Scales in geophysical flows Rupert Klein Mathematik & - PowerPoint PPT Presentation

Scales in geophysical flows Rupert Klein Mathematik & Informatik, Freie Universit¨ at Berlin CEMRACS 2019 “Geophysical Fluids, Gravity Flows” CIRM, Luminy, July 16, 2019

Motivation Scale analysis & distinguished limits Model hierarchy for atmospheric flows A puzzle R.K., Ann. Rev. Fluid Mech., 42 , 249–274 (2010)

Scale-Dependent Models 10 km / 20 min Changes in temperature latitude 1000 km / 2 days Winter (DJF) 10000 km / 1 season Thanks to: P.K. Taylor, Southampton Oceanogr. Inst.; P. N´ evir, Freie Universit¨ at Berlin; S. Rahmstorf, PIK, Potsdam

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 Compressible flow equations with general source terms � ∂ ∂t + � v · � ∇ + w ∂ � v � + � (2 Ω × v ) + 1 ρ ∇ || p = S v � , ∂z � ∂ ∂t + � v · � ∇ + w ∂ � w + (2 Ω × v ) ⊥ + 1 ∂p = S w − g , ∂z ρ ∂z � ∂ � ∂t + � v · � ∇ + w ∂ ρ + ρ ∇ · v = 0 , ∂z � ∂ � ∂t + � v · � ∇ + w ∂ Θ = S Θ , ∂z � p � R/c p = ρ Θ . p ref ρ ref T ref How do all the simplified models relate to this system?

Motivation Scale analysis & distinguished limits Model hierarchy for atmospheric flows A puzzle

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 Dimensionless parameters: RT ref p ref h sc ∼ 1 . 6 · 10 − 3 ∼ ε 3 h sc = ∼ 8 . 5 km Π 1 = = 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 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

F F = − cx m ¨ x distinguished limits for the harmonic oscillator F D ( t ) F R = − k ˙ x (D¨ amon)

m ¨ x + k ˙ x + cx = F 0 cos(Ω t ) x (0) = x 0 , x (0) = ˙ ˙ x 0 F F = − cx ε = m Ω 2 ≪ 1 m ¨ x c k Ω δ = ≪ 1 c F D ( t ) F R = − k ˙ x (D¨ amon) cx 0 = O (1) F 0 c ˙ x 0 = ? Ω F 0

F F = − cx Dimensionless representation m ¨ x ( t ) = F 0 x c y ( τ ) , τ = Ω t then F D ( t ) ε y ′′ + δ y ′ + y = cos( τ ) F R = − k ˙ x (D¨ amon) Is there a unique limit solution for ε = δ = 0 ?

m x’’ + k x’ + c x = F 0 * cos( Ω t), Exact Solution m x’’ + k x’ + c x = F 0 * cos( Ω t), Exact Solution 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 x [m] x[m] 0 0 0.2 0.2 0.4 0.4 0.6 0.6 reference solution with: m = k = 0 reference solution with: m = k = 0 0.8 0.8 x(t) with: k = 1; c = 25; m = 0.01; F0 = 0 x(t) with: k = 0.01; c = 25; m = 1; F0 = 0 1 1 0 1 2 3 4 5 6 0 10 20 30 40 50 60 time [s] time [s] ε = 0 . 0004 ε = 0 . 04 δ = 0 . 04 δ = 0 . 0004 The limit is path-dependent!

m x’’ + k x’ + c x = F 0 * cos( Ω t), Exact Solution m x’’ + k x’ + c x = F 0 * cos( Ω t), Exact Solution 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 x [m] x[m] 0 0 0.2 0.2 0.4 0.4 0.6 0.6 reference solution with: m = k = 0 reference solution with: m = k = 0 0.8 0.8 x(t) with: k = 1; c = 25; m = 0.01; F0 = 0 x(t) with: k = 0.01; c = 25; m = 1; F0 = 0 1 1 0 1 2 3 4 5 6 0 10 20 30 40 50 60 time [s] time [s] ε = 0 . 0004 ε = 0 . 04 δ = 0 . 04 δ = 0 . 0004 The limit is path-dependent!

δ II III I ε

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

Motivation Scale analysis & distinguished limits Model hierarchy for atmospheric flows A puzzle

Scale-Dependent Model Hierarchy Nondimensionalization ( x , z ) = 1 t = u ref ( x ′ , z ′ ) , t ′ h sc h sc � p ′ � , T ′ , ρ ′ RT ref ( u , w ) = 1 ( u ′ , w ′ ) , ( p, T, ρ ) = u ref p ref T ref p ref where u ref = 2 gh sc ∆Θ ( thermal wind scaling ) π Ω a T ref

Scale-Dependent Model Hierarchy 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 ∂ w + ε (2 Ω × v ) ⊥ + 1 ∂p = S w − 1 ε 3 , ε 3 ρ ∂z ∂z � ∂ � ∂t + v � · ∇ � + w ∂ ρ + ρ ∇ · v = 0 , ∂z � ∂ � ∂t + v � · ∇ � + w ∂ Θ = S Θ . ∂z

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 N 2 = g d Θ Example: the synoptic scale ∗ Θ dz � � √ gh sc L syn = Nh sc ∼ 1 g ∆Θ h sc = u ref ∆Θ h sc Ω Ω T ref h sc Ω h sc u ref T ref � 1 ∆Θ 2 = h sc h sc = h sc ε − 1 − 3 2 + 1 = Ro h sc ε 2 Ma T ref ∗ distance which an internal wave must travel until influenced at leading order by the Coriolis effect

Scale-Dependent Model Hierarchy Single-scale asymptotics m � � � φ i ( ε ) U ( i ) ( t, x , z ; ε ) + O U ( t, x , z ; ε ) = φ m ( ε ) i =0 Remark Generally, m < ∞ , and the series would not converge !

Scale-Dependent Model Hierarchy 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 ... and many more

Scale-Dependent Model Hierarchy [ 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 , 249–274 (2010)

Motivation Scale analysis & distinguished limits Model hierarchy for atmospheric flows A puzzle

Scale-Dependent Models Compressible flow equations without source terms D v � + ε (2 Ω × v ) � + 1 ε 3 ρ ∇ || p = 0 , Dt Dw + ε (2 Ω × v ) ⊥ + 1 ∂p = − 1 ε 3 , ε 3 ρ Dt ∂z Dρ + ρ ∇ · v = 0 , Dt D Θ = 0 . Dt where � ∂ � D ∂t + v � · ∇ � + w ∂ Dt = ∂z