DOI

1

Nonlinear dynamos at infinite magnetic Prandtl number - Alexakis, Alexandros

PHYSICAL REVIEW E 83, (2011)

Abstract : The dynamo instability is investigated in the limit of infinite magnetic Prandtl number. In this limit the fluid is assumed to be very viscous so that the inertial terms can be neglected and the flow is enslaved to the forcing. The forcing consist of an external forcing function that drives the dynamo flow and the resulting Lorentz force caused by the back reaction of the magnetic field. The flows under investigation are the Archontis flow and the ABC flow forced at two different scales. The investigation covers roughly 3 orders of magnitude of the magnetic Reynolds number above onset. All flows show a weak increase of the averaged magnetic energy as the magnetic Reynolds number is increased. Most of the magnetic energy is concentrated in flat elongated structures that produce a Lorentz force with small solenoidal projection so that the resulting magnetic field configuration is almost force free. Although the examined system has zero kinetic Reynolds number at sufficiently large magnetic Reynolds number the structures are unstable to small scale fluctuations that result in a chaotic temporal behavior.

PHYSICAL REVIEW E 83, (2011)

LPS

Abstract : The dynamo instability is investigated in the limit of infinite magnetic Prandtl number. In this limit the fluid is assumed to be very viscous so that the inertial terms can be neglected and the flow is enslaved to the forcing. The forcing consist of an external forcing function that drives the dynamo flow and the resulting Lorentz force caused by the back reaction of the magnetic field. The flows under investigation are the Archontis flow and the ABC flow forced at two different scales. The investigation covers roughly 3 orders of magnitude of the magnetic Reynolds number above onset. All flows show a weak increase of the averaged magnetic energy as the magnetic Reynolds number is increased. Most of the magnetic energy is concentrated in flat elongated structures that produce a Lorentz force with small solenoidal projection so that the resulting magnetic field configuration is almost force free. Although the examined system has zero kinetic Reynolds number at sufficiently large magnetic Reynolds number the structures are unstable to small scale fluctuations that result in a chaotic temporal behavior.

DOI

2

Searching for the fastest dynamo: Laminar ABC flows - Alexakis, Alexandros

PHYSICAL REVIEW E 84, (2011)

Abstract : The growth rate of the dynamo instability as a function of the magnetic Reynolds number R-M is investigated by means of numerical simulations for the family of the Arnold-Beltrami-Childress (ABC) flows and for two different forcing scales. For the ABC flows that are driven at the largest available length scale, it is found that, as the magnetic Reynolds number is increased: (a) The flow that results first in a dynamo is the 2 1/2-dimensional flow for which A = B and C = 0 (and all permutations). (b) The second type of flow that results in a dynamo is the one for which A = B similar or equal to 2C/5 (and permutations). (c) The most symmetric flow, A = B = C, is the third type of flow that results in a dynamo. (d) As R-M is increased, the A = B = C flow stops being a dynamo and transitions from a local maximum to a third-order saddle point. (e) At larger R-M, the A = B = C flow reestablishes itself as a dynamo but remains a saddle point. (f) At the largest examined R-M, the growth rate of the 2 1/2-dimensional flows starts to decay, the A = B = C flow comes close to a local maximum again, and the flow A = B similar or equal to 2C/5 (and permutations) results in the fastest dynamo with growth rate gamma similar or equal to 0.12 at the largest examined R-M. For the ABC flows that are driven at the second largest available length scale, it is found that (a) the 2 1/2-dimensional flows A = B, C = 0 (and permutations) are again the first flows that result in a dynamo with a decreased onset. (b) The most symmetric flow, A = B = C, is the second type of flow that results in a dynamo. It is, and it remains, a local maximum. (c) At larger R-M, the flow A = B similar or equal to 2C/5 (and permutations) appears as the third type of flow that results in a dynamo. As R-M is increased, it becomes the flow with the largest growth rate. The growth rates appear to have some correlation with the Lyapunov exponents, but constructive refolding of the field lines appears equally important in determining the fastest dynamo flow.

PHYSICAL REVIEW E 84, (2011)

LPS

Abstract : The growth rate of the dynamo instability as a function of the magnetic Reynolds number R-M is investigated by means of numerical simulations for the family of the Arnold-Beltrami-Childress (ABC) flows and for two different forcing scales. For the ABC flows that are driven at the largest available length scale, it is found that, as the magnetic Reynolds number is increased: (a) The flow that results first in a dynamo is the 2 1/2-dimensional flow for which A = B and C = 0 (and all permutations). (b) The second type of flow that results in a dynamo is the one for which A = B similar or equal to 2C/5 (and permutations). (c) The most symmetric flow, A = B = C, is the third type of flow that results in a dynamo. (d) As R-M is increased, the A = B = C flow stops being a dynamo and transitions from a local maximum to a third-order saddle point. (e) At larger R-M, the A = B = C flow reestablishes itself as a dynamo but remains a saddle point. (f) At the largest examined R-M, the growth rate of the 2 1/2-dimensional flows starts to decay, the A = B = C flow comes close to a local maximum again, and the flow A = B similar or equal to 2C/5 (and permutations) results in the fastest dynamo with growth rate gamma similar or equal to 0.12 at the largest examined R-M. For the ABC flows that are driven at the second largest available length scale, it is found that (a) the 2 1/2-dimensional flows A = B, C = 0 (and permutations) are again the first flows that result in a dynamo with a decreased onset. (b) The most symmetric flow, A = B = C, is the second type of flow that results in a dynamo. It is, and it remains, a local maximum. (c) At larger R-M, the flow A = B similar or equal to 2C/5 (and permutations) appears as the third type of flow that results in a dynamo. As R-M is increased, it becomes the flow with the largest growth rate. The growth rates appear to have some correlation with the Lyapunov exponents, but constructive refolding of the field lines appears equally important in determining the fastest dynamo flow.

DOI

3

Two-dimensional behavior of three-dimensional magnetohydrodynamic flow with a strong guiding field - Alexakis, Alexandros

PHYSICAL REVIEW E 84, (2011)

Abstract : The magnetohydrodynamic (MHD) equations in the presence of a guiding magnetic field are investigated by means of direct numerical simulations. The basis of the investigation consists of nine runs forced at the small scales. The results demonstrate that for a large enough uniform magnetic field the large scale flow behaves as a two-dimensional (2D) (non-MHD) fluid exhibiting an inverse cascade of energy in the direction perpendicular to the magnetic field, while the small scales behave like a three-dimensional (3D) MHD fluid cascading the energy forwards. The amplitude of the inverse cascade is sensitive to the magnetic field amplitude, the domain size, the forcing mechanism, and the forcing scale. All these dependences are demonstrated by the varying parameters of the simulations. Furthermore, in the case that the system is forced anisotropically in the small parallel scales an inverse cascade in the parallel direction is observed that is feeding the 2D modes k(parallel to) = 0.

PHYSICAL REVIEW E 84, (2011)

LPS

Abstract : The magnetohydrodynamic (MHD) equations in the presence of a guiding magnetic field are investigated by means of direct numerical simulations. The basis of the investigation consists of nine runs forced at the small scales. The results demonstrate that for a large enough uniform magnetic field the large scale flow behaves as a two-dimensional (2D) (non-MHD) fluid exhibiting an inverse cascade of energy in the direction perpendicular to the magnetic field, while the small scales behave like a three-dimensional (3D) MHD fluid cascading the energy forwards. The amplitude of the inverse cascade is sensitive to the magnetic field amplitude, the domain size, the forcing mechanism, and the forcing scale. All these dependences are demonstrated by the varying parameters of the simulations. Furthermore, in the case that the system is forced anisotropically in the small parallel scales an inverse cascade in the parallel direction is observed that is feeding the 2D modes k(parallel to) = 0.

DOI

4

Bounding the scalar dissipation scale for mixing flows in the presence of sources - Alexakis, A. and Tzella, A.

JOURNAL OF FLUID MECHANICS 688, 443-460 (2011)

Abstract : We investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source sink distribution. We focus on the spatial variation of the scalar field, described by the dissipation wavenumber, k(d), that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Peclet number (Pe >> 1) yield four distinct regimes for the scaling behaviour of kd, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of Pe and the ratio rho = l(u)/l(s), where l(e) and l(s) are, respectively, the characteristic length scales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their relation to previous bounds. Finally, we note some implications for three-dimensional turbulent flows.

JOURNAL OF FLUID MECHANICS 688, 443-460 (2011)

LPS

Abstract : We investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source sink distribution. We focus on the spatial variation of the scalar field, described by the dissipation wavenumber, k(d), that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Peclet number (Pe >> 1) yield four distinct regimes for the scaling behaviour of kd, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of Pe and the ratio rho = l(u)/l(s), where l(e) and l(s) are, respectively, the characteristic length scales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their relation to previous bounds. Finally, we note some implications for three-dimensional turbulent flows.