DOI

1

Critical behavior in the inverse to forward energy transition in two-dimensional magnetohydrodynamic flow - Seshasayanan, Kannabiran and Alexakis, Alexandros

PHYSICAL REVIEW E 93, (2016)

Abstract : We investigate the critical transition from an inverse cascade of energy to a forward energy cascade in a two-dimensional magnetohydrodynamic flow as the ratio of magnetic to mechanical forcing amplitude is varied. It is found that the critical transition is the result of two competing processes. The first process is due to hydrodynamic interactions and cascades the energy to the large scales. The second process couples small-scale magnetic fields to large-scale flows, transferring the energy back to the small scales via a nonlocal mechanism. At marginality the two cascades are both present and cancel each other. The phase space diagram of the transition is sketched.

PHYSICAL REVIEW E 93, (2016)

LPS

Abstract : We investigate the critical transition from an inverse cascade of energy to a forward energy cascade in a two-dimensional magnetohydrodynamic flow as the ratio of magnetic to mechanical forcing amplitude is varied. It is found that the critical transition is the result of two competing processes. The first process is due to hydrodynamic interactions and cascades the energy to the large scales. The second process couples small-scale magnetic fields to large-scale flows, transferring the energy back to the small scales via a nonlocal mechanism. At marginality the two cascades are both present and cancel each other. The phase space diagram of the transition is sketched.

DOI

2

Role of dissipation in flexural wave turbulence: From experimental spectrum to Kolmogorov-Zakharov spectrum - Miquel, Benjamin and Alexakis, Alexandros and Mordant, Nicolas

PHYSICAL REVIEW E 89, (2014)

Abstract : The weak turbulence theory has been applied to waves in thin elastic plates obeying the Foppl-Von Karman dynamical equations. Subsequent experiments have shown a strong discrepancy between the theoretical predictions and the measurements. Both the dynamical equations and the weak turbulence theory treatment require some restrictive hypotheses. Here a direct numerical simulation of the Foppl-Von Karman equations is performed and reproduces qualitatively and quantitatively the experimental results when the experimentally measured damping rate of waves gamma(k) = a + bk(2) is used. This confirms that the Foppl-Von Karman equations are a valid theoretical framework to describe our experiments. When we progressively tune the dissipation so that to localize it at the smallest scales, we observe a gradual transition between the experimental spectrum and the Kolmogorov-Zakharov prediction. Thus, it is shown that dissipation has a major influence on the scaling properties of stationary solutions of weakly nonlinear wave turbulence.

PHYSICAL REVIEW E 89, (2014)

LPS

Abstract : The weak turbulence theory has been applied to waves in thin elastic plates obeying the Foppl-Von Karman dynamical equations. Subsequent experiments have shown a strong discrepancy between the theoretical predictions and the measurements. Both the dynamical equations and the weak turbulence theory treatment require some restrictive hypotheses. Here a direct numerical simulation of the Foppl-Von Karman equations is performed and reproduces qualitatively and quantitatively the experimental results when the experimentally measured damping rate of waves gamma(k) = a + bk(2) is used. This confirms that the Foppl-Von Karman equations are a valid theoretical framework to describe our experiments. When we progressively tune the dissipation so that to localize it at the smallest scales, we observe a gradual transition between the experimental spectrum and the Kolmogorov-Zakharov prediction. Thus, it is shown that dissipation has a major influence on the scaling properties of stationary solutions of weakly nonlinear wave turbulence.

DOI

3

On the edge of an inverse cascade - Seshasayanan, Kannabiran and Benavides, Santiago Jose and Alexakis, Alexandros

PHYSICAL REVIEW E 90, (2014)

Abstract : We demonstrate that systems with a parameter-controlled inverse cascade can exhibit critical behavior for which at the critical value of the control parameter the inverse cascade stops. In the vicinity of such a critical point, standard phenomenological estimates for the energy balance will fail since the energy flux towards large length scales becomes zero. We demonstrate this using the computationally tractable model of two-dimensional (2D) magnetohydrodynamics in a periodic box. In the absence of any external magnetic forcing, the system reduces to hydrodynamic fluid turbulence with an inverse energy cascade. In the presence of strong magnetic forcing, the system behaves as 2D magnetohydrodynamic turbulence with forward energy cascade. As the amplitude of the magnetic forcing is varied, a critical value is met for which the energy flux towards the large scales becomes zero. Close to this point, the energy flux scales as a power law with the departure from the critical point and the normalized amplitude of the fluctuations diverges. Similar behavior is observed for the flux of the square vector potential for which no inverse flux is observed for weak magnetic forcing, while a finite inverse flux is observed for magnetic forcing above the critical point. We conjecture that this behavior is generic for systems of variable inverse cascade.

PHYSICAL REVIEW E 90, (2014)

LPS

Abstract : We demonstrate that systems with a parameter-controlled inverse cascade can exhibit critical behavior for which at the critical value of the control parameter the inverse cascade stops. In the vicinity of such a critical point, standard phenomenological estimates for the energy balance will fail since the energy flux towards large length scales becomes zero. We demonstrate this using the computationally tractable model of two-dimensional (2D) magnetohydrodynamics in a periodic box. In the absence of any external magnetic forcing, the system reduces to hydrodynamic fluid turbulence with an inverse energy cascade. In the presence of strong magnetic forcing, the system behaves as 2D magnetohydrodynamic turbulence with forward energy cascade. As the amplitude of the magnetic forcing is varied, a critical value is met for which the energy flux towards the large scales becomes zero. Close to this point, the energy flux scales as a power law with the departure from the critical point and the normalized amplitude of the fluctuations diverges. Similar behavior is observed for the flux of the square vector potential for which no inverse flux is observed for weak magnetic forcing, while a finite inverse flux is observed for magnetic forcing above the critical point. We conjecture that this behavior is generic for systems of variable inverse cascade.

DOI

4

Origins of the k(-2) spectrum in decaying Taylor-Green magnetohydrodynamic turbulent flows - Dallas, V. and Alexakis, A.

PHYSICAL REVIEW E 88, (2013)

Abstract : We investigate the origins of k(-2) spectrum in a decaying Taylor-Green magnetohydrodynamic flow with zero large scale magnetic flux that was reported by Lee et al. [Phys. Rev. E 81, 016318 ( 2010)]. So far, a possible candidate for this scaling exponent has been the weak turbulence phenomenology. From our numerical simulations, we observe that current sheets in the magnetic Taylor-Green flow are formed in regions of magnetic discontinuities. Based on this observation and by studying the influence of the current sheets on the energy spectrum, using a filtering technique, we argue that the discontinuities are responsible for the -2 power law scaling of the energy spectra of this flow.

PHYSICAL REVIEW E 88, (2013)

LPS

Abstract : We investigate the origins of k(-2) spectrum in a decaying Taylor-Green magnetohydrodynamic flow with zero large scale magnetic flux that was reported by Lee et al. [Phys. Rev. E 81, 016318 ( 2010)]. So far, a possible candidate for this scaling exponent has been the weak turbulence phenomenology. From our numerical simulations, we observe that current sheets in the magnetic Taylor-Green flow are formed in regions of magnetic discontinuities. Based on this observation and by studying the influence of the current sheets on the energy spectrum, using a filtering technique, we argue that the discontinuities are responsible for the -2 power law scaling of the energy spectra of this flow.

DOI

5

Symmetry breaking of decaying magnetohydrodynamic Taylor-Green flows and consequences for universality - Dallas, V. and Alexakis, A.

PHYSICAL REVIEW E 88, (2013)

Abstract : We investigate the evolution and stability of a decaying magnetohydrodynamic Taylor-Green flow, using pseudospectral simulations with resolutions up to 2048(3). The chosen flow has been shown to result in a steep total energy spectrum with power law behavior k(-2). We study the symmetry breaking of this flow by exciting perturbations of different amplitudes. It is shown that for any finite amplitude perturbation there is a high enough Reynolds number for which the perturbation will grow enough at the peak of dissipation rate resulting in a nonlinear feedback into the flow and subsequently break the Taylor-Green symmetries. In particular, we show that symmetry breaking at large scales occurs if the amplitude of the perturbation is sigma(crit) similar to Re-1 and at small scales occurs if sigma(crit) similar to Re-3/2. This symmetry breaking modifies the scaling laws of the energy spectra at the peak of dissipation rate away from the k(-2) scaling and towards the classical k(-5/3) and k(-3/2) power laws.

PHYSICAL REVIEW E 88, (2013)

LPS

Abstract : We investigate the evolution and stability of a decaying magnetohydrodynamic Taylor-Green flow, using pseudospectral simulations with resolutions up to 2048(3). The chosen flow has been shown to result in a steep total energy spectrum with power law behavior k(-2). We study the symmetry breaking of this flow by exciting perturbations of different amplitudes. It is shown that for any finite amplitude perturbation there is a high enough Reynolds number for which the perturbation will grow enough at the peak of dissipation rate resulting in a nonlinear feedback into the flow and subsequently break the Taylor-Green symmetries. In particular, we show that symmetry breaking at large scales occurs if the amplitude of the perturbation is sigma(crit) similar to Re-1 and at small scales occurs if sigma(crit) similar to Re-3/2. This symmetry breaking modifies the scaling laws of the energy spectra at the peak of dissipation rate away from the k(-2) scaling and towards the classical k(-5/3) and k(-3/2) power laws.

DOI

6

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

7

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

8

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

9

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.