Exploring inverse cascade scenarios in astrophysical turbulence a thesis submitted for the degree of doctor of philosohy in the Faculty of Science, Indian Institute of Science, Bangalore R. D. Prabhu [Ph.D Thesis]

Doctor of Philosophy Indian Institute of Science, Bangalore 1997

This thesis is an attempt to blend the latest observations of large scale coherent motions - coherent structures- observed in experiments with turbulent fluids, [6] with different astrophysical situations where turbulent flow is more of a rule than an exception. We highlight the need to re-invoke the once suggested [2] but long ignored role of turbulence in various astrophysical scenarios - viz .. the solar context, the galactic level, as well as on the cosmic scale- in the light of some of the latest developments in turbulence-studies. These studies have begun to emphasize the role of new invariants related to HELICITY and Helical fluctuations in the flow [73]. We feel that these new developments if incorporated in astrophysical fluid dynamics, may help in clarifying
some of the long standing problems in astrophysics related to large scale structures, and large scale motions which are well known. There is a well observed structural heirarchy in astrophysics viz. the way in which the fundamental blocks - galaxies, are clustered in groups and these groups in turn are re-grouped as superclusters over vast length scales ranging upto 100 Mpc
(lpc rv 3 x 1018cm). For example our Galaxy - Milky Way - belongs to the Local Group which consists of about 20 galaxies. A cluster may have upto 1000 galaxies. A typical linear extent of a cluster could be about 5 Mpc. A supercluster may be extending upto 50 Mpc. Further the 'Great Wall' is a linear structure of size 60 Mpc by 157 Mpc and is made up of several superclusters. The filamentary nature of matter distribution is well known. Using kolmogorovic arguments and the newly identified invariants related to helicity ( i.e the projection of vorticity along the velocity) and helical fluctuations viz. the helicity-helicity correlations (called the I - invariant) we can work out the inertial range stationary spectral behaviour for any turbulent medium whose net helicity is
zero but the helicity variance is a constant. This spectral dependence when translated into real space reflects the average energy that resides on each length scale. Thus the real space velocity fields also could carry a signature of this behaviour. This approach is the 3D analogue of the 2D case where apart from energy, enstrophy (i.e vorticity squared) is another invariant of the fluid flow. Upon inclusion of dissipation each of the invariants decay at different rates. The slowly decaying invariant generally cascades towards the larger scales and the faster one cascades to
the smaller scales. Levich et al. [73] , had evolved after a detailed study of the mesoscale atmospheric phenomena, a turbulent stationary spectrum which explains the energetics of the
cloud complexes over a range of scales. A similar spectrum was also employed to study the solar granulation scales by Krishan [16], which again could match with the observed and predicted energetics. Later, on a cosmic - scale too Krishan [8] and, Krishan and Sivaram [17] showed that the entire heirarchy of structures ranging from a galaxy to superclusters could be visualised as the consequence of similar turbulent processes operating over the whole range and leading to self-organized coherent states at various levels. To further verify the full spectrum in the astrophysical context we fit the real space velocity fields of various galaxies - known as 'rotation- curves'; with the predicted spectrum [9]. This exercise has yielded remarkable agreement with the spectra and
galactic-velocity fields. The parameters of turbulence so extracted from the fits are comparable with similar estimates made by other methods.
Next we test our proposed velocity-laws for the galactic velocity field, with the well known Tully-Fisher relation which highlights a tight correlation between the rotation velocity and the luminosity of a galaxy in various bands [87]. This is a statistical study wherein our interest is focussed on the correlations between the galactic luminosity with the 'turbulent' and 'gravity' components of the velocity field which our model could resolve after the proper fits are performed to the observed velocity fields. We confirm the normal trend of correlations with the total velocity first. For the individual correlations between the galactic luminosity and each of the velocity components , our study reveals an interesting feature. The turbulent component correlates reasonably better than the gravity component in the shorter wavebands. Whereas the gravity component correlates better than the turbulent component in the longer wavebands. This implies that the so called 'scatter' observed in the shorter wavelength bands could be a feature of galactic-turbulence which might be playing some constructive role in generating the observed large scale velocity field. Thus our study points out the necessity to re-consider the role of the self-organizing aspects of turbulent flows on galactic scales. Our results also convey the fact that our model is doing well in estimating the extent of gravity, and turbulence induced velocities. In an attempt to understand the generation of such large scale flows and other features of self-organization Frisch et al had performed a multi scale analysis of the Reynolds averaged set of Navier stokes equations with a well-defined forcing introduced on the small scales. They discovered that there indeed exists a large scale instability provided there is some small scale anisotropy in the flow. This is possibl if the turbulent medium lacks parity on small scales (i.e the statistical averages of the medium are NOT reflection invariant). Such a situation can be brought about by injecting helicity into the medium (on small scales), by rotation, or by compressibility effects or by using the specific forcing term used by Frisch et a1. in their analysis. This mechanism has its analogue in the DYNAMO MEOHANISM which is invoked for the generation of large scale magnetic fields, called the alpha effect. In fact the
equation for the evolution of VORTICITY, and the evolution of the MAGNETIC FIELD.. are similar in their structure. In order to prepare the ground for studying the mechanism in the context of an expanding universe we find a set of transformations which can help us in reducing the equations in the expanding frame to the normal Navier Stokes formulation [98] . This could then be used in the study of the formation of large scale structures in the universe which is indeed a long standing problem. Frisch et aI, and Sulem et al [27];[71] had performed extensive numerical simulation
of the incompressible Navier Stokes equation with a specific forcing term which produces a parity breaking velocity field on small scales. Later Druzhinin and Khomenko [101] also studied the same set of equations with compressibility effects We have performed a 32 x 32 x 32 numerical simulation of a compressible fluid (using spectral methods in a periodic box) on the IBM SP2 Convoy (as well as on the Power Challenge system) with parallelisation techniques implemented [102]. We have not used any empirical model to close our Reynolds averaged set of equations
for the large and the small scales . Instead we perform a spatial averaging over the entire real domain of velocity field at every time step. We choose a suitable length
scale over which to average, so that we get a statistically significant number of points over the grid. This we feel is a best approximation to the idea of ensemble averaging
We confirm that an inverse cascade of energy occurs when the fluid is forced on small scales with a forcing function which violates parity. We also study the evolution of helicity, vorticity, and density spectra in the simulation. We find that the compressibility of the medium also aids in generating a velocity field which lacks parity.