Internal wave dynamics in the atmosphere take-home messages Rupert - - PowerPoint PPT Presentation

Internal wave dynamics in the atmosphere take-home messages

Rupert Klein Mathematik & Informatik, Freie Universit¨ at Berlin

Sound-Proof Models

† e.g. Lipps & Hemler, JAS, 29, 2192–2210 (1982) ∗ Durran, JAS, 46, 1453–1461 (1989)

Compressible flow equations L ∼ hsc

ρt + ∇ · (ρv) = 0 (ρu)t + ∇ · (ρv ◦ u) + P∇

π = 0

(ρw)t + ∇ · (ρvw) + Pπz = −ρg Pt + ∇ · (Pv) = 0 drop term for: anelastic† (approx.) pseudo-incompressible∗ P = p

1 γ = ρθ ,

π = p/ΓP , Γ = cp/R , v = u + wk (u · k ≡ 0) Parameter range & length and time scales

• f asymptotic validity ?

Regimes of Validity ... Design Regime Characteristic inverse time scales

dimensional dimensionless advection : uref hsc 1 internal waves : N =

• g

θ dθ dz √ghsc uref

• hsc

θ dθ dz = 1 εν

• hsc

θ d θ dz sound :

• pref/ρref

hsc = √ghsc hsc √ghsc uref = 1 ε Realistic regime with three time scales θ = 1 + εµ θ(z) + . . . ⇒ hsc θ dθ dz = O(εµ) (ν = 1 − µ/2)

Analysis of internal wave spectra

− d dz

• 1

1 − εµω2/λ2

c2

1 θ P dW dz

• + λ2

θ P W = 1 ω2 λ2N 2 θ P W Internal wave modes

• ω2/λ2

c2

= O(1)

• pseudo-incompressible modes/EVals = compressible modes/EVals + O(εµ) †
• phase errors remain small over advection time scales for

µ > 2 3 Anelastic and pseudo-incompressible models remain relevant for stratifications 1 θ dθ dz < O(ε2/3) ⇒ ∆θ|hsc < ∼ 40 K not merely up to O(ε2) as in Ogura-Phillips (1962)

ε y′′ + δ y′ + y = cos(τ)

1 2 3 4 5 6 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 time [s] x[m] m x’’ + k x’ + c x = F

0 * cos(Ω t), Exact Solution

reference solution with: m = k = 0 x(t) with: k = 1; c = 25; m = 0.01; F0 = 0

ε = 0.0004 δ = 0.04

10 20 30 40 50 60 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 time [s] x [m] m x’’ + k x’ + c x = F

0 * cos(Ω t), Exact Solution

reference solution with: m = k = 0 x(t) with: k = 0.01; c = 25; m = 1; F0 =

ε = 0.04 δ = 0.0004 The limit is path-dependent!

1 2 3 4 5 6 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 time [s] x[m] m x’’ + k x’ + c x = F

0 * cos(Ω t), Exact Solution

reference solution with: m = k = 0 x(t) with: k = 1; c = 25; m = 0.01; F0 = 0

Matched asymptotic expansions ?

SFB 1114

CEMRACS 2019 “Geophysical Fluids, Gravity Flows” CIRM, Luminy, July 18, 2019

Strongly tilted atmospheric Vortices

Rupert Klein Mathematik & Informatik, Freie Universit¨ at Berlin

Thanks to ...

Eileen P¨ aschke (Deutscher Wetterdienst, Lindenberg) Ariane Papke (formely FU-Berlin) Patrick Marschalik (Fritz Haber Institute, Berlin) Antony Owinoh (†) Tom D¨

• rffel

(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

R.K., Ann. Rev. Fluid Mech., 42, 249–274 (2010)

Motivation Structure of atmospheric vortices I: two scales

(P¨ aschke et al., JFM, (2012))

Structure of atmospheric vortices II: cascade of scales

(D¨

• rffel et al., arXiv:1708.07674)

Conclusions

http://www.aoml.noaa.gov/hrd/tcfaq/A4.html

Tropical easterly african waves

Dunkerton et al., Atmos. Chem. Phys., 9, 5587–5646 (2009)

Developing tropical storm

(streamlines in co-moving frame and Okubo-Weiss-parameter (color))

T T T T T T

Ro = |v| fL ∼ 1 10

Developed hurricane

R∗

mw ≈ 50 . . . 200 km

uθ ≈ 30 . . . 60 m/s Rmw: radius of max. wind

Hurricane ”Rita“ Ro = uθ,max fRmw ∼ 10

Photo: Hurricane Rita from https://commons.wikimedia.org/wiki/File:HurricaneRita21Sept05a.jpg

Ensemble of Simulations of “Joaquin”-like Storms

• Gh. Alaka et al. (2019), WAF, submitted

Ensemble Tracks Vortex Tilts Storm Evolutions

Motivation Structure of atmospheric vortices I: two scales

(P¨ aschke et al., JFM, (2012))

Structure of atmospheric vortices II: cascade of scales

(D¨

• rffel et al., arXiv:1708.07674)

Conclusions

P¨ aschke, Marschalik, Owinoh, K., JFM, 701, 137–170 (2012) D¨

• rffel et al., preprint, arXiv:1708.07674 (2017)

Radial momentum balance regimes

− 1 ρ ∂p ∂r + fuθ = O

• 1
• geostrophic

Ro ≪ 1 typical “weather” u2

θ

r − 1 ρ ∂p ∂r + fuθ = O

• 1
• gradient wind

Ro = O (1) tropical storm incipient hurricane u2

θ

r − 1 ρ ∂p ∂r + fuθ = O

• 1
• cyclostrophic

Ro ≫ 1 hurricane

Photo: Hurricane Rita from https://commons.wikimedia.org/wiki/ File:HurricaneRita21Sept05a.jpg

http://www.aoml.noaa.gov/hrd/tcfaq/A4.html

Tropical easterly african waves

Dunkerton et al., Atmos. Chem. Phys., 9, 5587–5646 (2009)

Vortex tilt in the incipient hurricane stage

(Velocity potential) 200 hPa (∼ 12 km) 925 hPa (∼ 0.8 km)

∼ 200 km

Scaling regime for matched asymptotic expansions

centreline

Lsyn Lmes X(t, z) z x y

j

i k

hsc

Lsyn; |v| ∼ vsyn tsyn ∼ Lsyn/vsyn

• farfield: classical QG theory

Lmes = √εLsyn; vmes = 1 √ε

• core: gradient wind scaling

C ∼ (vL)syn; Rosyn ∼ vsyn fLsyn = O (ε) C ∼ (vL)mes; Romes ∼ vmes fLmes = O (1) Comparable levels of circulation C!

Photo: Hurricane Rita from https://commons.wikimedia.org/wiki/File:HurricaneRita21Sept05a.jpg

Vortex motion ⇒ precessing quasi-modes∗

∗effect of β-gyres; ∗akin to local-induction-approximation LIA ∗Grasso, Kallenbach, Montgomery, Reasor (1997, 2001, 2004)

Centerline evolution

(from the matching condition) ∂ X ∂τ = X · (∇

vQG) +

v∗

QG

• background advection

−

• ln 1

√ε + 1 2

• (k × χ)∗ + (k × Ψ)
• self-induced motion

χ = fct(total circulation, centerline geometry) Ψ = fct(core structure, centerline geometry, diabatic sources)

Vortex motion ⇒ precessing quasi-modes∗

∗

x i n k m

40 30 20 10 0 10 20 30 40

y in km

40 30 20 10 10 20 30 40

z in km

2 4 6 8 10

asymptotic theory

x i n k m

40 30 20 10 0 10 20 30 40

y in km

40 30 20 10 10 20 30 40

z in km

2 4 6 8 10

3D Simulation (EULAG∗)

Adiabatic lifting and WTG ( 0th & 1st circumferential Fourier modes: w = w0 + w11 cos θ + w12 sin θ + ... ) gradient wind balance (0th) and hydrostatics (1st) in the tilted vortex 1 ρ ∂p ∂r = u2

θ

r + f uθ , Θ1k = −1 ρ ∂p ∂r

• er · ∂

X ∂z

• 1k

,

• er ·

X = X cos θ + Y sin θ

• potential temperature transport (1st)

−(−1)kuθ r Θ1k∗ + w1k dΘ dz = QΘ,1k (k∗ = 3 − k) 1st-mode phase relation: vertical velocity – diabatic sources & vortex tilt w1k = 1 dΘ/dz  QΘ,1k +  er · ∂ X

⊥

∂z  

k

uθ r u2

θ

r + f uθ  

Spin-up by asynchronous heating

∂uθ,0 ∂τ + w0 ∂uθ,0 ∂z + ur,00 ∂uθ ∂r + uθ r + f

• standard axisymmetric balance

= − ur,∗ uθ r + f

• ur,∗ =
• w ∂

∂z

• er ·

X

• θ

er · X = X cos θ + Y sin θ w1k = 1 dΘ/dz

• QΘ,1k + ∂

∂z

• er ·

X

⊥ k

uθ r u2

θ

r + f uθ

Spin-up by asynchronous heating

∂uθ,0 ∂τ + w0 ∂uθ,0 ∂z + ur,00 ∂uθ ∂r + uθ r + f

• standard axisymmetric balance

= − ur,∗ uθ r + f

• ur,∗ =
• w ∂

∂z

• er ·

X

• θ

= 1 dΘ/dz

• QΘ,11

∂ X ∂z + QΘ,12 ∂ Y ∂z

• !!

er · X = X cos θ + Y sin θ w1k = 1 dΘ/dz

• QΘ,1k

WTG

+ ∂ ∂z

• er ·

X

⊥ k

uθ r u2

θ

r + f uθ

• adiabatic lifting

∗ Jones, Q.J.R. Met. Soc., 121, 821–851 (1995) ∗ Frank & Ritchie, Mon. Wea. Rev., 127, 2044–2061 (1999)

The adiabatic lifting in a tilted vortex∗∗

w Θ

∂X ∂z

w1k = 1 dΘ/dz

• QΘ,1k+ ∂

∂z

• er ·

X

⊥ k

uθ r u2

θ

r + f uθ

• figures adapted from Jones (1995)*

Lorenz, E. N., Generation of available potential energy and the intensity of the general circulation, Tech. Rep., UCLA, (1955)

Heating pattern for max intensification (APE-theory)∗

w Θ

∂X ∂z

w1k = 1 dΘ/dz

• QΘ,1k + ∂

∂z

• er ·

X

⊥ k

uθ r u2

θ

r + f uθ

• figures adapted from: Jones (1995), Q.J.R. Met. Soc., 121, 821–851

∗Thanks to Olivier Pauluis!

“Available Potential Energy” D¨

• rffel et al., preprint, arXiv:1708.07674 (2017)

Compatibility with Lorenz’ APE theory∗

• rek
• t +
• rur,0[ek + p′]
• r +
• rw0[ek + p′]
• z =

rρ N 2Θ

2

• Θ′

0QΘ,0 + Θ′ 1 · QΘ,1

• ek = ρu2

θ

2 Symmetric & asymmetric are equally important

(w = w0 + w11 cos θ + w12 sin θ + . . . )

Radial transport & tilting by asymmetric heating

Circumferential Fouriermodes of vertical velocity w1k = 1 dΘ/dz

• QΘ,1k + ∂

∂z

• er ·

X

⊥ k

uθ r u2

θ

r + f uθ

• er

ur,∗ ur,∗

ur,∗ =

• w ∂

∂z

• er ·

X

• θ

= 1 dΘ/dz

• QΘ,11

∂ X ∂z + QΘ,12 ∂ Y ∂z

∗ Ultimately leaves asymptotic regime!

D¨

• rffel et al., preprint, arXiv:1708.07674 (2017)

Recent results

Qualitative corroboration through 3D-numerics

0.0 2.5 5.0 7.5 10.0 12.5 15.0

time in h

5 10 15 20 25 30

max |uθ| in m/s

Artificial heating pattern:

w1k = 1 dΘ/dz

• − ∂

∂z

• er ·

X

• k

uθ r uθ2 r + f uθ

• + ∂

∂z

• er ·

X

⊥ k

uθ r u2

θ

r + f uθ

Motivation Structure of atmospheric vortices I: two scales

(P¨ aschke et al., JFM, (2012))

Structure of atmospheric vortices II: cascade of scales

(D¨

• rffel et al., arXiv:1708.07674)

Conclusions

Convective updrafts

level of free convection centreline boundary layer convergence convective updrafts

w ∼ √ CAPE ∼ 10...50m/s w < 1m/s

L = O(1)

= O(√ε)

d = O(ε)

Convection concentrates in narrow towers (area fraction σ ≪ 1) Essentially dry dynamics between towers Comparable average vertical mass fluxes

Calls for non-standard multiscale analysis

Spin-up by asymmetric convection

∂uθ,0 ∂τ + w0 ∂uθ,0 ∂z + ur,00 ∂uθ ∂r + uθ r + f

• standard axisymmetric balance

= − ur,∗ uθ r + f

• ur,∗ =
• w ∂

∂z

• er ·

X

• θ

= wupd,11 ∂ X ∂z + wupd,12 ∂ Y ∂z

!!

Area averaged updraft fluxes take role of heating-induced vertical velocities

Intensification & tilt destabilization

level of free convection centreline boundary layer convergence convective updrafts

∂X ∂z

Attenuation / tilt stabilization

level of free convection boundary layer convergence convective updrafts

centreline

∂X ∂z

“down-shear left” convection

Motivation Structure of atmospheric vortices I: two scales

(P¨ aschke et al., JFM, (2012))

Structure of atmospheric vortices II: cascade of scales

(D¨

• rffel et al., arXiv:1708.07674)

Conclusions

Spin-up by asymmetric heating

∂uθ,0 ∂τ + w0 ∂uθ,0 ∂z + ur,00 ∂uθ ∂r + uθ r + f

• standard axisymmetric balance

= − ur,∗ uθ r + f

• ur,∗ =
• w ∂

∂z

• er ·

X

• θ

= 1 dΘ/dz

• QΘ,11

∂ X ∂z + QΘ,12 ∂ Y ∂z

• er

ur,∗ ur,∗

Radial transport in a tilted vortex induced by asymmetric heating