The last two months have seen low levels of rainfall in Canberra and thus a decline in runoff. However, we have still had 246 mm of rainfall for the year, about average, and runoff of close to 33,000 megalitres.

The runoff total needs to be explained, however, as it is likely that we have had significantly less than this - perhaps as low as 28,000 megalitres. The reason is that I have factored in releases of 130 megalitres a day from Canberra dams over the last month or so. ACTEWAGL were definitely making releases of that magnitude over some of that period, and there were additional releases from smaller dams over one weekend. However, I am pretty certain that these releases stopped a couple of weeks or so ago. To ensure that I overestimate rather than underestimate runoff, I am working on the assumption that releases ceased as of 31 May.

It should be pointed out that even this overestimated runoff total is well below Canberra's average runoff for the first five months of the year. But it is better than last year: at the same point, we had had 21,000 megalitres of runoff.

## Tuesday, May 31, 2011

## Thursday, May 19, 2011

### Arctic ice volume

I have been doing some work on Arctic sea ice volume, trying to determine whether a second order polynomial function had a physical basis. And I have discovered that it does. While others have obviously already worked this out, it is new to me, and thus at least a little bit exciting. :)

To look at this, what I did was sit and think about what would happen in an Arctic that was melting, and write down a few things.

The first thing that I thought of was that there are two significant parts to the Arctic year - the melt and the freeze. Using the values generated by Frank http://snipt.org/xwgn, I determined that over the period of the model (and, yes, PIOMAS is *not* data, but a model, but it does not matter for the purposes of this exercise) there was an increase in the amount of ice melting each year and a decrease in the amount of ice freezing each year. This increase and decrease were each moving in a linear fashion. It was difficult for me to see how a second order polynomial function could emerge from these linear functions. Silly me, as we will see.

So I set up a model that mirrored these linear changes in melt and freeze, and then looked at the yearly totals at maximum and minimum that resulted. Graphing these totals, I found that the declines in each perfectly followed a second order polynomial function ... What an earth was going on here?

I tried various values for the change constants in both the melt and the freeze periods, but always ended up with second order polynomial functions. So I decided to investigate this function a little more by differentiating it and seeing if the resultant function related in any way to the change constants.

And, of course, it did. What I found was that the differentiated function for the decline in ice volume at the end of the melt season was - with X years - always:

- (Melt Constant + Freeze Constant)* X + (Melt Constant + Freeze Constant)/2

The differentiated function for the decline in ice volume at the end of the freeze season was - with X years - always:

- (Melt Constant + Freeze Constant)* X + (3*Melt Constant + Freeze Constant)/2

Why these particular functions? The constants in them result from the 1/2 years offset between the two seasons. The Melt Constant + Freeze Constant is simply the total yearly change - the two constants added.

So integrating this returns us to our second order polynomial. And why do we integrate? Because the reductions in ice volume in any year are *summed* to the reductions in ice volumes of all previous years. And a sum function is an integral.

In other words, we do not start from scratch each year: each year, we are melting from a lower volume of ice and freezing from a lower volume of ice.

Basically, what it means is that if melting and freezing change in a linear fashion then we get a second order polynomial function for the ice volume totals.

And is there a physical basis for such a linear increase and decrease? Of course: the linear increase in energy, as measured through linear temperature change, in the Arctic due to rising CO2.

Which points to a dramatic crash in Arctic ice volume, and thus area and extent, over the next few years. Indeed, using PIOMAS, further modelling suggests that zero volume will be reached at the end of the melt period in 2018 at the latest, with it occurring possibly as early as 2013.

My projections are:

Year Volume (cubic kilometres)

2011 -> 3744

2012 -> 2853

2013 -> 1935

2014 -> 990

2015 -> 18

2016 -> -981

2017 -> -2007

2018 -> -3060

(all values have a two deviation error range of +/- 2445)

To look at this, what I did was sit and think about what would happen in an Arctic that was melting, and write down a few things.

The first thing that I thought of was that there are two significant parts to the Arctic year - the melt and the freeze. Using the values generated by Frank http://snipt.org/xwgn, I determined that over the period of the model (and, yes, PIOMAS is *not* data, but a model, but it does not matter for the purposes of this exercise) there was an increase in the amount of ice melting each year and a decrease in the amount of ice freezing each year. This increase and decrease were each moving in a linear fashion. It was difficult for me to see how a second order polynomial function could emerge from these linear functions. Silly me, as we will see.

So I set up a model that mirrored these linear changes in melt and freeze, and then looked at the yearly totals at maximum and minimum that resulted. Graphing these totals, I found that the declines in each perfectly followed a second order polynomial function ... What an earth was going on here?

I tried various values for the change constants in both the melt and the freeze periods, but always ended up with second order polynomial functions. So I decided to investigate this function a little more by differentiating it and seeing if the resultant function related in any way to the change constants.

And, of course, it did. What I found was that the differentiated function for the decline in ice volume at the end of the melt season was - with X years - always:

- (Melt Constant + Freeze Constant)* X + (Melt Constant + Freeze Constant)/2

The differentiated function for the decline in ice volume at the end of the freeze season was - with X years - always:

- (Melt Constant + Freeze Constant)* X + (3*Melt Constant + Freeze Constant)/2

Why these particular functions? The constants in them result from the 1/2 years offset between the two seasons. The Melt Constant + Freeze Constant is simply the total yearly change - the two constants added.

So integrating this returns us to our second order polynomial. And why do we integrate? Because the reductions in ice volume in any year are *summed* to the reductions in ice volumes of all previous years. And a sum function is an integral.

In other words, we do not start from scratch each year: each year, we are melting from a lower volume of ice and freezing from a lower volume of ice.

Basically, what it means is that if melting and freezing change in a linear fashion then we get a second order polynomial function for the ice volume totals.

And is there a physical basis for such a linear increase and decrease? Of course: the linear increase in energy, as measured through linear temperature change, in the Arctic due to rising CO2.

Which points to a dramatic crash in Arctic ice volume, and thus area and extent, over the next few years. Indeed, using PIOMAS, further modelling suggests that zero volume will be reached at the end of the melt period in 2018 at the latest, with it occurring possibly as early as 2013.

My projections are:

Year Volume (cubic kilometres)

2011 -> 3744

2012 -> 2853

2013 -> 1935

2014 -> 990

2015 -> 18

2016 -> -981

2017 -> -2007

2018 -> -3060

(all values have a two deviation error range of +/- 2445)

## Tuesday, May 10, 2011

### Aerosol evolution: two scenarios

This is a post inspired by SteveF's work at Lucia's blog here:

http://rankexploits.com/musings/2011/a-simple-analysis-of-equilibrium-climate-sensitivity/#comment-75758

The above table from excel uses (I hope) SteveF's method to look at the evolution of aerosol forcings over time. In his simple analysis of equilibrium climate sensitivity, SteveF looked at the situation now and worked out what aerosol forcing would have to be if forcing caused an increase of .4207 degrees per watt per square metre and if forcing caused an increase of .81 degrees per square metre (and another higher scenario).

I have extended his analysis to cover the period 1970 to 2010. One of the thing that I noted in the comments to that thread was that the aerosol forcings under the higher sensitivity scenario are currently the same as they were after the Mount Pinatubo eruption. This seems unlikely. More reasonable is the lower sensitivity scenario, in which current sensitivity is about half of that after Mount Pinatubo erupted.

One interesting fact is that under the higher sensitivity scenario there is quite an upward trend over time in aerosol forcings. This does to some extent seem reasonable, imo, as the increase in CO2 emissions is directly associated with an increase in sulphur emissions. In fact, the correlation between well mixed greenhouse gas (WMGHG) forcings is high (r^2 value of 0.81). This makes sense to me.

Still not sure what it all means, but it is interesting to play with. :)

And I have realised that I may have missed one important component: solar forcings. I will check into that.

*Done a little checking. SteveF seems to simply use one value, but that could be because he is only looking at one year - he might change that value for each year.

*Re correlation, the lowest value for a statistically significant correlation, ignoring possible autocorrelation, which is relatively small, is 0.55 degrees per watt per square metre.

## Tuesday, May 3, 2011

### Hansen by logarithm

As I have been unable to find the linear graphs that I thought Hansen was using, I have recreated his numbers using the logarithmic model I described previously. After some fiddling around with the parameters, I have managed to create a reasonable match with observed temperatures and the observed rate of warming over the last 40 years using a climate sensitivity of 3.3 degrees per doubling. I homed in on this number because of a priori knowledge that Hansen's model E matches observations the best when such a sensitivity is used, so this is not an independent test.

I should again point out here that lower sensitivities require a faster response time and higher sensitivities require a lower response time.

My model predicts a rate of warming of .0187 degrees per year for the next 25 years, which equates to a bit less than half a degree of warming. At that point, we would be committed to a further one degree of warming, most of which would occur this century. If all human greenhouse gas emissions ceased at that point, total warming from preindustrial would be around 2.3 degrees by the time warming ceased.

I will be interested to see how my model compares with reality over the next little while.

I should again point out here that lower sensitivities require a faster response time and higher sensitivities require a lower response time.

My model predicts a rate of warming of .0187 degrees per year for the next 25 years, which equates to a bit less than half a degree of warming. At that point, we would be committed to a further one degree of warming, most of which would occur this century. If all human greenhouse gas emissions ceased at that point, total warming from preindustrial would be around 2.3 degrees by the time warming ceased.

I will be interested to see how my model compares with reality over the next little while.

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