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Lect-05-P4 - M.Sc in Computational Science Fundamentals of...

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M.Sc. in Computational Science Fundamentals of Atmospheric Modelling Peter Lynch, Met ´ Eireann Mathematical Computation Laboratory (Opp. Room 30) Dept. of Maths. Physics, UCD, Belfield. January–April, 2004. Lecture 5 Steady Vortical Flows 2 The Taylor-Proudman Theorem In deriving the Shallow Water Equations , we made the assumption that the horizontal velocity is independent of depth. Although dynamically consistent, this may seem an artificial limitation. However, we will show that rotation acts as a constraint on the flow , so that under certain circumstances, variations in the direction of the spin axis are resisted. Theorem. For incompressible, inviscid, hydrostatic, geostrophic flow on an f -plane, the velocity is independent of height. Proof. We assume the density is constant. We also ignore varia- tions of the Coriolis parameter f . We assume geostrophic and hydrostatic balance: f V = 1 ρ k × ∇ p , ∂p ∂z = - gρ . 3 Taking the curl of the geostrophic equation ( f constant): ∇ × ( k × V ) = 0 Let us compute the components in Cartesian coordinates: ∇× ( - v, u, 0) = i j k ∂/∂x ∂/∂y ∂/∂z - v u 0 = - ∂u ∂z , - ∂v ∂z , ∂u ∂x + ∂v ∂y = 0 . Combining this with the continuity equation ∇ · V = 0 we get ∂u ∂z = ∂v ∂z = ∂w ∂z = 0 . This result indicates that, under the assumptions of the theorem, the motion is quasi-two-dimensional , with velocity independent of depth. If the bottom is flat, w 0 . The result surprised G. I. Taylor, who wrote [Taylor, 1923]: “The idea appears fantastic, but the experiments . . . show that the motion does . . . approximate to this curious type”. 4
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The CSU Spin Tank The Taylor-Proudman Theorem can be demonstrated beau- tifully by means of spin-tank experiments. A Spin-tank or rotating dish-pan has been constructed at Colorado State University to demonstrate various geophys- ical fluid phenomena. It was built with a small budget ($3000) and is portable. A description of the spin-tank at CSU is given at http://einstein.atmos.colostate.edu/ mcnoldy/spintank/ At this site, a number of experiments are described. There are several MPEG loops showing the results of these exper- iments. See also the article in Bull. Amer. Met. Soc. , December, 2003. This Journal is freely available online. 5 Taylor-Proudman Column Schematic diagram of tank Taylor-Proudman Column 6 7 8
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9 Simple Steady-state Solutions 10 Polar Coordinates The flow will be assumed to be axially or circularly sym- metric, and it is convenient to introduce (cylindrical) polar coordinates . Let R and θ be the radial distance and azimuthal angle. Let U and V be radial and azimuthal components of velocity. 11 Axially Symmetric Steady Flow We will study some steady-state or time-independent solu- tions. We assume that the radial velocity vanishes: U 0 .
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