Energy and potential enstrophy flux constraints in quasi-geostrophic models

Transcription

1 Energy and potential enstrophy flux constraints in quasi-geostrophic models Eleftherios Gkioulekas University of Texas Rio Grande Valley August 25, 2017

2 Publications K.K. Tung and W.W. Orlando (2003a), J. Atmos. Sci. 60,, 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), Discr. Cont. Dyn. Sys. B 5, E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Sys. B 5, 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 (2012): J. Fluid Mech. 694, E. Gkioulekas (2014): Physica D 284, 27-41

3 Outline Flux inequality for 2D turbulence Flux inequality for multi-layer symmetric QG model Formulation Dissipation rate spectra symmetric streamfunction dissipation Flux inequality for two-layer QG model asymmetric streamfunction dissipation differential diffusion extrapolated Ekman term

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, 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 The Jacobian term J(ψ,ζ) describes the advection of ζ by ψ, and is defined as J(a, b) = a b x y b a x y.

5 Energy and enstrophy spectrum. I Two conserved quadratic invariants: energy E and enstrophy G defined as E(t) = 1 ψ(x, y, t)ζ(x, y, t) dxdy G(t) = 1 ζ 2 (x, y, t) dxdy. 2 2 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) H(k k 0 ) = dx 0 dk 0 R 2 R 4π 2 exp(ik 0 (x x 0 ))a(x 0 ) 2 Filtered inner product: a, b k = d dk R 2 dx a <k (x)b <k (x)

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 G(k) k = D E (k)+f E (k) = D G (k)+f G (k) In two-dimensional turbulence, the energy flux Π E (k) and the enstrophy flux Π G (k) are constrained by for all k not in the forcing range. k 2 Π E (k) Π G (k) 0

7 Energy and enstrophy spectrum. III Assuming a forced-dissipative configuration at steady state, and it follows that: k 2 Π E (k) Π G (k) = Π E (k) = Π G (k) = + k + k + k D E (q)dq, D G (q)dq, [k 2 D E (q) D G (q)]dq = + For the case of two-dimensional Navier-Stokes turbulence, D G (k) = k 2 D E (k), therefore (k, q) = k 2 D E (q) D G (q) = (k 2 q 2 )D E (q) 0 so we get k 2 Π E (k) Π G (k) < 0. k (k, q)dq.

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

9 Cascade Directions Proofs that energy goes mostly upscale in 2D turbulence: Fjørtøft (1953): Famous wrong argument. Merilee and Warn (1975): Noticed error in Fjørtøft Eyink (1996): Correct argument, but assumes inertial ranges. Gkioulekas and Tung (2007): Flux inequality for 2D Navier-Stokes Linear Cascade Superposition hypothesis: E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Sys. B 5, E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Sys. B 5, Under coexisting downscale cascades of energy and enstrophy: E(k) C 1 ε 2/3 uv k 5/3 + C 2 η 2/3 uv k 3 with η uv the downscale enstrophy flux and ε uv the downscale energy flux. Transition wavenumber: k t = η/ε.

10 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

11 Nastrom-Gage spectrum

12 Tung and Orlando spectrum

13 Generalized multi-layer model. I. Consider the generalized form of an n-layer model: q α t + J(ψ α, q α ) = d α + f α d α = β D αβ ψ β ˆq α (k, t) = β L αβ ( k )ˆψ β (k, t) We consider two types of forms for the dissipation term d α : Let Dα(k) be the spectrum of the operator D α Streamfunction dissipation: d α = +D αψ α = D αβ (k) = δ αβ D β (k) Symmetric streamfunction dissipation: (all layers have the same operator) d α = +Dψ α = D αβ (k) = δ αβ D(k)

14 Generalized multi-layer model. II 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) G(k) = α q α, q α k = αβγ L αβ (k)l αγ (k)c βγ (k) with C αβ (k) = ψ α,ψ β k. The energy dissipation rate spectrum D E (k) and the layer-by-layer potential enstrophy dissipation rate spectra D Gα (k) are given by D E (k) = 2 αβ D αβ (k)c αβ (k), D Gα (k) = 2 βγ L αβ (k)d αγ (k)c βγ (k).

15 Multi-layer QG model. I. In a multi-layer quasigeostrophic model, the relation between q α and ψ α reads q 1 = 2 ψ 1 +µ 1 k 2 R (ψ 2 ψ 1 ) q α = 2 ψ α λ α k 2 R (ψ α ψ α 1 )+µ α k 2 R (ψ α+1 ψ α ), for 1 < α < n q n = 2 ψ n λ n k 2 R (ψ n ψ n 1 ) Here λ α,µ α are the non-dimensional Froude numbers given by λ α = h 1 h α ρ 2 ρ 1 ρ α ρ α 1, for 1 < α n µ α = h 1 h α ρ 2 ρ 1 ρ α+1 ρ α, for 1 α < n with h 1, h 2,...,h n, the thickness of layers from top to bottom, in pressure coordinates. For h 1 = h 2 =... = h n, we note that λ α+1 = µ α for all 1 α < n.

16 Multi-layer QG model. II. The corresponding matrix L αβ (k) is given by: k 2 µ 1 kr 2, if α = 1 L αα (k) = k 2 (λ α +µ α )k 2 R, if 1 < α < n k 2 λ n kr 2, if α = n We define: L α,α+1 (k) = µ α k 2 R, for 1 α < n L α 1,α (k) = λ α k 2 R, for 1 < α n γ α (k, q) = k 2 + β L αβ (q) = k 2 q 2 < 0, for k < q We consider the case where h α = h for all layers. Then, L αβ (k) is symmetric, and our theoretical framework becomes applicable.

17 Multi-layer QG model. III. Let U α (k) = ψ α,ψ α k 0 for 1 α n. (streamfunction spectrum) Recall that: k 2 Π E (k) Π G (k) = + k [k 2 D E (q) D G (q)]dq = + k (k, q)dq. PROPOSITION 1: In a generalized n-layer model, under symmetric streamfunction dissipation d α = +Dψ α with spectrum D(k), we assume that L αβ (q) 0 when α β, and L αβ (q) = L βα (q), and γ α (k, q) 0 when k < q for all α. It follows that: (k, q) D(q) γ α (k, q)u α (q) 0 α It follows that under symmetric streamfunction, the flux inequality is satisfied. The case d α = Dq α presents unexpected challenges, and may violate the flux inequality in models with more than 2 layers.

18 The two-layer model. I The governing equations for the two-layer quasi-geostrophic model are ζ 1 t ζ 2 t T t + J(ψ 1,ζ 1 + f) = 2f h ω + d 1 + J(ψ 2,ζ 2 + f) = + 2f h ω + d [J(ψ 1, T)+ J(ψ 2, T)] = N2 ω + Q 0 f 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 the dissipation terms. The three equations are situated in three layers: ψ1 : At 250Atm, top streamfunction layer T: At 500Atm, temperature layer. ψ2 : At 750Atm, bottom streamfunction layer

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

20 The two-layer model. III We use the following asymmetric dissipation configuration: d 1 = ν( 1) p+1 2p+2 ψ 1, (1) d 2 = (ν + ν)( 1) p+1 2p+2 ψ 2 ν E 2 ψ s. (2) Differential hyperdiffusion: ν > 0. The Ekman term is given in terms of the streamfunction ψ s at the surface layer (p s = 1Atm) which is linearly extrapolated from ψ 1 and ψ 2 and it is given by ψ s = λψ 2 +µλψ 1, with λ and µ given by 0 < p 1 < p 2 < p s = 1 < µ < 0 λ = p s p 1 p 2 p 1 and µ = p 2 p s p s p 1. (3)

21 The two-layer model. IV The dissipation term configuration corresponds to setting the generalized dissipation operator spectrum D αβ (k) equal to [ ] D1 (k) 0 D(k) =, (4) µd(k) D 2 (k)+d(k) with D 1 (q), D 2 (q), and d(q) given by D 1 (k) = νk 2p+2 and D 2 (k) = (ν + ν)k 2p+2 and d(k) = λν E k 2. (5) The nonlinearity corresponds to an operator L αβ with spectrum L αβ (k) given by [ ] a(k) b(k) L(k) =, (6) b(k) a(k) with a(k) and b(k) given by a(k) = k 2 + k 2 R /2 and b(k) = k 2 R.

22 The two-layer model. V PROPOSITION 2:Assume streamfunction dissipation with both differential small-scale dissipation and extrapolated Ekman dissipation with 1 < µ < 0. Assume also that k 2 a(q) b(q) < 0, and b(q) < 0, and D(q) D 2 (q) D 1 (q) 0, and also that D 1 (q), D(q), and d(q) satisfy 2D 1 (q)+µd(q) D(q)+(µ+1)d(q) > Then it follows that (k, q) 0. b(q) k 2 a(q) b(q). (7) For dissipation term configurations, choose one of For µ = 0 and λ = 1: standard Ekman at ps = p 2. For µ = 1/3 and λ = 3/2: extrapolated Ekman at ps = 1Atm. and also one of For ν > 0: differential small-scale diffusion For ν = 0: no differential small-scale diffusion

23 The two-layer model. VI For µ = 0 and λ = 1: standard Ekman at p s = p 2 ; and ν = 0: no differential diffusion ν E 4νq 2p q2 k 2 kr 2 = (k, q) 0. For µ = 1/3 and λ = 3/2: extrapolated Ekman at p s = 1Atm; and ν > 0: with differential diffusion 0 < νq2p +ν E 4νq 2p < q2 k 2 ν E kr 2 = (k, q) 0. (8) Note that the hypothesis requires that νe < 4νq 2p (thank extrapolated Ekman) Increasing eitherνe or ν indicates a tendency towards violating the flux inequality. For ν > 0, the LHS of hypothesis will approach ν/(4ν) and remain bounded for large wavenumbers q (thank differential diffusion)

24 The two-layer model. VII For µ = 0 and λ = 1: standard Ekman at p s = p 2 ; and ν > 0: with differential diffusion νq 2p +ν E 4νq 2p < q2 k 2 k 2 R = (k, q) 0, (9) Hyperbolic blow-up is no longer possible. For ν > 0, the LHS of hypothesis will still approach ν/(4ν) and remain bounded for large wavenumbers q For µ = 1/3 and λ = 3/2: extrapolated Ekman at p s = 1Atm; and ν = 0: no differential diffusion ν E 4νq 2p < q 2 k 2 kr 2 = (k, q) 0. (10) +(q2 k 2 ) LHS vanishes with increasing q but RHS remains bounded. Condition is still tighter.

25 The two-layer model. VIII Sufficient conditions can be derived in terms of the streamfunction spectra U 1 (q) = ψ 1,ψ 1 k, U 2 (q) = ψ 2,ψ 2 k, and C 12 (q) = ψ 1,ψ 2 k Arithmetic-geometric mean inequality: 2 C 12 (q) U 1 (q)+u 2 (q) PROPOSITION 3: Assume streamfunction dissipation with ν = 0, and standard Ekman (i.e. µ = 0 and λ = 1) with d(k) > 0 and k 2 a(q) b(q) < 0 and b(q) < 0 and C 12 (q) U 2 (q). Then, it follows that (k, q) 0. PROPOSITION 4: Assume that b(q) < 0 and k 2 a(q) b(q) < 0. Assume also streamfunction dissipation with both differential small-scale dissipation and extrapolated Ekman dissipation with 1 < µ < If C 12 (q) 0, then (k, q) If C 12 (q) min{u 1 (q), U 2 (q)} and U 1 (q)+µu 2 (q) 0, then (k, q) 0. PROPOSITION 5: Assume that k 2 a(q) b(q) < 0 and b(q) < 0. We also assume streamfunction dissipation with extrapolated Ekman dissipation with 1 < µ < 0 and symmetric small-scale dissipation with D 1 (q) = D 2 (q). It follows that if C 12 (q) min{u 1 (q), U 2 (q)} then (k, q) 0.

26 Conclusions Under symmetric dissipation, the flux inequality is satisfied unconditionally for multi-layer QG models Under asymmetric dissipation, the flux inequality is satisfied only when the asymmetry satisfies restrictions. The restrictions given are sufficient but not necessary.

27 Thank you!