MET4305_13

MET4305_13 - Scaling the Mass Conservation Equation In...

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Scaling the Mass Conservation Equation In part, because the total density is dominated by the vertical stratifcation of the atmosphere, we decompose (separate) the density into base state (mean) that is height dependent and perturbation quantities which are a function of (x, y, z, t), where . We know that ρ 0 is a f(z) only , thus the above equation can be rewritten multiplying by 1/ ρ 0 we have When comparing the last two terms we know that , thus the 4th term is much smaller than the 5th term above. We now have what Holton has, namely t ---- ρ′ ρ 0 + () u ∇ρ ′ ρ 0 + ρ ρ 0 + u + +0 = ρ xyzt ,,, ρ 0 z () ρ + = ∂ρ 0 t ⁄∂ ρ 0 ,∂ x ρ 0 y , 0 = t u ∇ρ′ w d ρ 0 dz -------- ρ 0 + u 0 ++ + 1 ρ 0 ----- ∂ρ′ t ------- u +    w ρ 0 d ρ 0 ρ 0 u u 0 + small ρ′ ρ 0 0 « 1 ρ 0 t u + w ρ 0 d ρ 0 u 0 t u ρ 0 x v ρ 0 y , w ρ 0 z w ρ 0 d ρ 0 u x v y , w z ------ ρ 0 T v ----------- ρ 0 T h ----------- W ρ 0 H --------- U ρ 0 L U ρ 0 L W ρ 0 H d ρ 0 W ρ 0 U L --- W H 10 2 10 2 10 4 --------------------- 10 2 10 10 6 ---------------- 10 -7 10 -8 10 10 6 10 5 10 2 10 4 ---------- 10 6 ρ 0 ρ 0 0 e zH 1 ρ 0 d ρ 0 ρ 0 0 e H ρ 0 0 e ------------------------------- W H 10 6
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Thus, to a first order of magnitude for synoptic flows This means that for synoptic scale motions: the horizontal area of the parcels is conserved flow is quasi non-divergent A more accurate approximation keeps the vertical variation of density and terms as well, The equation above is referred to as the ‘ anelastic ’ mass conservation equation. We’ve neglected the terms. This is NOT an incompressible fluid, which is described by . The density is constant following the motion, while for an anelastic atmosphere, density changes as a function of height.
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This note was uploaded on 02/11/2012 for the course MET 4305 taught by Professor Lazarus during the Fall '09 term at FIT.

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MET4305_13 - Scaling the Mass Conservation Equation In...

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