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MET4305_13

# MET4305_13 - Scaling the Mass Conservation Equation In part...

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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 x y z t , , , ( ) ρ 0 z ( ) ρ′ x y z t , , , ( ) + = ∂ρ 0 t ∂ρ 0 , x ∂ρ 0 y , 0 = ( ) t ---- ρ′ u ∇ρ′ w d ρ 0 dz -------- ρ′ ρ 0 + ( )∇ u 0 + + + 1 ρ 0 ----- ∂ρ′ t ------- u ∇ρ′ + w ρ 0 ----- d ρ 0 dz -------- ρ′ ρ 0 ----- u u 0 + + + small ρ′ ρ 0 0 « 1 ρ 0 ----- ∂ρ′ t ------- u ∇ρ′ + w ρ 0 ----- d ρ 0 dz -------- u 0 + + ∂ρ′ t ------- u ρ 0 ----- ∂ρ′ x ------- v ρ 0 ----- ∂ρ′ y ------- , w ρ 0 ----- ∂ρ′ z ------- w ρ 0 ----- d ρ 0 dz -------- u x ----- v y ----- , w z ------ ρ′ ρ 0 T v ----------- ρ′ ρ 0 T h ----------- ρ′ W ρ 0 H --------- ρ′ U ρ 0 L --------- ρ′ U ρ 0 L --------- ρ′ W ρ 0 H --------- d ρ 0 dz -------- 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 z H 1 ρ 0 ----- d ρ 0 dz -------- ρ 0 0 ( ) e z H H ρ 0 0 ( ) e z H ------------------------------- W H ---- 10 6

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