The effect of asymmetric large-scale dissipation on energy and potential enstrophy injection in two-layer quasi-geostrophic turbulence

Transcription

1 The effect of asymmetric large-scale dissipation on energy and potential enstrophy injection in two-layer quasi-geostrophic turbulence Eleftherios Gkioulekas (1) and Ka-Kit Tung (2) (1) Department of Mathematics, University of Texas-Pan American (2) Department of Applied Mathematics, University of Washington p.1/22

2 Publications 1. K.K. Tung and W.T. Welch (2001), J. Atmos. Sci. 58, K.K. Tung and W.W. Orlando (2003a), J. Atmos. Sci K.K. Tung and W.W. Orlando (2003b), Discrete Contin. Dyn. Syst. Ser. B, 3, K.K. Tung (2004), J. Atmos. Sci., 61, E. Gkioulekas and K.K. Tung (2005), Discrete Contin. Dyn. Syst. Ser. B, 5, E. Gkioulekas and K.K. Tung (2005), Discrete Contin. Dyn. Syst. Ser. B, 5, E. Gkioulekas and K.K. Tung (2006), J. Low Temp. Phys., 145, [review] 8. E. Gkioulekas and K.K. Tung (2007), J. Fluid Mech., 576, E. Gkioulekas and K.K. Tung (2007), Discrete Contin. Dyn. Syst. Ser. B, 7, E. Gkioulekas (2010), J. Fluid Mech., submitted. [arxiv: v1 [nlin.cd]] p.2/22

3 Outline Review of 2D turbulence. The Nastrom-Gage spectrum The Tung-Orlando theory of double cascade. The two-layer quasi-geostrophic model. p.3/22

4 2D Navier-Stokes equations In 2D turbulence, the scalar vorticity ζ(x, y, t) is governed by ζ t + J(ψ, ζ) =d + f, (1) where ψ(x, y, t) is the streamfunction, and ζ(x, y, t) = 2 ψ(x, y, t), and d = [ν( ) κ + ν 1 ( ) m ]ζ (2) The Jacobian term J(ψ, ζ) describes the advection of ζ by ψ, and is defined as J(a, b) = a x b y b x a y. (3) p.4/22

5 Energy and enstrophy spectrum. I Two conserved quadratic invariants: energy E and enstrophy G defined as E(t) = 1 2 ψ(x, y, t)ζ(x, y, t) dxdy G(t) = 1 2 ζ 2 (x, y, t) dxdy. (4) Let a <k (x) be the field obtained from a(x) by setting to zero, in Fourier space, the components corresponding to wavenumbers with norm greater than k: a <k (x) = = dyp (k x y)a(y) (5) R 2 dx 0 R 2 dk 0 H(k k 0 ) 4π 2 exp(ik 0 (x x 0 ))a(x 0 ) (6) Filtered inner product: a, b k = d dk R 2 dx a <k (x)b <k (x) (7) p.5/22

6 Energy and enstrophy spectrum. II Energy spectrum: E(k) = ψ, ζ k Enstrophy spectrum G(k) = ζ,ζ k Consider the conservation laws for E(k) and G(k) : E(k) t G(k) t + Π E(k) k + Π E(k) k = D E (k)+f E (k) (8) = D G (k)+f G (k) (9) In two-dimensional turbulence, the energy flux Π E (k) and the enstrophy flux Π G (k) are constrained by k 2 Π E (k) Π G (k) < 0 (10) for all k not in the forcing range. p.6/22

7 KLB theory. ln E(k) C ir ε 2/3 ir k 5/3 k f = forcing wavenumber k ir = IR dissipation wavenumber k uv = UV dissipation wavenumber ε ir = upscale energy flux η uv = downscale ensrropht flux C uv η 2/3 uv k 3 [χ +ln(kl 0 )] 1/3 k ir k f k uv ln k p.7/22

8 Nastrom-Gage spectrum schematic ln E(k) k km 800km k 5/3 600km 1km k t 700km k R k 3 k 5/3 ln k k t p.8/22

9 Nastrom-Gage spectrum p.9/22

10 Interpretation of NG spectrum. The k 3 range is interpretated as downscale enstrophy cascade. The k 5/3 range used to be interpretated as an 2D inverse energy cascade forced at small scales by thunderstorms. This tortured interpretation followed from the perceived need to explain the Nastrom-Gage spectrum in terms of 2D turbulence. Tung and Orlando (formerly: Welch) challenged the Charney QG-2D equivalence: K.K. Tung and W.T. Welch (2001), J. Atmos. Sci. 58, K.K. Tung and W.W. Orlando (2003a), J. Atmos. Sci K.K. Tung and W.W. Orlando (2003b), Discrete Contin. Dyn. Syst. Ser. B, 3, New interpretation: A double downscale cascade of enstrophy and energy with enstrophy flux η uv and energy flux ε uv and transition from k 3 scaling to k 5/3 scaling at the transition wavenumber k t η uv /ε uv. p.10/22

11 Tung and Orlando spectrum p.11/22

12 More on Tung-Orlando theory Cho-Lindborg: Confirm downscale energy cascade. J.Y.N. Cho and E. Lindborg (2001), J. Geophys. Res. 106 D10, 10,223-10,232. J.Y.N. Cho and E. Lindborg (2001), J. Geophys. Res. 106 D10, 10,232-10,241. Smith-Tung debate: Transition not possible in 2D turbulence K.S. Smith (2004), J. Atmos. Sci. 61, K.K. Tung (2004), J. Atmos. Sci., 61, Gkioulekas-Tung superposition principle E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Syst. Ser. B, 5, E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Syst. Ser. B, 5, Transition may be possible in 2-layer QG with asymmetric dissipation stil open question. E. Gkioulekas and K.K. Tung (2007), Discr. Cont. Dyn. Syst. Ser. B, 7, p.12/22

13 The two-layer model. I The governing equations for the two-layer quasi-geostrophic model are ζ 1 t + J(ψ 1,ζ 1 + f) = 2f h ω + d 1 (11) ζ 2 t + J(ψ 2,ζ 2 + f) =+ 2f h ω + d 2 +2e 2 (12) T t [J(ψ 1,T)+J(ψ 2,T)] = N 2 f ω + Q 0 (13) where ζ 1 = 2 ψ 1 ; ζ 2 = 2 ψ 2 ; T =(2/h)(ψ 1 ψ 2 ). f is the Coriolis term; N the Brunt-Väisälä frequency; Q 0 is the thermal forcing on the temperature equation; d 1, d 2, e 2 the dissipation terms: d 1 =( 1) κ+1 ν 2κ ζ 1 (14) d 2 =( 1) κ+1 ν 2κ ζ 2 (15) e 2 = ν E ζ 2 (16) p.13/22

14 The two-layer model. II The potential vorticity is defined as with k R 2 2f/(hN) and it satisfies q 1 = 2 ψ 1 + f + k2 R 2 (ψ 2 ψ 1 ) (17) q 2 = 2 ψ 2 + f k2 R 2 (ψ 2 ψ 1 ) (18) with f 2 =(1/4)k 2 R hq 0 and f 2 =(1/4)k 2 R hq 0. q 1 t + J(ψ 1,q 1 )=f 1 + d 1 (19) q 2 t + J(ψ 2,q 2 )=f 2 + d 2 + e 2 (20) p.14/22

15 The two-layer model. III The two-layer model conserves total energy E and potential layer enstrophies G 1 and G 2 : E = G 1 = G 2 = [ ψ 1 (x, y, t)q 1 (x, y, t) ψ 2 (x, y, t)q 2 (x, y, t)] dxdy (21) q1 2 (x, y, t) dxdy (22) q2 2 (x, y, t) dxdy (23) We define: The energy spectrum E(k) = ψ 1,q 1 k + ψ 2,q 2 k Top-layer potential enstrophy spectrum G 1 (k) = q 1,q 1 k Bottom-layer potential enstrophy spectrum G 2 (k) = q 2,q 2 k Total potential enstrophy spectrum G(k) =G 1 (k)+g 2 (k) p.15/22

16 The two-layer model. IV Consider the generalized form of an n-layer model: q α t + J(ψ α,q α )= β D αβ ψ β + f α (24) ˆq α (k,t)= β L αβ ( k ) ˆψ β (k,t) (25) The energy spectrum E(k) and the potential enstrophy spectrum G(k) are given by: E(k) = α ψ α,q α k = αβ L αβ (k)c αβ (k) (26) G(k) = α q α,q α k = αβγ L αβ (k)l αγ (k)c βγ (k) (27) with C αβ (k) = ψ α,ψ β k. p.16/22

17 The two-layer model. V Let φ αβ (k) = f α,ψ β k, and D αβ(k) be the spectrum of the operator D αβ The energy forcing spectrum F E (k) and the potential enstrophy forcing spectrum F G (k) are: F E (k) =2 α φ αα (k) (28) F G (k) =2 αβ L αβ (k)φ αβ (k) (29) The energy dissipation spectrum D E (k) and the potential dissipation enstrophy spectrum D G (k) are related with C αβ (k) = ψ α,ψ β according to: k D E (k) =2 αβ D αβ (k)c αβ (k) (30) D G (k) =2 αβγ L αβ (k)d αγ (k)c βγ (k) (31) p.17/22

18 Forcing spectrum. I. For antisymmetric forcing f 1 = f and f 2 = f: F E (k) =2(Φ 1 (k) Φ 2 (k)) (32) F G (k) =(k 2 + k 2 R )F E(k) (33) with Φ 1 (k) = f, ψ 1 k and Φ 2 (k) = f, ψ 2 k. It follows that for k f k R : (η/ε) k 2 R. If both are dissipated at small scales, then k t k R. In 2D turbulence Ekman damping dissipates most of injected energy and some of injected enstrophy. Not true in two-layer QG model because the Ekman term appears only on bottom layer. p.18/22

19 Forcing spectrum. II. For Ekman-damped forcing: f 1 = f and f 2 = f ν E ψ 2, incorporating the damping effect to the forcing spectrum gives: F E (k) =2(Φ 1 (k) Φ 2 (k)) + 2ν E k 2 U 2 (k) (34) F G (k) =(k 2 + k 2 R )F E(k) ν E k 2 k 2 R (U 2(k)+C 12 (k)) (35) with U 1 (k) = ψ 1,ψ 1 k, U 2 (k) = ψ 2,ψ 2 k, and C 12 (k) = ψ 1,ψ 2 k. Note the F E (k) increases, because U 2 (k) > 0. If E K (k) E P (k) = C 12 (k) 0= F G (k) decreases. Thus, the tendency is to decrease k t. Layer interaction makes it non-obvious whether F G (k) increases or decreases. In SQG: E K (k)/e P (k) =1. In stratified 3D: E K (k)/e P (k) 3. (Lindborg (2009)) In full QG: E K (k)/e P (k) 2. (Charney theory Vallgren and Lindborg (2010)) p.19/22

20 Forcing spectrum. IV. Claim: Suppressing bottom-layer forcing tends to decrease k t. Let f 1 = ϕ and f 2 = µϕ with µ (0, 1) (suppression factor). Assume ϕ is random Gaussian with ϕ(x 1,t 1 )ϕ(x 2,t 2 ) =2Q(x 1, x 2 )δ(t 1 t 2 ) (36) fα (x 1,t 1 )f β (x 2,t 2 ) =2Q αβ (x 1, x 2 )δ(t 1 t 2 ) (37) Define the forcing spectrum: Q(k) = d dk Q αβ (k) = d dk dxdydz P (k x y)p (k x z)q(y, z) (38) dxdydz P (k x y)p (k x z)q αβ (y, z) (39) p.20/22

21 Forcing spectrum. V. It follows that the streamfunction-forcing spectrum reads: ϕ αβ (k) = f α,ψ β k = γ L 1 βγ (k)q αγ(k) (40) Energy and enstrophy forcing spectrum: F E (k) =2 α F G (k) =2 α ϕ αα (k) = 2Q(k)[2(1 + µ2 )k 2 +(1 µ) 2 k 2 R ] 2k 2 (k 2 + k 2 R ) (41) L αβ (k)ϕ αβ (k) =2(1+µ 2 )Q(k) (42) Consider the forcing range limit: k k R. For µ =1: F G (k) kr 2 F E(k) = k t k R For µ =0: F G (k) 2k 2 F E (k) = k t 2k f Thus: Baroclinically damped forcing contributes to inertial range transition. p.21/22

22 Conclusion The energy flux constraint prevents a transition from k 3 scaling to k 5/3 scaling in 2D turbulence. The energy flux constraint can be broken in two-layer QG turbulence under asymmetric dissipation. In the two-layer QG model, the rates of enstrophy over energy injection satisfy: (η/ε) =k 2 R Asymmetric Ekman dissipation increases ε and may decrease η, Suppressing the bottom layer forcing directly always decreases the ratio (η/ε). Open question: Can the injected energy and enstrophy be dissipated? p.22/22