural aspects of snow from the perspective
of formation which has been ignored so far. Implications of
this type of growth on the aerodynamic roughness length,
density, and the density correlation function of new snow
are discussed. Citation: Löwe, H., L. Egli, S. Bartlett,
M. Guala, and C. Manes (2007), On the evolution of the snow

surface during snowfall, Geophys. Res. Lett., 34, L21507,

doi:10.1029/2007GL031637.

1. Introduction

[2] The physical properties of the natural snowpack are
of great importance for many geophysical processes such as
heat transfer [Kaempfer et al., 2005; Sturm et al., 2002],
interaction with the turbulent boundary layer [Lehning et
al., 2002], wind erosion [Clifton et al., 2006], failure and
crack propagation [Sigrist and Schweizer, 2007; Heierli and
Zaiser, 2006]. However, the ice structure within a natural
snowpack is a non-equilibrium system which undergoes
metamorphic changes induced by many physical and chem-
ical processes [see, e.g., Arons and Colbeck, 1995;
Schweizer et al., 2003, and references therein]. Linking
the ice structure to its physical properties represents a major
challenge in snow physics.
[3] In contrast to the difficulty of predicting the complete

metamorphism dynamics of the ice structure some striking
features of its initial condition can be observed during
snowfall even by eye: due to cohesion a settling snow
crystal has the tendency of being attached to the snow
surface immediately at first contact rather than being re-
arranged to a position of minimum potential energy. The
immediate attachment of the crystal creates a small over-
hang of the surface and thus excess pore space directly
below the attached crystal. As an obvious consequence of

this attachment mechanism the snow surface develops its
irregular, rough appearance. Some degree of order within
this irregularity is sometimes revealed when a pattern of
bumps with well defined characteristic length scale is
clearly visible on the surface during snowfall. As a less
obvious consequence of the attachment mechanism, also
the ice structure below the snow surface should inherit
its structure from the way in which overhangs are buried
by successive, random attachment events. While the
initially generated ice structure is very early influenced by
the aforementioned metamorphism dynamics [Kaempfer
et al., 2005], the deposition process provides the initial
condition to the dynamics and therefore deserves special
attention.
[4] To the best of our knowledge the characteristics of

surface roughness and the ice structure of snow have never
been addressed from the most obvious perspective of
formation itself which is the purpose of this letter. We stress
that the purpose is not to give a comprehensive model of
snow deposition for ‘‘operational use’’. We rather demon-
strate the wide applicability of a simple but well behaved
theoretical framework.

2. Model of Snow Deposition

2.1. Ballistic Deposition

[5] Ballistic deposition (BD) was originally introduced as
a model of colloidal aggregation. For a comprehensive
overview on the theory presented below we refer to
Barabási and Stanley [1995] and Meakin [1998]. For
illustration of BD we consider the two-dimensional case
on a lattice where the surface height h(i, n) is defined at
discrete lattice sites i and discrete times n. At each time step,
a particle is released well above a randomly chosen site i. It
settles down on a straight vertical trajectory until it encoun-
ters an occupied neighboring lattice site. Thus, at each time
step the height of the surface is updated according to

h i; n þ 1ð Þ ¼ maxfh i þ 1; nð Þ; h i � 1; nð Þ; h i; nð Þ þ 1g: ð1Þ

The sticking rule (1) is illustrated in Figure 1a. The
restriction to a lattice can be abandoned and the general-
ization of (1) to the three-dimensional case is straightfor-
ward [see Meakin, 1998].

2.2. Continuum Description

[6] As an alternative to the particle based approach
within BD one may apply a related continuum description.
It is widely believed but still a matter of debate [cf. Katzav
and Schwartz, 2004, and references therein] that the uni-
versal properties of BD on length scales large compared to
the particle size can be recovered by a continuum growth
model, namely the Kardar-Parisi-Zhang (KPZ) equation

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[Kardar et al., 1986]. For a continuous surface h(x, t) in
three-dimensional space it reads

@

@t
h x; tð Þ ¼ nr2h x; tð Þ þ ljrh x; tð Þj2 þ x x; tð Þ: ð2Þ

Time is denoted by t and the gradient by r. The individual
terms in (2) can be attributed to particular surface processes:
The term x(x, t) = h0 + h(x, t) constitutes the mass supply. It
includes a deterministic growth velocity h0 and a fluctuating
term h(x, t) which is zero-mean, Gaussian white noise in
space and time. The diffusion term in (2) includes local
surface relaxation processes. Most important for the relation
of the KPZ equation to BD is the non-linear term of strength
l. It is the leading-order term for growth normal to the
surface [Barabási and Stanley, 1995] which is the analog of
an overhang in BD.
[7] From the description of a settling snow crystal given

in the introduction it is clear that a minimal model of snow
deposition should at least include two effects: 1) sticking
(which covers both, sintering and interlocking) as the origin
of excess pore space and 2) the stochastic nature of the
deposition mass flux. Both models, BD and KPZ include
these effects. Since we focus on implications and applica-
tions of these effects on snow structure we will apply either
of them likewise. Obvious limitations of the model can in
principle be investigated within extensions of BD/KPZ: the
influence of imperfect sticking (e.g., as a consequence of
crystal shapes) can be addressed by competitive growth
models [Braunstein and Lam, 2005]; non-uniform particle
shapes or sizes have been studied by Silveira and Reis
[2007]; oblique particle incidence (e.g. as a result of steady,
low winds) has been studied by Meakin and Krug [1992]. In
contrast, the dynamical evolution of the ice structure below
the surface (metamorphism, stress induced re-arrangement)
cannot easily be regarded as a generalization of BD/KPZ.

3. Surface Roughness

[8] In general, surface roughness can be characterized
by studying temporal and spatial correlations of the height
h(x, t) of a growing surface in terms of its dynamic, two-
point correlation function [Barabási and Stanley, 1995]

G x; tð Þ :¼ h x0 þ x; tð Þ � h x0; tð Þð Þ
2
: ð3Þ

Here, x, x0 denote two-dimensional position vectors on the
substrate. Assuming statistical homogeneity and isotropy of
the growth process G(x, t) is independent of x0 and solely a
function of the magnitude x:= jxj. The overbar in (3)
denotes an ensemble average over realizations of the depo-
sition process. For illustration, the correlation function (3)
for BD/KPZ type growth is schematically plotted as a
function of x in Figure 1b). Such a behaviour on a double
logarithmic scale implies that G(x,t) follows the dynamic
scaling form G(x, t) 
 x2a g(x/t1/z) [Barabási and Stanley,
1995; Meakin, 1998]. Here g(s) is a scaling function which
is constant for s � 1 and decreases algebraically g(s) 

s
�2a

for s � 1. This implies that G(x, t) 
 x2a is
independent of t if x � t1/z and approaches a constant
G(x, t) 
 t2a/z for x � t1/z when plotted as a function of x
(cf. Figure 1b)). The exponents a, z are universal and
commonly referred to as roughness- and dynamic exponent,
respectively.
[9] The origin of scaling is the combination of random

deposition events with a nearest neighbor sticking rule:
Initially, different parts of the surface are uncorrelated. After
time t regions of correlated surface heights have formed
within a spatial extent x* 
 t1/z from the sticking rule (1) or,
likewise, by the non-linear term in (2). Typical numerical
estimates for the scaling exponents a,z for KPZ and BD are
found to be in a range 0.3 < a < 0.4 and 1.36 < z < 1.65 [cf.
Katzav and Schwartz, 2004, and references therein].

4. Measurements of the Snow Surface

[10] Quantitative roughness measurements of snow sur-
face outlines at different times were carried out by means of
digital photography during several snowfalls in winter
2006/07. Pictures were taken using a 7.0 megapixel digital
camera and a scaled target which was carefully inserted
within the snow (see Figure 2 for an example). In order to
avoid the formation of roughness due to i) wind-induced
snow erosion, ii) anomalous deposition processes due to a
predominant wind direction, or iii) finite size geometry
effects, images of the snow surface outlines were taken
during snow falls in the absence of wind, starting from a flat
solid surface with dimensions significantly larger than the
largest resolved scale which is 20 cm. The snow rough-
ness measured in these experiments can thus be regarded (in
good approximation) solely as the result of the deposition
process. From each image, roughness outlines were identi-

Figure 1. Ballistic deposition: (a) Nearest neighbor stick-
ing for deposition at site i creates an overhang and thus
excess pore space which remains inaccessible for other
particles. (b) Schematic plot of the correlation function G(x, t)
(see equation (3)) for different times t = 1,10,100,1000
(arbitrary units).

Figure 2. Examples of images taken to estimate snow
roughness elevations showing (a) an early and (b) a later
stage of the evolution. Estimated outlines are displayed in
red. The magnification nicely displays an overhang.

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fied by defining an optimal grey scale threshold value
separating the pixels relating to the snow from those to
the target. Overall, it was noted that, when inserted into the
snowpack, the thin metal target produced very sharp cuts
which preserved the shape of the foreground roughness
outlines. The accuracy obtained for the roughness elevations
depended solely on the camera resolution and the physical
size covered by each image. Such an accuracy was estimat-
ed to be 0.1 mm which is fine enough to investigate
roughness properties belonging to the sub-crystal scale.
[11] As a typical drawback of natural experiments we

acknowledge that some observed snow falls, or parts of
them, are not documented here, due to the occurrence of
non negligible wind during the snowfall or to significant
changes of the crystal sizes in time. The results presented
below are thus representative of the time evolution of the

rough snow surface during snowfall where BD/KPZ should
be applicable.

5. Results and Discussion

[12] Qualitatively, the measured correlation function
G(x,t) (see equation (3)) behaves similarly to what is
expected for a BD/KPZ surface. Quantitatively, G(x,t)
is plotted for the three sets of experiments as a function
of x for different times in Figure 3. The roughness exponent
a is obtained by fitting the curves in the ascending range to
a power law. The mean values for each set are given in
Figure 3, the individual values are a = 0.36, 0.46, 0.43, a =
0.34, 0.45 and a = 0.37, 0.38 for Figures 3a, 3b and 3c
respectively which are in the range of the predictions from
BD/KPZ (see section 3).
[13] Instead, a reliable estimation of the dynamic expo-

nent z from the standard deviation s(t) 
 ta/z [Barabási and
Stanley, 1995] is presently unfeasible. The reason for this is
the difficulty to encounter a sufficiently long period of
persistent, ideal snowfall conditions (long duration, persis-
tent shapes of the crystals, without wind) which allows to
measure the correlation function at many times and estimate
the exponent from a single experiment. Likewise, an alter-
native approach, namely the estimation of z from data
collapse of different experiments could only be achieved
by explicitly measuring the number flux of settling crystals:
the fundamental length scale l0 of a single experiment is
given by l0

3
= v/n where v is the velocity of the mean surface

height in units of ms
�1

and n is the mean number flux of
settling crystals in units of m

�2
s
�1
. By definition l0

3
is the

average volume occupied by a single crystal after its
incorporation into the snowpack. Thus, l0 covers branched
crystals as well as those for which sticking can be accom-
panied by interpenetration. For this reason l0 is an effective
crystal size and not the exact one, which is an ambiguous
quantity for non-spherical shapes. The fundamental time
scale is then given by t0 = l0/v. Only by rescaling the
variance by l0 and time by t0 all data can be combined
consistently in a single plot. However, measuring n was
clearly beyond the scope of our experimental setup.
[14] For completeness we simply plotted s of all three

experiments as a function of time from the beginning of
each experiment in one plot (see Figure 4). As a guide to the
eye we added a line of slope a/z = 0.24. The increase of the
variance is also evident from Figure 3 and follows qualita-
tively that of BD/KPZ type growth (cf. Figure 1b).

6. Applications

[15] We demonstrate below that, apparently, different
aspects involving the structure of snow can be explained
from a unifying perspective of BD/KPZ type growth.

6.1. Aerodynamic Roughness Length of New Snow

[16] It has been observed that the aerodynamic roughness
length z0 which determines the logarithmic velocity profile
in turbulent boundary layer flows over snow varies even for
apparently similar samples of new snow [Clifton et al.,
2006]. By dimensional analysis one would expect z0 to scale
on the standard deviation of the surface height. Such a
dependence has also been suggested by Lancaster et al.
[1991] for desert surfaces. Hence, variations of the rough-

Figure 3. Correlation function for three different snow
events. Arrows indicate increasing time, or increasing
heights of the mean snow surface, corresponding to (a) 15,
20, 25 mm, (b) 70, 95 mm, and (c) 50, 80 mm.

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ness length z0 of new snow might be attributed to variations
of the standard deviation s as predicted by BD/KPZ.

6.2. Slope Dependent Density of New Snow

[17] It has been observed that the density of new-snow
depends on the inclination angle of the slope onto which it
is deposited [Endo et al., 1998]. This slope dependence is
partly attributed to the mechanism of the deposition process.
Such a slope dependence can be predicted by BD/KPZ as
follows. For constant vertical mass flux q onto an inclined
substrate with slope m = tan q and inclination angle q one
can use the KPZ equation (2) to infer v(m) = v(0) + lm2

[Barabási and Stanley, 1995] for the velocity of the mean
surface height in terms of the strength l of the non-linear
term. Thus, the velocity of the mean surface height normal
to the surface increases when the inclination angle is
increased and the mean density of the deposit given by
r(m) = q/v(m) decreases. For small slopes m this amounts to
r(m)  r(0)(1 � lm2). Here r(0) = q/v(0) is the density of
the deposit for a flat substrate. On one hand this can explain
the aforementioned claim of Endo et al. [1998]. On the
other hand it provides an experimental method to estimate l
by fitting the density as a function of slope to a parabola
[Barabási and Stanley, 1995].

6.3. Anisotropy of Density Correlations in New Snow

[18] It has been observed that the ice structure of snow
might exhibit an anisotropy [Flin et al., 2004] as a result of
metamorphism dynamics under isothermal conditions. A
different anisotropy naturally emerges from BD/KPZ solely
as a result of the deposition process and irrespective of
possible anisotropic crystal shapes. To explain this we will
extend recent two-dimensional results on properties of
deposits below the growing surface from Katzav et al.
[2006].

[19] Fluctuations of the density around the ensemble
mean r in a porous medium are commonly studied by
means of the two-point density correlation function

C x; x3ð Þ :¼ r x; x3ð Þ � rð Þ r 0; 0ð Þ � �ð Þ; ð4Þ

where x3 is the vertical component of the three-dimensional
lag vector (x, x3) and again x = (x1, x2). We note that the
density r(x, x3) has to be understood as a microscopic
quantity. It is given by the indicator function of the porous
medium, i.e. r(x, x3) = rice if (x, x3) lies in the ice phase and
r(x, x3) = 0 if (x, x3) lies in the pore space.
[20] Katzav et al. [2006] calculated the correlation function

(4) from the time dependent structure factor of the surface.
Their results can easily be generalized to three dimensional
space by employing the scaling form equation (A.11) of
Barabási and Stanley [1995] for the structure factor,
yielding

C x; x3ð Þ 
 x
�4 1�að Þ=z
3 f jxj=x

1=z
3

� �
: ð5Þ

[21] Here, f(s) is a scaling function which vanishes for
s � 1 and approaches a constant for s � 1. In other words,
for x3 � jxj, that is, in a thin vertical strip the correlation
function has an algebraic tail C(x, x3) 
 x3�4(1�a)/z which is
missing for x3 � jxj, that is, in a thin horizontal strip.
Using, e.g., a = 0.4 and z = 1.6, the power law decay is
governed by an exponent �1.5. The anisotropy, a result of
formation, becomes manifest in (5) since it is not solely a
function of the magnitude (x1

2
+ x2

2
+ x3

2
) of the lag.

[22] The great importance of density correlation functions
stems from the fact that they arise in rigorous expressions
for the effective transport and mechanical properties of
porous media [Torquato, 2002].

7. Conclusions

[23] With simple experiments and analytical predictions,
we suggest that Ballistic Deposition/Kardar-Parisi-Zhang
(BD/KPZ) processes i) well describe the growth of the
snow surface during snowfall, ii) allow for the interpretation
of previously observed behaviour of the density of new
snow and iii) can predict density correlations within new
snow. Despite the fact that the BD/KPZ model cannot
always be applied to natural snowfalls (limitations are
discussed), we emphasize its consequences on the aerody-
namic roughness length or the anisotropy of snow as a
porous medium. It remains an open question, however, how
long the signatures of BD/KPZ persist after snowfall.

[24] Acknowledgments. We thank Charles Fierz and Michi Lehning
for a careful reading of the manuscript.

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�����������������������
S. Bartlett, L. Egli, M. Guala, H. Löwe, and C. Manes, WSL, Swiss

Federal Institute for Snow and Avalanche Research SLF, Flüelastr. 11,
CH-7260 Davos Dorf, Switzerland. (loewe@slf.ch)

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