The concept of similitude is commonly employed in the fields of fluid dynamics and engineering but rarely used in cryospheric research. Here we apply this method to the problem of ice flow to examine the dynamic similitude of isothermal ice sheets in shallow-shelf approximation against the scaling of their geometry and physical parameters. Carrying out a dimensional analysis of the stress balance we obtain dimensionless numbers that characterize the flow. Requiring that these numbers remain the same under scaling we obtain conditions that relate the geometric scaling factors, the parameters for the ice softness, surface mass balance and basal friction as well as the ice-sheet intrinsic response time to each other. We demonstrate that these scaling laws are the same for both the (two-dimensional) flow-line case and the three-dimensional case. The theoretically predicted ice-sheet scaling behavior agrees with results from numerical simulations that we conduct in flow-line and three-dimensional conceptual setups. We further investigate analytically the implications of geometric scaling of ice sheets for their response time. With this study we provide a framework which, under several assumptions, allows for a fundamental comparison of the ice-dynamic behavior across different scales. It proves to be useful in the design of conceptual numerical model setups and could also be helpful for designing laboratory glacier experiments. The concept might also be applied to real-world systems, e.g., to examine the response times of glaciers, ice streams or ice sheets to climatic perturbations.

In the fields of fluid dynamics and engineering, scaling laws are used to
perform experiments with spatially reduced models in water channels or wind
tunnels to predict the behavior of the associated full-scale system

The similitude concept is also applied to some extent in the field of
glaciology: in laboratory glacier experiments dimensionless numbers like the
Reynold's, Froude and Ramberg numbers are used to check for (dynamic)
similarity between the geometrically scaled model (based on the properties of
the analogue ice material) and the real-world system

Schematic of the similitude-analysis method carried out in this
study. A reference system (blue ice sheet and bed topography) with geometric
scales

Here we apply the concept of similitude to the dynamics of idealized ice
sheets based on the shallow-shelf approximation

A dimensional analysis of the ice-dynamic equations is often carried out to
compare the magnitudes of the different acting forces and thus to derive
physically motivated approximations, as done when deriving the SSA from the
full Stokes stress balance

The paper is structured as follows: in the next section the governing equations in SSA are non-dimensionalized to derive ice-sheet scaling laws for one and two horizontal dimensions, respectively. Afterwards the analytically predicted ice-sheet scaling behavior is compared with results from numerical modeling. To this extent conceptual experiments are designed in two and three spatial dimensions. Steady states as well as the transient response to perturbation of the simulated ice sheet are analyzed for a systematic variation of the scaling parameters which are prescribed according to the scaling laws. We then examine analytically the implications of the scaling conditions for the response times of ice sheets considering the geometric scaling factors and basal friction parameter as independent variables. Eventually we discuss the results and conclude.

Here we derive scaling laws that determine how the geometry, response time
and the involved physical parameters for ice softness, surface mass balance
and basal friction have to relate in order to satisfy similitude between
different ice sheets. This is visualized conceptually in
Fig.

The problem addressed here is the one of an isothermal ice-sheet in SSA

The evolution equation for the ice thickness, i.e., the ice thickness
equation (ITE), which results out of mass conservation

In the flow-line case the geometry of an ice sheet can be scaled in
horizontal (

Since we neglect the

Now we bring these two equations into non-dimensionalized form by introducing
the dimensionless variables

The two governing Eqs. (

We can link the ratios of surface mass balance and ice softness by combining
Eqs. (

Using Eqs. (

Results of an application of the derived scaling laws in numerical flow-line
simulations are given in Sect.

Parameter values as prescribed in the unscaled reference simulations
for the flow-line setup (2-D) and the three-dimensional channel setup (3-D),
respectively. For the scaling experiments the bed geometry (

Scaling ratios as used for our numerical
simulations. Prescribed scaling ratios are highlighted in bold and the others
result from Eqs. (

The two-dimensional SSA (Eq.

Starting again from the two-dimensional SSA (Eq.

We investigate ice-sheet scaling also in a three-dimensional setup in the next section.

We compare our analytical findings with results from numerical simulations
applying the Parallel Ice Sheet Model in conceptual geometric setups. The
model is the same as used in

Halving the horizontal and/or vertical length scales of the reference
topography we obtain three geometric configurations which are shortened in
vertical

Ice-sheet profiles at three different stages of the flow-line
simulations 2-D

Steady-state ice-sheet profiles for cross section along the
centerline (

Bed topography of the three-dimensional channel setup, here shown in
the scaled version with

The experiments are designed to perturb an ice sheet in equilibrium, triggering a marine ice-sheet instability that unfolds unaffected by the ceased perturbation. The speed of unstable grounding-line retreat and the equilibrium ice-sheet profiles before and after the instability serve as a measure to compare the scaling of the dynamic response and the steady-state geometry, respectively.

All of our simulations show a similar pattern of grounding-line evolution
after perturbation (Figs.

The simulations clearly differ in the timescale of the evolution of the marine ice-sheet instability (MISI) which
can be measured by the grounding-line retreat rate

Based on the scaling laws derived in Sect.

Time series of grounding-line position for the
reference and three geometrically scaled flow-line experiments with

Time series of centerline grounding-line position (along

To be able to follow physically motivated curves through the phase space we
link the horizontal and the vertical scale. In idealized flow-line
experiments

Scaling of

Scaling of response time

Response-time scaling for hypersurfaces through the

An exponent of

Assuming Vialov conditions under identical friction, the scaling of the
response time (Eq.

The response-time scaling considered here is a function of the basal friction
exponent

Fixing the horizontal scale, i.e., going along

Requiring the response-time scaling law (Eq.

Carrying out a dimensional analysis of the stress balance in SSA and the
equation of mass conservation we derive ice-sheet scaling conditions for the
vertical and horizontal length scales, the response time and the relevant
physical parameters which determine ice-sheet behavior
(Eqs.

Specifically, we find that the scaling relations derived for the SSA in flow
line (Eqs.

Our analysis also shows that although the full SSA accounts for both
horizontal dimensions there can only exist one timescale

To non-dimensionalize the SSA stress balance we introduce scales for
ice-sheet length, thickness and response time without assuming typical numerical
values for these scales. We thus do not neglect further terms in the SSA
stress balance by comparing orders of magnitudes of acting stresses

The presented scaling conditions can provide rules in the design of model
setups for numerical simulations as well as laboratory experiments to obtain
parameter sets that leave the ice-sheet geometry (shape and extent)
unchanged. For instance, a doubling of the basal-friction parameter under
identical surface mass balance requires the ice softness to be reduced to 1

For the numerical simulations conducted in this study we apply parameter
configurations that halve the geometric scale in horizontal and/or vertical
direction with respect to the reference. The resulting ice-sheet response
times range over 3 orders of magnitude (see Table

In contrast to the flow-line configuration the three-dimensional setup
inherently accounts for the buttressing effect in the initial steady-state
simulation due to the presence of a confined ice shelf

To analytically investigate the implications of geometric scaling for the
ice-sheet response time we make the simplifying assumption of identical basal
friction (

Assuming a relation between the horizontal and vertical scale of the form

In place of prescribing basal friction, the assumption of identical surface
mass balance (

Our approach includes several assumptions (shallow stress balance, isothermal ice, choice of sliding law, parameter constraints) and thus simplifies the problem of ice flow. At the same time it allows for the fundamental scaling analysis conducted here which incorporates the relevant physics of fast ice flow and results in scaling conditions that relate important physical parameters of an ice sheet to each other. A similitude analysis based on a less simplified stress balance than the one used here would certainly better account for the complexity of real-world systems, but that is beyond the scope of the current study. All statements on the ice-sheet scaling behavior made here therefore need to be considered in light of the idealized character of the underlying simplified SSA stress balance.

The SSA is of vertically integrated form and thus in particular does not
account for variations of ice-sheet velocity within the ice column. The
assumption of uniform ice softness further reduces complexity, neglecting the
dependency of the ice softness on ice temperature, which typically varies in
horizontal and vertical direction. The applied Weertman-type sliding law
(Eq.

Our analytic exploration of the derived ice-sheet scaling behavior applies several constraints to the parameter space and is thus far from being holistic but is aimed to allow for (simplified) statements on the influence of geometric scaling on response time. The set of scaling conditions presented here shall provide a model which allows for a fundamental comparison of the large-scale scaling of the geometry and relevant parameters that determine ice-sheet dynamics. In particular the response-time scaling conditions might be suitable to analyze speed of the transient response to climatic perturbations of the polar ice sheets that took place in the past or might become relevant for the future.

Here we show that the scaling conditions derived above by dimensional
analysis under the concept of similitude are consistent with the
boundary-layer theory which was introduced by

According to the boundary-layer theory the ice-sheet surface slope is given
by

The boundary-layer method considers the ITE in steady state (

A central result of the boundary-layer theory is an analytic solution for the
grounding-line flux as a function of ice thickness at the grounding line

Setting

Thus the same three independent equations that determine the ice-sheet scaling behavior and were derived by the means of similarity analysis in the previous section also result from boundary-layer theory.

Introducing the dimensionless velocities

We thus found that in order to fulfill the required scaling similarity in the
considered two-dimensional SSA-case there can only exist one horizontal
length scale and one timescale (as opposed to one in each horizontal
direction, as assumed initially). All the scaling relations derived for the
flow-line SSA case (Eqs.

For the two-dimensional simulations, we use the symmetric flow-line geometry
and the sequence of experiments described in

For three parameter sets the simulations deviate from the above described
scenario. In two simulations (2-D

For the three-dimensional experiments we extend our flow-line geometry by
introducing a second horizontal dimension (

Spinning up the model we obtain a symmetric ice sheet in equilibrium with a
stable bay-shaped grounding line. Along the centerline of the setup (

For an alternative comparison of grounding-line retreat rates between the
different scaling experiments described in Sect.

Scaled time series of grounding-line position for the reference and
three geometrically scaled flow-line experiments with

Scaled time series of centerline grounding-line position (along

We would like to thank three anonymous reviewers for their very helpful comments that helped to improve the manuscript. The research leading to these results has received funding from the Deutsche Forschungsgemeinschaft (DFG) under priority program 1158. Edited by: F. Pattyn Reviewed by: three anonymous referees