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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

- 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.

Chaos Theory and Predictability

- “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!

Chaos Theory and Predictability

- 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!

Chaos Theory and Predictability

- 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”.

Chaos Theory and Predictability

- 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!

Chaos Theory and Predictability

- 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’)

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- Particular Solution:

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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”

Chaos Theory and Predictability

- 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)

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

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

Chaos Theory and Predictability

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

Chaos Theory and Predictability

- 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

Chaos Theory and Predictability

How we see it…….

Chaos Theory and Predictability

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.

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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:

Chaos Theory and Predictability

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

Chaos Theory and Predictability

- A set of ellipses in the phase space.

Chaos Theory and Predictability

- 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”.

Chaos Theory and Predictability

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!

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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!)

Chaos Theory and Predictability

- Example of an elliptic equation in meteorology:

Chaos Theory and Predictability

- Taken from Lupo et al. 2001 (MWR)

Chaos Theory and Predictability

- 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”.

Chaos Theory and Predictability

- Example: forced Pendulum (J. Hansen)

Chaos Theory and Predictability

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

Chaos Theory and Predictability

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

Chaos Theory and Predictability

- 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

Chaos Theory and Predictability

- 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?

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- X = 0.5 X = 0.50001

Chaos Theory and Predictability

- What’s the diff?

Chaos Theory and Predictability

- 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

Chaos Theory and Predictability

- 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!

Chaos Theory and Predictability

- 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!

Chaos Theory and Predictability

- 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!).

Chaos Theory and Predictability

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

Chaos Theory and Predictability

- Fractal geometry – “self similar”
- Norwegian Model L. Lemon

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- Simple differential equation:
- with the solution as:

Chaos Theory and Predictability

- 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!

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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!

Chaos Theory and Predictability

- 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!

Chaos Theory and Predictability

- 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).

Chaos Theory and Predictability

- Lorenz (1960)
- Tellus

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- Lorenz (1963)
- solution:

Chaos Theory and Predictability

- 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:

Chaos Theory and Predictability

- The “Butterfly”

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

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

Chaos Theory and Predictability

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

Chaos Theory and Predictability

- 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

Chaos Theory and Predictability

- 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).

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

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.

Chaos Theory and Predictability

- 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):

Chaos Theory and Predictability

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

Chaos Theory and Predictability

- 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).

Chaos Theory and Predictability

- 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)

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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!

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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:

Chaos Theory and Predictability

- 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?

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

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

Chaos Theory and Predictability

- The End!

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

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

Chaos Theory and Predictability

- 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.

Chaos Theory and Predictability

- 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

Chaos Theory and Predictability

- Questions?
- Comments?
- Criticisms?
- lupoa@missouri.edu

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