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Chaos Theory and Predictability. Anthony R. Lupo Department of Soil, Environmental, and Atmospheric Sciences 302 E ABNR Building University of Missouri Columbia, MO 65211. Chaos Theory and Predictability. Some popular images…………. Chaos Theory and Predictability.

Chaos Theory and Predictability

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Chaos Theory and Predictability

Anthony R. Lupo

Department of Soil, Environmental, and Atmospheric Sciences

302 E ABNR Building

University of Missouri

Columbia, MO 65211

- Some popular images…………..

- Any attempt at weather “forecasting is immoral and damaging to the character of a meteorologist” - Max Margules (1904) (1856 – 1920)
- Margules work forms the
foundation of modern

Energetics analysis.

- “Chaotic” or non-linear dynamics Is perhaps one of the most important “discovery” or way of relating to and/or describing natural systems in the 20th century!
- “Caoz” Chaos and order are opposites in the Greek language - like good versus evil.
- Important in the sense that we’ll describe the behavior of “non-linear” systems!

- Physical systems can be classified as:
Deterministic laws of motion are known and orderly (future can be directly determined from past)

- Stochastic / random no laws of motion, we can only use probability to predict the location of parcels, we cannot predict future states of the system without statistics. Only give probabilities!

- Chaotic systems We know the laws of motion, but these systems exhibit “random” behavior, due to non-linear mechanisms. Their behavior may be irregular, and may be described statistically.
- E. Lorenz and B. Saltzman Chaos is “order without periodicity”.

- Classifying linear systems
- If I have a linear set of equations represented as:
(1)

And ‘b’ is the vector to be determined. We’ll assume the solutions are non-trivial.

- Q: What does that mean again for b?
- A: b is not 0!

- Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots (source: Mathworld) (l)
- Eigenvectors are a special set of vectors associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic vectors (vector ‘b’)

- Thus we can easily solve this problem since we can substitute this into the equations (1) from before and we get:
Solve, and so, now the general solution is:

- Values of ‘c’ are constants of course. The vectors b1 and b2 are called “eigenvectors” of the eigenvaluesl1 and l2.

- Particular Solution:

- One Dimensional Non-linear dynamics
- We will examine this because it provides a nice basis for learning the topic and then applying to higher dimensional systems.
- However, this can provide useful analysis of atmospheric systems as well (time series analysis). Bengtssen (1985) Tellus – Blocking. Federov et al. (2003) BAMS for El Nino. Mokhov et al. (1998, 2000, 2004). Mokhov et al. (2004) for El Nino via SSTs (see also Mokhov and Smirnov, 2006), but also for temperatures in the stratosphere. Lupo et al (2006) temperature and precip records. Lupo and Kunz (2005), and Hussain et al. (2007) height fields, blocking.

- First order dynamic system:
- (Leibnitz notation is “x –dot”?)
- If x is a real function, then the first derivative will represent a(n) (imaginary) “flow” or “velocity” along the x – axis. Thus, we will plot x versus “x – dot”

- Draw:
- Then, the sign of “f(x)” determines the sign of the one – dimensional phase velocity.
- Flow to the right (left): f(x) > 0 (f(x) < 0)

- Two Dimensional Non-linear dynamics
- Note here that each equation has an ‘x’ and a ‘y’ in it. Thus, the first deriviatives of x and y, depend on x and y. This is an example of non-linearity. What if in the first equation ‘Ax’ was a constant? What kind of function would we have?
- Solutions to this are trajectories moving in the (x,y) phase plane.

- Coupled set: If the set of equations above are functions of x and y, or f(x,y).
- Uncoupled set: If the set of equations above are functions of x and y separately.

- Definitions
- Bifurcation point: In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied.

- Example: “pitchfork” bifurcation (subcritical)
- Solution has three roots, x=0, x2 = r

- The devil is in the details?............

- An attractor is a set of states (points in the phase space), invariant under the dynamics, towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct basins of attraction

How we see it…….

Mathematics looks at Equation of Motion (NS) is space such that:

- Closed or compact space such that boundaries are closed and that within the space divergence = 0
- Complete set div = 0 and all the “interesting” sequences of vectors in space, the support space solutions are zero.

- Ok, let’s look at a simple harmonic oscillator (pendulum):
- Where m = mass and k = Hooke’s constant.
- When we divide through by mass, we get a Sturm – Liouville type equation.

- One way to solve this is to make the problem “self adjoint” or to set up a couplet of first order equations like so let:

- Then divide these two equations by each other to get:
- What kind of figure is this?

- A set of ellipses in the phase space.

- Here it is convenient that the origin is the center!
- At the center, the “flow” is still, and since the first derivative of x is positive, we consider the “flow” to be anticyclonic (NH) “clockwise” around the origin. The eigenvalues are:
- Now as the flow does not approach or repel from the center, we can classify this as “neutrally stable”.

Thus, the system behaves well close to certain “fixed points”, which are at least neutrally stable.

- System is forever predictable in a dynamic sense, and well behaved.
- we could move to an area where the behavior changes, a bifurcation point which is called a “separatrix”.
- Beyond this, system is unpredictable, or less so, and can only use statistical methods. It’s unstable!

- Hopf’s Bifurcation:
- Hopf (1942) demonstrated that systems of non-linear differential equations (of higher order that 2) can have peculiar behavior.
- These type of systems can change behavior from one type of behavior (e.g., stable spiral to a stable limit cycle), this type is a supercritical Hopf bifurcation.

- Hopf’s Bifurcation:
- Subcritical Hopf Bifurcations have a very different behavior and these we will explore in connection with Lorenz’s equations, which describe the atmosphere’s behavior in a simplistic way. With this type of behavior, the trajectories can “jump” to another attractor which may be a fixed point, limit cycle, or a “strange” attractor (chaotic attractor – occurs in 3 – D only!)

- Example of an elliptic equation in meteorology:

- Taken from Lupo et al. 2001 (MWR)

- Ok, let’s modify the equation above:
- d is now the “damping constant”, so let’s “damp“ (“add energy to”) this expression d > 0 (d< 0).
- Then the oscillator loses (gains) energy and the determinant of the quadratic solution is also less (greater) than zero! So trajectories spin toward (away from) the center. This is a(n) “(un)stable spiral”.

- Example: forced Pendulum (J. Hansen)

- Another Example: behavior of a temperature series for Des Moines, IA (taken from Birk et. al. 2010)

- Another Example: behavior of 500 hPa heights in the N. Hemi. (taken from Lupo et. al. 2007, Izvestia)

- Sensitive Dependence on Initial Conditions (SDIC – not a federal program ).
- Start with the simple system :
- A iterative-type equation used often to demonstrate population dynamics:
- Experiment with k = 0.5, 1.0, 1.5, 1.6, 1.7, 2.0

- For each, use the following xn and graph side-by-side to compare the behavior of the system.
- Xn = -0.5, Xn = -0.50001
- Try to find: “period 2” attractor or attracting point: behavior1 behavior2 behavior 1 behavior2, and a “period 4” attractor.
- Period 2 behaves like the large-scale flow?

- Examine the initial conditions. One can be taken to be a “measurement” and the other, a “deviation” or “error”, whether it’s “generated” or real. It’s a point in the ball-park of the original.
- Asside: Heisenberg’s Uncertainty Principle All measurements are subject to a certain level of uncertainty.

- X = 0.5X = 0.50001

- What’s the diff?

- The differences that emerge illustrate the concept of Sensitive Dependence on Initial Conditions (SDIC). This is an important concept in Dynamic systems. This is also the concept behind ENSEMBLE FORECASTING!
- Toth, Z., and E. Kalnay, 1993: Ensemble forecasting at NCEP: The generation of perturbations. Bull.Amer. Meteor. Soc., 74, 2317 – 2330.
- Toth, Z., and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea.Rev., 125, 3297 – 3319.
- Tracton, M.S., and E. Kalnay, 1993: Ensemble forecasting at the National Meteorological Center: Practical Aspects. Wea. and Forecasting, 8, 379 – 39

- The basic laws of geophysical fluid dynamics describe fluid motions, they are a highly non-linear set of differentials and/or differential equations.
- e.g.,
- Given the proper set of initial and/or boundary conditions, perfect resolution, infinite computer power, and precise measurements, all future states of the atmosphere can be predicted forever!

- Given that this is not the case, these equations have an infinite set of solutions, thus anything in the phase space is possible.
- In spite of this, the same “solutions” appear time and time again!

- Note: We will define Degrees of Freedom here this will mean the number of coordinates in the phase space.
- Advances in this area have involved taking expressions with an infinite number of degrees of freedom and replacing them with expressions of finite degrees of freedom.
- For the equation of motion, whether we talk about math or meteorology, we usually examine the N-S equations in 2-D sense. Mathematically, this is one of the Million dollar problems to solve in 3-d (no “uniqueness” of solutions!).

- “Chaotic” Systems:
- 1. A system that displays SDIC.
- 2. Possesses “Fractal” dimensionality

- Fractal geometry – “self similar”
- Norwegian ModelL. Lemon

- Fractals:
- Fractal geometry was developed by Benoit Mandelbrot (1983) in his book the Fractal Geometry of Nature. Fractal comes from “Fractus” – broken and irregular.
- Fractals are precisely a defining characteristic of the strange attractor and distinguishes these from familiar attractors.

- 3. Dissipative system:
- Lyapunov Exponents - defined as the average rates of exponential divergence or convergence of nearby trajectories.
- They are also in a very real sense, they provide a quantitative measure of SDIC. Let’s introduce the concept using the simplest type of differential equation.

- Simple differential equation:
- with the solution as:

- Thus, after some large time interval “t”, the distance e(t) between two points initially separated by e(0) is:
- Thus, the SIGN of the exponent l here is of crucial importance!!!!
- A positive value for –l infers that trajectories separate at an exponential rate, while a negative value implies convergence as t infinity!

- Well, we can use our simple differential equation to get the value of the exponent as:
- So, in the general case of our differential equation, we can think of a (particular) solution as a point on the phase space, and the neighboring points as encompassing an n-dimensional ball of radius e(0)!
- With an increase in time, the ball will become an ellipsoid in non-uniform flow, and will continue to “deform” as time approaches infinity.

- There must be, by definition, as many Lyapunov exponents as there are dimensions in the phase space.
- Again, positive values represent divergence, while negative values indicate convergence of trajectories, which represent the exponential approach to the initial state of the Attractor!

- There must also, by definition, be one exponent equal to zero (which means the solution is unity) or corresponds to the direction along the trajectory, or the change in relative divergence/convergence is not exponential.
- Now, for a dissipative system, all the trajectories must add up to be negative!

- Lorenz (1960), Tellus:
- First Low Order Model (LOM) in meteorology, derived using “Galerkin” methods, which approximate solutions using finite series. (e.g. Haltiner and Williams, 1980).

- Lorenz (1960)
- Tellus

- Lorenz (1963), J. Atmos. Sci., 20, 130 - 142
- Investigated Rayleigh – Bernard (RB) convection, a classical problem in physics.
- We need to scale the primitive equations (use Boussinesq approx), then use Galerkin Techniques again.

- Lorenz (1963)
- solution:

- Lorenz (1963) – then using the initial conditions: s = 10.0 , b = 8/3, r = 28.0, and a non-dimensional time step of 0.0005.
- Then using 50 lines of FORTRAN code, and the “leapfrog” method, we can produce:

- The “Butterfly”

- We cannot solve Lorenz’s (1963) LOM unless we examine steady state conditions; that is dx/dt, dy/dt, and dz/dt all equal zero.
- The “trivial solution” x = y = z = 0, is the state of no convection.

- But, if we solve the equations, we get some interesting roots; (0 < r < 1).

- But when r > 1, we get convection and chaotic motions:

- Predictability:
- SDIC in the flow exists in set A if there exists error > 0, such that for any and any neighborhood U of x, there exists and t > 0. such that

- In ‘plane’ English: there will always be SDIC in a system (it’s intrinsic to many systems). Possible outcomes are larger than the error in specifying correct state!
- SDIC means that trajectories are “unpredictable”, even if the dynamics of a system are well-known (deterministic).

- Thus, if you wish to compute trajectories of X in a system displaying SDIC, after some time t, you will accumulate error in the prediction regardless of increases in computing power!
- There is always resolution and measurement error to contend with as well. This will further muddy the waters.

Singular Values and Vectors

- Is the factor by which initial error will grow for infinitesimal errors over a finite time at a particular location (singular vectors, as the name implies, give the direction).
- Can be numerically estimated using linear theory. Singular values/vectors are dependent upon the choice of norm; they are critically state dependent.

- Thus, after some large time interval “t”, the distance e(t) between two points initially separated by e(0) is (from slide 48 and 49):

- Thus, if the “error” doubles, or the ratio between one trajectory and another:
- and the time to accomplish this is:

- This is the basis for stating that the predictability of various phenomena is about the size of it’s growth period. For extratropical cyclones this is approximately 0.5 – 3 days.
- For the planetary scale, the time period is roughly 10 – 14 days (evolution of large-scale troughs and ridges).

- This is why we say that 10 – 14 days is the of time is the limit of dynamic weather prediction.
- In atmospheric science, we know that this is the time period for the evolution of Rossby – inertia waves, which are the result of the very size and rotation rate of the planet earth! (f = 2Wsinf)

- Now, the question is, if we know exactly the initial state (is it possible to know this?) of the atmosphere at some time t, can we make perfect forecasts?
- This question is central to the contention that the atmosphere contains a certain amount of inherent unpredictability.

- Laplace argued that given the entire and precise state of the universe at any one instant, the entire cosmos could be predicted forever and uniquely, by Newton’s Laws of motion. He was a firm believer in determinism.

- But, can we know the exact initial state? Let’s revisit Heisenberg!
- Exact solutions do exist, so in theory we can find them.
- What we can never do –even in principle - is specify the exact initial conditions!

- Measurement error and predictability:
- If we solve for t (as we did earlier for error-doubling) :
- Where h is the sum of the positive Lyapunov exponents.

- Suppose our uncertainty is at a level of 10-5, then:
- Now, let’s improve the accuracy by 5 orders of magnitude, or 10-10:

- Then, we should be able to infer that:
- Or, this increase in precision only doubles the “forecast” time. Thus, input error, will swell very quickly! Should we be pessimistic?

- Not a great “return” on investment! Pessimistic about our prospects on forecasting? From a selfish standpoint, no because this demonstrates that we cannot turn over weather forecasting to computers.
- From a scientific standpoint, no as well, because we just need to realize that forecasting beyond a certain limit at a certain scale is inevitable. As long as we realize the limitations, we can make good forecasts.

- One beneficial issue has been stimulated for operational meteorology by Chaos Theory, and that is ‘how do we express “uncertainty” in forecasts’?
- Example:

- The End!

- Overtime!
- Fractal Dimension:
- We’re used to integer whole numbers for dimensionality, but the Fractal can have a dimensionality that is not a whole number. For example, the Koch Snowflake (1904) dimension is 1.26.

- What? How can you have 1.25 dimensions? But the snowflake fills up space more efficiently than a smooth curve or line (1-D) and is less efficient than an area (2-d). So a dimension between one and two captures this concept.

- Example: (Sierpinski Gasket, 1915)
- Has a Fractal (Hausdorf) dimension of 1.59

- Hausdorf dimension:
- d = ln(N(e)) / [ln(L) – ln(e)]
- N(e) = is the smallest number of “cubes” (Euclidian shapes) needed to cover the space.
- Here it is: 3n or makes 3 copies of itself with each iteration.

- The denominator is: ln( L / e) where L = 1 (full space) and e is copy scale factor ((1/2)n length of full space with each iteration).
- So we get: d = n ln(3) / n ln (2) = 1.59

- Questions?
- Comments?
- Criticisms?
- [email protected]