Chapter 9Land Ice, Climate, and Modeling

Yonggang Liu

William M. Connolley

Lisiecki and Raymo (2005, Paleoceanography)

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

Land Ice Sheet Drew Tremendous Attention During the Past Decade

Paolo et al. (2015, Science) NOAA Arctic Report Card for 2013

How Does Ice Sheet Affect the Climate?

Answer: mainly through albedo feedback

But sea level rise has a substantial impact on the human society

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

Surface and Basal Mass Balance

Winkelmann et al. (2012, Nature)

Dynamics (to get mass flux q)

Deviatoric stress

Strain rate

Pressure

Constitutional relationship

where

Stokes’ Equations

What’s the difference from themomentum equation for theatmosphere and ocean

T’ is a tensor, which makes the Stokes’ Equation hard to solve

Are all these components important for the ice sheet?

http://mail.colonial.net/~hkaiter/Glaciers_Ice_Ages.html

Ice sheet is very thin compared to its horizontal scale; It is moving slowly

Winkelmann et al. (2012, Nature)

Additionally, in most part of the ice sheet, vertical shear dominates the motion

Shallow-Ice Approximation

Where s and b are surface and bottom elevation, respectively. C(T) is the basalsliding 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

Alpine Glacier (http://mail.colonial.net/~hkaiter/Glaciers_Ice_Ages.html)

Lateral friction is large, obviously cannot ignored

Other examples: fast flowing ice streams

Full Stokes’ Equation needs to be solved!

Shallow-Shelf Approximation

Assume vertical shear is 0

Winkelmann et al. (2012, Nature)

The following equations result

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 andgeothermal heating at the bottom of the ice sheet may cause melting there.These require that the temperature evolution of the ice sheet need to besolved.

Where

is the frictional heating

MacAyeal (1997)

Summary