DOI

21

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

22

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

23

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.

DOI

24

Stratified shear flow instabilities at large Richardson numbers - Alexakis, Alexandros

PHYSICS OF FLUIDS 21, (2009)

Abstract : Numerical simulations of stratified shear flow instabilities are performed in two dimensions in the Boussinesq limit. The density variation length scale is chosen to be four times smaller than the velocity variation length scale so that Holmboe or Kelvin-Helmholtz unstable modes are present depending on the choice of the global Richardson number Ri. Three different values of Ri were examined Ri=0.2, 2, and 20. The flows for the three examined values are all unstable due to different modes, namely: the Kelvin-Helmholtz mode for Ri=0.2, the first Holmboe mode for Ri=2, and the second Holmboe mode for Ri=20 that has been discovered recently and this is the first time that it is examined in the nonlinear stage. It is found that the amplitude of the velocity perturbation of the second Holmboe mode at the nonlinear stage is smaller but comparable to first Holmboe mode. The increase in the potential energy, however, due to the second Holmboe modes is greater than that of the first mode. The Kelvin-Helmholtz mode is larger by two orders of magnitude in kinetic energy than the Holmboe modes and about ten times larger in potential energy than the Holmboe modes. The effect of increasing Prandtl number is also investigated, and a weak dependence on the Prandtl number is observed. The results in this paper suggest that although mixing is suppressed at large Richardson numbers it is not negligible, and turbulent mixing processes in strongly stratified environments cannot be excluded.

PHYSICS OF FLUIDS 21, (2009)

LPS

Abstract : Numerical simulations of stratified shear flow instabilities are performed in two dimensions in the Boussinesq limit. The density variation length scale is chosen to be four times smaller than the velocity variation length scale so that Holmboe or Kelvin-Helmholtz unstable modes are present depending on the choice of the global Richardson number Ri. Three different values of Ri were examined Ri=0.2, 2, and 20. The flows for the three examined values are all unstable due to different modes, namely: the Kelvin-Helmholtz mode for Ri=0.2, the first Holmboe mode for Ri=2, and the second Holmboe mode for Ri=20 that has been discovered recently and this is the first time that it is examined in the nonlinear stage. It is found that the amplitude of the velocity perturbation of the second Holmboe mode at the nonlinear stage is smaller but comparable to first Holmboe mode. The increase in the potential energy, however, due to the second Holmboe modes is greater than that of the first mode. The Kelvin-Helmholtz mode is larger by two orders of magnitude in kinetic energy than the Holmboe modes and about ten times larger in potential energy than the Holmboe modes. The effect of increasing Prandtl number is also investigated, and a weak dependence on the Prandtl number is observed. The results in this paper suggest that although mixing is suppressed at large Richardson numbers it is not negligible, and turbulent mixing processes in strongly stratified environments cannot be excluded.

DOI

25

Planar bifurcation subject to multiplicative noise: Role of symmetry - Alexakis, Alexandros and Petrelis, Francois

PHYSICAL REVIEW E 80, (2009)

Abstract : The effect of multiplicative noise on a system described by two modes close to a bifurcation point is investigated. The bifurcation is assumed stationary and noise acts as random coupling between these modes. An analytic formula that predicts the onset of instability is derived, and the domain of existence of on-off intermittency is calculated based on an eigenvalue problem. This approach, confirmed by numerical simulations of the Langevin equations, allows quantifying the possible effects of the noise. The stability and the on-off behavior are shown to be very sensitive to deviations of the deterministic system from the case where both modes grow with equal rate and the system displays a continuous symmetry associated to rotation in phase space. In general, a noise term that breaks this continuous symmetry will increase the domain of instability of the system while a noise term that preserves the symmetry reduces the domain of instability.

PHYSICAL REVIEW E 80, (2009)

LPS

Abstract : The effect of multiplicative noise on a system described by two modes close to a bifurcation point is investigated. The bifurcation is assumed stationary and noise acts as random coupling between these modes. An analytic formula that predicts the onset of instability is derived, and the domain of existence of on-off intermittency is calculated based on an eigenvalue problem. This approach, confirmed by numerical simulations of the Langevin equations, allows quantifying the possible effects of the noise. The stability and the on-off behavior are shown to be very sensitive to deviations of the deterministic system from the case where both modes grow with equal rate and the system displays a continuous symmetry associated to rotation in phase space. In general, a noise term that breaks this continuous symmetry will increase the domain of instability of the system while a noise term that preserves the symmetry reduces the domain of instability.