Anthony R. Lupo Department of Soil, Environmental, in addition to Atmospheric Sciences 302

Anthony R. Lupo Department of Soil, Environmental, in addition to Atmospheric Sciences 302 www.phwiki.com

Anthony R. Lupo Department of Soil, Environmental, in addition to Atmospheric Sciences 302

Curran, Donna, Food Editor has reference to this Academic Journal, PHwiki organized this Journal Anthony R. Lupo Department of Soil, Environmental, in addition to Atmospheric Sciences 302 E ABNR Building University of Missouri Columbia, MO 65211 Some popular images Any attempt at weather “ as long as ecasting is immoral in addition to damaging to the character of a meteorologist” – Max Margules (1904) (1856 – 1920) Margules work as long as ms the foundation of modern Energetics analysis.

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“Chaotic” or non-linear dynamics Is perhaps one of the most important “discovery” or way of relating to in addition to /or describing natural systems in the 20th century! “Caoz” Chaos in addition to 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 in addition to orderly (future can be directly determined from past) Stochastic / r in addition to om 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 “r in addition to om” behavior, due to non-linear mechanisms. Their behavior may be irregular, in addition to may be described statistically. E. Lorenz in addition to 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 as long as 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 be as long as e in addition to we get: Solve, in addition to so, now the general solution is: Values of ‘c’ are constants of course. The vectors b1 in addition to b2 are called “eigenvectors” of the eigenvalues l1 in addition to l2. Particular Solution: One Dimensional Non-linear dynamics We will examine this because it provides a nice basis as long as learning the topic in addition to 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 as long as El Nino. Mokhov et al. (1998, 2000, 2004). Mokhov et al. (2004) as long as El Nino via SSTs (see also Mokhov in addition to Smirnov, 2006), but also as long as temperatures in the stratosphere. Lupo et al (2006) temperature in addition to precip records. Lupo in addition to Kunz (2005), in addition to 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’ in addition to a ‘y’ in it. Thus, the first deriviatives of x in addition to y, depend on x in addition to 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 in addition to y, or f(x,y). Uncoupled set: If the set of equations above are functions of x in addition to 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 as long as a nonlinear system as some parameter is varied. Example: “pitch as long as k” 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 in addition to that within the space divergence = 0 Complete set div = 0 in addition to 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 in addition to 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: Curran, Donna Palm Springs Life's Desert Guide Food Editor www.phwiki.com

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, in addition to 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 as long as ever predictable in a dynamic sense, in addition to 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, in addition to 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.

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) in addition to 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 lupoa@missouri.edu

Curran, Donna Food Editor

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