Multiphasic equations for a NWP system

Transcription

1 Mutiphasic equations for a NWP system CNRM/GMME/Méso-NH 21 novembre / 37

2 Contents Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice 2 / 37

3 Resoved versus parametrised processes AROME isotropic turbuence dry thermics non precipitating shaow convection NH dynamics + expicit microphysics «arge scae forcing» (couping) sub grid processes expicit deep convection «stratiform» couds and precipitations scaes ALADIN «arge scae forcing» isotropic turbuence dry thermics shaow convection deep convection gravity wave drag «stratiform» couds and precipitations ARPEGE 3 / 37

4 Couds in NWP modes Large scae physics For «arge scae» mode, we usuay suppose(d) that a diagnostic representation of condensates is enough (no condensates in the air state forcasted by the mode). We use diagnostic formuation from the mean water vapor content when needed (radiation, output) As the condensates (coud and rain) are not pronostic, they are not directy known by the dynamics : no advection, no inertia, no weight of condensates 4 / 37

5 Couds in NWP modes Coud scae physics In order to sove expicity couds formation and ife cyce, we need a NH-dynamics, but aso a finer description of microphysics processes. Expicit (pronostic) evoution equations for condensates (mutiphasic system) Interaction between condensates and the dynamics (advection, weight of condensed phases) Reaistic sedimentation (precipitation) with «finite» faing veocity (precipitating species are a componant of an air parce) Detaied microphysics with a compete phase transition description (see J. P. Pinty s tak) 5 / 37

6 Large scae physics / coud scae physics in practice Water phase changes in a «arge scae» physics equations for gaseous species ony cond/evap with the pseudo-adiabatic hypothesis (no consensed phase in the atmosphere, precipitations are known ony through their consequences on T and q v ) ρ v = P Water phase changes in a coud scae physics equations for gases and condensates parametrisation of detaied microphysica processes (but there may sti be probem with unresoved couds) ρ v = (ρ c + ρ r ) + P 6 / 37

7 Large scae physics / coud scae physics Our worry To have a consistant set of equations with hypotheses and approximations «under contro» and vaid for pseudo-adiabatic or mutiphasic, (δm = 0 or δm = 1), hydrostatic or non-hydrostatic NWP mode. Warnings It is a «burning» subject with recent references ony 7 / 37

8 The mutiphasic system of Arome instant t instant t+dt dry air (0) rain (3) vapor (1) coud (2) snow (5) ice (4) graupe (6) 8 / 37

9 Questions Variabes in a mutiphasic atmospheric parce Which ρ? Which p and T? Which wind? What veocities shoud we manipuate? Advection of what by what? How can we treat condensates? As individuas wih their own evoution aws? As continuous components of a mixture? 9 / 37

10 The oca scae (our scae of the continuum) A continuum of dropets and drops ρ c = m c V gaz ρ = k ρ r = m r V gaz ρ k Loca equiibrium hypothesis T eq = T c = T r = T gaz v eq = v c = v r = v gaz 10 / 37

11 Barycentric parameter at the oca scae Barycentric formuation ρψ = ρ ψ or ψ = q ψ oca departure : ψ = ψ ψ Consequences of equiibrium hypotheses v = q v = ( q ) v eq = v eq h = ( ) ( ) q h == q h + q c p T = h + c p T with c p = q c p 11 / 37

12 Averaging from the oca scae to the mode scae Voumic averaging ψ = 1 V with ψ = ψ ψ and ψ = 0 V ψdv Mass-weighted averaging ψ = 1 m with ψ = ψ ψ and ψ = 0 m ψdm 12 / 37

13 Averaging in a mutiphasic system Averaging are weighted by the voume or the mass of the mixture Voumic averaging ψ = 1 V ψ = 1 V V V ψdv ψ dv 13 / 37

14 Averaging in a mutiphasic system Mass-weighted averaging of a barycentric parameter ψ = 1 ψdm = 1 ρψdv m m m V Mass-weighted averaging of a parameter for the species ψ = 1 ψ dm = 1 ρψ dv m m m V with ψ = ψ + ψ but and ψ = ψ + ψ ψ = ψ ψ = ψ 0 14 / 37

15 Exampe instant t instant t+dt Ĝ ˆ w a ŵ ˆ w r 15 / 37

16 at oca scae An equation for perfect gaz ony p = p a + e = ρ a R a T + ρ v R v T or p = ρr h T with R h = q a R a + q v Rv and q a = ρ a /ρ, q v = ρ v /ρ 16 / 37

17 at the mode scae Mean state equations for dry air and water vapor Hypothesis p a = ρ q a R a T + Ra ρq a T e = ρ q v R v T + Rv ρq v T To conserve the shape of the gaz state aw, we have to negect non-inear terms p = p a + e = ρ(r a q a + R v q v ) T 17 / 37

18 Tota mass conservation Budget of mass in any geometric voum V m = ( V ρdv) = (ρ u. n)ds + S V ρ dv No mass source at the oca scae ρ = 0 18 / 37

19 Continuity equation Euerian form of the oca mutiphasic continuity equation ρ = div(ρ u) Euerian form of the avegarge mutiphasic continuity equation ρ = div(ρ u) = div(ρ u) 19 / 37

20 An aternative : dry air conservation (Bannon, 2002) Budget of dry air in a geometric voum V m a = ( V ρ adv ) = ρ a u a. nds + No dry air source ρ a = 0 Euerian form of the dry air continuity equation S V ρ a dv ρ a = div(ρ a u a ) 20 / 37

21 Budget equation for any oca variabe ψ Budget for any geometric voume V V (ρ ψ ) dv = S (ρ ψ u. n) ds + V Ṡ dv Loca form of the budget equation (ρψ) = div[ (ρ ψ u )] + Ṡ 21 / 37

22 Advection term Barycentric advection ρ (ψ) + ρ u. grad(ψ) = [ (ρ ψ w )] + }{{ } (+) with ψ = q ψ and (ρ w ) = 0 Aternative : Dry air advection (Bannon, 2002) ρ a (ψ) + ρ a u a. grad(ψ) = [ (ρ ψ w )] + }{{ } (+) with ψ = r ψ but (ρ w ) 0 Ṡ Ṡ 22 / 37

23 Turbuent / organised diffusive terms Budget equation for any modee scae variabe (ρ b ψ) = div(ρ ψ u) div(ρψ u ) [P ρbq ψ b b w ] [P bq ρψ w ] [P bψ ρq w ] [P b w ρq ψ ] Ṡ + [P ρq ψ w ] Simpified form ( ψ) ρ + ρ u. grad( ψ) }{{} advection = div(ρψ u ) }{{} turbuent [ ρ q ψ w ] } {{ } organised + Ṡ 23 / 37

24 Mass budget for species Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Mass of variations in any voume V m = ρ u. nds + S V ρ dv with m = ρ V = V ρ dv = V ρq dv and ρ the source/sink 24 / 37

25 Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Mode scae equation for q Euerian form ρ q Lagrangian form ) = div (ρ q u (ρ q w ) ( ) ρq w + ρ ρ D q Dt ( q = ρ ) ( ) ) + u. (ρ q w ρq grad( q w ) = + ρ 25 / 37

26 Barycentric momentum budget Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Variations of one componant of the barycentric momentum for any voume V ( V ( ρ u α ) dv ) ( ) = ρ u α u. nds S ( ) + ρ g α dv + V V ( ) ρ [2Ω u ] α dv + [ τ s. n] α ds S 26 / 37

27 The stess tensor Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Hypothesis on the stress tensor We suppose that τ s may be written as the sum of two tensors : τ s = p δ + σ with p the tota pressure of the gas and σ the viscous stress tensor («moecuar diffusion»). 27 / 37

28 Barycentric momentum budget Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Variations of one componant of the barycentric momentum for any voume V [ (ρu α ) V ] dv = + + V V V V ( ) div ρ u α u dv (ρg α ) dv ( ρ[2ω u] α) dv [ grad(p)] α dv + V div[ σ α s ]dv 28 / 37

29 Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Barycentric zona momentum equation at mode scae Euerian budget (ρû) ( ) ) = div ρû u div (ρu u +2ρΩ sin(ϕ) v p x +div(σ u) «Lagrangian» form ρ Dû Dt = ρ[ û + u. grad(û)] ) = div (ρ u u + 2ρΩ sin(ϕ) v p x + div(σ u) 29 / 37

30 Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Barycentric NH vertica momentum equation at mode scae «Lagrangian» form ρ (ŵ) ) + ρ u. grad(ŵ) = div (ρw u ( ) ρ w 2 ρg p + div(σ w) 30 / 37

31 Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Barycentric NH vertica momentum equation at mode scae The second term on the right hand side of the ast equation may be decomposed in : ( ) ( ) ρ w 2 ρq w 2 = ( ρ q w 2 = 2 ( ρq ( ρq ) w 2 ) w w 2 ) ( ρ q w ) w ( ρq ( w ) 2) 31 / 37

32 Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Barycentric NH vertica momentum budget We keep ony the first term of this deveopment. The average NH equation for the vertica veocity is then finay : ρ (ŵ) ) + ρ u. grad(ŵ) = div (ρw u ) ( ρ q w 2 ρg p + div(σ w) 32 / 37

33 Tota energy budget Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Method The cassica method to deduce the thermodynamic equation is to substract the equation for the kinetic energy from the equation for the tota energy (kinetic energy + interna energy). In practice do this operation at the oca scae appy the average operator on the resuting thermodynamic equation if we negect some non inear terms, the exact form of the mean thermodynamics equation may depend on the form of the oca equation to which the averaging is appied. 33 / 37

34 Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Hydrostatic thermodynamic equations at mode scae Negecting a the term with a perturbation of c p or a perturbation of w k and the presso-correations ρĉ p T ) + div (ρĉ p T u = p + u. gradp +div(ĉ p ρt u ) [ )] k (ρc pk q k T w k +ɛ + div( J Q ) +L v (T = 0) ρ + L i (T = 0) ρ i 34 / 37

35 Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Hydrostatic temperature equation at mode scae ρĉ p D T Dt = Dp Dt +div(ĉ p ρt u ) k ( +ɛ + div( J Q ) ρ k w k c pk T +L v (T ) ρ + L i (T ) ρ i ) 35 / 37

36 NH tota energy budget at oca scae Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice [ (ei + ẽ c ) ρ ] + u. grad(e i + ẽ c ) = ( k (ρ kei k + p k ) w k ) ρẽ c (w) pdiv( u) + ɛ + div( J Q ) 36 / 37

37 Practica appications for Arome Individua mass species equations Momentum equation Thermodynamics equations Mutiphasics systems in practice Tendency and diagnostic (budget) interface Because of the «finite» faing veocity of the precipitation, the current budget equations used in the Arpege/Aadin physics to compute the physica tendencies from fuxes and in the diagnostics (DDH) is not vaid. A more generaised set of equations/interface as to be coded (B. Catry et a) and a nice set of fuxes have to be recovered from the microphysic processes (T. Kovavic et a). Barycentric departure fuxes The «new» terms («barycentric departure fuxes») in the equations shoud be coded and anaysed (computationa cost compared to order of magnitude). 37 / 37