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Weather Modeling Joseph Lutgen May 15 2009

Abstract This document is an essay of purely researched material. The essay focuses on weather modeling using differential equations. A basic understanding of differential equations is suggested before reading the material herein.

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Introduction

Differential equations plays an important role in weather modeling in modern times. In the 1920’s the weather was predicted mostly using historical events. Data on barometric pressure, wind, temperature and other factors were recorded. Patterns were observed and research was done to see what happened in the past to forecast the future weather. This method was inaccurate due to the unpredictability of air currents. Weather can only be predicted on a small time scale because of it’s chaotic nature. The first documented case of using calculations to predict weather was by a man named Lewis Fry Richardson in the early 20th century. Although his forecasts were completely inaccurate, they led to more meteorologists looking at math to predict weather. Lewis used physics to determine rules about barometric pressure in that air currents would flow toward the lower pressure zones. Along with wind velocity and other factors he made mathematical models using differential equations with only paper and pen. Currently computers have the capability of performing thousands of computations per second making modeling more efficient. Another man by the name of Ed Lorenz was a meteorologist who developed a system of three ordinary differential equations that were used to predict the weather back in the early 1

1960’s. Although these equations proved to be ineffective, they did show the same pattern of unpredictability that occurs within the weather. Weather has many contributable factors. One such factor is the heat produced from the sun being varied over the surface of the earth. An additional factor is the differences of air temperature over the surface of the earth. Vorticity also plays a major part in weather modeling. Due to the earth’s rotation, the geostrophic wind patterns are changed making twists and spirals which are high and low pressure areas. Another variability in weather is due to pressure in the atmosphere decreasing as height increases. When air pressure decreases the temperature will drop. Since precipitation is dependent on air pressure and air pressure is so chaotic, it is difficult to predict weather more than a week into the future.

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Lorenz Ordinary Differential Equations

Ed Lorenz found three ordinary differential equations by condensing a large set of partial differential equations that were used to predict weather. These equations were proved to be inapplicable for normal weather predictions, but were similar to weather patterns in the way that they are completely unpredictable on a long time scale. These equations did prove applicable in depicting simplified atmospheric turbulence beneath a thunderhead. These equations were later classified as the Lorenz System. The three equations are as follows. x0 = σ(y − x) y 0 = −xz + rx − y z 0 = xy − bz The figures below show Lorenz curves with different values of σ, r, and b.

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2.1

Finding Equilibrium in Lorenz attractors

To find the equilibrium solutions for the Lorenz attractor, the initial equations are set to 0. 0 = σ(y − x) 0 = −xz + rx − y 0 = xy − bz 2.1.1

X-nullclines

To find the x-null cline, the equation below must be solved for y. 0 = σ(y − x) Since σ is constant, y=x 2.1.2

Y-nullclines

Since y = x, and 0 = −xz + rx − y, then 0 = −xz + rx − x x(r − 1 − z) = 0 So, x = 0 or z = r − 1 2.1.3

Z-nullclines

Now that y = x, and either x = 0 or z = r − 1 −bz + xy = 0 −bz + x2 = 0 So, z = 0 or −r(r − 1) + x2 = 0 → x =

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p

b(r − 1)

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Equilibrium

The equilibrium points are determined to be at (0, 0, 0) p p ( b(r − 1), b(r − 1), r − 1) p p (− b(r − 1), − b(r − 1), r − 1) Now that the nullclines and equilibrium points have been found, graphs can be made to show the three planes and equilibrium points created by the nullclines of this system of ordinary differential equations. The following graphs depict different combinations of σ, b, and r.

Figure 1: σ < r < b

Figure 2: r < σ < b

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2.3

Figure 3: r < b < σ

Figure 4: σ < b < r

Figure 5: b < σ < r

Figure 6: b < r < σ

Classifying Equilibrium Points

To classify the equilibrium points in the Lorenz system, it first needs to be simplified using the Jacobian. Remember that the equilibrium points are. (0, 0, 0) p p ( b(r − 1), b(r − 1), r − 1) p p (− b(r − 1), − b(r − 1), r − 1) 6

The real part of the eigenvalue for the jacobian will determine the classification of stable or unstable for the certain parameters of σ, b, and r. If all real parts of the eigenvalues are negative, the system is stable. If any of the real parts of the eigenvalues are positive, the system is unstable. When r < 1, there is only one equilibrium point which is at (0, 0, 0). When this happens, the system is stable because it converges to zero. The Lorenz system that is intriguing is when r > 470/19. At r > 470/19 the system is unstable. Using Matlab, a program can be made to quickly determine the eigenvalues and eigenvector’s for any initial set of conditions σ, b, and r. An example of such a program is in the following link. This essay isn’t strictly on the Lorenz system, therefore classifications for each equilibrium point will be based on σ = 10, b = 8/3, and r being varied. 2.3.1

Classification when σ = 10, b = 8/3, and r = .5

As mentioned above, when r is less than one there is only the equilibrium point at the origin. This point is stable according to the calculations done by the program mentioned.

Figure 7: X,Y,and Z vs T when r = 1/2

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2.3.2

Classification when σ = 10, b = 8/3, and r = 15

Using a modified version of the previous program with given values of σ = 10, b = 8/3, and r = 15 the eigenvalues for each equilibrium point can be obtained. It turns out that the only unstable equilibrium point is at the origin.

Figure 8: X,Y,and Z vs T when r = 15

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2.3.3

Classification when σ = 10, b = 8/3, and r = 28

Using the same program with r = 28 all equilibrium points become unstable. This is where Ed Lorenz found the most fascinating results with respect to chaotic nature.

Figure 9: X,Y,and Z vs T when r = 28 The second program in the following link is the modified program mentioned above.

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Fluid Flow with Respect to the Atmosphere A Little Background in Vorticity

Vorticity, in respect to atmospheric sciences, is the rotation of air around a vertical axis. The relationship between linear velocity and angular velocity is c = ωr. The derivative of this equation with respect to r is. dc =ω dr Fluids may turn at varying speeds in time and space. However, not all particles are alike giving them different curvature qualities. 9

If rotation is positive going counterclockwise as in the figure above, then the

Figure 10: Fluid Particles in xy plane following equations apply.

−u = ω2 y − du dy = ω2

v = ω1 x dv dx = ω1

The average between the two angular velocity’s ω 1 , and ω 2 is

1 1 dv du (ω 1 + ω 2 ) = ( − ) 2 2 dx dy The vorticity is defined as twice the average angular velocity therefore vorticity(ζ) is

ζ=

dv du − dx dy

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The earth has it’s own counterclockwise vorticity f = 2Ωsin(θ), therefore we must add the two to find the absolute vorticity. An example of relative vorticity (ζ) is horizontal wind shear even though it flows in a linear manner.

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A Little Weather Modeling

In the book, An Introduction to Dynamic Meteorology, models are created and included as MatLab files on a cd. The following are a few of the files with descriptions on what is going on. Figure 11 describes the different temperatures based on height and pressure in a tropical surrounding.

Figure 11: Temperature Change based on Pressure and Height

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Figure 12 is vorticity in the x-y plane.

Figure 12: Vorticity and Velocity in the x-y Plane

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Figure 13 shows geopotential in respect to pressure and a mixed winds system of vector form.

Figure 13: Geopotential and Mixed Layer Winds

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Figure 14 shows geostrophic wind along with vorticity for a comparison of real particle motion and general wind direction to give a first approximation to model wind.

Figure 14: Geostrophic Wind and Vorticity

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Conclusion

The subject of weather and differential equations both partial and ordinary are contained throughout many works of literature. To condense any of these books into an essay would be a feat unimaginable, therefore reading materials in this subject may provide more understanding and background than the condensed version presented here. Particularly, An Introduction to Dynamic Meteorology was an invaluable resource for this project and is actually a text book used in Iowa state and the University of Kansas. If a greater understanding of meteorology was desired, this book would be highly recommended. There were many partial differential equations that were used to predict the weather on a short time scale but because this essay is based on ordinary differential equations, the partial differential equations were not discussed. Overall, because of the nature of the weather, the numerous factors that come into play, and it’s predictability to be unpredictable it is very difficult to predict weather over long periods of time. In the future there may be better methods to where predictions could be made months ahead of time.

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References [1] ”Ordinary Diferential Equations Chaos.” astro.uk. ¡http://www.astro.ku.dk/comp-phys/Notes/10.pdf¿

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[2] Byers, Horace Robert. General Meteorology. 4. New york: McGraw-Hill Book Company, 1974. Print. [3] Polking, John, Albert Boggess, and David Arnold. Differential Equations with Boundary Value Problems. 2nd ed. New Jersey: Pearson Education, Inc., 2005. Print. [4] Hayes, Brian. ”Calculating the weather.(Book review).” American Scientist 95.3 (May-June 2007): 271(3). General OneFile. Gale. Flathead Valley Community College. 25 Apr. 2009 ¡http://find.galegroup.com/ips/start.do?prodId=IPS¿. [5] Holton, James R.. An Introduction to Dynamic Meteorology. 4. San Diego: Elsevier Academic Press, 2004. Print. [6] Hurricane Andrew [Online image] Available

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