by Robert Ellison
Climate sensitivity is large in the vicinity of tipping points but moderate otherwise.
‘Prediction of weather and climate are necessarily uncertain: our observations of weather and climate are uncertain, the models into which we assimilate this data and predict the future are uncertain, and external effects such as volcanoes and anthropogenic greenhouse emissions are also uncertain. Fundamentally, therefore, therefore we should think of weather and climate predictions in terms of equations whose basic prognostic variables are probability densities ρ(X,t) where X denotes some climatic variable and t denoted time. In this way, ρ(X,t)dV represents the probability that, at time t, the true value of X lies in some small volume dV of state space.’ (Predicting Weather and Climate – Palmer and Hagedorn eds – 2006)
The schematic below makes the point visually. Model solutions diverge as a result of sensitive dependence to variable starting points and boundary conditions – to a boundary defined by the topology of the global strange attractor – the state space – for the particular set of equations. Model solutions bifurcate chaotically. In climate a bifurcation is commonly called abrupt climate change – which are seen both in reconstructions of past climate changes and in modern records. These difficult ideas are fundamental principles of climate science and have very practical implications for the evolution of climate this century.
Figure 1: Probabilistic model schematic (Source: Julia Slingo and Tim Palmer – 2011- Uncertainty in weather and climate prediction)
Julia Slingo is the head of the British Met Office and Tim Palmer is the head of the European Centre for Mid-Range Forecasting. The schematic shows small differences – within the limits of precision for data on both initial and boundary conditions – leading to diverging trajectories of solutions and a range of ultimate solutions that are both irreducibly imprecise and unknowable beforehand. The probabilistic forecast relies on running the model hundreds of times using small variations in initial and boundary conditions to at least estimate where the ballpark is. Even then there is no guarantee that it maps one to one to actual climate.
The trajectories of the multiple feasible solutions generated from slightly different starting and boundary conditions include the potential for chaotic bifurcation. Here is a schematic showing bifurcation in complex systems.
Catastrophe theory originated with the work of the French mathematician René Thom in studying the dramatic changes in landslides and earthquakes. A small change in a control variable – μ – at a threshold causes a nonlinear shift called a bifurcation in the system.
The global strange attractor is simply the solution space possible for a set of nonlinear equations. The famous ‘butterfly’ plot of Edward Lorenz is the classic example. Lorenz used simplified Navier-Stokes equations for fluid motion – which are the core equations for weather and climate modelling – to build a convection model for the atmosphere.
The nonlinear Lorenz equations are:
dx/dt = P(y – x)
dy/dt = Rx – y – xz
dz/dt = xy – By
where P is the Prandtl number representing the ratio of the fluid viscosity to its thermal conductivity, R represents the difference in temperature between the top and bottom of the system, and B is the ratio of the width to height of the box used to hold the system. The values Lorenz used are P = 10, R = 28, B = 8/3. It simply gives values for changes in x, y and z with changes in time t – allowing the system of equations to be solved through time in three dimensions from an initial starting point.
The figure above was created using an executable Java file downloaded here. The wings are known as attractor basins and the solution unpredictably shifts from one basin to the other through a saddle node bifurcation. Using slightly different inputs changes the trajectory of the solution through time – it diverges unpredictably from the original solution. This is math that is not in Kansas anymore.
Lorenz started his calculation in the middle of a run by inputting values truncated to three decimal places in place of the original six. By all that was known – it should not have made much of a difference. The rest is history in the discovery of chaos theory as the third great idea – along with relativity and quantum mechanics – of 20th century physics. It has applications in ecology, physiology, economics, electronics, weather, climate, planetary orbits and much else. In climate it is driving a new math of networks in complexity theory.
Michael Ghil is the Distinguished Research Professor of Atmospheric and Oceanic Sciences, University of California, Los Angeles (UCLA). Ghil explored the idea with an energy balance climate model that follows the evolution of global surface-air temperature with changes in the global radiative balance. The plot originates from work for Ghil’s Ph.D. thesis in 1975 and was reproduced in a 2013 World Scientific Review article to illustrate this other paradigm of climate sensitivity.
The model has two stable states with two saddle node bifurcation points – the latter at the transitions from the blue to red lines from above and below. The control function – μ – here is a normalized insolation value. The current position is μ =1 with a global mean temperature of 287.7K.
It caused a bit of consternation in the 1970’s when it was realised that a very small decrease in insolation is sufficient to cause a rapid transition to an icy planet. Slightly below current insolation levels – or with higher albedo – there is a climate tipping point. Climate sensitivity – γ – is the tangent to the curve as shown. Climate sensitivity increases as you move down the upper curve to the left and becomes arbitrarily large as the system approaches bifurcation.
Figure 4: Bifurcation diagram for the solutions of an energy-balance model (EBM), showing the global-mean temperature (T) vs. the fractional changeof insolation (μ) at the top of the atmosphere.(Source: Michael Ghil, A Mathematical Theory of Climate Sensitivity or, How to Deal With Both Anthropogenic Forcing and Natural Variability?)
The details of the model can be found here.
Climate sensitivity is large in the vicinity of tipping points but moderate otherwise. A variable sensitivity – as must be the case in such a finely balanced system as the Earth’s climate. While this is necessarily a simplified approach to a new mathematics of climate sensitivity – it is nonetheless a vastly more sophisticated concept – and overwhelmingly likely to be truer – than back of the envelope constant high or low sensitivity calculations.
In climate a bifurcation – equivalently a phase transition, a catastrophe (in the sense of René Thom), or a tipping point – is commonly referred to as an abrupt climate change.
‘Technically, an abrupt climate change occurs when the climate system is forced to cross some threshold, triggering a transition to a new state at a rate determined by the climate system itself and faster than the cause. Chaotic processes in the climate system may allow the cause of such an abrupt climate change to be undetectably small . . .
Modern climate records include abrupt changes that are smaller and briefer than in paleoclimate records but show that abrupt climate change is not restricted to the distant past.’ US National Academy of Sciences, 2002, Abrupt Climate Change: Inevitable Surprises.
The science of abrupt change on decadal scales in the modern record explain observations that have been a puzzle for decades – notably the ‘Great Pacific Climate Shift’ of 1976/1977.
‘It is hypothesized that persistent and consistent trends among several climate modes act to ‘kick’ the climate state, altering the pattern and magnitude of air-sea interaction between the atmosphere and the underlying ocean… These climate mode trend phases indeed behaved anomalously three times during the 20th century, immediately following the synchronization events of the 1910s, 1940s, and 1970s. This combination of the synchronization of these dynamical modes in the climate, followed immediately afterward by significant increase in the fraction of strong trends (coupling) without exception marked shifts in the 20th century climate state. These shifts were accompanied by breaks in the global mean temperature trend with respect to time, presumably associated with either discontinuities in the global radiative budget due to the global reorganization of clouds and water vapor or dramatic changes in the uptake of heat by the deep ocean. Similar behavior has been found in coupled ocean/atmosphere models, indicating such behavior may be a hallmark of terrestrial-like climate systems [Tsonis et al., 2007].
The subject of decadal to inter-decadal climate variability is of intrinsic importance not only scientifically but also for society as a whole. Interpreting past variability and making informed projections about potential future variability requires (i) identifying the dynamical processes internal to the climate system that underlie such variability [see, e.g., Mantua et al., 1997; Zhang et al., 1997, 2007; Knight et al., 2005; Dima and Lohmann, 2007], and (ii) recognizing the chain of events that mark the onset of large amplitude variability events, i.e., shifts in the climate state. Such shifts mark changes in the qualitative behavior of climate modes of variability, as well as breaks in trends of hemispheric and global mean temperature. The most celebrated of these shifts in the instrumental record occurred in 1976/77. That particular winter ushered in an extended period in which the tropical Pacific Ocean was warmer than normal, with strong El Niño-Southern Oscillation (ENSO) events occurring after that time, contrasting with the weaker ENSO variability in the decades before [Hoerling et al., 2004; Huang et al., 2005]. Global mean surface temperature also experienced a trend break, transitioning from cooling in the decades prior to 1976/77 to the strong warming that characterized the remainder of the century.’ Kyle L. Swanson and Anastasios A. Tsonis, 2009, Has the climate recently shifted?
These multi-decadal climate shifts correspond precisely to changes in Pacific Ocean circulation, in global hydrological patterns and in changes in the trajectory of surface temperature. The multi-decadal Pacific Ocean pattern can be seen in El Niño-Southern Oscillation (ENSO) proxies for up to 1000 years. In theory we then have a mechanism – albeit a complex one – that better explains climate data than the old forcing paradigm.
The abrupt shifts in Pacific circulation involve changes in the Pacific Decadal Oscillation (PDO) in the north-eastern Pacific and coincident changes in the frequency and intensity of ENSO events. Increased frequency and intensity of La Niña occur with a cool mode PDO and vice versa. It is associated with changes in wind, currents and cloud that change the energy dynamic of the planet. Cool decadal modes cool the planetary surface and warm modes add to the surface temperatures. (e.g. http://earthobservatory.nasa.gov/IOTD/view.php?id=8703)
Although a significant factor in global climate on the scale of decades – the Pacific Ocean modes are part of a global climate system that is variable at many scales in time and space.
‘The global climate system is composed of a number of subsystems – atmosphere, biosphere, cryosphere, hydrosphere and lithosphere – each of which has distinct characteristic times, from days and weeks to centuries and millennia. Each subsystem, moreover, has its own internal variability, all other things being constant, over a fairly broad range of time scales. These ranges overlap between one subsystem and another. The interactions between the subsystems thus give rise to climate variability on all time scales.’ Michael Ghil
The theory suggests that the system is pushed past a threshold at which stage the components start to interact chaotically in multiple and changing negative and positive feedbacks – as tremendous energies cascade through powerful subsystems. It produces extremes of weather at phase transitions that Didier Sornette has called dragon-kings. Climate in this theory is an emergent property of the shift in global energies as the system settles down into a new climate state.
In the way of true science – it suggests at least decadal predictability. The current cool Pacific Ocean state seems more likely than not to persist for 20 to 40 years from 2002. There are scientific bonus points for having predicted this a decade or more ago – as some did. The flip side is that – beyond the next few decades – the future evolution of the global mean surface temperature may hold surprises on both the warm and cold ends of the spectrum.
Biosketch. My degrees are in environmental science and engineering – but my bent is writing – technical, journalistic and creative. For a number of years I have been looking for a breakthrough way of expressing these ideas. A way of breasting the fog to explain these ideas in ways that are comprehensible to someone with a moderate education from either side of the divide. Is this it? I don’t know. They are intrinsically difficult if fundamental climate concepts that most people have not fully appreciated to date – and I have done my best. I am not a sceptic. Although if you question or challenge what are most commonly simplistic narratives of the climate war – you are inevitably metaphorically tarred and feathered. So I suppose I must be. That this happens to people like Judith Curry, Anastasion Tsonis and Lennart Bergstrom is a dismal episode in a querulous science.
JC note: This is a guest post, that has evolved from comments made at Climate Etc. by Chief Hydrologist/Generalissimo Skippy/Robert Ellison. As with all guest posts, keep your comments relevant and civil.