Trees, volcanoes and climate

..large eruptions in the tropics and high latitudes were primary drivers of interannual-to-decadal temperature variability in the Northern Hemisphere during the past 2,500 years. Overall, cooling was proportional to the magnitude of volcanic forcing and persisted for up to ten years after some of the largest eruptive episodes. Our revised timescale now firmly implicates volcanic eruptions as catalysts in the major century pandemics, famines, and socioeconomic disruptions in Eurasia and Mesoamerica..[1]


  • a large volcanic eruption injects tens of millions of tons of sulphur dioxide into the stratosphere, causing global or hemispheric dimming. The cooling effect may last for several years.
  • the box-and-whisker graph above shows distributions of the annual growth of trees at two subarctic (taiga) forest sites in Quebec and  Russia and at a subalpine forest in France, from 1300 to the present day. These sites are close to the tree line where temperature is likely to have been the limiting factor for growth. The graph suggests that volcanic cooling had a significant impact on forest growth.
  • tree ring chronologies for each site were created by detrending raw data from individual trees and averaging. While this procedure loses information about climatic shifts lasting longer than a few decades, shorter term variability (for example, from volcanoes) is retained.
  • sulfur deposition in Antarctic ice cores was taken as the measure of volcanic aerosols. A simple annual average from 21 carefully dated Antarctic ices cores was taken[1]. To allow for the fact that aerosol effects can last for more than one year, an exponentially weighted moving average with decay constant of 3 years was computed. Years in which this average exceeded a threshold (40ng/g) were deemed “volcanic”. There were 30 such years since 1300, an average of 4 per century. Interestingly there have been no such years since 1821.
  • the list of “volcanic” years according to the above criterion since 1300 are: 1347 1459 1460 1461 1462 1463 1464 1465 1466 1600 1601 1602 1603 1604 1643 1644 1696 1697 1698 1810 1811 1812 1813 1815 1816 1817 1818 1819 1820 1821. Most of these correspond to years following well-known historical eruptions such as Kuwae 1453, Huaynaputina 1600, Tambora 1815 and the mystery 1809 volcano.
  • strangely, the impact of volcanoes appears weaker if Greenland ice core data is used instead of Antarctic data. Furthermore, the relation between tree-ring and ice core data appears much weaker in the pre-1300 data. For example, there is little evidence of the very large 1257 Samalas eruption in the growth indices at the sites selected (below).





Calculation Details

Tree ring indices were calculated from raw research data (.rwl files) archived by NOAA paleoclimate. For example, the Quebec l1 data is available here:

Growth index chronologies were calculated in R using the dendrochronology dplR package. Simple negative exponential detrending was used, which attempts to capture mean biological growth rates over the life of a tree (typically conifer ~ 100 years). For example, the raw Quebec l1 site index chronology (Gennaretti et al ) was calculated as follows:

Antarctic sulfate ice core data described in Sigl et al are available in this spreadsheet.

[1] Timing and climate forcing of volcanic eruptions for the past 2,500 years. Sigl, M., Winstrup, M., McConnell, J. R., Welten, K. C., Plunkett, G., Ludlow, F., … Woodruff, T. E. (2015).  Nature, 523(7562), 543–549. DOI: 10.1038/nature14565

Braess Paradox in AC Power Networks

Adding an extra road in a road network can increase traffic congestion. This counter-intuitive outcome is called a Braess Paradox.  Braess Paradoxes, where increased connectivity degrades the performance of a network, has been shown to occur in electric power transmission networks.[1]

An AC power network consists of “buses” (vertices or nodes) and “transmission lines” (edges or links). Buses are of two types – “generators” (power sources) and “loads” (power sinks). To a good approximation, power flowing along a link from node i to node j in a high voltage AC transmission network is given by

(1)   \begin{equation*}P_{ij} = K_{ij} \sin(\theta_j-\theta_i)\end{equation*}

where \theta_i is the voltage phase at node i and K_{ij} is the capacity of the transmission line linking i and j. Maximum power flows when the phase difference is 90^\circ.

The capacity of the transmission network as a whole is the maximum total power consumption that it allows. Assume for simplicity that all transmission lines have the same capacity K and that all generators and all loads are identical. This simplification means that, if there are N_g generators and N_c cities, each generator produces power g and each load consumes power g N_g/N_c. Given a network topology and g/K, the set of \theta‘s which satisfy Equation 1 can easily be found, at least for smaller networks. Eventually, when g/K is increased to a critical value, no solution is possible. The capacity of the network has been reached.

Figure 1 shows a 5-node 3-generator 5-link network operating at it’s critical capacity. Each generator produces power K and each load consumes 1.5K. Power flows are indicated in red, with arrows  on links giving the directions of flow. Node arrows indicate phase angles. Note the 30^\circ phase difference between generators A and B and between the cities.

Figure 1

Suppose that a new link is added between A & B, coupling the phases of these two generators.  The large phase angle between A & B of Figure 1 produces a current which cannot be sustained without disrupting the phases on all of the other nodes.

The new critical state with phases reorganised to maximise capacity in the new network is shown in Figure 2. Generators A & B are now nearly in phase, with only a small current flowing between them. The capacity of the network is reduced to 2.91K, 3\% lower than the less connected network of Figure 1.

Figure 2

A more extreme case is shown below, where adding an additional three links to a simple power network degrades the capacity of the network by 1\over 3.

Figure 3

Calculations using R‘s excellent igraph and nleqslv packages.

[1]  Braess’s paradox in oscillator networks, desynchronization and power outageD. Witthaut and M. Timme, 2012.