Stratified Convection Driven by Internal Heating (a convective amplitudes talk) Nick Featherstone Collaborators: Brad Hindman Mark Miesch Juri Toomre
The Rossby Number typical velocity v Rotational Timescale Ro 2 L Convective Timescale rotation rate typical length scale What is Ro in the Sun? What is v? Why do we care?
Meridional Circulation: Where is the Sun? Ω Ω sun Featherstone & Miesch 2015 Mass Flux Zhao et al. (2012) v lat Convection Simulations Helioseismology Fast Convection Slow Convection High Ro Low Ro Single-celled MC Multi-celled MC
The location of the dynamo may depend on Ro Strong Rotational Influence Weaker Rotational Influence v r Nelson et al. 2013 B Alternative to an Interface Dynamo?
Local Helioseismic Inversion vs. Simulation ASH Simulated velocities are too high?! Hanasoge et al (2010, 2012) How do you get the flux out with such weak flows? Observations How can you drive DR with weak convection?
Convection Models The whole story? Maybe not Ring-Diagram Analysis (Greer et al. 2015) This is exciting! Disagreeing Observations Time-Distance (Hanasoge et al. 2012) Something new under the Sun?
But Something strange might still be going on Simple numerical experiment: Step 1. Take one solar convection model. Step 2. Fix the rotation rate and luminosity Step 3. Decrease diffusivities, so that convection becomes more turbulent and hopefully more solar-like
Differential Rotation & Frogurt 2 0 90 (nhz) 90 (nhz) 60 (nhz) High diffusion (should be bad) - frame - + Fast Equator 0 EQ Fast Poles Low diffusion (should be good) 0 2 0 Featherstone & Miesch 2015
Solar Simulations Commonly Observed Effect Guerrero et al. 2013 See also Kapyla et al. 2014, Featherstone & Miesch 2015 Planetary & Stellar Convection Gastine et al. 2014
Solar Simulations Commonly Observed Effect High-Rayleigh number convection yields: large velocities & anti-solar differential rotation Guerrero et al. 2013 See also Kapyla et al. 2014, Featherstone & Miesch 2015 Planetary & Stellar Convection Gastine et al. 2014
What sets the amplitude of convection motions? Some ingredients of solar convection Density Stratification Radiative heating This is a complicated system Radiative cooling Rotation Magnetism Stable radiative zone/convective overshoot Tachocline Highly turbulent (high Ra & Re)
What sets the amplitude of convection motions? Some ingredients of solar convection Density Stratification Radiative heating Radiative cooling I am a simple man Rotation How Magnetism does the interaction of these bare-bones ingredients Stable radiative affect the zone/convective amplitude of convective overshoot motions? Tachocline Highly Turbulent (high Ra ) Set up series of convection models to investigate
Modeling Convection: Rayleigh-Bénard Setup Cold Plate T z = 1 Temperature Profiles Ra = 2 10 8-2.8 10 13 z = 0 Hot Plate Verzicco & Sreenivasan (2008) Energetics Input: Conduction Throughput: Rayleigh Number Output: Conduction
Modeling Convection: Rayleigh-Bénard Setup Cold Plate z = 1 Temperature Profiles Ra = 2 10 8-2.8 10 13 T z = 0 Hot Plate Verzicco & Sreenivasan (2008) Energetics Input: Conduction Throughput: Rayleigh Number Output: Conduction
Modeling Solar Convection I: Internal Heating Cold Plate Temperature Profiles Ra Internal Heating T Insulating Plate 0.72 1 r/r Energetics Input: Radiative Heating Throughput: Solar Luminosity Output: Conduction
Modeling Solar Convection I: Internal Heating Cold Plate Internal Heating Electrolytic Solution AC Current Insulating Plate Kulacki & Goldstein 1972 YOU CAN ALSO DO THIS IN THE LAB!
Modeling Solar Convection II: Spherical Geometry Cold Plate Insulated Core Internal Heating Hart et al. 1986 YOU CAN ALSO DO THIS IN THE (SPACE) LAB! Strong stratification still tough
The Model: Stratified Convection in Spherical Geometry Physics: Anelastic MHD P P T T Numerics Horizontal Discretization: Spherical harmonics u Wrˆ Zrˆ + F H Rayleigh Convection Code Highly Scalable Open-source 2016 email me: feathern@gmail.com Radial Discretization: Chebyshev Polynomials Time-Stepping: Adams-Bashforth & Crank-Nicolson
The Numerical Experiment: Vary Density Stratification: N : 1, 2, 3, 4 Vary Thermal Diffusivity: : 5x10 11 to 2x10 13 cm 2 s -1 Pr =1 Maintain Shell Aspect Ratio: 3 scale-height cases look like model S Flux-based Ra Ra= g L sun D2 ρ Tc p κ 2 υ Ra in the Sun: 10 22 Measure: Kinetic Energy Ra = 10 5 64 x 192 x 384 v r 53 models v r Ra : 5x10 9 256 x 1536 x 3072
Non-dimensionalization Length Scale Density ρ Temperature T Timescale D Shell Depth V ρdv V dv V TdV V dv τ D2 κ KE 1 2V V ρv2 dv Plot KE KE vs. Ra Kinetic Energy KE ρκ2 2D 2 Rayleigh Number Ra g L sund 2 ρ Tc p κ 3
Non-dimensional Kinetic Energy Non-dimensional View of Kinetic Energy Scaling KE KE ~ Ra0.69 4+ decades in Ra Rayleigh Number 0.69 is very close to 2/3 this is interesting
Kinetic Energy Scaling If KE ~ KE Ra 2/3 Then KE~ g ρ1/2 L sun TDc p 2/3 KE scaling is independent of diffusion Free-fall scaling Kinetic Energy KE ρκ2 2D 2 Rayleigh Number Ra g L sund 2 ρ Tc p κ 3
Dimensional View of Kinetic Energy Scaling Kinetic energy trends toward constant value this is interesting Why?
Why is this interesting? If the kinetic energy is constant, the integral of its power spectrum is conserved. As Ra is increased, high-wavenumber power increases. High-wavenumber power comes at the expense of low-wavenumber power.
Deep Shell Power Spectrum 3 N High Ra Low Ra Mid Ra Spherical Harmonic Degree l
Mid Shell Power Spectrum 3 N High Ra Low Ra Mid Ra Spherical Harmonic Degree l
Upper Shell Power Spectrum 3 N High Ra Low Ra Mid Ra Spherical Harmonic Degree l
Closing Points Rotation and Magnetism affect critical Ra -- Are most rotating simulations in this regime? Is this regime even relevantfor rotating convection, and is it computationally feasible to attain? Is it possible that a similar effect is at work in surface convection simulations?