This document discusses modeling of land ice sheets and their impact on climate. It summarizes different approaches that can be used to model ice sheet dynamics, including the shallow-ice approximation and shallow-shelf approximation, which make simplifying assumptions about stress and shear. It also discusses the importance of surface and basal mass balance, temperature dynamics, and friction in modeling ice sheets accurately. The key properties of ice sheets that are important to model are their mass and thickness over time.

1. Chapter 9
Land Ice, Climate, and Modeling
Yonggang Liu
William M. Connolley

2. Lisiecki and Raymo (2005, Paleoceanography)

3. Petit et al. (1999, Nature)
Figure by Robert A. Rohde

5. Land Ice Sheet Drew Tremendous Attention During the Past Decade
Paolo et al. (2015, Science)NOAA Arctic Report Card for 2013

6. How Does Ice Sheet Affect the Climate?
Answer: mainly through albedo feedback
But sea level rise has a substantial impact on the human society

7. What Property of the Ice Sheet Do We Care
About the Most?
Answer: its mass or thickness
Where h is the thickness of ice sheet, A dot and B dot are the
surface and basal mass balance, respectively, q = uh is the mass
flux

8. Surface and Basal Mass Balance
Winkelmann et al. (2012, Nature)

9. Dynamics (to get mass flux q)
Deviatoric stress
Strain rate
Pressure
Constitutional relationship
where
Stokes’ Equations
What’s the difference from the
momentum equation for the
atmosphere and ocean

10. T’ is a tensor, which makes the Stokes’ Equation hard to solve
Are all these components important for the ice sheet?

11. http://mail.colonial.net/~hkaiter/Glaciers_Ice_Ages.html
Ice sheet is very thin compared to its horizontal scale;
It is moving slowly

12. Winkelmann et al. (2012, Nature)
Additionally, in most part of the ice sheet, vertical shear
dominates the motion

13. Shallow-Ice Approximation
Where s and b are surface and bottom elevation, respectively. C(T) is the basal
sliding constant and B dot is the basal melting rate.
Then we get decoupled equations of u, v and w (Greve, 1995)
, andAssume
Note the temperature dependence

14. Alpine Glacier
(http://mail.colonial.net/~hkaiter/Glaciers_Ice_Age
s.html)
Lateral friction is large, obviously cannot ignored
Other examples: fast flowing ice streams
Full Stokes’ Equation needs to be solved!

15. Shallow-Shelf Approximation
Assume vertical shear is 0
Winkelmann et al. (2012, Nature)
The following equations result

16. Thermal dynamics
What assumption has been made?
Boundary conditions
T = Ta
Q = Qb At the bottom
At the surface
The viscosity of ice is dependent on temperature, and frictional heating and
geothermal heating at the bottom of the ice sheet may cause melting there.
These require that the temperature evolution of the ice sheet need to be
solved.
Where
is the frictional heating

17. MacAyeal (1997)

18. Summary