Increased Climate Variability An Empirical Fact to . . . Is More - PowerPoint PPT Presentation

Formulation of the . . . A Simplified System- . . . Towards the Second . . . An Empirical Fact . . . Increased Climate Variability An Empirical Fact to . . . Is More Visible Than Global Estimating the . . . Estimating the . . . Warming: A General Analyzing the Ratio . . . Discussion System-Theory Explanation Home Page Title Page L. Octavio Lerma, Craig Tweedie, and Vladik Kreinovich ◭◭ ◮◮ ◭ ◮ Cyber-ShARE Center University of Texas at El Paso Page 1 of 15 500 W. University El Paso, TX 79968, USA Go Back lolerma@episd.org, ctweedie@utep.edu, Full Screen vladik@utep.edu Close Quit

Formulation of the . . . A Simplified System- . . . 1. Outline Towards the Second . . . • Global warming is a statistically confirmed long-term An Empirical Fact . . . phenomenon. An Empirical Fact to . . . Estimating the . . . • Somewhat surprisingly, its most visible consequence is: Estimating the . . . – not the warming itself but Analyzing the Ratio . . . – the increased climate variability. Discussion Home Page • In this talk, we explain why increased climate variabil- ity is more visible than the global warming itself. Title Page ◭◭ ◮◮ • In this explanation, use general system theory ideas. ◭ ◮ Page 2 of 15 Go Back Full Screen Close Quit

Formulation of the . . . A Simplified System- . . . 2. Formulation of the Problem Towards the Second . . . • Global warming usually means statistically significant An Empirical Fact . . . long-term increase in the average temperature. An Empirical Fact to . . . Estimating the . . . • Researchers have analyzed the expected future conse- Estimating the . . . quences of global warming: Analyzing the Ratio . . . – increase in temperature, Discussion – melting of glaciers, Home Page – raising sea level, etc. Title Page • A natural hypothesis was that at present, we would see ◭◭ ◮◮ the same effects, but at a smaller magnitude. ◭ ◮ • This turned out not to be the case. Page 3 of 15 • Some places do have the warmest summers and the Go Back warmest winters in record. Full Screen • However, other places have the coldest summers and the coldest winters on record. Close Quit

Formulation of the . . . A Simplified System- . . . 3. Formulation of the Problem (cont-d) Towards the Second . . . • What we actually observe is unusually high deviations An Empirical Fact . . . from the average. An Empirical Fact to . . . Estimating the . . . • This phenomenon is called increased climate variabil- Estimating the . . . ity . Analyzing the Ratio . . . • A natural question is: why is increased climate vari- Discussion ability more visible than global warming? Home Page • A usual answer is that the increased climate variability Title Page is what computer models predict. ◭◭ ◮◮ • However, the existing models of climate change are still ◭ ◮ very crude. Page 4 of 15 • None of these models explains why temperature in- Go Back crease has slowed down in the last two decades. Full Screen • It is therefore desirable to provide more reliable expla- nations. Close Quit

Formulation of the . . . A Simplified System- . . . 4. A Simplified System-Theory Model Towards the Second . . . • Let us consider the simplest model, in which the state An Empirical Fact . . . of the Earth is described by a single parameter x . An Empirical Fact to . . . Estimating the . . . • In our case, x can be an average Earth temperature or Estimating the . . . the temperature at a certain location. Analyzing the Ratio . . . • We want to describe how x changes with time. Discussion • In the first approximation, dx Home Page dt = u ( t ), where u ( t ) are Title Page external forces. ◭◭ ◮◮ • We know that, on average, these forces lead to a global warming, i.e., to the increase of x ( t ). ◭ ◮ Page 5 of 15 • Thus, the average value u 0 of u ( t ) is positive. Go Back def • We assume that the random deviations r ( t ) = u ( t ) − u 0 Full Screen are i.i.d., with some standard deviation σ 0 . Close Quit

Formulation of the . . . A Simplified System- . . . 5. Towards the Second Approximation Towards the Second . . . • Most natural systems are resistant to change: other- An Empirical Fact . . . wise, they would not have survived. An Empirical Fact to . . . def Estimating the . . . • So, when y = x − x 0 � = 0, a force brings y back to 0: dy Estimating the . . . dt = f ( y ); f ( y ) < 0 for y > 0, f ( y ) > 0 for y < 0. Analyzing the Ratio . . . • Since the system is stable, y is small, so we keep only Discussion linear terms in the Taylor expansion of f ( y ): Home Page f ( y ) = − k · y, so dy Title Page dt = − k · y + u 0 + r ( t ) . ◭◭ ◮◮ • Since this equation is linear, its solution can be repre- ◭ ◮ sented as y ( t ) = y s ( t ) + y r ( t ), where Page 6 of 15 dy s dy r dt = − k · y s + u 0 ; dt = − k · y r + r ( t ) . Go Back • Here, y s ( t ) is the systematic change (global warming). Full Screen • y r ( t ) is the random change (climate variability). Close Quit

Formulation of the . . . A Simplified System- . . . 6. An Empirical Fact That Needs to Be Explained Towards the Second . . . • At present, the climate variability becomes more visi- An Empirical Fact . . . ble than the global warming itself. An Empirical Fact to . . . Estimating the . . . • In other words, the ratio y r ( t ) /y s ( t ) is much higher Estimating the . . . than it will be in the future. Analyzing the Ratio . . . • The change in y is determined by two factors: Discussion – the external force u ( t ) and Home Page – the parameter k that describes how resistant is our Title Page system to this force. ◭◭ ◮◮ • Some part of global warming may be caused by the ◭ ◮ variations in Solar radiation. Page 7 of 15 • Climatologists agree that global warming is mostly caused Go Back by greenhouse effect etc., which lowers resistance k . Full Screen • What causes numerous debates is which proportion of the global warming is caused by human activities. Close Quit

Formulation of the . . . A Simplified System- . . . 7. An Empirical Fact to Be Explained (cont-d) Towards the Second . . . • Since decrease in k is the main effect, in the 1st ap- An Empirical Fact . . . proximation, we consider only this effect. An Empirical Fact to . . . Estimating the . . . • In this case, we need to explain why the ratio y r ( t ) /y s ( t ) Estimating the . . . is higher now when k is higher. Analyzing the Ratio . . . • To gauge how far the random variable y r ( t ) deviates Discussion from 0, we can use its standard deviation σ ( t ). Home Page • So, we fix values u 0 and σ 0 , st. dev. of r ( t ). Title Page • For each k , we form the solutions y s ( t ) and y r ( t ) cor- ◭◭ ◮◮ responding to y s (0) = 0 and y r (0) = 0. ◭ ◮ • We then estimate the standard deviation σ ( t ) of y r ( t ). Page 8 of 15 • We want to prove that, when k decreases, the ratio Go Back σ ( t ) /y s ( t ) also decreases. Full Screen Close Quit

Formulation of the . . . A Simplified System- . . . 8. Estimating the Systematic Deviation y s ( t ) Towards the Second . . . • We need to solve the equation dy s An Empirical Fact . . . dt = − k · y s + u 0 . An Empirical Fact to . . . • If we move all the terms containing y s ( t ) to the left- Estimating the . . . hand side, we get dy s ( t ) Estimating the . . . + k · y s ( t ) = u 0 . dt Analyzing the Ratio . . . def • For an auxiliary variable z ( t ) = y s ( t ) · exp( k · t ), we get Discussion Home Page dz ( t ) = dy s ( t ) · exp( k · t ) + y s ( t ) · exp( k · t ) · k = Title Page dt dt � dy s ( t ) � ◭◭ ◮◮ exp( k · t ) · + k · y s ( t ) . dt ◭ ◮ • Thus, dz ( t ) = u 0 · exp( k · t ) , so z ( t ) = u 0 · exp( k · t ) − 1 Page 9 of 15 , dt k Go Back and y s ( t ) = exp( − k · t ) · z ( t ) = u 0 · 1 − exp( − k · t ) Full Screen . k Close Quit

Formulation of the . . . A Simplified System- . . . 9. Estimating the Random Component y r ( t ) Towards the Second . . . • For the random component, we similarly get An Empirical Fact . . . � t An Empirical Fact to . . . y r ( t ) = exp( − k · t ) · r ( s ) · exp( k · s ) ds, so Estimating the . . . 0 Estimating the . . . � t � t y r ( t ) 2 = exp( − 2 k · t ) · Analyzing the Ratio . . . ds dv r ( s ) · r ( v ) · exp( k · s ) · exp( k · v ) , 0 0 Discussion and σ 2 ( t ) = E [ y r ( t ) 2 ] = Home Page � t � t Title Page exp( − 2 k · t ) · ds dv E [ r ( s ) · r ( v )] · exp( k · s ) · exp( k · v ) . 0 0 ◭◭ ◮◮ • Here, E [ r ( s ) · r ( v )] = E [ r ( s )] · E [ r ( v )] = 0 and E [ r 2 ( s )] = ◭ ◮ σ 2 0 , so � t Page 10 of 15 σ 2 ( t ) = E [ y r ( t ) 2 ] = exp( − 2 k · t ) · ds σ 2 0 · exp( k · s ) · exp( k · s ) . Go Back 0 Full Screen 0 · 1 − exp( − 2 k · t ) • Thus, σ 2 ( t ) = σ 2 . 2 k Close Quit