WSN16 Evening school, 29 July 2016, Chinese University of Hong Kong Introduction to Mesoscale Modelling Kazuo Saito ksaito@mri-jma.go.jp Meteorological Research Institute 1. Fundamental equations for atmosphere 2. Waves in mesoscale atmosphere 3. Classification of nonhydrostatic models 4. Numerics of JMA-NHM
1. Fundamental equations for atmosphere Six variables which describe the state of dry atmosphere: three velocity components, pressure, temperature and density Prognostic equations • Momentum equation （ three wind components: u , v and w ） • Continuity equation （ pressure: p ） • Thermodynamic equation (temperature: T ) Diagnostic equation • State equation (density: ρ ) In the case of moist atmosphere, preservation of water substances and the phase change must be considered (cloud micro-physics).
Momentum equation • Momentum equation （ three components ） ∂ du 1 p + = . dif u ρ ∂ d t x ∂ : partial derivative symbol ∂ dv 1 p + = dif . v ρ ∂ d t y ∂ dw 1 p + + = g dif . w ρ ∂ d t z ・ Newton’s law of motion: （ Force ） = （ mass × acceleration ） → Navier-Sokes’ equation for fluid ： （ acceleration ） = （ pressure gradient force per unit mass ） ( +diffusion+gravity force for vertical direction ）
Momentum equation • Momentum equation （ three components ） ∂ du 1 p + = . dif u ρ ∂ d t x ∂ : partial derivative symbol ∂ dv 1 p + = dif . v ρ ∂ d t y ∂ dw 1 p + + = g dif . w ρ ∂ d t z ・ Nwton’s law of motion: （ Force ） = （ mass × acceleration ） → Navier-Sokes’ equation for fluid ： （ acceleration ） = （ pressure gradient force per unit mass ） ( +diffusion+gravity acceleration for vertical direction ）
地軸 Centrifugal force Universal gravitation Gravity force Rotating spheroid
Hydrostatic equilibrium In case the aspect ratio of the atmospheric motion is much smaller than unity, the equation for vertical motion low pressure ∂ dw 1 p + + = g dif . w ρ ∂ Pressure gradient force d t z Isobaric plain acceleration Gravity force can be replaced by hydrostatic equilibrium ∂ 1 p + g = high pressure 0 ρ ∂ z Vertical pressure gradient This relation equilibrium (balance) of force usually balances with forces between vertical pressure the gravity force (hydrostatic gradient force and gravity force. equilibrium)
Coriolis’ force Effect of Coriolis’ force is added on the earth Northward motion shifts eastward in Northern North Pole hemisphere due to the difference High latitude of speeds of earth rotation. Earth rotation Low latitude Earth rotation In vector formulation, Coriolis’ force is given by vector product of angular velocity of the earth rotation vector Ω and wind vector V : d V 1 = − Ω × − ∇ + + ( 2 V ) p g F , ρ dt
Continuity equation Continuity equation （ law of mass preservation ） ∂ρ ∂ρ ∂ρ ∂ρ u v w + + + = 0 ∂ ∂ ∂ ∂ t x y z ‥ local time tendency of density=differences of mass flux through surrounding boundaries Mass flux (density x wind speed)
From the following relationship, ∂ρ ∂ρ ∂ρ ∂ρ ρ ∂ ∂ u v w + + + + ρ + + = u v w ( ) 0 ∂ ∂ ∂ ∂ ∂ ∂ ∂ t x y z x y z we obtain the continuity equation in advective form: ρ ∂ ∂ ∂ 1 d u v w + + + = 0 ρ ∂ ∂ ∂ d t x y z .. flow dependent time tendency of density is given by local convergence
State equation for ideal gasses Boyle-Charles’s combined law for ideal gas with a molecular weight of m R T = ρ * = M R * p T m m V R* ： universal gas constant (=8.314J/mol/K) In case of dry air (represented by subscript d ), by Dalton’s law for partial pressure, ∑ M i * M * ∑ R ∑ R = = = = ρ i i p p T T RT d i d V m V m i i i d i is index to represent gas component such as nitrogen, oxygen, and argon, and m d the weigh-average molecular of dry air (28.966 g/mol) ∑ M i = i m d M ∑ i m i i R (= R*/m d ) ： gas constant for dry air (=287.05 J/Kg/K)
Diagnostic equation for density State equation for dry air = ρ p RT Using the specific volume α (inverse of density ρ) α = p RT If we define the non-dimensional pressure (Exner function) π and the potential temperature θ p R / C T π = θ = ( ) p , π p 0 （ p 0 =1000hPa, C p is the specific heat of dry air in constant pressure; 7 R /2=1004.7J/Kg/K ） p p C / C ρ = 0 v p ( ) θ R p 0 where C v is the specific heat of dry air in constant volume; C v = C p –R =5 R /2=717.6J/Kg/K
Pressure-height equation for hydrostatic atmosphere Applying the state equation to hydrostatic equilibrium low pressure Pressure gradient force 1 dp Isobaric plain + g = 0 acceleration ρ d z Gravity force 1 dp g high pressure = − → p d z RT Veridical pressure gradient force usually d g = − → (log p ) balances with the gravity d z RT force (hydrostatic equilibrium) gz − = → RT p p e m 0 We obtain well-known barometric height formula (pressure-height equation)
Thermodynamic equation （ Conservation of potential temperature ） First law of thermodynamics = + α dQ dI pd ‥ change in the internal energy of a closed system is equal to the amount of heat supplied to the system minus the amount of work done by the system on its surroundings Let non-adiabatic heating rate Q , = + α = + − α Qdt C dT pd ( C R ) dT dp v v = π θ C d P θπ C p R α = α = π = θ π R p RT , dp d ( p ) C d 0 p p Thus, θ d Q = π d t C P ‥ Conservation law (prognostic equation) of potential temperature
State equation for moist air Partial pressure of moist air M M M m R * R = + = + = + d v v d p p p ( ) T ( M ) T d v d m m V m V d v v R = = + ρ + ρ ( M 1 . 61 M ) T ( 1 . 61 ) RT d v d v V = = ρ + ρ ρ + = ρ ( 0 . 61 ) RT ( 1 0 . 61 q ) RT RT v a v a v a ρ a ： density of moist air, T v ： virtual temperature, q v : Specific humidity Virtual potential temperature is defined by replacing temperature by virtual temperature in definition of potential temperature + T ( 1 0 . 61 q ) T θ ≡ = = + θ v v ( 1 0 . 61 q ) π π v v
Thermodynamic equation for moist air Specific heat of water vapor in constant pressure C pv =1854J/Kg/K Specific heat of water vapor in constant volume C vv = C pv –R*/m v =1390J/Kg/K First law of thermodynamics for moist air is given by + ( C rC ) = ≅ + p pv dQ dT C ( 1 0 . 85 q ) dT + p v 1 r where r is the mixing ratio. Likewise, specific heat of moist air in constant volume is + ( C rC ) = ≅ + v vv C C ( 1 0 . 94 q ) + vv v v 1 r Potential temperature of moist air may be modified as + R ( 1 0 . 61 q ) R v − ( 1 0 . 24 q ) + v p R / C p p C ( 1 0 . 85 q ) C θ = = ≅ m pm v ( ) ( ) p ( ) p T T T 0 0 0 moist p p p The difference between θ and θ moist is less than 0.1K, and can be ignored as p p C / C ρ = 0 v p ( ) θ a R p v 0
Flux form equation Transport of water vapor ∂ ∂ ∂ ∂ q q q q + + + = u v w M ∂ ∂ ∂ ∂ t x y z ‥ local tie tendency of q =advection + moisture source Combining with the continuity equation, ∂ρ ∂ρ ∂ρ ∂ρ u v w + + + = 0 ∂ ∂ ∂ ∂ t x y z We obtain the following flux form equation, ∂ρ ∂ρ ∂ρ ∂ρ q uq vq wq + + + = ρ M ∂ ∂ ∂ ∂ t x y z
Perturbation of the density From state equation, perturbation of density can be divided into the following two terms: C v p p p ρ = = C 0 p ' ( ) ' { ( ) } ' θπ θ R R p 0 C C − v v θ 1 p p ' p C p p ' = C − + C 0 0 v p p ( ) ( ) ( ) θ θ 2 R p R C p p 0 p 0 0 θ θ C ' p ' ' p ' = − ρ + ρ = − ρ + v ( A . 11 ) θ θ 2 C p C p S where C S = p C RT ( A . 12 ) C v Since potential temperature is invariant for total derivation , ' = ρ dp d 2 C S dt dt
2. Waves in mesoscale atmosphere Starting from the following 2-dimensional basic equations, we derivate three wave solutions in mesoscale atmosphere. ∂ 1 du p = − ( 1 ) ρ ∂ dt x ∂ dw 1 p = − − g ( 2 ) ρ ∂ dt z ∂ρ ∂ρ ∂ρ u w + + = 0 ( 3 ) ∂ ∂ ∂ t x z θ d = 0 ( 4 ) dt
1) Sound waves If we linearize the equation (1)-(3) by u=u’, w=w’, θ= θ + θ ’ , ρ=ρ 0 + ρ ‘ The system becomes ∂ ∂ ' u 1 p ' + = ( A . 1 ' ' ) 0 ∂ ρ ∂ x t 0 ∂ ∂ w ' p ' 1 + = 0 ( A . 2 ' ' ) ∂ ρ ∂ t z 0 ∂ρ ∂ ∂ ' u ' w ' + ρ + = ( ) 0 ( A . 3 ' ' ) ∂ ∂ ∂ 0 t x z Time tendency of the density is replaced by the pressure tendency as ∂ ∂ ∂ p ' u ' w ' + ρ + = 2 C ( ) 0 ( A . 13 ) ∂ ∂ ∂ S 0 t x z
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