Internal wave dynamics in the atmosphere take-home messages Rupert Klein Mathematik & Informatik, Freie Universit¨ at Berlin
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)
Regimes of Validity ... Design Regime Characteristic inverse time scales dimensional dimensionless u ref advection : 1 h sc � � � √ gh sc d � g dθ h sc dz = 1 dθ h sc θ internal waves : N = dz u ref ε ν dz θ θ θ � √ gh sc √ gh sc p ref /ρ ref = 1 sound : = h sc h sc u ref ε Realistic regime with three time scales h sc dθ θ = 1 + ε µ � dz = O ( ε µ ) θ ( z ) + . . . ⇒ ( ν = 1 − µ / 2) θ
Analysis of internal wave spectra � � + λ 2 λ 2 N 2 − d 1 1 dW 1 θ P W = θ P W 1 − ε µω 2 /λ 2 ω 2 dz dz θ P c 2 � � ω 2 /λ 2 Internal wave modes = O (1) c 2 • pseudo-incompressible modes/EVals = compressible modes/EVals + O ( ε µ ) † µ > 2 • phase errors remain small over advection time scales for 3 Anelastic and pseudo-incompressible models remain relevant for stratifications 1 dθ ∆ θ | h sc < dz < O ( ε 2 / 3 ) ⇒ ∼ 40 K 0 θ not merely up to O ( ε 2 ) as in Ogura-Phillips (1962)
ε y ′′ + δ y ′ + y = cos( τ ) 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 1 0.8 0.6 0.4 0.2 x[m] 0 0.2 0.4 0.6 reference solution with: m = k = 0 0.8 x(t) with: k = 1; c = 25; m = 0.01; F0 = 0 1 0 1 2 3 4 5 6 time [s] Matched asymptotic expansions ?
SFB 1114 Strongly tilted atmospheric Vortices Rupert Klein Mathematik & Informatik, Freie Universit¨ at Berlin CEMRACS 2019 “Geophysical Fluids, Gravity Flows” CIRM, Luminy, July 18, 2019
Thanks to ... Eileen P¨ aschke (Deutscher Wetterdienst, Lindenberg) Ariane Papke (formely FU-Berlin) Patrick Marschalik (Fritz Haber Institute, Berlin) Antony Owinoh ( † ) Tom D¨ orffel (FU-Berlin) Sabine Hittmeir (Univ. of Vienna) Piotr Smolarkiewicz (ECMWF, Reading) Boualem Khouider (Univ. of Victoria) Mike Montgomery ( Naval Postgraduate School, Monterey) Roger Smith (Ludwig-Maximilians Univ., M¨ unchen) MetStröm CRC 1114 Scaling Cascades in Complex Systems
Motivation Structure of atmospheric vortices I: two scales (P¨ aschke et al., JFM, (2012)) Structure of atmospheric vortices II: cascade of scales (D¨ orffel et al., arXiv:1708.07674) Conclusions R.K., Ann. Rev. Fluid Mech., 42 , 249–274 (2010)
Tropical easterly african waves http://www.aoml.noaa.gov/hrd/tcfaq/A4.html
Developing tropical storm (streamlines in co-moving frame and Okubo-Weiss-parameter (color)) T T T T T T Ro = | v | fL ∼ 1 10 Dunkerton et al., Atmos. Chem. Phys., 9 , 5587–5646 (2009)
Photo: Hurricane Rita from https://commons.wikimedia.org/wiki/File:HurricaneRita21Sept05a.jpg Developed hurricane R ∗ mw ≈ 50 . . . 200 km u θ ≈ 30 . . . 60 m / s R mw : radius of max. wind Hurricane ”Rita“ Ro = u θ, max ∼ 10 fR mw
Ensemble of Simulations of “Joaquin”-like Storms Ensemble Tracks Vortex Tilts Storm Evolutions Gh. Alaka et al. (2019), WAF, submitted
Motivation Structure of atmospheric vortices I: two scales (P¨ aschke et al., JFM, (2012)) Structure of atmospheric vortices II: cascade of scales (D¨ orffel et al., arXiv:1708.07674) Conclusions
File:HurricaneRita21Sept05a.jpg https://commons.wikimedia.org/wiki/ Photo: Hurricane Rita from Radial momentum balance regimes � � − 1 ∂p ∂r + fu θ = O 1 geostrophic Ro ≪ 1 typical “weather” ρ u 2 � � − 1 ∂p tropical storm θ ∂r + fu θ = O 1 gradient wind Ro = O (1) incipient hurricane r ρ � � u 2 − 1 ∂p θ + fu θ = O 1 cyclostrophic Ro ≫ 1 hurricane r ρ ∂r P¨ aschke, Marschalik, Owinoh, K., JFM, 701 , 137–170 (2012) D¨ orffel et al., preprint, arXiv:1708.07674 (2017)
Tropical easterly african waves http://www.aoml.noaa.gov/hrd/tcfaq/A4.html
Vortex tilt in the incipient hurricane stage (Velocity potential) 200 hPa ( ∼ 12 km) 925 hPa ( ∼ 0.8 km) ∼ 200 km Dunkerton et al., Atmos. Chem. Phys., 9 , 5587–5646 (2009)
Photo: Hurricane Rita from https://commons.wikimedia.org/wiki/File:HurricaneRita21Sept05a.jpg Scaling regime for matched asymptotic expansions L mes z h sc y X ( t , z ) k j centreline i L syn x L mes = √ ε L syn ; v mes = 1 √ ε L syn ; | v � | ∼ v syn t syn ∼ L syn /v syn � �� � � �� � farfield: classical QG theory core: gradient wind scaling C ∼ ( vL ) syn ; Ro syn ∼ v syn C ∼ ( vL ) mes ; Ro mes ∼ v mes = O ( ε ) = O (1) fL syn fL mes Comparable levels of circulation C !
Vortex motion ⇒ precessing quasi-modes ∗ Centerline evolution (from the matching condition) � � ∂ � X ln 1 √ ε + 1 ( k × χ ) ∗ + ( k × Ψ ) = � v ∗ � v QG ) + � X · ( ∇ − QG ∂ τ 2 � �� � � �� � self-induced motion background advection χ = fct( total circulation, centerline geometry ) Ψ = fct( core structure , centerline geometry, diabatic sources ) ∗ effect of β -gyres; ∗ akin to local-induction-approximation LIA ∗ Grasso, Kallenbach, Montgomery, Reasor (1997, 2001, 2004)
Vortex motion ⇒ precessing quasi-modes ∗ 10 10 8 8 z in km z in km 6 6 4 4 2 2 0 0 40 40 30 30 20 20 10 10 y in km y in km 40 30 20 10 0 40 30 20 10 0 0 0 10 10 20 20 x x i 10 n i 10 n k 30 k 30 m 20 m 20 30 30 40 40 40 40 3D Simulation (EULAG ∗ ) asymptotic theory ∗
Adiabatic lifting and WTG ( 0th & 1st circumferential Fourier modes: w = w 0 + w 1 1 cos θ + w 1 2 sin θ + ... ) gradient wind balance (0th) and hydrostatics (1st) in the tilted vortex � � e r · ∂ � ∂r = u 2 1 ∂p Θ 1 k = − 1 ∂p X θ r + f u θ , , ρ ρ ∂r ∂z 1 k � � e r · � X = � X cos θ + � Y sin θ potential temperature transport (1st) − ( − 1) k u θ d Θ ( k ∗ = 3 − k ) r Θ 1 k ∗ + w 1 k dz = Q Θ , 1 k 1st-mode phase relation: vertical velocity – diabatic sources & vortex tilt � � u 2 ⊥ e r · ∂ � 1 u θ X Q Θ , 1 k + θ w 1 k = r + f u θ ∂z r d Θ /dz k
Spin-up by asynchronous heating � ∂u θ � � u θ � ∂u θ, 0 ∂u θ, 0 ∂r + u θ + w 0 ∂z + u r, 00 r + f = − u r, ∗ r + f ∂ τ � �� � standard axisymmetric balance � �� � w ∂ e r · � u r, ∗ = X ∂z θ e r · � X = � X cos θ + � Y sin θ � � u 2 �� � ⊥ � Q Θ , 1 k + ∂ 1 u θ e r · � θ w 1 k = r + f u θ X ∂z r d Θ /dz k
Spin-up by asynchronous heating � ∂u θ � � u θ � ∂u θ, 0 ∂u θ, 0 ∂r + u θ + w 0 ∂z + u r, 00 r + f = − u r, ∗ r + f ∂ τ � �� � standard axisymmetric balance � � � �� � ∂ � ∂ � w ∂ 1 X Y !! e r · � u r, ∗ = = Q Θ , 1 1 ∂z + Q Θ , 1 2 X ∂z ∂z d Θ /dz θ e r · � X = � X cos θ + � Y sin θ � � � u 2 � � ⊥ � 1 + ∂ u θ e r · � θ w 1 k = Q Θ , 1 k r + f u θ X ∂z r d Θ /dz k � �� � � �� � WTG adiabatic lifting
figures adapted from Jones (1995) * The adiabatic lifting in a tilted vortex ∗∗ w Θ ∂ X ∂z � � u 2 �� � ⊥ � 1 Q Θ , 1 k + ∂ u θ e r · � θ w 1 k = r + f u θ X ∂z r d Θ /dz k ∗ Jones, Q.J.R. Met. Soc., 121 , 821–851 (1995) ∗ Frank & Ritchie, Mon. Wea. Rev., 127 , 2044–2061 (1999)
figures adapted from : Jones (1995), Q.J.R. Met. Soc., 121 , 821–851 Heating pattern for max intensification (APE-theory) ∗ w Θ ∂ X ∂z � � u 2 �� � ⊥ � 1 Q Θ , 1 k + ∂ u θ e r · � θ w 1 k = r + f u θ X ∂z r d Θ /dz k Lorenz, E. N., Generation of available potential energy and the intensity of the general circulation, Tech. Rep. , UCLA, (1955)
Compatibility with Lorenz’ APE theory ∗ � � � � � � � � r ρ ru r, 0 [ e k + p ′ ] rw 0 [ e k + p ′ ] Θ ′ 0 Q Θ , 0 + Θ ′ re k t + r + z = 1 · Q Θ , 1 2 N 2 Θ e k = ρu 2 θ 2 Symmetric & asymmetric are equally important ∗ Thanks to Olivier Pauluis! “Available Potential Energy” D¨ orffel et al., preprint, arXiv:1708.07674 (2017)
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