# Department of MathematicsSpring 2002 MATH 462 - Fluid Dynamics Week 0

### A Mathematical View of Fluid Motions

Fluid dynamics, the study of the motion of liquids and gases, is one of the classical applications of applied mathematics. Sciences such as aerodynamics, hydrodynamics, meteorology and oceanography, to name a few, draw heavily on the mathematics of fluid mechanics for their quantitative underpinnings. The central theme of this class is the development of the mathematics for understanding the basic variables that describe the motion of fluids: flow velocity, pressure and density.

Fluid dynamics is an application of the mathematics of partial differential equations. The core aims of this class are: deriving the equations of motion from basic physical principles, learning differential equation techniques for finding special solutions, and most importantly, interpreting such solutions in the context of understanding the science of fluids. Computer visualization will be an important accompaniment to the lectures and assigned work. The rudiments of numerical computing and graphics will be introduced through the use and modification of downloadable Matlab scripts.

The ultimate objective is to use mathematics to reveal, in a quantitive way, some everyday mysteries of the motions of liquids and gases. Why does water swirl as it drains from the bathtub? Why do radiator pipes make a lot of banging sounds? Why does a curve-ball curve?

weekly class hours & locations:
mondays: 2:30-4:20pm, room AQ5037
wednesdays: 2:30-3:20pm, room AQ5018

organizational meeting:
first class session, monday 07 january 2002
2:30pm, room AQ5037

The purpose of this meeting will be to discuss the interests of the class members, explain the computing environments, and answer general questions about the course. Registered students who miss this meeting should arrange an alternative meeting with the instructor ASAP.

Ka-Kit Tung, University of Washington
friday 04 january 2002
2:30pm, room K9509

Turbulent Energy Spectrum in the Atmosphere for Scales of Motion from 10 to 10,000 Kilometers

For over 30 years and despite many attempt, often using various turbulence theories of increasing sophistication, the problem of the atmosphere's turbulence energy spectrum as a function of scales of motion remains unsolved. At issue is the task of understanding how energy input from the sun is "injected" into the earth's atmosphere--and at what scales---and how such energy is then redistributed to the large (planetary) scales and to the small (meso- and micro-) scales. Such an understanding is invaluable in areas of current research involving predictability of weather and climate modeling.

Nastrom, Gage and Jasperson (1984)[1] analyzed wind measurements taken by instruments mounted on commercial aircrafts (Boeing 747's) during six thousand commercial flights from 1975 to 1979, and found a robust spectrum of atmosphere's energy. This spectrum has since been confirmed by other measurements and the spectrum shape is known to be insensitive to longitude, latitude, season, over land or over ocean. As a function of wavenumber k it shows a k^-3 shape between 3000 km to 600km and then a smooth transition to a k^-5/3 spectrum from 300 km to 2 km. When coupled with other global analyses, we know that the k^-3 part of the spectrum extends to the synoptic scales, peaks at zonal wavenumber 5 and attains slightly lower values at the planetary scales. This is the spectrum we are attempting to simulate numerically.

Direct numerical simulations [2] have been used to study the nonlinear interactions among various wavenumbers and to validate or refute various theories of turbulence. As the scales of motion to be resolved span over almost 5 orders of magnitude and the power of the energy range over almost 10 orders, the challenge confronting computer simulations has been termed "intimidating"[3]. Theoretically, the shape of the spectrum appears, on one hand, to be tentalizingly close to the prediction of Kraichnan in 1976 [4] based on two-dimensional turbulence, and, on the other hand, to be "paradoxical"[5], as various pieces of the 2D turbulence puzzle do not quite fit. A main problem, we argue, is the lack of a unifying theory, or even a conceptual picture, spanning the necessary scales of motion from energy injection at the synptic scales to dissipation at the micro-scales. Therefore, the numerical simulation has to encompass the full 5 orders of magnitudes of scales. In this talk I will discuss a conceptual theory for atmospheric turbulence, and present some lower resolution numerical results in support of the theory.

References:

• [1]Nastom, G. D., K.S. Gage, and W. H. Jasperson, 1984, Nature, 310, 36-38.
• [2]Maltrud, M.E. and G.K. Vallis, 1991, J. Fluid Mechanics, 288, 321-342.
• [3]Lily, D.K. 1989, J. Atmospheric Sci., 46, 2026-2030.
• [4]Kraichnan, R.H., 1967, Physics of Fluids, 10, 1417-1423.
• [5]Frisch, U., 1995, Turbulence, Cambridge University Press.

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