Atmospheric turbulence spectrum: competing theories

Transcription

1 Atmospheric turbulence spectrum: competing theories Explanations for Nastrom & Gage, JAS, 42, 1985 by Tung & Orlando, JAS, 60, 2003 Tulloch & Smith, PNAS, 103, 2006 (k p where p = 3, 5/3,...) as told by Navid C CASPO theory seminar, 6 Nov. 2015

2 Atmospheric energy spectrum by Nastrom & Gage km

3 Atmospheric energy spectrum by Nastrom & Gage 1985 synoptic scales 1000 km < < 5200 km mesoscales 2 km < < 600 km

4 Some possible (early) explanations k 5/3 forward cascade predicted by 3D isotropic homogeneous arguments internal gravity waves tropopause km > > 3D effects only for scales <2 km ruled out by Gage & Nastrom 1986 k 3 forward enstrophy inertial range (as in KBL theory) Charney 1971 synoptic scales are definitely not inertial (baroclinic instability energetic scales) (several problems )

5 Tung & Orlando 2003 reproduced an energy spectrum similar to Gage & Nastrom with a 2-layer QG system Tulloch & Smith 2006 (argued that Tung & Orlando are wrong and) reproduced an energy spectrum similar to Gage & Nastrom with a finite height SQG system

6 Tung & Orlando 2003 two-layer QG system tq 1 + J(ψ 1 Uy, q 1 )= 0ν s ( x 2 + y 2 ) 10 ψ 1 tq 2 + J(ψ 2 + Uy, q 2 )= ν E ( 2 x + 2 y )ψ 2 ν s ( 2 x + 2 y ) 10 ψ 2 q 1 =( 2 x + 2 y )ψ 1 k 2 R 2 (ψ 1 ψ 2 )+βy q 2 =( 2 x + 2 y )ψ 2 + k2 R 2 (ψ 1 ψ 2 )+βy k R, U, β, ν E, ν s, domain size fixed parameters of the problem

7 Tung & Orlando 2003 hyperdiffusion for numerical stability two-layer QG system tq 1 + J(ψ 1 Uy, q 1 )= ν s ( 2 x + 2 y ) 10 ψ 1 tq 2 + J(ψ 2 + Uy, q 2 )= ν E ( 2 x + 2 y )ψ 2 ν s ( 2 x + 2 y ) 10 ψ 2 q 1 =( 2 x + 2 y )ψ 1 k 2 R 2 (ψ 1 ψ 2 )+βy q 2 =( 2 x + 2 y )ψ 2 + k2 R 2 (ψ 1 ψ 2 )+βy k R, U, β, ν E, ν s, domain size fixed parameters of the problem

8 Tung & Orlando 2003 adjusted parameters to resemble midlatitude tropopause k 3 k 5/3

9 Tung & Orlando 2003 m 2 ε

10 Tung & Orlando 2003 m 2 ε k 3 k 5/3 k t = η/ε 600 km

11 Tung & Orlando 2003 enstrophy/energy fluxes based on their model k 3 k 5/3

12 Tulloch & Smith 2006 finite depth SQG system tθ + J(ψ, θ) =0, z = 0 θ = z ψ q =( 2 x + 2 y + σ 2 2 z )ψ = 0, z < 0 z = zψ = 0, z = σ = N/f H z = 0 H 10 km z = H

13 Tulloch & Smith 2006 finite depth SQG system tθ + J(ψ, θ) =0, z = 0 θ = z ψ q =( 2 x + 2 y + σ 2 2 z )ψ = 0, z < 0 zψ = 0, z = σ = N/f H z = 0 H 10 km z = H

14 Tulloch & Smith 2006 finite depth SQG system tθ + J(ψ, θ) =0 F, z = 0 θ = z ψ q =( 2 x + 2 y + σ 2 2 z )ψ = 0, z < 0 zψ = 0, z = σ = N/f H plus a 2/3 filter for numerical stability z = 0 H 10 km z = H

15 Tulloch & Smith 2006 relation between θ and ψ in fsqg θ = z ψ ˆψ(k, z = 0) = ˆθ(k, 0) σktanh (σhk) k (σh) 1 ˆθ(k, 0) σ 2 k 2 H ˆψ(k, 0) k (σh) 1 ˆθ(k, 0) σkˆψ(k, 0) (2D-like) (SQG-like)

16 Tulloch & Smith 2006 what does that imply for energy spectra? 1 2 ψ2 P(k) dk E = k 2 P(k) dk E(k) dk ε kt (k) [k 5 P(k)] 1/2 = const. T = 1 2 θ2 T (k) dk = [σktanh(k/k t )] 2 P(k) dk some fiddling around E(k) ε 2/3 [σ tanh(σhk)] k 3 4/3 k 5/3 k (σh) 1 k 5/3 k (σh) 1 regime transition at scale k t f/(nh)

17 Tulloch & Smith 2006 k 3 k 5/3 k t = 50 and they checked their prediction of kt by doing several simulations with varying σ

18 Discussion The Nastrom & Gage observed atmospheric spectrum is essentially captured by the very simple models presented TO03 predict that k t = η/ε while TS06 that k t = f/(nh) (known params) Nastrom & Gage spectrum is a bit less steep than -3 at synoptic scales while TO03 and TS06 are more steep there There is also the work of Cho & Lindborg J Geophys Res 2001 that computed 3-correllators from observations (4/5-law) δu L δu L δu L = 4 5 ε(ξ ξ ) the sign of this term determines whether there is forward or inverse energy cascade TS06 speculate that perhaps fully a continuously stratified QG model might resolve any inadequacies their simple model has see paper by Deusebio & Lindborg JFM 2013

19 part of the whole story Tung & Orlando JAS 2003 Smith comment on TO03, JAS 2004 Tung reply, JAS 2004 Gkioulekas & Tung later papers by Gkioulekas Tulloch & Smith PNAS 2006 Tulloch & Smith 2009 (baroclinic+sqg) Lindborg comment on TS06 & TS09, JAS 2009 Smith & Tulloch reply, JAS 2009

20 thanks