Geostrophic turbulence and the formation of large scale structure

Geostrophic turbulence and the formation of large scale structure Edgar Knobloch University of California, Berkeley, CA 9472, USA knobloch@berkeley.edu http://tardis.berkeley.edu Ian Grooms, Keith Julien, Antonio Rubio, Geoff Vasil and Jeff Weiss IPAM, 31 October 214 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 1 / 36

Motivation: Balanced flows (hydrostatic) Rotational constraint Ro = U f L 1 Stable stratification Fr = U NL 1 Wide aspect ratios H L 1 Modis image of Gulf Stream, SST 4/18/25, NASA. Observations inform mathematical approximations (QG). Sunday, September 3, 12 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 2 / 36

Motivation: Balanced flows (nonhydrostatic) Rotational constraint Ro = U f L 1 Weak stratification Fr = U NL = O(1) Columnar flows H L > 1 Jones and Marshall (JPO 1993) Observations inform mathematical approximations (NHBGE). Sunday, September 3, 12 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 3 / 36

Rotating convection with H = 6cm, T = 2.6 deg C, Ω = 14.3rpm. From Sakai JFM 333, 85 (1997) Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 4 / 36

Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 5 / 36

Life in an asymptotic wedge... Sunday, September 3, 12 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 6 / 36

NHBGE captures large Ta - low Ro regime for Rayleigh-Benard convection Sunday, September 3, 12 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 7 / 36

Basic equations where D t u + 1 1 ẑ u = P p ΓT ẑ + Ro Re 2 u D t T = 1 Pe 2 T u =, Ro = U 2ΩL, P = P ρ U 2, UL Re = ν, UL Pe = κ, Γ = gα T L U 2 and L and U are arbitrary horizontal length and velocity scales to be selected depending on the process of interest. We suppose that Ro ɛ 1 and H/L = ɛ 1 with t ɛ 2 t + τ, x ɛ 1 x, y ɛ 1 y, z ɛ 1 z + Z. The slow spatial scale Z is required by the boundary conditions. Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 8 / 36

Asymptotics An asymptotic expansion in ɛ with u v W = O(1), and T = T + ɛθ leads at O(ɛ 1 ) to geostrophic balance: ẑ u = p, u =, u = ( ψ y, ψ x ), ψ p At O(1) the vertical vorticity ω 2 ψ and vertical velocity W satisfy t ω + J[ψ, ω] Z W = Re 1 2 ω t W + J[ψ, W ] + Z ψ = Γθ + Re 1 2 W Fluctuating buoyancy equation at O(ɛ 1 ): Mean buoyancy equation at O(ɛ 1 ): t θ + J[ψ, θ] + W Z T = Pe 1 2 θ τ T + Z W θ = Pe 1 ZZ T. Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 9 / 36

Asymptotics These equations constitute a closed reduced system of equations referred to as NHBGE. In these equations the overbar denotes horizontal average, followed by an average over fast time, and J(f, g) f x g y f y g x. The boundary conditions are w = ψ Z = θ =, T = 1, on Z =, w = ψ Z = θ =, T =, on Z = 1 corresponding to stress-free boundaries. The resulting inviscid dispersion relation for modes of the form exp i(λt + k x + k z z) is λ 2 reduced = k2 z k 2, cf. λ 2 NS = k 2 z k 2 + E 2/3 k 2 z Thus fast inertial waves on O(E 1/3 H) vertical scales are filtered out. Remark: With the choice L/H = E 1/3 corresponding to the preferred linear theory scale at large rotation rates the expansion parameter is ɛ = E 1/3. Here E ν/2ωh 2 is the Ekman number. Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 1 / 36

Regimes described by the reduced system The reduced equations describe four distinct dynamical regimes, depending on the values of the Rayleigh number Ra gα TH 3 /νκ and the Prandtl number σ ν/κ. Cellular convection (C) Convective Taylor columns (T) Convective plumes (P) Geostrophic turbulence (G) 16 14 12 1 RaE 4/3 8 6 4 3 2 1 G G G G G G G P P P P P P P P P P T P P T T T P P T T T P P T T T C C C C C 1 3 7 15 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 11 / 36 σ P P P P T T P P T T T T

Volume renders of θ for RaE 4/3 = 2, 4, 8, 12, 16 and σ = 7 (left) and RaE 4/3 = 16 and σ = 1, 3, 7, 15, (right) Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 12 / 36

Geostrophic turbulence Movie for RaE 4/3 =1, Pr=1 Sunday, September 3, 12 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 13 / 36

Geostrophic turbulence Volume render of θ for RaE 4/3 = 16 and σ =.3 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 14 / 36

and this relation is not closed. Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 15 / 36 An exact result The reduced system saturates in a statistically steady state with W θ Z T = 1 σ θ 2 Z T = σw θ Nu, where Nu is the Nusselt number. Thus Z T = 1 2 Nu ± 1 [ ] 1/2 Nu 2 4 θ 2 2. Thus the thermal dissipation rate is bounded: θ 2 Nu 2 /4. Since the sign refers to the thermal boundary layer and the + sign to the bulk the transition between these two regions occurs where θ 2 = Nu 2 /4. At this location, hereafter Z = δ ɛ, we have equipartition between conduction and convection: Z T = σw θ = Nu/2. Remark: In nonrotating RB convection we have instead W θ Z T + 1 2 Z W θ 2 = 1 σ θ 2 1 σ ( Z θ) 2

Boundary layer instability We use the rescaled equations to analyze the stability of the mean thermal profile as determined from a nonlinear two-point eigenvalue problem for the Nusselt number Nu given R, and look for the onset of convective instability in the boundary layer. 25 2 SM BL 1.8 1.9 15 Nu 1.6 z.4.8 z.7 5 1 2 3 4 5 RaE 4/3.2 T SM T BL -1 -.8 -.6 -.4 -.2 T.6 ζ w θ.5-15 -1-5 5 1 15 In contrast to RB convection the boundary layer becomes convectively unstable already at small supercriticality. Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 16 / 36

Boundary layer structure in geostrophic turbulence We postulate a scaling relation for the structure of the boundary layer of the form τ = R bτ t, η = R bη Z, λ = R bλ x, ψ = R bψ Ψ, ω = R b ψ+2 bλ Ω, w = R bw W, θ = R bθ Θ, Z T = R bη η T ɛ, where R RaE 4/3 R c. Our simulations indicate that all terms remain in play as R increases, leading to the relations λ = ŵ = s, η = 3s, τ = Ω = 2s, = 1 s, ψ =, θ = 3s 1, where s >. Thus Nu R 4s 1 and we have a one parameter family of scaling solutions. Comparison with the measured turbulent scaling law Nu R 3/2 implies that s = 5/8 and yields predictions of all other quantities. Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 17 / 36

Measured turbulent scaling law: Nu σ 1/2 R 3/2.1 (Nu-1)R -3/2.1 Nu-1~R 2.2 Nu-1~2(R-R c )/R c Nu-1~σ -1/2 R 3/2 σ=.3 σ=.5 σ=.7 σ=1 1 2 4 8 12 16 R=RaE 4/3 Edgar Compensated Knobloch (UC Berkeley) plot of Nu Rapidly 1 as Rotating a function Convectionod R. The curves 31 ctober for214 σ 18 1 / 36

Theoretical interpretation We postulate a relation of the form Nu 1 C(σ)Ra α E β valid for Ra Ra c. In nonrotating convection.28 α.31 (Grossmann and Lohse (2), Xu et al (2), Ahlers and Xu (21)) unless the boundary layers become turbulent when α.38. Here the mean temperature gradient at midheight decreases to zero as Ra increases. Thus heat transport is limited by the efficiency of the boundary layers. In rapidly rotating convection the mean temperature gradient at midheight saturates as Ra increases and heat transport is limited by the efficiency of the turbulent interior. For Ra c Ra Ra t we expect Nu to depend only on Ra/Ra c. Thus Nu 1 (RaE 4/3 ) α. If we suppose that the heat flux is independent of microscopic diffusion coefficients, then α = 3/2 (β = 2), and Nu 1 C 1 σ 1/2 Ra 3/2 E 2 with C 1.4 ±.25 from simulations. Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 19 / 36

Evidence that heat flux in bulk determines Nu 1 1 Nu ε θ int ε θ bl 83% 81% 8% 78% 78% - z T z=1/2.1 σ=1 σ=3 σ=7 σ=15 σ= 1 2 4 8 1216 RaE 4/3 1 1 59% 41% 74% 17% 22% 2%19% 69% 22% 26% 31 1 2 4 8 1216 R (a) Midheight gradient. (b) Contributions (in percentage form) to Nu = ( Z T ) 2 + θ 2 from the bulk (Eθ int ) and the boundary layers ). Heat transport is limited by the efficiency of the turbulent interior. (E BL θ Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 2 / 36

Evidence for s = 5/8 1 σ=.3 σ=.5 σ=.7 σ=1 1 σ=.3 σ=.5 σ=.7 σ=1.1 1 2 4 8 1216.1 1 2 4 8 12 16 R Compensated plots of (a) θe 1/3 R 7/8, (b) W σe 1/3 R 5/8 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 21 / 36

Evidence for s = 5/8 5 1 1 σ=.3 σ=.5 σ=.7 σ=1.5 σ=.3 σ=.5 σ=.7 σ=1 1 2 4 8 1216 R 1 2 4 8 1216 R Compensated plots of (a) boundary layer width ηr 15/8, (b) associated temperature drop δt R 3/8 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 22 / 36

Transition to nonrotating scaling As Ra increases at fixed rotation rate we expect a transition to nonrotating scaling. This is a consequence of increasing convective Rossby number Ro conv E Ra/σ. We conjecture that the transition to nonrotating scaling is triggered by the loss of geostrophic balance in the boundary layer, i.e., when the boundary layer Rossby number Ro loc E loc Ra 1/2 loc 1. Here E loc = Eη 2 and Ra loc = Ra( T loc / T )η 3 = Ra Nu η 4, where η R 3s H is the boundary layer width. Since Nu R 4s 1 and Ra = R E 4/3, it follows that E loc = E R 6s ε 3 and Ra loc = E 4/3 R 8s ε 4, where ε E 1/3 R 2s. It follows that Ro loc ε and hence that Ro loc 1 when E 1/3 R 2s 1. Since s = 5/8 this occurs when Ra reaches Ra t E 8/5, or equivalently when Ro = Ro t E 1/5. Since E 1 the transition Rossby number Ro t 1, i.e., the transition from the 3/2 scaling law occurs in the rapidly rotating regime. There is some evidence for the validity of these predictions as shown next. Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 23 / 36

Nusselt number scaling (from King et al 212) Ro t! as E! King et al NATURE (29) and King et al JFM (212) Sunday, September 3, 12 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 24 / 36

Depth averaged vorticity Sunday, September 3, 12 Julien et al, GAFD 16, 392 428 (212) Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 25 / 36

Julien et al, GAFD 16, 392 428 (212) Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 26 / 36

Evolution of barotropic mode Sunday, September 3, 12 Julien et al, GAFD 16, 392 428 (212) Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 27 / 36

Spontaneous formation of large scale vortices Rubio et al, PRL 112, 14451 (214) Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 28 / 36

(a) t = 1, (b) t = 1, (c) t = 37.5, (d) t = 1 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 29 / 36

1 2-3 1-5/3 K bc (k), K bt (k) 1-2 1-4 1-6 1-8 t =1 t = 1 t = 1 k c -3 1 1 1 1 2 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 3 / 36

Barotropic/baroclinic vorticity equations Let ω = ω + ω, ψ = ψ + ψ, where... denotes a depth average. Then and ω t + J[ ψ, ω ] + J[ψ, ω ] = 2 ω ω t + J[ ψ, ω ] + J[ψ, ω ] + J[ψ, ω ] DW = 2 ω Thus the baroclinic-baroclinic term acts as a source term for the barotropic mode. Without this term the barotropic flow is identical to 2D hydrodynamics and an inverse energy cascade to large scales is expected. In fact this is so even in the presence of this term, and leads to a k 3 pile up at large scales, eg., Smith and Waleffe, Phys. Fluids 11, 168 (1999). However, the fluctuation equation is fully 3D and hence exhibits the usual energy spectrum expected from Kolmogorov theory. k 5/3 The emergence of a coherent structure from a turbulent state has been termed spectral condensation (PRL 95, 26391 (25); 11, 19454 (28); 112, 14451 (214)). Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 31 / 36

Formation of large scale vortices: spectral description The growth of barotropic kinetic energy at horizontal wave number k obeys t K bt (k) = T k + F k + D k, where T k pq T kpq and F k pq F kpq represent, respectively, the symmetrized transfer of energy between Fourier modes within the barotropic component and the transfer of energy between baroclinic and barotropic modes; D k k 2 K bt is the viscous dissipation of the barotropic mode. Moreover, T kpq = b pq Re [ ψ k ψ p ψ q ] δ k+p+q,, F kpq = b pq Re [ ψ k ψ pψ q ] δ k+p+q,, b pq = b qp 1 2 (p2 q 2 )(p x q y p y q x ). These transfer rates can be computed from the simulations. Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 32 / 36

Time evolution of baroclinic and barotropic modes 1 2 I II III K bc (t), K bt (k,t) 1 1-2 1-4 bc bt 1 bt 2 bt 3 bt 4 1-1 1 1 1 1 2 Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 33 / 36

Transfer rates T k and F k at three successive times Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 34 / 36

Unreasonable effectiveness of asymptotics (E = 1 7 ) RAPIDLY ROTATING CONVECTION: DNS SHOW REGIMES SIMILAR TO REDUCED EQUATIONS 2 Results 2.2 Convective-Taylor-Column Regime 2.1 Columnar Regime Convective Taylor Columns Cellular Figure 1: Side- and topview of the temperatue-deviation δt in the columnar regime; red: positive deviation, blue: negative deviation 2.3 Plume Regime Kurtosis of δt ; E = 1 7, P r = 1, E 4/3 Ra = 11 1 2.4 GT Regime Kurtosis of δt ; E = 1 7, P r = 15, E 4/3 Ra = 5 1 Skewness of δt ; E = 1 7, P r = 15, E 4/3 Ra = 5 1 1.9.9.8.8.7.7.7.6.6.6.6.5.4.3.5.4.5.4.3.3.2.2.2.1.1 2 4 6 8 1 Kurtosis Box Height.9.8.7 Box Height.9.8 Box Height Box Height Figure 5: Side- and topview of the temperatue-deviation δt in the convective taylor-column regime; red: positive deviation, blue: negative deviation Skewness of δt ; E = 1 7, P r = 1, E 4/3 Ra = 11.2.2.5.4.3.2.1.4.1.4 Skewness 5 1 15 1 2 Kurtosis Figure 2: kurtosis (left) and skewness (right) profiles of the temperature deviation δt in the columnar regime Plumes Figure 9: Side- and topview of the temperatue-deviation δt in the plume regime; red: positive deviation, blue: negative deviation.5.8.7.7.7.6 2.6.4.6.5.5 4.4.4.3.3 4/3.2.2.1.1 Box Height.9.8 Box Height.9.9.8 Box Height Box Height Skewness of δt ; E = 1 7, P r = 1, E 4/3 Ra = 9 1.9.8.5 1 Geostrophic turbulence 1 Skewness of δt ; E = 1 7, P r = 3, E 4/3 Ra = 7 1.7.5 Figure 13: Side- and topview of the temperatue-deviation δt in the geostrophic turbulence regime; red: positive deviation, blue: negative deviation Kurtosis of δt ; E = 1 7, P r = 1, E 4/3 Ra = 9 Kurtosis of δt ; E = 1 7, P r = 3, E 4/3 Ra = 7 1 Skewness Figure 6: kurtosis (left) and skewness (right) profiles of the temperature deviation δt in the convective taylor column regime.6.5.4.3 Cellular regime: Pr = 1, E Ra = 11; CTC Regime: Pr = 15, E 4/3 Ra = 15; Plume Regime: Pr = 3, E 4/3 Ra = 5; GT Regime: Pr = 1, E 4/3 Ra = 9.3.2.1 2 4 6 Kurtosis 8 1 1.1 2 4 6 Kurtosis.5 8 1.4.2.2.4 Skewness 1 Skewness Figure 1: kurtosis (left) and skewness (right) profiles of the teperature deviation δt in the plume regime Edgar Knobloch (UC Berkeley).2.5 Figure 14: kurtosis (left) and skewness (right) profiles of the teperature deviation δt in the plume regime Rapidly Rotating Convection 31 ctober 214 35 / 36

Conclusions Heat transport in rapidly rotating convection (E 1): At large Ra the Nusselt number scales as Nu 1 C 1 σ 1/2 Ra 3/2 E 2 with C 1.4 ±.25. This scaling is a consequence of inefficient heat transport in the turbulent bulk This is a result of the saturation of the midheight mean temperature gradient as Ra increases The scaling is a consequence of the scaling behavior of the boundary layers at large Ra Transition from this scaling occurs when the local Rossby number in the boundary layer becomes of order unity, i.e., at Ra t E 8/5 as E, or equivalently when Ro = Ro t E 1/5. Geostrophic turbulence in this system is unstable to a large scale barotropic (vortical) mode The spectra of the barotropic and baroclinic components of the HKE are consistent with the Kraichnan and Kolmogorov pictures (2D conserves energy and enstrophy, 3D conserves energy only) Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 36 / 36

Conclusions (ctd) Baroclinic-baroclinic forcing injects energy directly to the largest scales despite the small scale nature of the baroclinic fields Certain aspects of these predictions have been confirmed in simulations of the primitive equations by Stellmach (212), Favier et al (PF 26, 9665, 214) and Guervilly et al (JFM 758, 47, 214). The fact that a fully 3D turbulent flow exhibits a large scale instability may have geophysical and astrophysical implications References: K Julien, E Knobloch and J Werne, Th. Comp. Fl. Dyn. 11, 251 261 (1998) M Sprague, K Julien, E Knobloch and J Werne, JFM 551, 141 174 (26) K Julien, E Knobloch, R Milliff and J Werne, JFM 555, 233 274 (26) I Grooms, K Julien, J B Weiss and E Knobloch, PRL 14, 22451 (21) K Julien, A M Rubio, I Grooms and E Knobloch, GAFD 16, 392 428 (212) K Julien, E Knobloch, A M Rubio and G M Vasil, PRL 19, 25453 (212) A M Rubio, K Julien, E Knobloch and J B Weiss, PRL 112, 14451 (214) Edgar Knobloch (UC Berkeley) Rapidly Rotating Convection 31 ctober 214 37 / 36