# User:Eugene M. Izhikevich/Proposed/Solar granulation

Prof. Åke Nordlund accepted the invitation on 8 December 2006. The article is still incomplete, and not yet ready for review.

The Solar Granulation is a characteristic cellular brightness pattern visible on the solar surface, on scales from a fraction of an arc second to several arc seconds, corresponding to physical scales from a few hundred km to several thousand km.

## Solar Granulation

Images of the Sun taken with sufficient angular resolution reveal a characteristic cellular brightness pattern, which is called the Solar Granulation. The pattern was discovered already by Herschel [1], who interpreted the pattern as being due to "hot clouds" floating over a cooler solar surface. Dawes [2] was the one who coined the term "granules", contesting the description of Nasmyth [3], who described them as having a shape similar to "willow leaves". The first good photographs, published by Janssen (~1896) ended the controversy, in favor of Dawes' description.

The granulation pattern is associated with heat transport by convection; hot gas rises to the surface of the Sun, where it cools by emitting the light that we see. At the solar surface the mass density decreases very rapidly with height, dropping with a factor two over over only couple of hundred km. To conserve mass ascending gas must therefore expand, while descending gas must contract. It is because of this expansion that the ascending, hot gas forms cellular structures; the expanding gas carries heat with it, and before the gas has time to cool it has expanded and formed the characteristic granular bright cells. Typical velocities of ascent and expansion are 2-4 km per second.

The heat capacity of gas at the solar surface is such that it takes temperature fluctuations of the order of several thousand degrees and velocity amplitudes of several km per second to account for the solar luminosity (the amount of energy per unit area and time that reaches the solar surface). Below the surface, where the density grows rapidly with depth, the heat capacity rapidly becomes much larger, and much smaller velocities and temperature fluctuations are then sufficient to carry the same luminosity. As the horizontal temperature fluctuations decrease, the changes of temperature with depth become more and more adiabatic; i.e., the temperature increases with increasing density and pressure in nearly the same manner as when gas is compressed under energetically isolated conditions.

The scales over which the density and pressure change with a factor of e are called the density scale height and the pressure scale height, respectively. These are approximately proportional to temperature, and they therefore increase rapidly below the solar surface. This allows larger scale horizontal motion patterns to exist, without coming in conflict with mass conservation. If one imagines the flows to be roughly cylindrical, the ratio of (circular) horizontal area to vertical (cylindrical) area is D / 4 H, for a cylinder section of height H and diameter D. The mass that flows up through a circular cross section with a certain ascent velocity V can thus expand with the same velocity across a cylinder surface with height H = D/4. If the expansion velocity is larger than the ascent velocity the vertical cylinder surface can be correspondingly smaller.

### Meso- and super-granulation

Consecutively larger scale and lower amplitude motions are thus allowed to exists below the solar surface. At the visible solar surface these larger scale motions are primarily revealed by their advection of the smaller scale motions. Cellular motions with scales of the order of 10,000 km are called meso-granules, while motions with scales of the order of 30,000 km are called super-granules. Larger scale motions (e.g. with scales ~100,000 km) exist; these are traditionally referred to as giant cells.

The classification into distinct scales is of historical origin; super-granulation was discover in the 1960s, as a long lived pattern of largely horizontal velocities, which turned out to correspond closely to the so called magnetic network; a network pattern of stronger than average magnetic field, created by the horizontal advection of (mainly) vertical magnetic fields at the solar surface.

Mesogranulation was discovered in the early 1980s, from the advection of the smaller scale granulation pattern, and from the advection of small scale magnetic features.

Since the scales of the motions corresponding to granulation, meso-granulation, and super-granulation only differ by about a factor of three it is not surprising to find that there is actually a continuous range of motions, with velocity amplitudes that decrease smoothly with increasing horizontal scale. Observations (e.g. with the MDI instrument on the SOHO satellite) as well as supercomputer models confirm the existence of such a smooth spectrum of motions. The velocity amplitude is approximately inversely proportional to the horizontal scale, ranging from several km/s at granular scales over a few hundred m/s on supergranulation scales, down to tens of m/s on scales of hundreds of thousands of km.

### Velocity and temperature fluctuations

The temperature and velocity amplitudes on the scale of granulation are difficult to measure directly, both because some of the fluctuation power is not spatially resolved, and because the very strong temperature sensitivity of the opacity (the light absorbtivity of the gas) complicates the analysis. However, these amplitudes may be deduced from supercomputer simulations of the solar surface, since there exist diagnostics (fingerprints) that are independent of the spatial resolution. One such diagnostic is the asymmetry of spectral lines formed near the solar surface; the spectral line bisectors of such spectral lines are usually C-shaped, with the 'belly' of the C towards the blue side. Another, related fingerprint is a net blue-shift of the spectral lines; the 'belly' of the C is typically shifted to the blue with several hundred meters per second, relative to the wavelength expected from laboratory measurements (corrected for gravitational red shift).

Together these diagnostics function as fingerprints that can be matched between observations and supercomputer simulations. The actual velocity and temperature fluctuations can then be deduced from the numerical models, in a manner that is essentially independent of the spatial resolution of the observations. At horizontal scales of the order of 1000 km the velocity amplitudes are several km per second, and the temperature fluctuations are several thousand degrees. Nevertheless, because the hotter gas is so much more opaque, the resulting root mean square radiation intensity fluctuations are only some 15-20%.

### Modern observations

... images from La Palma, and velocity amplitude spectra from SOHO/MDI and from TRACE ...

### Supercomputer simulations

... surface images, and 3-D renderings of flow patterns ...