Internal wave dynamics in the atmosphere Rupert Klein Mathematik - PowerPoint PPT Presentation

Internal wave dynamics in the atmosphere Rupert Klein Mathematik & Informatik, Freie Universit¨ at Berlin CEMRACS 2019 “Geophysical Fluids, Gravity Flows” CIRM, Luminy, July 17, 2019

Thanks to ... Ulrich Achatz (Goethe-Universit¨ at, Frankfurt) Didier Bresch (Universit´ e de Savoie, Chamb´ ery) Omar Knio (KAUST, Saudi Arabia) Fabian Senf (IAP, K¨ uhlungsborn) Piotr Smolarkiewicz (ECMWF, Reading, UK) Olivier Pauluis (Courant Institute, NYU, New York) Martin G¨ otze (formerly FU-Berlin) Dennis Jentsch (formerly FU-Berlin) MetStröm CRC 1114 Scaling Cascades in Complex Systems

Scale-dependent models for atmospheric motions Background on sound-proof models Formal asymptotic regime of validity Steps towards a rigorous proof Summary

Scale-Dependent Models 10 km / 20 min Changes in temperature latitude 1000 km / 2 days Winter (DJF) 10000 km / 1 season

Scale-Dependent Models u t + u · ∇ u + w u z + ∇ π = S u w t + u · ∇ w + ww z + π z = − θ ′ + S w θ ′ t + u · ∇ θ ′ + wθ ′ z = S ′ θ ∇ · ( ρ 0 u ) + ( ρ 0 w ) z = 0 θ = 1 + ε 4 θ ′ ( x , z, t ) + o ( ε 4 ) ( ∂ τ + u (0) · ∇ ) q = 0 Anelastic Boussinesque Model � � ρ (0) q = ζ (0) + Ω 0 βη + Ω 0 ∂ d Θ /dz θ (3) ρ (0) ∂z 10 km / 20 min θ (3) = − ∂π (3) u (0) = 1 ζ (0) = ∇ 2 π (3) , k × ∇ π (3) ∂z , Ω 0 ∂Q T ∂t + ∇ · F T = S T ∂Q q ∂t + ∇ · F q = S q Quasi-geostrophic theory � � � H a � H a � ( u ′ ϕ ′ ) + D ϕ � � Q ϕ = ρ ϕ dz , F ϕ = ρ u ϕ + dz , ϕ ∈ { T, q } z s z s � � � � − z − z s T = T s ( t, x ) + Γ( t, x ) min( z, H T ) − z s , q = q s ( t, x ) exp H q 1000 km / 2 days � � � � � z − z − γz T dz ′ ρ = ρ ∗ exp , p = p ∗ exp + p 0 ( t, x ) + gρ ∗ h sc h sc T ∗ 0 u = u g + u a , fρ ∗ k × u g = −∇ x p u α = α ∇ p 0 V. Petoukhov et al., CLIMBER-2 ... , Climate Dynamics, 16, (2000) EMIC - equations (CLIMBER-2) 10000 km / 1 season

Scale-Dependent Models ∼ 6 · 10 6 m Earth’s radius a 10 − 4 s − 1 Earth’s rotation rate Ω ∼ 9 . 81 ms − 2 Acceleration of gravity g ∼ 10 5 kgm − 1 s − 2 Sea level pressure p ref ∼ H 2 O freezing temperature T ref ∼ 273 K L q vs ∼ 4 · 10 4 J kg − 1 K − 1 Latent heat of water vapor 287 m 2 s − 2 K − 1 Dry gas constant R ∼ Dry isentropic exponent γ ∼ 1 . 4 Distinguished limit: RT ref p ref h sc ∼ 1 . 6 · 10 − 3 ∼ ε 3 Π 1 = h sc = = ∼ 8 . 5 km g ρ ref g a � � L q vs ∼ 1 . 5 · 10 − 1 ∼ c ref = RT ref = gh sc ∼ 300 m / s where Π 2 = ε c p T ref γR ∼ 4 . 7 · 10 − 1 ∼ √ ε c ref c p = Π 3 = γ − 1 Ω a

Scale-Dependent Model Hierarchy Classical length scales and dimensionless numbers L mes = ε − 1 h sc Fr int ∼ ε L syn = ε − 2 h sc ∼ ε − 1 Ro h sc Ro L Ro ∼ ε L Ob = ε − 5 / 2 h sc ∼ ε 3 / 2 = ε − 3 h sc Ma L p Remark: There aren’t the low Mach number limit equations. Asymptotic results depend on the adopted distinguished limit and scalings of length, time and initial data !

Scale-Dependent Models Compressible flow equations with general source terms � ∂ � ∂t + � v · � ∇ + w ∂ � v + ε � (2 Ω × v ) + 1 ε 3 ρ ∇ || p = S � v , ∂z � ∂ � ∂t + � v · � ∇ + w ∂ + ε (2 Ω × v ) ⊥ + 1 ∂p = S w − 1 w ε 3 , ε 3 ρ ∂z ∂z � ∂ � ∂t + � v · � ∇ + w ∂ ρ + ρ ∇ · v = 0 , ∂z � ∂ � ∂t + � v · � ∇ + w ∂ = S Θ Θ ∂z Θ = p 1 /γ ρ Asymptotic single-scale ansatz m � � � φ i ( ε ) U ( i ) ( t, x , z ; ε ) + O U ( t, x , z ; ε ) = φ m ( ε ) i =0

Scale-Dependent Models Recovered classical single-scale models: U ( i ) = U ( i ) ( t ε , x , z Linear small scale internal gravity waves ε ) U ( i ) = U ( i ) ( t, x , z ) Anelastic & pseudo-incompressible models U ( i ) = U ( i ) ( ε t, ε 2 x , z ) Linear large scale internal gravity waves U ( i ) = U ( i ) ( ε 2 t, ε 2 x , z ) Mid-latitude Quasi-Geostrophic Flow U ( i ) = U ( i ) ( ε 2 t, ε 2 x , z ) Equatorial Weak Temperature Gradients U ( i ) = U ( i ) ( ε 2 t, ε − 1 ξ ( ε 2 x ) , z ) Semi-geostrophic flow U ( i ) = U ( i ) ( ε 3 / 2 t, ε 5 / 2 x, ε 5 / 2 y, z ) Kelvin, Yanai, Rossby, and gravity Waves These all share one distinguished limit ⇒ Starting point for multiscale analyses!

Scale-Dependent Models [ h sc / u ref ] 1/ 3 PG 1/ 5/2 1/ 2 QG inertial waves HPE WTG 1/ +Coriolis anelastic / pseudo-incompressible +Coriolis internal waves WTG 1 HPE Boussi- acoustic waves nesq Obukhov advection scale 1/ 5/2 1/ 2 1/ 3 1 1/ [ h sc ] bulk convective meso synoptic planetary micro R.K., Ann. Rev. Fluid Mech., 42 , 2010

What about the puzzle? Compressible flow equations distinguished limit Fr int ∼ ε ∼ ε − 1 Ro h sc D � v ε (2 Ω × v ) � + 1 + ε 3 ρ ∇ || p = 0 , Dt Ro L Ro ∼ ε Dw ε (2 Ω × v ) ⊥ + 1 ∂z = − 1 ∂p ∼ ε 3 / 2 Ma ε 3 , Dt + ε 3 ρ � ∂ � length / time scalings ∂t + v � · ∇ � + w ∂ x ′ ρ + ρ ∇ · v = 0 , x = ∂z h sc � ∂ � ∂t + v � · ∇ � + w ∂ z ′ Θ = 0 z = ∂z h sc t ′ Θ = p 1 /γ t = h sc /u ref ρ

One possible solution Leading orders (2) , (5) ⇒ ∇ || ρ = ∇ || Θ = 0 (6) ∇ || p = 0 (1) ∂ z p = − ρ (2) (4) & Θ = const ⇒ (4) (7) ρ t + ∇ · ( ρ v ) = 0 (3) D Θ (3) ⇒ ∇ · ( ρ v ) = 0 (8) Dt = 0 (4) Θ = p 1 /γ ρ . (5) ⇓ Anelastic & pseudo-incompressible ∗ models � ∂ � D ∂t + v � · ∇ � + w ∂ (key aspect: weak stratification) Dt = ∂z ∗ also called “soundproof models”

Scale-dependent models for atmospheric motions Background on sound-proof models Formal asymptotic regime of validity Steps towards a rigorous proof Summary

Motivation ... Numerics Why not simply solve the full compressible flow equations? (a) = 10 min (b) = 10 sec 2 2 10 10 external external Lz=80km Lz=80km Lz=8km Lz=8km 0 0 Lz=800m Lz=800m 10 10 wave fequency [s 1 ] Lz=80m wave fequency [s 1 ] Lz=80m 2 2 10 10 4 4 10 10 marked: regularized marked: regularized unmarked: exact dispersion unmarked: exact dispersion 6 6 10 10 0 2 4 0 2 4 10 10 10 10 10 10 horizontal length scale [km] horizontal length scale [km] Dispersion relations for acoustic, Lamb, and internal waves From: Hundertmark & Reich, Q.J.R. Meteorol. Soc. 133 , 1575–1587 (2007)

Motivation ... Numerics Why not simply solve the full compressible equations? Linear Acoustics, simple wave initial data, periodic domain (integration: implicit midpoint rule, staggered grid, 512 grid pts., CFL = 10 ) 1 1 0.5 0.5 t = 0 p 0 p 0 -0.5 -0.5 -1 -1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 x x 1 1 0.5 0.5 t = 3 p 0 p 0 -0.5 -0.5 -1 -1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 x x Ideas: “super-implicit” scheme Slave short waves ( c ∆ t/ℓ > 1 ) to long waves ( c ∆ t/ℓ ≤ 1 ) non-standard multi grid with pseudo-incompressible limit behavior projection method ∗ see, e.g., Reich et al. (2007)

Motivation ... Numerics Central questions: How to characerize a fully compressible flow at sub-acoustic time scales? What should be the “required” limit behaviour of a numerical flow solver? The answers depend on the scaling regimes considered!

Scaling regimes z Stratosphere h ~10 km Troposphere sc θ T ref

Scaling regimes L ~ h sc l << L ~ hsc L << h sc anelastic & Boussinesq pseudo-incompressible psinc + WKB R.K., TCFD, 2009; R.K. et al., JAS, 2010; Achatz et al., JFM, 2010

Sound-Proof Models Compressible flow equations L ∼ h sc ρ t + ∇ · ( ρ v ) = 0 drop term for: ( ρ u ) t + ∇ · ( ρ v ◦ u ) + P ∇ � π = 0 anelastic † (approx.) ( ρw ) t + ∇ · ( ρ v w ) + Pπ z = − ρg pseudo-incompressible ∗ P t + ∇ · ( P v ) = 0 1 γ = ρθ , P = p π = p/ Γ P , Γ = c p /R , v = u + w k ( u · k ≡ 0) Parameter range & length and time scales of asymptotic validity ? † e.g. Lipps & Hemler, JAS, 29 , 2192–2210 (1982) ∗ Durran, JAS, 46 , 1453–1461 (1989)

Scale-dependent models for atmospheric motions Background on sound-proof models Formal asymptotic regime of validity Steps towards a rigorous proof Summary

From here on: ε is the Mach number

Regimes of Validity ... Design Regime Characteristic inverse time scales dimensional dimensionless u ref advection : 1 h sc � � � √ gh sc g dθ h sc dz = 1 dθ h sc dθ internal waves : N = dz u ref dz ε θ θ θ � √ gh sc √ gh sc p ref /ρ ref = 1 sound : = h sc h sc u ref ε Ogura & Phillips’ regime ∗ with two time scales � h sc dθ � h sc θ = 1 + ε 2 � dz = O ( ε 2 ) θ ( z ) + . . . ⇒ ⇒ ∆ θ z =0 < 1 K θ

Regimes of Validity ... Design Regime Characteristic inverse time scales dimensional dimensionless u ref advection : 1 h sc � � � √ gh sc d � g dθ h sc dθ dz = 1 h sc θ internal waves : N = dz u ref dz ε θ θ θ � √ gh sc √ gh sc p ref /ρ ref = 1 sound : = h sc h sc u ref ε Ogura & Phillips’ regime ∗ with two time scales � h sc dθ � h sc θ = 1 + ε 2 � dz = O ( ε 2 ) θ ( z ) + . . . ⇒ ⇒ ∆ θ z =0 < 1 K θ ∗ Ogura & Phillips (1962)