Ch4: Basics of Stratified Fluid ∂
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AOS611Ch.4, Z. Liu, 02/16/2016 1 Ch4: Basics of Stratified Fluid Sec. 4.1: Basic Equations Stratification will introduce new physics. We first derive the equations. Denoting rotation vector as Ω , gravity potential , u u,v, w, i x j y k z . The momentum equations are du 2Ω u p F . (4.1.1) dt The mass equation is d u 0 or u 0 (4.1.2) dt t The equation of state in general is p, T, S... (4.1.3) In the ocean, neglecting salinity, the equation of state is m 1T T0 (4.1.4) 1 where is the coefficient of thermal expansion and is a constant m T P reference density, which can be chosen as the average density. In the atmosphere, using the perfect gas, the equation of state is P (4.1.5) RT where R is the gas constant. The thermodynamic equation describes the internal energy change. For the ocean, which is incompressible, the thermodynamic equation is dT ~ C Q k 2T (4.1.6) p dt ~ where k is the thermal conductivity, Q is the heating rate per unit mass. This can be rewritten as dT J k 2T (4.1.6a) dt ~ Q k where J and k . Cp Cp Copyright 2014, Zhengyu Liu AOS611Ch.4, Z. Liu, 02/16/2016 2 The atmosphere is compressible, so the thermodynamic equation is dT 1 dp J k 2T (4.1.7) dt C p dt p 0 R , Define the potential temperature as T , where C we have p p p0 ln lnT ln lnT ln p const p d dT dp T p T p0 T d dT dp dT dp T p p p d p0 2 J k T (4.1.8) dt p T RT 1 where we have used (4.1.7) and the ideal gas law (4.1.5), such that . p pC p C p Therefore, in the local Cartesian coordinate, we have the full set of equations for the stratified ocean and atmosphere as: 1 1 u u u v u w u fv p F t x y z x x 1 1 v u v v v w v fu p F (4.1.8) t x y z y y 1 1 w u w v w w w g p F t x y z z z t u x v y w z xu y v z w (4.1.9) 2 tT u xT v yT w zT J k T oceam (4.1.10a) p 0 2 (4.1.10b) t u x v y w z J k T atmosphere p m 1T T0 ocean (4.1.11a) P atmosphere (4.1.11b) RT Copyright 2014, Zhengyu Liu AOS611Ch.4, Z. Liu, 02/16/2016 3 where a beta-plane is used such that f f0 y . The characteristics of the ocean and atmosphere also allow the equations to be further simplified for each system. 1. Oceanic Equations 0 The ocean is almost incompressible. Therefore, m 0 (x,y,z,t) with 1. The m d0 incompressibility has three consequences. First in the mass equation, dt 0 1 m , and therefore (4.1.2) reduces to volume conservation u 0 (4.1.12) Second, the thermodynamic equation and the equation of state can be combined together. Eqn.(4.1.11a) can be written as 0 mT ' where T ' T T0 . The thermodynamic equation (4.1.10a) can therefore be written as 2 t 0 u 0 k 0 J S0 (4.1.13) Third, large scale oceanic process also satisfy D/L<<1 and in turn the hydrostatic approximation (as in the case of shallow water in section 1.1). The vertical momentum equation at the leading order can be shown as p g z Defining p Pz p0 (x, y, z,t) , where P g m z is the static pressure due to the average density, the rest of perturbation pressure p0 satisfies p 0 g (4.1.14) z 0 Fourth, the momentum equations can be further simplified using the Boussinesq approximation, such that the density is a constant except when it represents the buoyancy forcing in the vertical momentum equation. The horizontal pressure gradient forcing can be approximated as 1 1 1 h p h p0 h p0 m where h i x j y is the horizontal gradient. Finally, after the four more approximations, we have the ocean equations as: Copyright 2014, Zhengyu Liu AOS611Ch.4, Z. Liu, 02/16/2016 4 1 1 t u u u fv x p0 Fx m m 1 1 t v u v fu y p0 Fy m m 1 g0 z p0 (4.1.15) m m xu y v z w 0 t 0 u 0 S0 2. Atmospheric Equations Unlike the ocean, the atmosphere is very compressible. The mass equation (4.1.9) therefore has to remain a predictive equation in general. This equation, however, can be simplified for large scale D/L<<1 processes by using the hydrostatic approximationp z p g . The mass per unit area contained between the pressure surface p and p+p is p p z g The mass of a material element is: p+p xyp m xyz g d Following the flow, the mass conservation m 0 becomes dt d 1 d d d m yp xxpyxyp 0 dt g dt dt dt xyp 1 dx 1 dy 1 dp g x dt y dt p dt xyp xu yv p 0 g dp where . The continuity equation for the atmosphere is therefore dt xu yv p 0 (4.1.16) This is a great simplification due to the p-coordinate. But, for other purposes, it turns out not to be very convenient to use the p-coordinate. We can use the log p which combines both the advantages of the p and z coordinates. Copyright 2014, Zhengyu Liu AOS611Ch.4, Z. Liu, 02/16/2016 5 Since p RT , we have p g g p z RT z z g dz dz RT z H z 0 0 s p poe poe Since T is roughly constant (in Kelvin) with height, say T≈Ts, p decreases roughly exponentially with a scale height z H s p poe , e-folding decaying Hs with a scale height RT H s . (4.1.17) s g p (the ocean can be taken as the case of an infinite scale height). So wave-like motions, which have a simple form in z, have a more cumbersome mathematical structure in p. To avoid the problem, we use the height-like vertical coordinate: p Z H s ln (4.1.18) po or Z H s p poe , (4.1.18a) where H is constant ( so Z is still in p-coordinate). We could in principle choose H to be anything, but Z will be like the real height if we choose H to be a typical value of the scale height Hs for the region of interest. For example, if we choose Ts 250 K, then R H T 7.3Km. s g s Copyright 2014, Zhengyu Liu AOS611Ch.4, Z. Liu, 02/16/2016 6 For US standard atmosphere ( Po 1013.25hPa) p(hPa) z(km) z (km) 1000 0.111 0.096 850 1.457 1.282 700 3.012 2.700 500 5.534 5.156 300 9.164 8.885 200 11.784 11.884 100 16.18 16.904 50 20.576 21.965 30 23.849 25.694 10 31.055 33.714 Now, we will write Zg for the real geometric height (wg for vertical velocity in Zg ). dp dt Z H ln p H ln po dp dZ H p dZ H dp H w dt p dt p d H d H d pw 1 d pw dp p dZ p dZ H p dZ The continuity equation in the log p or Z-coordinate can be derived from (4.1.16) as. 1 u v pw 0 (4.1.19) x y p Z Furthermore, in the p- (or Z-) coordinate, we also have to change the form of the horizontal pressure gradient. Take the u-equation as an example. d 1 1 u fv x p y,z Fx dt g Copyright 2014, Zhengyu Liu AOS611Ch.4, Z. Liu, 02/16/2016 7 C Z=const, or p=const Z Zg=const g B A x p p p p g(Z g Z g ) g(Z g Z g ) p A B A C A C B C , x Z g x x x x x Z Here, we have used pB pC , Z gA Z gB , the hydrostatic approximation p gZ g , so that p A pC g(Z gA Z gC ) and the definition of geopotential height as gZ g . 1 This also shows another advantage of the p- (or log p-) coordinate: it gets rid of the factor in the pressure gradient term and therefore acts similar to the Bousinessq approximation in the ocean. Similarly, we have 1 p y y Z g Z Finally the hydrostatic approximation is p Z g 1 g Z g p g 1 p p T g Z H Ts The complete set of the atmospheric equations are therefore Copyright 2014, Zhengyu Liu AOS611Ch.4, Z. Liu, 02/16/2016 8 F u u u fv x t x F v u v fu y t y 1 u v pw 0 (4.1.20) x y p Z T g p Z g g Ts Ts po s t u Qa Z p p where o , H s and o 2 .