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One fascinating aspect of the climate system is that its variability is governed by simple mathematical relationships. A recent paper in Geophysics Reviews explains how these mathematical relationships — in the form of power laws– structure a large part of climate variability in terms of time, space and intensity. Understanding how climate variability is structured helps us make robust climate reconstructions and skillful future predictions. Here, two authors give a brief overview of what a power law is, how it can be used to improve our understanding of the climate system and the physical mechanisms behind it.
How are the different aspects of the climate system interrelated?
Climate variables, such as temperature, precipitation, total ozone, relative humidity, and sea level change, are highly interconnected, both over long periods of time and over great spatial distances. It shows in the datasets: there are correlations in the averages of the data over long intervals or between distant areas.
The majority of climatic correlations can be described by a simple mathematical relation called “scale”.
An interesting conclusion is that the majority of correlations in climate can be described by a simple mathematical relationship called a “scale,” which says that the means at different scales are related by a simple mathematical function: a power law.
In practice, this means that, for example, the duration of temperature fluctuations increases systematically, as the temperature fluctuation increases.
In addition, the likelihood of an event occurring also decreases as the event becomes intense. This is similar to Richter’s Law for Earthquakes, which says that small earthquakes happen very often while very large earthquakes rarely occur.
How can mathematical tools be applied to understand these relationships?
Power laws appear as straight lines in graphs where both axes have a logarithmic scale and the slopes indicate the exponents of the scale. They are familiar to many scientists in undergraduate physics classes.
A well-known example is the relationship between the angular frequency of a pendulum and its length: the angular frequency is proportional to the inverse square root of the length.
Another example is how the gravitational force between two bodies depends on their distance from each other: the gravitational force is proportional to the inverse of the distance squared.
It is fascinating that power laws also exist in climate time series, as shown in the graphs to the right, where scale exponents measure the persistence of climate anomalies and the likelihood of extreme events of different intensity.
Which climatic variables exhibit scaling behavior?
Scaling behavior is pervasive in the climate system. It has been detected in many climatic variables, using in situ observation recordings, paleo-reconstructions and model simulations.
Most important is surface air temperature, where the most pronounced scaling behavior is found in oceanic and coastal regions, as shown in the map below. However, other variables, such as precipitation, river runoff, ozone levels, humidity, and sea level height also have this property, suggesting that scaling is a very common feature of the climate system.
What are the physical mechanisms behind the scaling?
There are a few possible explanations at the scale of the climate system, all caused by the underlying nonlinear nature of the equations of motion. A great example is turbulence: the scaling behavior in which energy is distributed between spatial scales was discovered by AN Kolmogorov in 1941.
The timescale can be explained by the nonlinear nature of the equations governing the climate system and the interaction of components of the climate system with different timescales such as the atmosphere (with a typical timescale ranging from a few seconds to a few weeks), the ocean (with time scales from weeks to decades) and ice caps (evolving over time scales ranging from decades to millennia).
How can scaling be applied to improve climate modeling and forecasting?
Using scaling allows us to more quickly calculate the Earth system’s response to greenhouse gas emissions and improve our understanding of predictability and climate models. Recently a Seasonal to interannual stochastic prediction system was developed with this in mind. Its forecasting accuracy compares favorably with long-term operational forecasting models based on traditional climate models.
What are some of the unresolved questions that require additional research, data or modeling?
Through scale analysis, we now have a better understanding of the climate system and appreciate that it consists of different time scales characterized by different scale relationships. While for the weather (up to 2 weeks) and climatic (over 30 years) time scales, the variability increases sharply with the time scale; this is not the case for the intermediate periods (2 weeks to 30 years) where the increase is quite small. How this affects predictability on these timescales is an open question. It should be noted that an inclusion of scale behaviors in traditional climate models may also be useful (e.g. for improving sub-network scale parameterizations), but more effort is still needed to this regard.
How far scaling can help rebuild the past climate is another area of ​​ongoing research. Scaling of paleoclimatic data has received much attention and has been used to assess how well climate models reproduce observed long-term climate variability. While the variability of global average temperature on interannual to millennial time scales appears to be consistent between climate models and climate reconstructions, the large gap in slow climate variability at regional scales calls for further research on temporal structures and of climate variability as well as on improving the interpretation and quality of paleoclimatic records. The PAGE group “Climate variability across scalesIs currently working on this issue.
A better appreciation of how scale relates climate variability at different scales has already improved our understanding of the climate system, but there is still a lot of exciting theoretical and applied work to be done in this area.
—Christian LE Franzke ([email protected]; 
Quote:
Franzke, CLE, Yuan, N. (2020), Is climate variability organized ?, Eos, 101, https://doi.org/10.1029/2020EO145366. Posted on June 11, 2020.
Text © 2020. The authors. CC BY-NC-ND 3.0
Unless otherwise indicated, images are subject to copyright. Any reuse without the express permission of the copyright holder is prohibited.
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