Stephan Stellmach, University of Münster / Germany

Transcription

1 NUMERICAL SIMULATION OF DOUBLE DIFFUSIVE CONVECTION IN OCEANS, STARS AND PLANETARY INTERIORS Stephan Stellmach, University of Münster / Germany with: Pascale Garaud, Adienne Traxler, Nic Brummell, Timour Radko, Erica Rosenblum, Giovanni Mirouh, Timo Kleinwechter, Jan Verhoeven, Matthias Lischper

2 MOTIVATION: WHY DO WE CARE? Many natural objects host stably stratified region. In the dynamo context, for example Mercury s core [Christensen (26), Christensen & Wicht (28), Manglik et. al. (2)] Saturn s metallic hydrogen region [Christensen & Wicht (28), Leconte & Cahbrier (22)] Earth s core? [e.g. Lister & Buffett (998), Braginsky (984)] In these regions, both compositional and thermal gradients play a role, typically with opposite effects on stability. In such cases, may occur. double-diffusive convection

3 DOUBLE-DIFFUSIVE INSTABILITIES Double-diffusive Instabilities occur in stably stratified systems if density depends on at least two components (typically temperature and composition) with opposite effects on stability and if these diffuse at different rates (temperature fast, composition slow). Fingering regime thermohaline regime in astrophysics Diffusive regime semi-convective regime in astrophysics z ρ T C z C T ρ Chemical field perturbations Chemical field perturbations

4 Double-Diffusive Instabilities II Important Parameters: Pr = ν κ τ = κ S κ T R ρ = α T ( T ( T ) ad ) α S S no chemical gradient overturning convection T =( T ) ad stable system z T unstable chemical gradient stable chemical gradient R ρ = overturning convection diffusive regime R ρ = Pr+ Pr+ τ fingering regime stable system R ρ = R ρ = τ stable system z T z T

5 MOTIVATION: WHY DO WE CARE? Many natural objects host stably stratified region. In the dynamo context, for example Mercury s core [Christensen (26), Christensen & Wicht (28), Manglik et. al. (2)] Saturn s metallic hydrogen region [Christensen & Wicht (28), Leconte & Cahbrier (22)] Earth s core? [e.g. Lister & Buffett (998), Braginsky (984)] In these regions, both compositional and thermal gradients play a role, typically with opposite effects on stability. In such cases, double-diffusive transport may be important. Other examples in Nature, where this is (or might be) the case: Classical example: Ocean s thermocline Planet-bearing stars [(Vauclair (24), Garaud (2)] Exoplanets? Jupiter? [Chabrier & Baraffe (27), Leconte & Charbrier, 22] low-mass RGB stars [(Charbonnel & Zahn 27; Stancliffe 2; Denissenkov 2]

6 SOMETIMES, SMALL SCALE TRUBULENCE DRIVES LARGE-SCALE DYNAMICS... Movie!

7 WHAT KIND OF LARGE-SCALE FLOWS CAN BE DRIVEN BY DOUBE DIFFUSION? Fingers: gravity waves conv. layers Intrusions

8 WHAT KIND OF LARGE-SCALE FLOWS CAN BE DRIVEN BY DOUBE DIFFUSION? Fingers: Oscillatory modes: gravity waves conv. layers Intrusions

9 GOALS / QUESTIONS Thorough understanding of the mechanisms behind large-scale dynamics Reliable prediction of which large-scale features are expected in different systems and situations What do we expect in the ocean, in giant planets, in stars? Material properties differ strongly from system to system, e.g. Ocean Giant planets Stellar interiors Pr = ν κ τ = κ S κ T ~7 O( -2 ) O( -6 ) ~. O( -2 ) O( -6 ) How do the large-scale features evolve with time in the systems? (Merging,...) Is there a statistically stationary equilibrium state? If so, what are its properties? Spatial scale? Heat, material and momentum transport? What are the implications for planetary evolution, interior models, dynamos,...?

10 GOAL: 3D MULTI-SCALE SIMULATIONS Perfrom simulations that resolve i) the small-scale double-diffusive instabilities ii) the large scale structure dynamics.

11 MODELS & NUMERICAL METHODS triply periodic Boussinesq model, driven by prescribed horizontal vertical gradients in T,C (very simple but very efficient!) fully spectral primitive variables IMEX AB3/BDF3 time stepping or Low-Storage 4th order Runge-Kutta with CN2 substeps or integrating factors classical Patterson-Orzang algorithm, operation count ~ log2(n) N 3 Plane-layer (Rayleigh-Bénard) model, technique similar to spectral spherical dynamos! Rotation, magnetic fields, density stratification (anelastic equations by Gough (969)) Poloidal/toroidal formulation (derived from curl and double-curl) Fourier/Chebyshev-tau scheme or Fourier/high-order FD scheme (Chebyshev Matrices become ill-conditioned at high resolution > O()!) IMEX AB/BDF time stepping, implicit Coriolis Force (trivial here, since no coupling over m!) operation count in all cases ~ log 2 (N) N 3

12 PARALLELIZATION APPROACHES Classical domain decomposition Transpose based spectral transform z x =% 6#,59= z y x x z y y! ) +

13 SCALING ON IBM BLUE GENE/L Strong Scaling speedup x x x 384 ideal # ( mpi tasks ) time per time step [s] x x x 384 /x # ( mpi tasks ) Weak Scaling time per time step [s] x x x 384 linear data volume per task (GPU) time per time step [s] / log 2 (N max ) x x x 384 linear data volume per task (GPU)

14 FIRST 3D DNS OF LAYER FORMATION R=., tau=/3, Pr =7 536x536x52 grid points Simulation reveals layering! Layering can be traced down to a particular mean-field instability proposed first by Radko (23) Density Concentration! (Concentration) Fingers Waves Layers from Stellmach, Traxler, Garaud, Brummell & Radko (2)

15 FINGERS CAN T GENERATE LAYERS IN GIANT PLANETS AND STARTS rescaled heat flux (Nu T -)/(! 3/2 Pr /2 ) Set Set 2 Set 3 Set 4 Set 5 Set 6 Traxler, Garaud & Stellmach (2) r rescaled compos. flux (Nu µ -)/(Pr/!) / r Set Set 2 Set 3 Set 4 Set 5 Set 6 Set Pr τ /3 /3 2 /3 / 3 / /3 4 / / 5 / /3 6 /3 / Computed flux parameterizations show that the instability which generates layers in the ocean has a negative growth rate in stars and giant planets!

16 NuT = (.7 ±.5) Pr τ τ ( r) R (7) BUT LAYERS IN GIANT PLANETS & STARS CAN BE FORMED IN THE SEMI-CONVECTIVE REGIME: where the power index ξ.25 ±.5. The large uncertainty on the index comes from the uncertainty on the measurements themselves, compounded with the short range of Pr/τ values available. However, the exact value of ξ does not matter much for the γtot predictions. Figure?? compares out empirical fit for NuT given by equation (7) to the actual data. Mirouh, Garaud, Stellmach, Traxler & Wood (22) NuT -,theory 22 Pr = o =.3 Pr = o =. Pr = o =.3 Pr = o =. up the limiting diffusive value at r =. This which explains the observed oblique asymptote Pr =.3, o =. remark, incidentally, also explains why noo =layers can form in the fingering regime. There, Pr =.,.3 Pr =.3, o =.3 fingering a similar argument applies and implies that = τ 2. This limiting Pr =.3, o =.3γtot = τ Rc. diffusive. value pulls up the end of the γtot (R ) curve to very large value, to such an extent Nuof T-,data that it prevents the presence a region where γ descreases with R. Mean-field parameterization is shown to follow certain 7. in Table XXX. Comparison between expression (7) and the data presented scaling laws. Pr = o =. tot Fig. Pr = τ =.3.9 Pr = o =.3 Pr = o =. Pr =.,()o we =.3 () and Pr =.3, o =.3.8 Using (), can now estimate γtot semi-analytically. Figure??two compares these Generally regimes:.7 with the same data as presented in Figure 6b, for all runs with Pr.. As it turns estimates.6 prediction is only qualitatively good for runs with Pr =.3, with errors up to 4%. out, the Regime : (moderate to high Rρ-) For all.5possible (Pr, τ ) pairs with Pr <.3 on the other hand, the model is quantitatively no layers, negligible transport quite.4 accurate, and adequately predicts both the behavior for small r (relevant to estimate.3 growth timescales) and the position of the minimum (relevant to estimate the the layer -) Regime II: (small R ρ.2 transition between layered and non-layered convection). Fig. Note. that the large r behavior Example of simulation results for Pr = τ =.3, for R =.5 (top row) and R Radko-Instability layers, considerable (bottom Thedescribed figures on above. the leftcauses are snapshots of the compositional perturbation field. is actually very easy to understand and predict regardless of therow). model t =becomes 76 for negligible, the R =.5 case total and t = 46 for R = 5 case. Note the vast difference in transport Indeed,it corresponds to the case where turbulent transport so the amplitude of the perturbations for the two cases: for reference, the total compositional cont buoyancy flux ratio is dominated by the diffusive transport. Mathematically speaking, r atot - see across the domain is µ = 5 for R =.5 and µ = 5 for R = 5. The figures on τ Nu(2) τ ( τ )the corresponding temporal also Rosenblum, Garaud, Traxler & Stellmach µ evolution of the non-dimensional turbulent fluxes wt Large-Scale 3D confirm NuT = Nuµ = γtot = = τ R =right show r+τ (8)simulations wµ. Pr τ data presented in Table Fig. 8. Comparison between the model R prediction for γtot (r) and+the NuT XXX. Runs with Pr =.3 are omitted, as the fit is not as good. predictive power of the theory!

17 FIRST 3D INTRUSION SIMULATIONS Anisotropic interfacial salt fingers Kleinwechter & Stellmach (in preparation) R=2, tau=/6, Pr=7 a=.3 Waves at the diffusive interface saturate the intrusion (similar to 2d) 536x288x96 grid > 2x 6 time steps

18 BEYOND DOUBLE DIFFUSION: SHEAR FLOW GENERATION IN ANESTAIC TURBULENCE Shear flow generation in anelastic rotating turbulence ( grid)

19 RAPIDLY ROTATING CONVECTION: 2 Results DNS SHOW REGIMES SIMILAR TO REDUCED EQUATIONS 2.2 Convective-Taylor-Column Regime 2. Columnar Regime Cellular Figure : Side- and topview of the temperatue-deviation δt in the columnar regime; red: positive deviation, blue: negative deviation 2.3 Plume Regime 2.4 GT Regime Skewness of δt ; E = 7, P r =, E 4/3 Ra = Kurtosis of δt ; E = 7, P r = 5, E 4/3 Ra = 5 Skewness of δt ; E = 7, P r = 5, E 4/3 Ra = Box Height Box Height Box Height Box Height Kurtosis of δt ; E = 7, P r =, E 4/3 Ra = Convective Taylor Columns Figure 5: Side- and topview of the temperatue-deviation δt in the convective taylor-column regime; red: positive deviation, blue: negative deviation Kurtosis Skewness 5 5 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 2 Kurtosis of δt ; E =.9, P r = 3, E 4/3 Skewness of δt ; E = 7, P r =, E 4/3 Ra = Skewness of δt ; E = Ra = 7, P r = 3, E Ra = 7 Geostrophic turbulence 4/3.5 Figure 3: Side- and topview of the temperatue-deviation δt in the geostrophic turbulence regime; red: positive deviation, blue: negative deviation 7 Skewness Figure 6: kurtosis (left) and skewness (right) profiles of the temperature deviation δt in the convective taylor column regime Kurtosis of δt ; E = 7, P r =, E 4/3 Ra = 9 7.5

20 THANK YOU!