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
Publications 1. K.K. Tung and W.T. Welch (2001), J. Atmos. Sci. 58, 2009-2012. 2. K.K. Tung and W.W. Orlando (2003a), J. Atmos. Sci. 60 824-835. 3. K.K. Tung and W.W. Orlando (2003b), Discrete Contin. Dyn. Syst. Ser. B, 3, 145-162. 4. K.K. Tung (2004), J. Atmos. Sci., 61, 943-948. 5. E. Gkioulekas and K.K. Tung (2005), Discrete Contin. Dyn. Syst. Ser. B, 5, 79-102 6. E. Gkioulekas and K.K. Tung (2005), Discrete Contin. Dyn. Syst. Ser. B, 5, 103-124. 7. E. Gkioulekas and K.K. Tung (2006), J. Low Temp. Phys., 145, 25-57 [review] 8. E. Gkioulekas and K.K. Tung (2007), J. Fluid Mech., 576, 173-189. 9. E. Gkioulekas and K.K. Tung (2007), Discrete Contin. Dyn. Syst. Ser. B, 7, 293-314 10. E. Gkioulekas (2010), J. Fluid Mech., submitted. [arxiv:1011.3163v1 [nlin.cd]] p.2/22
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
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
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
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
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
Nastrom-Gage spectrum schematic ln E(k) k 3 3000km 800km k 5/3 600km 1km k t 700km k R k 3 k 5/3 ln k k t p.8/22
Nastrom-Gage spectrum p.9/22
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, 2009-2012. K.K. Tung and W.W. Orlando (2003a), J. Atmos. Sci. 60 824-835. K.K. Tung and W.W. Orlando (2003b), Discrete Contin. Dyn. Syst. Ser. B, 3, 145-162. 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
Tung and Orlando spectrum p.11/22
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, 937-942 K.K. Tung (2004), J. Atmos. Sci., 61, 943-948. Gkioulekas-Tung superposition principle E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Syst. Ser. B, 5, 79-102 E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Syst. Ser. B, 5, 103-124. 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, 293-314 p.12/22
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 + 1 2 [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
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
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
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
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
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
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
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
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
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