Integration of Passive Tracers in a Three-Dimensional Ice Sheet Model - Towards an Interdisciplinary Approach in Earth System Science - page 154

u

expansion

w

Fig. 5 Ice sheet equilibrium topography of the EISMINT Fixed Margin experiment compared to
the analytical description of the Nye-Vialov ice sheet. The shaded areas illustrate the combined
effect of compression and layer thinning (not to scale). The box depicts constant horizontal
velocities along the vertical axis and constant vertical velocities along the horizontal axis close to
the ice divide

2.2.2 Analytical Comparison to the Nye-Vialov Solution

The Nye-Vialov solution (Rybak and Huybrechts 2003 ) for an idealized ice sheet
(Eq. 3 ) was used to compare the performance of the Lagrangian and Eulerian tracer
advection approach (see Fig. 6 ). Utilizing the analytical age-depth relationship for
an ice sheet without basal melting, no horizontal advection, and constant vertical
strain rate we can directly compare both advection methods to a pre-de

ned age-

depth relationship (Eq. 4 ):

1
2 ð n þ 1 Þ

1
2 ð n þ 1 Þ

20M
A

1
q g

1
2

H 0 ¼ L ð 3 Þ

H 0
G

z
H 0

A ¼ ln ð 4 Þ

where H 0 is the ice sheet thickness at the divide, M the mass balance ( a ), A is the
time (years) it would take to get from the surface (divide position) to the location z
within the ice sheet, g gravitational force (9.81 ms − 2 ) and

ice density
(910 Kgm − 3 ), L denotes the characteristic length of the ice sheet, G is surface
accumulation and z the depth inside the ice sheet. The basic assumption behind the
Nye-Vialov solution is that ice inside glaciers or ice caps undergoes compression by
overburden pressure (compaction) and, increasing with depth, layer thinning due to
lateral extension. The horizontal velocity is assumed to be constant trough any
vertical transect through the ice sheet while the vertical velocity is constant along
any horizontal line (strip shaped ice sheet). In the standard con

q

guration of 50 km
lateral resolution and 41 vertical sigma layers (terrain following coordinates) the
difference between both advection methods to the analytical solution is negligible in
the upper domain of the ice divide. However, the solutions diverge in the lower part