I have experiemented with using matlabs built-in ode solvers to integrate a Shallow ice . Surprisingly the ode solvers are able to take much greater timesteps than what theoretical stability criteria would suggest. I am really impressed by how well it works although I will not recommend using this approach:

• There is a discontinuity where the height can not go below zero. This is very hard for the ode solvers to deal with.
  
• Tiny instabilities result in oscillations (I've set the tolerances such that they are visible in the video on the right). However, this can in practice lead to an accumulation bias due to the altitude-accumulation feedback.
  

The main advantage of the ode solver approach is that the code can be made extremely short and readable. It could therefore serve as a nice introduction to the Shallow Ice Approximation.

Mail me if you would like a piece of the code. It is short and easy to read.

brown is the bedrock. It is a modification of matlabs peaks function.
transparent white is the glacier surface.

Here's a very short version of Conways Game of Life for Matlab. In fact it is so short that it would fit in a tweet:

a=rand(128)>0.8;
for ii=1:500
  spy(a);
  drawnow;
  a=abs(filter2(ones(3),a)*2-6-a)<2;
end

And here's a High-life version:

a=rand(256)>0.8;
for ii=1:500
  spy(a);
  drawnow;
  f=filter2(ones(3),a);
  a=(f-6)+2*a==0|f==3;
end

Cellular automata are fun. Here's an automaton which generates labyrinths (although i am not sure if it technically fits the classification).



And here's the matlab code which generates the above video:

c=-ones(5); c(2:4,2:4)=2; c(3,3)=13; c=c/8;
A=rand(300,400)>.995;
for ii=1:600
  A=max(min(filter2(c,A),1),0);
  imagesc(A==0);
  drawnow;
end


This is based on a music visualisation for WinAmp AVS i made years ago.

One graph that is often put forward to illustrate the link between global temperature is the Archer 2006 book graph (shown below). It looks as if there is a simple linear relationship between temperature and sea level with 20m/degC! From that you would guess that we would be facing maybe 50m of sea level rise. However, there are several weaknesses with this graph.

1. No uncertainties (which are huge)
2. Eocene sea level is a very poor analogy to present day climate because of tectonic effects (e.g. check this map).

Furthermore all the more recent and much more relevant data from the last interglacials have been ignored. That raises the question if the points in this illustration have been picked to simplify the message.


Rohling et al. 2009 has plotted sea level against Antarctic Temperature over the past 5 glacial cycles. From that you can see that the relationship between temperature and sea level is not linear. During glacials where ice volume was large the sea level response to a was also large. In interglacials, with much less ice volume, the sea level response is much smaller.

From the graph it is estimated that for present day we get 3-5m of sea level rise per degC of antarctic warming. Taking polar amplification of (here taken to be 2x) into account we can convert that into a 6-10 m/degC of global average temperature change. (Note that the actual polar amplification valid for dome C is probably somewhat smaller than 2.)



I think that the Archer graph is a nice illustrative cartoon. However, it is very unfortunate that it is being picked up in official climate reports like the german WBGU 2006. I have had to defend some of my own work because it was in conflict with a present day slope of 20m/degC.

In this figure below i put the rohling curve on a 1961-1990 baseline, converted to global temperatures and also added projections (from Grinsted et al. 2009). I used 1.5 as the polar amplification because we are talking about the Antarctic.


The cyan variations include variations of the form:

• sea level was within 1m of equilibrium in 1850 (standard error).
  
• polar amplification of 1-2.3 (5-95%). [polaramp=1.5*exp(randn/4)] Note: this is the polar amplification at DomeC.
  
• Rohling to 1961-1990 baseline correction of 1degC_global with a standard error of 1degC. I am not 100% sure what reference they used but it seems to be somewhere during the current interglacial. 
  

[EDIT: Perhaps the pliocene can be used as an analogy, but i question the present quality of the data. A quick google gave me this quote: "Geologic estimates of maximum Pliocene sea level thus range from +5 to +40 m relative to present, with +25 m typically used by the modeling community." from PLIOMAX: Pliocene maximum sea level project. Combine that with uncertainties in temperature and i wonder if it can give any useful constraint on these types of graphs.

EDIT2: Found a mistake in the polaramlification. Thanks to Bart Verheggen for letting me know of the mistake.

EDIT3: Added my take on the Archer Graph based on Rohlings curve. I focused on warmer climate.]

There has been a lot of blogging about yamal. Some of it concerns if RCS introduces spurious signals. It is clear that it is very important to try to get an unbiased estimate of the growth curve.

Here i have calculated an very simple alternative to RCS. The idea is that if you only use tree rings of a particular age, then you do not need to make any corrections for the growth curve. The procedure:

• Select a narrow age window. In my case i choose a 10 year wide window.
  
• For each tree
• calculate the log(mean(ringwidth)+0.05). for all the data in the age-window.
  
• calculate the average year of growth for that age window.
  
• Center the reconstruction for that window.
  
• Repeat for different age windows.

The age intervals i chose were: 1-9, 10-19, ... ,390-399. It might be wise to disregard:

• The youngest time slices where the growth curve signal is strongest.
• The oldest time slices where there will be so few data that the record will be full of gaps. I am no expert on trees, but i suppose that really old trees can start to behave rather anomalously. For that reason it might be prudent to remove the oldest ring-widths.
  

I have made several ad-hoc choices here, but this page is only meant to serve as an illustration.

The colored dots show all the ringwidth estimates using this approach. Color shows the central age for the age window i have chosen. Colored lines is a 50 year smooth of the points for a given age window (note i have not used a very clever interpolation/gapfilling scheme). The blue line shows 50 year average of all dots.

The interactive discussion of the differences between the two models can be found on realclimate.org.

References

• Grinsted, A., J. C. Moore, and S. Jevrejeva (2009), Reconstructing sea level from paleo and projected temperatures 200 to 2100 AD, Clim. Dyn., doi:10.1007/s00382-008-0507-2
• Rahmstorf, S., A. Cazenave, J. A. Church, J. E. Hansen, R. F. Keeling, D. E. Parker & R. C. J. Somerville, Recent Climate Observations Compared to Projections, Science 316, p709, DOI: 10.1126/science.1136843 (2007)
• Esper, J., E.R. Cook, and F.H. Schweingruber, 2002, Low-Frequency Signals in Long Tree-Ring Chronologies for Reconstructing Past Temperature Variability, Science, Volume 295, Number 5563, 22 March 2002.
• Jones, P.D. and M.E. Mann. 2004. Climate Over Past Millennia. Reviews of Geophysics 42, RG2002. http://www.ncdc.noaa.gov/paleo/pubs/jones2004/jones2004.html
• Mann, M.E. and P.D. Jones, 2003, Global Surface Temperatures over the Past Two Millennia, Geophysical Research Letters, Vol. 30, No. 15, 1820, August 2003. doi: 10.1029/2003GL017814.
• Moberg, A., Sonechkin, D. M., Holmgren, K., Datsenko, N. M. & Karlén W. Highly variable Northern Hemisphere temperatures reconstructed from low- and high-resolution proxy data Nature, 433, 613-617 (2005)
• Sivan, D., et al., 2004: Ancient coastal wells of Caesarea Maritima, Israel, an indicator for sea level changes during the last 2000 years. Earth Planet. Sci. Lett., 222, 315–330.
• Lambeck, K., et al., 2004: Sea level in Roman time in the Central Mediterranean and implications for recent change. Earth Planet. Sci. Lett., 224, 563–575.
• Donnelly, J.P., P. Cleary, P. Newby, and R. Ettinger, 2004: Coupling instrumental and geological records of sea-level change: Evidence from southern New England of an increase in the rate of sea-level rise in the late 19th century. Geophys. Res. Lett., 31(5), L05203, doi:10.1029/2003GL018933.
  
• Gehrels, W.R., et al., 2004: Late Holocene sea-level changes and isostatic crustal movements in Atlantic Canada. Quat. Int., 120, 79–89 Bindoff, N.L., J. Willebrand, V. Artale, A, Cazenave, J. Gregory, S. Gulev, K. Hanawa, C. Le Quéré, S. Levitus, Y. Nojiri, C.K. Shum, L.D. Talley and A. Unnikrishnan: Observations: Oceanic Climate Change and Sea Level. In: Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor and H.L. Miller (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA. (2007)