lect_08 - Momentum Equations in Spherical Coordinates For a...

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1 Momentum Equations in Spherical Coordinates • For a variety of reasons, it is useful to express the vector momentum equation for a rotating earth as a set of scalar component equations. • The use of latitude-longitude coordinates to describe positions on earth’s surface makes it convenient to write the momentum equations in spherical coordinates. • The coordinate axes are ( λ , φ ,z) where is longitude, is latitude, and z is height. Orientation of Coordinate Axes d cos a dx = The x- and y-axes are customarily defined to point east and north, respectively, such that and d a dy = Thus the horizontal velocity components are dt dy v , dt dx u = =
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2 A Complication of Spherical Coordinates When the x and y coordinates are defined in this way, the coordinate system is not strictly Cartesian, because the directions of the unit vectors depend on their position on the earth’s surface. This dependence on position can be accounted for mathematically (see Holton 2.3) by adding terms to each component of the total derivative: k ˆ a v u dt dw j ˆ a vw a tan u dt dv i ˆ a uw a tan uv dt du dt V d + + + + + + = 2 2 2 φ r r F g p V dt V d r r r r r + + × = ρ Ω 1 2 K K K = + = + + = + a v u dt dw a vw a u dt dv a uw a uv dt du 2 2 2 tan tan Vector momentum equation in rotating coordinates Total derivative
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3 r F g p V dt V d r r r r r + + × = ρ Ω 1 2 Vector momentum equation in rotating coordinates K K K + Ω = + + Ω = + + + Ω Ω = + φ cos 2 sin 2 tan cos 2 sin 2 tan 2 2 2 u a v u dt dw u a vw a u dt dv w v a uw a uv dt du
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This note was uploaded on 10/25/2011 for the course ENVSCI 324 taught by Professor Broccoli during the Spring '11 term at Rutgers.

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lect_08 - Momentum Equations in Spherical Coordinates For a...

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