Research: PhD Thesis

A little rabbit was sitting in a field, scribbling on a pad of paper, when a fox came along.

"What are you doing, little rabbit?"
"'I'm working on my dissertation," said the rabbit.
"Really?" said the fox. "And what is your topic?"
""Oh, the topic doesn't matter," said the rabbit.
"No, tell me , " begged the fox.

"If you must know," said the rabbit, "I'm advancing a theory that rabbits can eat many quite large animals - including, for instance, foxes. "

"Surely you have no experimental evidence for that, " scoffed the fox.

"Yes, I do," said the rabbit, "and if you'd like to step inside this cave for a moment I'll be glad to show you. " So the fox followed the rabbit into the cave. About half an hour passed. Then the rabbit came back out, brushing a tuft of fox fur off his chin, and began once more to scribble on his pad of paper.

News spreads quickly in the forest, and it wasn't long before a curious wolf came along. "I hear you're writing a thesis, little rabbit.' said the wolf.

"Yes , " said the rabbit, scribbling away.
"And the topic?" a sked the wolf.

"Not that it matters, but I'm presenting some evidence that rabbits can e a t larger animals - including, for example, wolves." The wolf howled with laughter. " I see you don't believe me , " said the rabbit. "Perhaps you would like to step inside this cave and see my experimental apparatus." Licking her chops, the wolf followed the rabbit into the cave. About half an hour passed before the rabbit came out of the cave with his pad of paper , munching on what looked like the end of a long gray tail.

Then along came a big brown bear. "Wha t's this I hear about your thesis topic?" he demanded. ''I can't imagine why you all keep pestering me about my topic," said the rabbit irritably. "As if the topic made any difference at all . " The bear sniggered behind his paw. "Something about rabbits eating bigger animals was what I heard- and apparatus inside the cave. "

"That's r ight , " snapped the rabbit, putting down his pencil. "And if you want to see it I'll gladly show you. " Into the cave they went, and a half hour later the rabbit came out again picking his teeth with a big bear claw.

By now all the animals in the forest were getting nervous about the rabbit's project, and a little mouse was elected to sneak up and peek into the cave when the rabbit's back was turned. There she discovered that the mystery of the rabbit's thesis had not only a solution but also a moral. The mystery's solution is that the cave contained an enormous lion. And the moral is that your thesis topic really doesn't matter - as long as you have the right thesis advisor.

B. Devine and J.E. Cohen in Absolute Zero Gravity.

Studies on static and dynamic multiscaling

I submitted my PhD thesis in the fall of the year 2004 in Physics Department of Indian Institute of Science, Bangalore and received my degree in the summer of the year 2005. The thesis also received the Martin Forster medal. My supervisor was Rahul Pandit. You can download a pdf file of the thesis here.

The following papers contains material from the thesis:

Dynamic multiscaling in fluid turbulence: an overview, Dhrubaditya Mitra and Rahul Pandit, Physica A 318 179 (2003).
This is my first paper. Unfortunately I did not submit it to the ArXiv and later I lost the tex file. For those of you who do not have subscription to view Physia A (isn't that practically everybody ?) and keen to read the paper (that would be practically nobody ? ) here is a pdf file.
Statistics of active vs. passive advections in magnetohydrodynamic turbulence , Thomas Gilbert and Dhrubaditya Mitra, Phys Rev. E 69 057301 (2004).
The Varieties of Dynamic Multiscaling in Fluid Turbulence , Dhrubaditya Mitra, and Rahul Pandit, Phys Rev. Lett 93 024501 (2004).
Is Multiscaling an Artifact in the Stochastically Forced Burgers Equation? , Dhrubaditya Mitra, Jeremie Bec, Rahul Pandit, and Uriel Frisch, Phys Rev. Lett 94 194501 (2005).
Dynamics of Passive-Scalar Turbulence , Dhrubaditya Mitra, and Rahul Pandit, Phys Rev. Lett 95 144501 (2005).

The abstract of the thesis is reproduced below.

The physics of turbulence is the study of the chaotic and irregular behaviour driven fluids. It is ubiquitous in cosmic, terrestrial and laboratory environments. To describe how the properties of a simple incompressible fluid it is sufficient to know its velocity at all points in space and as a function of time. The equation of motion for the velocity of such a fluid is the incompressible Navier-Stokes equation. In more complicated cases, for example if the temperature of the fluid also fluctuates in space and time, the Navier-Stokes equation must be supplemented by additional equations. Incompressible fluid turbulence is the study of solutions of the Navier-Stokes equation at very high Reynolds numbers, , the dimensionless control parameter for this problem. The chaotic nature of these solutions leads us to characterise them by their statistical properties. For example, statistical properties of fluid turbulence are characterised often by structure functions of velocity. For intermediate range of length scales, that is the inertial range, these structure functions show multiscaling. Most studies concentrate on equal-time structure functions which describe the equal-time statisticsl properties of the turbulent fluid. Dynamic properties can be measured by more general time-dependent structure functions. A major challenge in the field of fluid turbulence is to understand the multiscaling properties of both the equal-time and time-dependent structure functions of velocity starting from the Navier-Stokes equation. In this thesis we use numerical and analytical techniques to study scaling and multiscaling of equal-time and time-dependent structure functions in turbulence not only i fluids but also in advection of passive-scalars and passive vectors, and in randomly forced Burgers equation. The thesis is organised as follows:

In Chapter 1 we give an overview of the physics of homogeneous and isotropic turbulence. We describe some of the progresses made in experimental, theoretical and computational studies. We begin by summarising the scaling theory of structure functions, originally proposed by A.N. Kolmogorov in 1941. We further discuss recent experimental and numerical evidence in favour of multiscaling. We give a short introduction to the multifractal model. We also introduce turbulence in the passive-scalar problem and in the Burgers equation.
In Chapter 2 we present our new results on dynamic multiscaling in fluid turbulence. We show that different ways of extracting time scales from time-dependent velocity structure functions lead to different dynamic-multiscaling exponents in fluid turbulence. These exponents are related to equal-time multiscaling exponents by different classes of bridge relations which we derive. We check this explicitly by detailed numerical simulations of the GOY shell model for fluid turbulence.
In Chapter 3 we study dynamic multiscaling for passive-scalar and passive-vector turbulence. An example of a passive-scalar field is the temperature field in the fluid, if the gradients of temperature are small, such that the effects of convection in the system are neglibible. By using analytical methods, we find simple dynamic scaling for the Kraichnan model for passive scalar advection and also for a Kraichnan-type shell model for passive-scalar turbulence. Although the passive-scalar field in the Kraichnan case is known to show equal-time multiscaling for structure functions, we find simple dynamic scaling of the time-dependent structure function. These results are further supported by our numerical simulations on the shell models. In the Kraichnan model the advecting velocity is a random Gaussian variable which is white-in-time. We also consider a passive-scalar shell model advected by velocity from the GOY shell model of fluid turbulence. In this case the advecting velocity field is not Gaussian but has multifractal properties and we find dynamic multiscaling for the time-dependent passive-scalar structure functions. We discuss the implication of our study for passive-vector turbulence. We also touch upon the issues of the kinematic dynamo in the context of passive-vector shell models.
In Chapter 4 we present preliminary results from our study of dynamic multiscaling for time-dependent velocity structure functions obtained from the Navier-Stokes equation. We present the first numerical algorithm for a pseudo-spectral, direct numerical simulation that yields quasi-Lagrangian velocity. We calculate the equal-time, quasi-Lagrangian velocity structure functions from (512 cubed and 256 cubed) two-third dealiased pseudo-spectral simulations. We also present preliminary data for time-dependent Eulerian and quasi-Lagrangian structure functions.
In Chapter 5 we study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum of one over wave number. From very-high-resolution numerical simulations, in the limit of vanishing viscosity, we provide strong evidence for multiscaling of velocity structure functions, as opposed to bifractal scaling suggested by earlier, low-resolution studies. We discuss possible artifacts that can make bifractal scaling appear like genuine multiscaling and show that they are not present in our work. We discuss the implications of our study for multiscaling in fluid turbulence.

Last modified: Wed Apr 25 22:28:34 CEST 2012

Dhrubaditya Mitra

B. Devine and J.E. Cohen in Absolute Zero Gravity.

Studies on static and dynamic multiscaling