Microscopic laws vs. Macroscopic laws: Perspectives from kinetic theory and hydrodynamics

Reductionism is a prevalent viewpoint in science according to which all physical phenomena can be understood from fundamental laws of physics. Anderson [Science, 177, 393 (1972)], Laughlin and Pines [PNAS, 97, 28 (2000)], and others have countered this viewpoint and argued in favour hierarchical structure of the universe and laws. In this paper we advance the latter perspective by showing that some of the complex flow properties derived using hydrodynamic equations (macroscopic laws) are very difficult, if not impossible, to describe in microscopic framework---kinetic theory. These properties include Kolmogorov's theory of turbulence, turbulence dissipation and diffusion, and dynamic pressure. We also provide several other examples of hierarchical description

Variable energy flux in quasi-static magnetohydrodynamic turbulence

Experiments and numerical simulations show that the energy spectrum of the magnetohydrodynamic turbulence in the quasi-static limit deviates from Kolmogorov's $k^{-5/3}$ spectrum as the external magnetic field, or equivalently the interaction parameter, is increased. To explain this phenomena, we construct an analytical turbulence model with variable energy flux that arises due to the Lorentz-force induced dissipation. The energy spectra computed using our model for various interaction parameters are in qualitative agreement with earlier experimental and numerical results.Comment: 12 page

Contrasting turbulence in stably stratified flows and thermal convection

In this paper, the properties of stably stratified turbulence (SST) and turbulent thermal convection are contrasted. A key difference between these flows is the sign of the kinetic energy feed by buoyancy, $\mathcal{F}_B$. For SST, $\mathcal{F}_B < 0$ due to its stable nature; consequently, the kinetic energy flux $\Pi_u(k)$ decreases with wavenumber $k$ that leads to a steep kinetic energy spectrum, $E_u(k) \sim k^{-11/5}$. Turbulent convection is unstable, hence $\mathcal{F}_B > 0$ that leads to an increase of $\Pi_u(k)$ with $k$; this increase however is marginal due to relatively weak buoyancy, hence $E_u(k) \sim k^{-5/3}$, similar to that in hydrodynamic turbulence. This paper also describes the conserved fluxes for the above systems.Comment: to appear in Physica Script

On Generation of magnetic field in astrophysical bodies

In this letter we compute energy transfer rates from velocity field to magnetic field in MHD turbulence using field-theoretic method. The striking result of our field theoretic calculation is that there is a large energy transfer rate from the large-scale velocity field to the large-scale magnetic field. We claim that the growth of large-scale magnetic energy is primarily due to this transfer. We reached the above conclusion without any linear approximation like that in $\alpha$-dynamo.Comment: 4 pages, Revtex, 1 figur

Mean magnetic field renormalization and Kolmogorov's energy spectrum in MHD turbulence

In this paper we construct a self-consistent renormalization group procedure for MHD turbulence in which small wavenumber modes are averaged out, and effective mean magnetic field at large wavenumbers is obtained. In this scheme the mean magnetic field scales as $k^{-1/3}$, while the energy spectrum scales as $k^{-5/3}$ similar to that in fluid turbulence. We also deduce from the formalism that the magnitude of cascade rate decreases as the strength of the mean magnetic field is increased.Comment: 14 pages REVTEX, 1 postscript figure, Submitted to Phys. Plasma

Turbulent Heating in the Solar Wind and in the Solar Corona

In this paper we calculate the turbulent heating rates in the solar wind using the Kolmogorov-like MHD turbulence phenomenology with Kolmogorov's constants calculated by {\it Verma and Bhattacharjee }[1995b,c]. We find that the turbulent heating can not account for the total heating of the nonAlfv\'enic streams in the solar wind. We show that dissipation due to thermal conduction is also a potential heating source. Regarding the Alfv\'enic streams, the predicted turbulent heating rates using the constants of {\it Verma and Bhattacharjee }[1995c] are higher than the observed heating rates; the predicted dissipation rates are probably overestimates because Alfv\'enic streams have not reached steady-state. We also compare the predicted turbulent heating rates in the solar corona with the observations; the Kolmogorov-like phenomenology predicts dissipation rates comparable to the observed heating rates in the corona [{\it Hollweg, }% 1984], but Dobrowoly et al.'s generalized Kraichnan model yields heating rates much less than that required.Comment: Latex, 22 pages, no figures, submitted to Journal of Geophysical Researc

Introduction to Statstical Theory of Fluid Turbulence

This is a brief introduction to the statistical theory of fluid turbulence, with emphasis on field-theoretic treatment of renormalized viscosity and energy fluxes.Comment: 40 pages; The labels in Feynman diagram missing due to tex problems. Look at my webpage http://home.iitk.ac.in/~mkv//prof/publications.html for proper Feynman diagram

Turbulent drag reduction: A universal perspective from energy fluxes

Injection of dilute polymer in a turbulent flow suppresses frictional drag. This challenging and technologically important problem remains primarily unresolved due to the complex nature of the flow. An important factor in the drag reduction is the energy transfer from the velocity field to the polymers. In this paper we quantify this process using energy fluxes, as well as show its universality in diverse flows such as magnetohydrodynamics, quasi-static magnetohydrodynamics, and bubbly turbulence. We show that in such flows, the transfer from kinetic energy to elastic energy leads to a reduction in kinetic energy flux compared to the corresponding hydrodynamic turbulence. This leads to a reduction in nonlinearity of the velocity field that results into a more ordered flow and a suppression of turbulent drag

Anisotropy in Quasi-Static Magnetohydrodynamic Turbulence

In this review we summarise the current status of the quasi-static magnetohydrodynamic turbulence. The energy spectrum is steeper than Kolmogorov's $k^{-5/3}$ spectrum due to the decrease of the kinetic energy flux with wavenumber $k$ as a result of Joule dissipation. The spectral index decreases with the increase of interaction parameter. The flow is quasi two-dimensional with strong ${\bf U}_\perp$ at small $k$ and weak $U_\parallel$ at large $k$, where ${\bf U}_\perp$ and $U_\parallel$ are the perpendicular and parallel components of velocity relative to the external magnetic field. For small $k$, the energy flux of ${\bf U}_\perp$ is negative, but for large $k$, the energy flux of $U_\parallel$ is positive. Pressure mediates the energy transfer from ${\bf U}_\perp$ to $U_\parallel$.Comment: Review articl

Direct Interaction Approximation of Magnetohydrodynamic Turbulence

In this paper we apply Kraichnan's direct interaction approximation, which is a one loop perturbation expansion, to magnetohydrodynamic turbulence. By substituting the energy spectra both from kolmogorov-like MHD turbulence phenomenology and a generalization of Dobrowolny et al.'s model we obtain Kolmogorov's and Kraichnan's constant for MHD turbulence. We find that the constants depend of the Alfv\'en ratio and normalized cross helicity; the dependence has been studied here. We also demonstrate the inverse cascade of magnetic energy for Kolmogorov-like models. Our results are in general agreement with the earlier simulation results except for large normalized cross helicity.Comment: Revtex, 32 pages, submitted to Phys. Rev. E, 3 figures: available upon request at [email protected]