Numerical Calculation of Convection with Reduced Speed of Sound Technique

Context. The anelastic approximation is often adopted in numerical calculation with low Mach number, such as stellar internal convection. This approximation requires frequent global communication, because of an elliptic partial differential equation. Frequent global communication is negative factor for the parallel computing with a large number of CPUs. Aims. The main purpose of this paper is to test the validity of a method that artificially reduces the speed of sound for the compressible fluid equations in the context of stellar internal convection. The reduction of speed of sound allows for larger time steps in spite of low Mach number, while the numerical scheme remains fully explicit and the mathematical system is hyperbolic and thus does not require frequent global communication. Methods. Two and three dimensional compressible hydrodynamic equations are solved numerically. Some statistical quantities of solutions computed with different effective Mach numbers (due to reduction of speed of sound) are compared to test the validity of our approach. Results. Numerical simulations with artificially reduced speed of sound are a valid approach as long as the effective Mach number (based on the reduced speed of sound) remains less than 0.7.Comment: 16 pages, 10 figures, accepted to A&

The Role of Subsurface Flows in Solar Surface Convection: Modeling the Spectrum of Supergranular and Larger Scale Flows

We model the solar horizontal velocity power spectrum at scales larger than granulation using a two-component approximation to the mass continuity equation. The model takes four times the density scale height as the integral (driving) scale of the vertical motions at each depth. Scales larger than this decay with height from the deeper layers. Those smaller are assumed to follow a Kolomogorov turbulent cascade, with the total power in the vertical convective motions matching that required to transport the solar luminosity in a mixing length formulation. These model components are validated using large scale radiative hydrodynamic simulations. We reach two primary conclusions: 1. The model predicts significantly more power at low wavenumbers than is observed in the solar photospheric horizontal velocity spectrum. 2. Ionization plays a minor role in shaping the observed solar velocity spectrum by reducing convective amplitudes in the regions of partial helium ionization. The excess low wavenumber power is also seen in the fully nonlinear three-dimensional radiative hydrodynamic simulations employing a realistic equation of state. This adds to other recent evidence suggesting that the amplitudes of large scale convective motions in the Sun are significantly lower than expected. Employing the same feature tracking algorithm used with observational data on the simulation output, we show that the observed low wavenumber power can be reproduced in hydrodynamic models if the amplitudes of large scale modes in the deep layers are artificially reduced. Since the large scale modes have reduced amplitudes, modes on the scale of supergranulation and smaller remain important to convective heat flux even in the deep layers, suggesting that small scale convective correlations are maintained through the bulk of the solar convection zone.Comment: 36 pages, 6 figure

Modeling of differential rotation in rapidly rotating solar-type stars

We investigate differential rotation in rapidly rotating solar-type stars by means of an axisymmetric mean field model that was previously applied to the sun. This allows us to calculate the latitudinal entropy gradient with a rea- sonable physical basis. Our conclusions are as follows: (1) Differential rotation approaches the Taylor-Proudman state when stellar rotation is faster than so- lar rotation. (2) Entropy gradient generated by the attached subadiabatic layer beneath the convection zone becomes relatively small with a large stellar angu- lar velocity. (3) Turbulent viscosity and turbulent angular momentum transport determine the spatial difference of angular velocity $\Delta \Omega$. (4) The results of our mean field model can explain observations of stellar differential rotation.Comment: 36 pages, 10 figures, accepted to Ap

Production of Biodiesel from Waste Oil via Catalytic Distillation

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Decorrelation Times of Photospheric Fields and Flows

We use autocorrelation to investigate evolution in flow fields inferred by applying Fourier Local Correlation Tracking (FLCT) to a sequence of high-resolution (0.3 \arcsec), high-cadence ($\simeq 2$ min) line-of-sight magnetograms of NOAA active region (AR) 10930 recorded by the Narrowband Filter Imager (NFI) of the Solar Optical Telescope (SOT) aboard the {\em Hinode} satellite over 12--13 December 2006. To baseline the timescales of flow evolution, we also autocorrelated the magnetograms, at several spatial binnings, to characterize the lifetimes of active region magnetic structures versus spatial scale. Autocorrelation of flow maps can be used to optimize tracking parameters, to understand tracking algorithms' susceptibility to noise, and to estimate flow lifetimes. Tracking parameters varied include: time interval $\Delta t$ between magnetogram pairs tracked, spatial binning applied to the magnetograms, and windowing parameter $\sigma$ used in FLCT. Flow structures vary over a range of spatial and temporal scales (including unresolved scales), so tracked flows represent a local average of the flow over a particular range of space and time. We define flow lifetime to be the flow decorrelation time, $\tau$. For $\Delta t > \tau$, tracking results represent the average velocity over one or more flow lifetimes. We analyze lifetimes of flow components, divergences, and curls as functions of magnetic field strength and spatial scale. We find a significant trend of increasing lifetimes of flow components, divergences, and curls with field strength, consistent with Lorentz forces partially governing flows in the active photosphere, as well as strong trends of increasing flow lifetime and decreasing magnitudes with increases in both spatial scale and $\Delta t$.Comment: 48 pages, 20 figures, submitted to the Astrophysical Journal; full-resolution images in manuscript (8MB) at http://solarmuri.ssl.berkeley.edu/~welsch/public/manuscripts/flow_lifetimes_v2.pd

Local helioseismology of sunspot regions: comparison of ring-diagram and time-distance results

Local helioseismology provides unique information about the subsurface structure and dynamics of sunspots and active regions. However, because of complexity of sunspot regions local helioseismology diagnostics require careful analysis of systematic uncertainties and physical interpretation of the inversion results. We present new results of comparison of the ring-diagram analysis and time-distance helioseismology for active region NOAA 9787, for which a previous comparison showed significant differences in the subsurface sound-speed structure, and discuss systematic uncertainties of the measurements and inversions. Our results show that both the ring-diagram and time-distance techniques give qualitatively similar results, revealing a characteristic two-layer seismic sound-speed structure consistent with the results for other active regions. However, a quantitative comparison of the inversion results is not straightforward. It must take into account differences in the sensitivity, spatial resolution and the averaging kernels. In particular, because of the acoustic power suppression, the contribution of the sunspot seismic structure to the ring-diagram signal can be substantially reduced. We show that taking into account this effect reduces the difference in the depth of transition between the negative and positive sound-speed variations inferred by these methods. Further detailed analysis of the sensitivity, resolution and averaging properties of the local helioseismology methods is necessary for consolidation of the inversion results. It seems to be important that both methods indicate that the seismic structure of sunspots is rather deep and extends to at least 20 Mm below the surface, putting constraints on theoretical models of sunspots.Comment: 10 pages, 10 figures, submitted to Journal of Physics: Conference Series (JPCS) GONG 2010 - SoHO 24 "A new era of seismology of the Sun and solar-like stars", June 27 - July 2, 2010 Aix-en-Provence, Franc