Navier-Stokes Equation - DocsLib

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Home , Baroclinity, Coriolis force

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IV/2 Dynamics: Introduction

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IV/3 Pressure gradient force

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IV/4 Pressure gradient force

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y 3UHVVXUH IRUFHDUHD y )RUFHIURPOHIW =⋅ ⋅ Fpdydzleft ∂p F=− p + dx dy ⋅ dz right ∂x

∂∂pp y VXP RI IRUFHV FFF= + =−⋅⋅⋅=−⋅ dxdydzdV pleftrightx ∂∂xx ∂∂ y )RUFHSHUXQLWPDVV −⋅pdV =−⋅1 p ∂∂ρ xdmm x ρ = m m V

K 11K y *HQHUDO f=− ∇ p =− ⋅ grad p p ρ mmρ 1RWHXQLWLV 1NJ IV/5 Pressure gradient force

,93UHVVXUHJUDGLHQWIRUFH FRQWLQXHG K 11K f=− ∇ p =− ⋅ grad p p ρ mmρ

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IV/6 Thermal circulation

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IV/7 Thermal circulation

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IV/8 baroclinity and thermal circulation

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IV/9 Gravity force

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K K y 1HZWRQ ODZ =⋅ Fmgg KKK y JUDYLW\IRUFH =− =− ⋅ fggkg 1RWHXQLWLV 1NJ

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Î*HRSRWHQWLDO φ =⋅= ⋅ gzo g(,) hh ϕ 0 KKKKKK  f =−∇ ⋅φ =−∇ ⋅ = 0 gz hφ  φ ⋅ gzo IV/10 Friction

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IV/11 Coriolis force

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KK Î&RULROLVIRUFHLVDYLUWXDOIRUFH +=* FFcc0

IV/12 Coriolis force

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IV/13 Coriolis force

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()ϕϕ+∆ − ϕ ϕ avfv=Ω⋅2sinϕ ⋅= ⋅ cos cos → dcos ∆ϕϕd f ≡Ω⋅2 sinϕ Coriolis parameter

IV/14 Coriolis force

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K K K =− ⋅ Ω× y &RULROLV IRUFH fvC 2 ()

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IV/15 Coriolis force

,9&RULROLVIRUFH FRQWLQXHG K K K =− ⋅ Ω× y /HWµVORRN LQWRWKHYDULRXVWHUPVFRQWDLQHGLQWKH&RULROLVIRUFH fvC 2 () u K  vv=   &DOFXODWHGOLNHDGHWHUPLQDQW RIDPDWUL[ w KK iuΩ i0 u KKK K K Kx K K K ffvgkjvgkj+ =Ω×−⋅=−22cos Ω −⋅=− Ω⋅ϕ vgk −⋅= CgKK y ΩΩ⋅ϕ kwz ksin w K =Ω2sincos()vwi ⋅ϕ − ⋅ ⋅ K ϕ wuv , −Ω() ⋅ϕ ⋅ + 2sinuj 2QO\KRUL]RQWDO w ≈ 0 ()KK FRPSRQHQW RI &RULROLV 2Ω⋅ug +Ω2cosukgk ⋅ϕ ⋅ − ⋅ = IRUFHLPSRUWDQW 2sinΩ⋅v ⋅ ϕ ()KKK () ≈Ω2viujgku ⋅ sinϕ ⋅−Ω 2 ⋅ sin ⋅− ⋅ =−Ω⋅⋅ 2 sin  IV/16 ϕ −g ϕ Coriolis force

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KKK K 2sinΩ⋅ϕ vv fvf=−2 ⋅() Ω× ≈() = = f = CCH −Ω2sinϕ ⋅uu − KK K()K =⋅−⋅=−× fv i fu j f k vH 2QO\KRUL]RQWDOYHORFLW\FRPSRQHQWLPSRUWDQW &RULROLVSDUDPHWHU K u v =  f ≡Ω⋅2sinϕ H v K k K KKK cab=× vH K K × K K kvH −× kvH y &RULROLV IRUFHGLUHFWVWKH ZLQGWRWKH ULJKW IURPLWV

RULJLQDOGLUHFWLRQJLYHQE\ Y+ LQWKH1+ WRWKHOHIW LQ6+ IV/17 Navier-Stokes Equation

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IV/18 Navier-Stokes Equation

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IV/19 Navier-Stokes Equation

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IV/20 Scale analysis

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IV/21 Scale analysis

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IV/22 Geostrophic approximation

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KK += ()ffCp ()0 HHKKKK K K ⇒ −ρ −−11 ∇− Ω× =− ∇− × = mHmHpv2( ) pfkv ( ) 0 ρ K K ()fv⊥ CHH ()K K fv⊥ pHH

K 1 K K y *HRVWURSKLF ZLQG FRPSRQHQW vkp=×∇⋅() gsρ ⋅ H m f IV/23 Geostrophic approximation

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KKK ++ = I FHQWULIXJDOIRUFH fffCpZ0 ] IV/24 Cyclostrophic wind

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IV/25 Ageostrophic wind

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IV/26 ageostrophic wind

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IV/27 ageostrophic wind

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IV/28 Vertical motion

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