Energy and potential enstrophy flux constraints in quasi-geostrophic models Eleftherios Gkioulekas University of Texas Rio Grande Valley August 25, 2017
Publications 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. K.K. Tung (2004), J. Atmos. Sci., 61, 943-948. E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Sys. B 5, 79-102 E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Sys. B 5, 103-124. E. Gkioulekas and K.K. Tung (2007): J. Fluid Mech., 576, 173-189. E. Gkioulekas and K.K. Tung (2007): Discrete Contin. Dyn. Syst. Ser. B, 7, 293-314 E. Gkioulekas (2012): J. Fluid Mech. 694, 493-523 E. Gkioulekas (2014): Physica D 284, 27-41
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
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.
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)
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
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.
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
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, 79-102 E. Gkioulekas and K.K. Tung (2005), Discr. Cont. Dyn. Sys. B 5, 103-124. 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.
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
Nastrom-Gage spectrum
Tung and Orlando spectrum
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)
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).
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.
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.
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.
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 2 + 1 2 [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
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.
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)
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.
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
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)
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.
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 < 0. 1. If C 12 (q) 0, then (k, q) 0. 2. 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.
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.
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