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361. NEQNET: first two weeks of April

Posted by Dmitry

at April 18, 2009

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Well, it seems that other two weeks have passed… What’s up? First of all, it looks like I figured out why energy seems to dissipate inhomogeneously in a turbulent flow with very large Reynolds numbers (kindly see the list N4 below). Apart from this topic, other things that I was interested in during these two weeks are listed here:

1. String theory, field theory, quantum gravity

1.1. What is twistor. Everybody seems to get a bit crazy about twistor formalism in string theory and Y.-M. lately, so I’ve decided it’s time finally for me to learn what is it all about…

1.2. Twistors and non-linear differential equations. Curved spacetime, where I continue discussing twistors. While I explain how using the language of twistors allows to express solutions of linear differential equations (and free massless fields) in the previous post, here I discuss how twistors help dealing with non-linear differential equations.

1.3. Twistors: getting more formal, where I give a formal definition of twistor as a straight line in projective space and discuss symmetries of CP^3 .

1.4. Thermal equilibrium in special relativity by David Cubero (U. of Sevilla). Consider a single ultrarelativistic (say, massless) particle coupled to a heat bath. The motion of this particle is Brownian, and the magnitude of the random force acting on it is defined by the temperature of the bath. What happens with Langevin and Fokker-Planck equations if we describe the motion of the particle using proper time instead of world time? Post gives an answer to this question.

2. Cosmology

2.1. Cosmological parameters in the context of time varying w by Rahul Biswas (U. of Illinois). How much will our estimations of cosmological parameters change if we allow the effective equation of state to change with time (or with redshift )?

3. Nuclear physics

3.1. Thermonuclear fusion: some basic facts about thermonuclear reactions. Since I’ve included thermonuclear fusion in my list of 10 most important open problems in physics, it would be only fair for me to present “a bit deeper than basic level” review of physics behind it. Here I discuss the very basic of thermonuclear fusion like Coulomb barrier and a couple of strategies that can help us pursue the problem.

3.2. Thermonuclear fusion. Coulomb barrier and reaction rates (and the second part). Continuation of the previous post, devoted to estimation of reaction rates.

3.3. Introduction into thermonuclear reactors. That’s what it is: I classify different reactors according to the scheme of confinement of plasma they use.

4. Hydrodynamics, turbulence

4.1. Vortex line representation. Cauchy invariant. The goal of this post is to demonstrate the nature of infinite number of (non-local) integrals of motion that exist for Eulerian flows. As it turns out, the physics behind them is vorticity field frozen into the motion of the ideal incompressible fluid.

4.2. Vortex line representation. Clebsch variables. In this post, I derive equations of motion for vortex lines frozen into Eulerian flow.

5. Fun and stuff

5.1. Interview with Bogzabraloff brothers: science and religion. Two of the deepest thinkers of our time, Bogzabraloff brothers, explain their views on interplay between science and religion.

5.2. How much should you publish? Really, this question is not as meaningless as it sounds. How much should you publish in order to be considered competitive on the physics job market? Let us figure out together.

5.3. The question of quality. Maybe number of your publications is less important than the quality of your research. But how to measure the quality of your research quantitatively? Is the total number of citations a metrics that is good enough to estimate it? What is h-factor and how good is that metrics?

5.4. Followup: BumpTop. As it turns out, BumpTop finally goes out of the private beta stage – you can download it for free even if you don’t have an invite code.

5.5. Trading olympiad. Interactive Brokers has just announced that the 2009 IB Collegiate Trading Olympiad is now over. The winner gets 200000$ – a nice addition to his fellowship.

5.6. 48 years ago. Space opened its doors for Major Yuri Gagarin on Apr 21 1961. Did it really close its doors for us then?

There is a couple of other rather exiting things I was dealing with during these 14 days but I am not ready to disclose it to you yet… To stay in touch, you can subscribe to NEQNET daily email updates or RSS feed.

Entered Apprentice, This blog

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360. ISS Tour: video of the day

Posted by Dmitry

at April 18, 2009

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A great International Space Station tour by Col. Michael Fincke, commander of 18th expedition. The view from station’s windows will blow your mind. Michael is a lot of fun, while Yuri Lonchakov is way too serious (I saw several videos of him at this point, and it seems that he is always like that )

Actually, Col. Fincke is another figure in NASA I highly admire. He speaks Russian, maybe not really fluently but at the level that a native speaker can actually understand him. Apart from Sewickley Academy, MIT and Stanford he has also attended Moscow State Aviation Institute (on summer exchange program between MAI and MIT).

Entered Apprentice, Fun

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359. Michael Griffin to fill professor’s position in Alabama

Posted by Dmitry

at April 17, 2009

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Michael Griffin

The University of Alabama in Huntsville (UAHuntsville) has named Michael Griffin, one of the world’s leading aerospace engineers, as an eminent scholar and a professor of mechanical and aerospace engineering.

The announcement was made today by President David Williams.

In 2005, Dr. Griffin was appointed as the 11th NASA Administrator, serving in that role until earlier this year. He was previously Head of the Space Department at the Applied Physics Laboratory (APL) of the Johns Hopkins University, and he played a leading role in numerous other space missions through employment at the APL, the Jet Propulsion Laboratory and Computer Science Corporation.

Prior to joining APL, he served in many executive positions with aerospace-related companies and he has held several academic appointments. Dr. Griffin has served as an adjunct professor at the University of Maryland, the Johns Hopkins University, and George Washington University.

“Michael Griffin is recognized worldwide as a leading authority on aerospace engineering and as a visionary for American space flight,” Williams said. “We believe his contributions to this university and the Huntsville community will be of tremendous value. This appointment adds new dimensions to historic areas of strength, making his appointment an investment in the future of UAHuntsville.

“Dr. Griffin’s appointment as a professor and eminent scholar will help raise the visibility of our aerospace engineering program to an even higher level. His achievements, both from a technical standpoint and as an academician, make him a valuable addition to our campus and provide great opportunities for this university. I’m delighted that Mike has decided to come and educate our students and collaborate with our faculty and with the rest of the Huntsville aerospace community.”

Griffin looks forward to his new role at UAHuntsville.

“Everyone I have talked to in Huntsville, beginning with Dave Williams and his senior staff at UAHuntsville, university trustees, and local industry and community leaders, has been incredibly gracious in welcoming Rebecca and me to the Huntsville community,” he said. “For my part, in my new role at the university and in the larger community, I look forward to the opportunity to show how glad we are to be here.

“I intend to remain actively involved in all aspects of U.S. defense and civil space programs,” he added, including such activities as teaching, research and helping develop the next generation of aerospace designers and leaders.

Dr. Griffin will be filling the university’s eminent scholar position in engineering. An endowment for this appointment was established 20 years ago but the position was never occupied. Dollars invested in this endowment have been growing during the past two decades. “The university has carefully conserved the resources in its Eminent Scholar Foundation awaiting the right opportunity to invest that endowment,” Williams said. “This is that right opportunity.”

Dr. Griffin is expected to assume his duties in mid-May.

I really admire Griffin and wish him the best of luck. NASA will be a different place without him, so will be US Space Program. He should be probably awfully tired and disappointed by his conflicts with new NASA administration and will certainly have a time to relax somewhat in Alabama. UAHuntsville is a great place, I doubt though he will be there for long

Full press release can be found on the UAHuntsville web-site.

Entered Apprentice, Personal

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358. Thermonuclear reactors. Inertial confinement

Posted by Dmitry

at April 17, 2009

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I am currently keeping studying thermonuclear fusion and reactors a bit and, I should admit, I’m absolutely in love with HiPER and inertial confinement as an idea – it is so much more elegant than magnetic confinement used in Tokamaks… But before I’ll turn to the discussion of inertial confinement reactors, let me finish with generalities and trivialities (I’ll need them anyway for further reference).

Every thermonuclear reactor is characterized by its fusion energy gain factor equal to the ratio of reactor’s power to power spent for starting and sustaining the nuclear reaction. So far, we were unable to build a reactor with $Q\ge{}1$ , while what be useful for us in practise to achieve is the gain factor of the order of 20. Why we were so unsuccessful so far (Russians started working on thermonuclear fusion and building Tokamaks from 1960s)? We will discuss associated difficulties in one of the next posts.

It seems that the simplest option for us is to build a reactor working on d-t fuel, the corresponding reaction rate being larger than for any other thermonuclear reactions. The next (but actually more attractive as you’ll see below) possibility is a reactor working on d- ${}^{3}He$ fuel. In this case, neutrons can only appear in subsequent d-d and d-t reactions, and associated danger is much lower compared to the usual fission nuclear reactors: there is no need to develop the whole industry dealing with radioactive waste, etc. The main problem associated with d- ${}^{3}He$ thermonuclear reactor is almost absolute absence of ${}^{3}He$ in Nature. It sounds somewhat funny but we may end up delivering it to Earth from Moon…

(the guy on the video is way too funny )

Ok, finished with generalities and trivialities and let me now get to the interesting part and explain how inertial confinement works physically. There is a lot of interesting and simple associated physics, so probably explanations will take couple of posts but, I think, it’s definitely worth studying.

Contrary to usual idea of magnetic confinement of plasma, plasma in reactors with interial confinement is not really confined – it propagates freely. Conditions for the thermonuclear reaction to start are achieved on the stage of compression of plasma. The systems with inertial confinement are initially designed to be out of equilibrium (a time scale $t_{IC}$ exists characterizing inertial confinement).

Imagine that the plasma of d, t nuclei and electrons with densities $n_{D}$ , $n_{T}$ , $n_{e}$ correspondingly is localized within the sphere of radius . The number of d-t fusion reactions in the corresponding spherical volume is given by

$\frac{dN}{dt}=n_{T}n_{D}\langle\sigma v\rangle_{d-t}\frac{4}{3}\pi R^{3}\sim\rho^{2}\langle\sigma v\rangle_{d-t}V$ ,

where $\rho=mn$ for given species (d or t). (The d-t reaction rate $\langle\sigma v\rangle_{d-t}$ is given as usual by averaging over Maxwell distribution for the given temperature of plasma). Since plasma is not confined, reaction can be only effective during a characteristic kinematic time

$t_{IC}\sim\frac{R}{v}$ ,

where is the velocity of the plasma during the state of compression. In the very first approximation, we can estimate it as a speed of sound in the plasma

$v\sim c_{s}=\sqrt{\left(\frac{dp}{d\rho}\right)_{T}.}$

Therefore, a characteristic number of d (or t) nuclei that entered the reaction as

$\frac{N}{N_{D}}\sim\langle\sigma v\rangle_{d-t}\rho t_{IC}$ .

Using the ideal gas approximation for plasma we find

$c_{s}\sim\left(\frac{T(1+Z)}{M}\right)^{1/2}$ ,

where is atomic number of ions in plasma and is their mass. Then,

$\frac{N}{N_{D}}\sim\langle\sigma v\rangle_{d-t}T^{-1/3}\rho R$ .

To be continued…

Fellow Craft, Nuclear physics

364. Thermonuclear reactors. More on inertial confinement

355. Introduction into thermonuclear reactors

365. Inertial confinement – using lasers for compression

385. NEQNET: Last two weeks of April

357. Vortex line representation. Coulomb interaction of vortex lines

Posted by Dmitry

at April 16, 2009

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After brief introduction into vortex line representation we are probably ready to discuss the interaction of vortex lines between each other. But before I proceed to the actual derivation, let me focus for a bit on not so terribly popular (but powerful) formulation of ideal hydrodynamics – Hamiltonian formulation.

The Lagrangian of incompressible fluid (I set $\rho=1$ for simplicity) is

${\cal L}=\frac{1}{2}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}|\mathbf{v}_{\mathbf{k}}|^{2}$ , (1)

and, as usual, we define the canonical momentum as

$\mathbf{p}=\frac{\delta{\cal L}}{\delta\mathbf{v}}$ .

Vorticity field $\mathbf{\Omega}(\mathbf{r},t)$ can be written in terms of momentum as

$\mathbf{\Omega}={\rm curl}\mathbf{p}$ ,

and the Hamiltonian is

${\cal H}(\mathbf{\Omega})=\left(\int d\mathbf{r}\left(\frac{\delta{\cal L}}{\delta\mathbf{v}}\cdot\mathbf{v}\right)-{\cal L}\right)|_{\mathbf{v}=\mathbf{v}(\mathbf{\Omega})}$ .

It is easy to show that the e.o.m. for the vorticity field is given by

$\frac{\partial\mathbf{\Omega}}{\partial t}={\rm curl}\left({\rm curl}\left(\frac{\delta{\cal H}}{\delta\mathbf{\Omega}}\times\mathbf{\Omega}\right)\right)$ .

Exercise: check it out explicitly.

The Hamiltonian written in terms of vorticity field has a remarkably simple form:

${\cal H}_{M}(\mathbf{\Omega})=\frac{1}{2}\int\frac{d\mathbf{k}}{(2\pi)^{3}}\frac{|\mathbf{\Omega}_{\mathbf{k}}|^{2}}{k^{2}}$ ,

i.e.,

${\cal H}_{M}(\mathbf{\Omega})=\frac{1}{2}\int\int\frac{(\mathbf{\Omega}(\mathbf{r}_{1})\cdot\mathbf{\Omega}(\mathbf{r}_{2}))}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}d\mathbf{r}_{1}d\mathbf{r}_{2}$ .

Although the Coulomb interaction is present , probably you don’t quite see yet how separate vortex filaments interact with each other. To show this explicitly, let me finally use the vortex line representation. I express vorticity field as

$\mathbf{\Omega}(\mathbf{r},t)=\int d^{2}\nu\oint\delta(\mathbf{r}-\mathbf{R}(\nu,l,t))\frac{\partial\mathbf{R}}{\partial l}dl$ ,

where $\nu$ are 2d Lagrangian coordinates marking vortex lines and is affine parameter along the given line. Substituting (2) into (1), I finally find

${\cal H}=\frac{\Gamma^{2}}{2}\oint\oint\frac{(\mathbf{R}'(l_{1})\cdot\mathbf{R}'(l_{2}))}{|\mathbf{R}(l_{1})-\mathbf{R}(l_{2})|}dl_{1}dl_{2}$ ,

where $\Gamma$ is circulation (it is conserved due to the Kelvin theorem I have discussed in the previous post).

Let us discuss the physics of this Hamiltonian a bit.

1. Suppose that we have just a single vortex filament. In this case, $\mathbf{R}$ does not depend on line marker $\nu$ at all, and the circulation $\Gamma$ is simply given by the integral $\int d^{2}\nu$ , presumed to be finite. The overall flow is potential (that is, $\mathbf{p}=\nabla\Phi$ ) – the fluid is circulating around the center of the filament. Dynamics is still quite non-trivial, though: small pieces of the filament interact with each other by Coulomb interaction.

2. This Hamiltonian describes an infinitely thin vortex filament – string , and its self-energy is clearly infinite due to the Coulomb-like divergence. If we want to deal with it in a practical fashion, we will have to regularize it somehow (starting presumably from the initial Lagrangian (1) or introducing viscosity).

3. If we wait for some time, we will find that the self-induced velocity of the vortex filament becomes infinite as well (unless, as I said above, we don’t regularize the Coulomb Hamiltonian) This shows that viscosity is very important in turbulence – even if we start in the regime where the Reynolds number is extremely large (so that the viscosity is effectively zero), finite time collapse of the vortex lines will lead to the appearance of the localized regions in the flow, where dissipation should be huge. That’s what we were talking about in the post about four puzzles in physics of turbulence.

Hydrodynamics, Master Mason, Turbulence