EXPANDING THE CALCULUS HORIZON. Hurricane Modeling

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1 EXPANDING THE CALCULUS HORIZON Hurricane Modeling Each ear population centers throughout the world are ravaged b hurricanes, and it is the mission of the National Hurricane Center to minimize the damage and loss of life b issuing warnings and forecasts of hurricanes developing in the Caribbean, Atlantic, Gulf of Meico, and Eastern Pacific regions. Your assignment as a trainee at the Center is to construct a simple mathematical model of a hurricane using basic principles of fluid flow and properties of vector fields. Modeling Assumptions You have been notified of a developing hurricane in the Bahamas (designated hurricane Isaac) and have been asked to construct a model of its velocit field. Because hurricanes are complicated three-dimensional fluid flows, ou will have to make man simplifing assumptions about the structure of a hurricane and the properties of the fluid flow. Accordingl, ou decide to model the moisture in Isaac as an ideal fluid, meaning that it is incompressible and its viscosit can be ignored. An incompressible fluid is one in which the densit of the fluid is the same at all points and cannot be altered b compressive forces. Eperience has shown that water can be regarded as incompressible but water vapor cannot. However, incompressibilit is a reasonable assumption for a basic hurricane model because a hurricane is not restricted to a closed container that would produce compressive forces. All fluids have a certain amount of viscosit, which is a resistance to flow oil and molasses have a high viscosit, whereas water has almost none at subsonic speeds. Thus, it is reasonable to ignore viscosit in a basic model. Net, ou decide to assume that the flow is in a stead state, meaning that the velocit of the fluid at an point does not var with time. This is reasonable over ver short time periods for hurricanes that move and change slowl. Finall, although hurricanes are three-dimensional flows, ou decide to model a two-dimensional horizontal cross section, so ou make the simplifing assumption that the fluid in the cross section flows horizontall.

2 Epanding the Calculus Horizon The photograph of Isaac shown at the beginning of this module reveals a tpical pattern of a Caribbean hurricane a counterclockwise swirl of fluid around the ee through which the fluid eits the flow in the form of rain. The lower pressure in the ee causes an inward-rushing air mass, and circular winds around the ee contribute to the swirling effect. Your first objective is to find an eplicit formula for Isaac s velocit field F(, ), so ou begin b introducing a rectangular coordinate sstem with its origin at the ee and its -ais pointing north. Moreover, based on the hurricane picture and our knowledge of meteorological theor, ou decide to build up the velocit field for Isaac from the velocit fields of simpler flows a counterclockwise vorte flow F (, ) in which fluid flows counterclockwise in concentric circles around the ee and a sink flow F 2 (, ) in which the fluid flows in straight lines toward a sink at the ee. Once ou find eplicit formulas for F (, ) and F 2 (, ), our plan is to use the superposition principle from fluid dnamics to epress the velocit field for Isaac as F(, ) = F (, ) + F 2 (, ). Modeling a Vorte Flow A counterclockwise vorte flow of an ideal fluid around the origin has four defining characteristics (Figure a on the following page): The velocit vector at a point (, ) is tangent to the circle that is centered at the origin and passes through the point (, ). The direction of the velocit vector at a point (, ) indicates a counterclockwise motion. The speed of the fluid is constant on circles centered at the origin. The speed of the fluid along a circle is inversel proportional to the radius of the circle (and hence the speed approaches + as the radius of the circle approaches 0). In fluid dnamics, the strength k of a vorte flow is defined to be 2π times the speed of the fluid along the unit circle. If the strength of a vorte flow is known, then the speed of the fluid along an other circle can be found from the fact that speed is inversel proportional to the radius of the circle. Thus, our first objective is to find a formula for a vorte flow F (, ) with a specified strength k. Eercise Show that k F (, ) = (i j) is a model for the velocit field of a counterclockwise vorte flow around the origin of strength k b confirming that (a) F (, ) has the four properties required of a counterclockwise vorte flow around the origin; (b) k is 2π times the speed of the fluid along the unit circle. Eercise 2 strength 2π. Use a graphing utilit that can generate vector fields to generate a vorte flow of Modeling a Sink Flow A uniform sink flow of an ideal fluid toward the origin has four defining characteristics (Figure b): The velocit vector at ever point (, ) is directed toward the origin. The speed of the fluid is the same at all points on a circle centered at the origin. The speed of the fluid at a point is inversel proportional to its distance from the origin (from which it follows that the speed approaches + as the distance from the origin approaches 0). There is a sink at the origin at which fluid leaves the flow. As with a vorte flow, the strength q of a uniform sink flow is defined to be 2π times the speed of the fluid at points on the unit circle. If the strength of a sink flow is known, then the speed of the

3 Topics in Vector Calculus fluid at an point in the flow can be found using the fact that the speed is inversel proportional to the distance from the origin. Thus, our net objective is to find a formula for a uniform sink flow F 2 (, ) with a specified strength q. Eercise 3 Show that q F 2 (, ) = (i + j) is a model for the velocit field of a uniform sink flow toward the origin of strength q b confirming the following facts: (a) F 2 (, ) has the four properties required of a uniform sink flow toward the origin. [A reasonable phsical argument to confirm the eistence of the sink will suffice.] (b) q is 2π times the speed of the fluid at points on the unit circle. Eercise 4 Use a graphing utilit that can generate vector fields to generate a uniform sink flow of strength 2π. Figure (a) (b) A Basic Hurricane Model It now follows from Eercises and 3 that the vector field F(, ) for a hurricane model that combines a vorte flow around the origin of strength k and a uniform sink flow toward the origin of strength q is F(, ) = [(q + k)i + (q k)j] () Eercise 5 (a) Figure 2 shows a vector field for a hurricane with vorte strength k = 2π and sink strength q = 2π. Use a graphing utilit that can generate vector fields to produce a reasonable facsimile of this figure. (b) Make a conjecture about the effect of increasing k and keeping q fied, and check our conjecture using a graphing utilit. (c) Make a conjecture about the effect of increasing q and keeping k fied, and check our conjecture using a graphing utilit.

4 Epanding the Calculus Horizon Figure 2 Modeling Hurricane Isaac You are now read to appl Formula () to obtain a model of the vector field F(, ) of hurricane Isaac. You need some observational data to determine the constants k and q, so ou call the Technical Support Branch of the Center for the latest information on hurricane Isaac. The report that 20 km from the ee the wind velocit has a component of 5 km/h toward the ee and a counterclockwise tangential component of 45 km/h. Eercise 6 (a) Find the strengths k and q of the vorte and sink for hurricane Isaac. (b) Find the vector field F(, ) for hurricane Isaac. (c) Estimate the size of hurricane Isaac b finding a radius beond which the wind speed is less than 5 km/h. Streamlines for the Basic Hurricane Model The paths followed b the fluid particles in a fluid flow are called the streamlines of the flow. Thus, the vectors F(, ) in the velocit field of a fluid flow are tangent to the streamlines. If the streamlines can be represented as the level curves of some function ψ(,), then the function ψ is called a stream function for the flow. Since ψ is normal to the level curves ψ(,) = c, it follows that ψ is normal to the streamlines; and this in turn implies that ψ F = 0 (2) Your plan is to use this equation to find the stream function and then the streamlines of the basic hurricane model. Since the vorte and sink flows that produce the basic hurricane model have a central smmetr, intuition suggests that polar coordinates ma lead to simpler equations for the streamlines than rectangular coordinates. Thus, ou decide to epress the velocit vector F at a point (r, θ) in terms of the orthogonal unit vectors u r = cos θi + sin θ j and u θ = sin θi + cos θ j The vector u r, called the radial unit vector, points awa from the origin, and the vector u θ, called the transverse unit vector, is obtained b rotating u r counterclockwise 90 (Figure 3). Eercise 7 Show that the vector field for the basic hurricane model given in () can be epressed in terms of u r and u θ as F = 2πr (qu r ku θ )

5 Topics in Vector Calculus u u = sin ui + cos uj u = p/2 F u u r = cos ui + sin uj u u u r r u (r, u) u = 0 F decomposed into radial and transverse components at (r, u). Figure 3 It follows from Eercise 75 of Section 4.6 that the gradient of the stream function can be epressed in terms of u r and u θ as ψ = ψ r u r + ψ r θ u θ Eercise 8 satisfied if Eercise 9 Confirm that for the basic hurricane model the orthogonalit condition in (2) is ψ r = k r and ψ θ = q B integrating the equations in Eercise 8, show that ψ = k ln r + qθ is a stream function for the basic hurricane model. Eercise 0 the form Show that the streamlines for the basic hurricane model are logarithmic spirals of r = Ke qθ/k (K > 0) Eercise Use a graphing utilit to generate some tpical streamlines for the basic hurricane model with vorte strength 2π and sink strength 2π. Streamlines for Hurricane Isaac Eercise 2 In Eercise 6 ou found the strengths k and q of the vorte and sink for hurricane Isaac. Use that information to find a formula for the famil of streamlines for Isaac; and then use a graphing utilit to graph the streamline that passes through the point that is 20 km from the ee in the direction that is 45 NE from the ee.... Module b: Josef S. Torok, Rochester Institute of Technolog Howard Anton, Dreel Universit