Archontis dynamo
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Author: Dr. Vasilis D. Archontis, University of St. Andrews
Dr. Vasilis D. Archontis accepted the invitation on 4 July 2007 (self-imposed deadline: 4 September 2007).
This article will briefly cover: the concept of a laminar dynamo that firstly saturates at equipartition but later on undergoes transitions between turbulent and laminar solutions.
Contents |
Introduction
Magnetic fields are ubiquitous in astrophysical bodies throughout the cosmos. They vary in scale and strength from strong magnetic fields in e.g. solar active regions to relatively weak magnetic fields in planets and galaxies. These fields are controlled by the motions of plasmas through a transfer of kinetic energy to magnetic energy: the resulting amplification and regeneration of the magnetic field is called dynamo action (Cowling 1934, Moffatt 1978,Parker 1979 ). The study of dynamo action in the limit of infinitely high magnetic Reynolds number is the subject of fast dynamo theory and is relevant to most astrophysical systems, where the diffusion timescales are typically much larger than the advection timescales.
Considering the importance of the magnetic forces relative to the motion of the fluid, one divides dynamo action into two regimes: the linear kinematic regime in which the flow amplifies the magnetic field exponentially (e.g. by stretching the magnetic field lines) and the non-linear regime where the magnetic field becomes strong enough to modify the initial flow topology through the Lorentz force, and consequently halts the exponential growth. The dynamos saturate either because of a suppression of the stretching ability of the flows and a reduction of the field dissipation or via a balance between vigorous stretching and strong dissipation.
In many astrophysical systems the stretching of the magnetic field lines, which drives the initial exponential amplification of the magnetic energy, is achieved by turbulent flows. An interesting issue is then the equilibrium field strength at which this turbulent dynamo action saturates. Moreover, if the generated magnetic field at saturation has magnetic energy equal to the kinetic energy of the flow (equipartition), then this dynamo is able to generate strong magnetic fields that are comparable with those observed in many astrophysical systems.
The first known example of a dynamo that saturates at equipartition in a
laminar manner, for relatively high Reynolds numbers is the dynamo found
by Archontis, 2000. The initial flow involved in these studies is a non
scale-seperated flow, similar to the ABC flows but with no cosine terms and it was
found to produce fast dynamo action (Galloway and Proctor, 1992).
A parametetric study of the saturation level of this dynamo (Dorch and Archontis, 2004) showed
that if the magnetic Prandtl number is different from unity the velocity
and magnetic fields are not identical but they are still proportional. The magnetic
Prandtl number is defined as the ratio of the magnetic to the fluid Reynolds number (
), where
and
with 
being the characteristic velocity and length scales, the magnetic diffusivity
and the viscosity respectively.
Cameron and Galloway, 2006a and
Cameron and Galloway, 2006b studied the equipartition
solution for large values of 
and they found that there
is no evidence that the laminar behaviour at equipartition is not asymptotic and
concluded that this dynamo yield fields strong enough to
be astrophysically relevant. They also discovered new classes of dynamos, which
saturate also at close equipartition. The forcing and flow in these dynamos are
not so symmetric and, thus, one may argue that are more generic examples of equipartition dynamos.
Finally, Archontis at.al, 2007 found that that the equipartition laminar mode
is not a time-independent solution, but it suffers from transitions
between turbulent and equipartition modes during the evolution of the system.
Flow topology
The initial velocity field that produces fast dynamo action belongs to the well-known class of three-dimensional, steady ABC flows and is given by
- (1)
.
The total kinetic helicity of this flow is equal to zero. Also, although locally helicity density is not zero, there is no overall integrated helicity in the sytem.
The flow contains significant regions of chaotic particle trajectories and contains two different types of stagnation points. There are stagnation points where the streamlines are diverging in a two-dimensional plane with a spiraling form and are converging in an axis perpendicular to the plane (a-type), and stagnation points with the opposite topology of the streamlines (b-type). The diverging streamlines from one type of the stagnation points become converging for the other and vice versa. Similarly to the velocity field, the vorticity has eight null points which are found in the middle of the straight heteroclinic connections of the stagnation points of the flow. It is important to understand how the network of streamlines and vortex lines connect individual points with very small velocity and vorticity magnitude because the evolution of vorticity in association with the topology of the flow is a key process of the dynamo action. Fig.1 shows how stagnation and vortex null points are connected along a main diagonal of the computational box.
Dynamo action: kinematic regime
Numerical experiments have been performed (Archontis et.al, 2007) to study the action
and saturation of this dynamo. In a first series of experiments the
fluid Reynolds number was small (
), the reason being that the flow
becomes unstable for moderate values of Re.
Thus, very small values of Re ensure that
the flow remains smooth and stable during the exponential growth of
the magnetic energy and one may follow closely the time evolution of
individual structures in space. In the experiments, the full 3D compressible
MHD equations were solved on a periodic Cartesian mesh. The initial magnetic field was
random and weak. Due to dynamo action, the magnetic field first growed exponentially and then
saturated in the dynamical regime.
During the kinematic dynamo action, the magnetic field has the form of strong and curved sheets that are wrapped around the strong vorticity structures. The fieldlines from these strong sheets are stretched and twisted by the flow and finally pile up to neighboring sheets with similar polarity (see Fig.2). In fact, most of the work done against the Lorentz force occurs between the strong flux sheets and the strong velocity field which surrounds them. The weak magnetic field has the form of rounded sheets that are folded at the a-type stagnation points. The folding of magnetic field lines in the plane of divergence of an a-type stagnation point is shown in Fig.2. Constructive folding of the magnetic field also occurs close to the center of the weak velocity channels. There, the magnetic field lines are folded in the same way as in the a-type stagnation points. The stretching and folding of the fieldlines increase the magnetic energy of the system.
An interesting question now is what happens when the exponential amplification of the magnetic energy stops and the strength of the magnetic field becomes large enough for it to react back on the flow. The resulting Lorentz force on the fluid is perpendicular to the magnetic field and acts in the vertical direction. Moreover, it is found that the regions of maximum Lorentz force are correlated with high amplitude vorticity regions and the force acts typically perpendicular to the vorticity. When the Lorentz force becomes strong enough the vortex strength in the high vorticity regions is reduced and the elongated vortex tubes shrink together with the velocity channels of low pressure that feed them. In fact, they retreat towards the initial position of the b-type stagnation points of the flow. The vorticity now between the magnetic flux sheets becomes very low and it almost adapts the properties of an a-type vortex null-point. Thus, it seems that the dynamical coupling of the vorticity and the magnetic field, which has not been reported in previous studies of this dynamo, is of potential importance to a better understanding of the magnetic energy amplification and the subsequent saturation.
It is also of great importance to find where in the computational domain the relative orientation of the magnetic and velocity field is such that dynamo stops or still works effectively. Nearly perfect alignment---and thus no stretching of the magnetic field lines by the flow---is achieved where the two fields are strong. Joule dissipation is also at minimum there and thus the size and the strength of the relevant structures remain unchanged in time. The sites of imperfect alignment are the a-type stagnation points of the flow and their close vicinity. The work done against the Lorentz force at these sites is balanced by high Joule dissipation. Thus, a total balance between work and dissipation is achieved in most of the volume. The total kinetic and magnetic energies become equal and this solution tends to relax to a unique time-independent state.
Dynamo action: saturation regime
In another series of experiments Archontis et.al, 2007 studied the saturation regime of the dynamo for higher values of Re and Rm.
Fig.3 shows the time evolution of the kinetic and the magnetic energy in an experiment
with 
(case 1). At the beginning of the simulation, the magnetic energy increases
exponentially with time. Then, it saturates reaching equipartition values. This is a laminar solution
with the magnetic field being equal in direction and magnitude to the velocity field. In fact, one may show
analytically that this is only an exact solution to the MHD equations in the absense of forcing and
dissipation. The exact equipartition phase lasts until 
. Then, another phase begins
where turbulence kicks in and the two energies deviate from equipartition.
.
and 
.Fig.4 shows the evolution of the energies for and (case 2).
Similar to the experiment shown in Fig.1 the dynamo shows a laminar equipartition solution in
saturation until 
. Later on, the energies undergo oscillations with large
amplitudes. Although the solution becomes unstable the two energies follow a similar time evolution, which
deviates slightly from equipartition, mostly close to the minima of the oscillations. A striking result is
is that the magnetic energy goes through a turbulent phase (after 
) but later on comes
out as a laminar equipartition solution for short periods of time. However, the solution does not stay at equipartition
but eventually becomes turbulent again. This transition of the magnetic energy between the two states
is repeatedly observed during experiments with various values of Reynolds numbers and it shows the very
complex and time-dependent evolution of the fields during the saturation of the system.
In case 1, the dynamo action follows first an equipartition and then a turbulent phase during saturation. The equipartition phase was reported in the work by Cameron and Galloway, 2006a who used the Elsasser variables, to show the source of the variation in the energies for the U=B solution of this dynamo. They found that (U+B) is large and time-independent and, thus, most of the variations between the velocity and magnetic fields comes from (U-B).
A complimentary study of these variables (Archontis et.al, 2007) revealed that both variables are important.
Fig.5 shows the yellow isosurface, which is 80 percent of the maximum value of U-B, and the
blue isosurface that is 10 percent of the maximum value of U+B at 
.
The yellow structures adopt a tube-like shape in most of the volume, although there are also sites where
U-B forms curved sheets (close to the center of the box).
On the other hand, the structures with small values of the difference between the field vectors are concentrated in between the
strong tube and sheet-like structures and connect them at their ends. It is worthwhile to mention that the small U+B
values coincide, in space, with the minima of cross helicity, which is a measurement of the alignment
between the velocity and magnetic fields. Thus, it seems that most of the work done (although it is small due to equipartition)
in the saturation regime occurs at the places with small values of U+B. In fact, visualization of magnetic fieldlines
(red lines in the computational volume) show that significant twist and folding occurs exactly there.
The fieldlines are running in parallel and are added to the tubes and sheets, but they are following highly twisted spiral paths
when they approach the minima of U+B. Eventually, they are folded with fieldlines which are coming from
neighboring structures and have similar polarity. This process increases the magnetic energy of the system.
In case 2, the magnetic energy jumps from a turbulent state to another regime where it evolves with less-irregular oscillations at close equipartition with the kinetic energy. Thus, it is interesting to visualize the structure of the magnetic field when the magnetic energy has escaped from the turbulent phase and returns to the energy equipartition level.
Fig.6 and Fig.7 show the three-dimensional topology of the full magnetic field vector at two times:
and 
, when the magnetic energy goes from a minimum to a maximum value respectively.
Shown in the snapshots is the isosurfaces of magnetic field strength with a value of 70 per cent of the maximum.
Fig.6 shows that when the magnetic energy is minimum the field is concentrated into
small-scale coherent tubes and sheets. Fig.7 shows that, later on, most of the structures
are folded constructively in various positions inside the computational volume.
For example, in the closest lower and the upper corner of the box, large scale structures are formed due to twisting and
folding of many small scale magnetic field tubes and sheets. The fieldlines adopt a spiral shape at the
sites of folding, indicating the twisting motion of the structures prior to the folding that
increases the magnetic energy of the system. The following decrease of the magnetic energy is
accompanied by less twist and folding of the magnetic structures, which eventually take again the form
of individual tubes and sheets. We assume that the same physical processes occur when the magnetic energy
decays or grows during the temporal evolution of the system in this phase of saturation.
Discussion
An interesting result of the aforementioned experiments is that the dynamo under consideration equilibrates first with approximately equal (scaled) magnetic and velocity fields, which are very similar to the initial flow topology. It does so even for high Reynolds numbers. This solution is an equipartition solution and, as opposed to generic turbulent dynamos in astrophysics, this dynamo has laminar flows and a correspondingly smooth magnetic topology. Thus, it is illustrated that turbulence is not a necessary condition for a dynamo to generate strong magnetic fields, such as those that are observed in many astrophysical contexts.
Also, it has been shown that after many diffusive time scales the laminar equipartition solution looses its time independent nature. Firstly, the temporal evolution of the energies shows clear oscillations. It seems that the period and amplitude of these oscillations is independent of the magnetic or viscous Reynold numbers. On the other hand, it is likely that these oscillations indicate the operation of another dynamo mode, which is different from the time-independent equipartition mode. Also, it is worthwhile to mention that we find indications of breaking of the strong symmetries of the flow during this phase. It is interesting that the disruption of the three-fold symmetries is more effective in experiments with high Reynold numbers and, thus, the system deviates from its initial configuration even before the appearance of the fully developed turbulent phase.
After a period of oscillatory behaviour the system enters a turbulent regime. However, the turbulent mode is not the final mode at saturation, because the system finds again the way to adopt states with high energy levels where kinetic and magnetic energies become almost equal. We find that the saturation of this dynamo suffers from transitions between these modes and, thus, exhibits an intricate dynamical behaviour characterized by laminar and turbulent phases.
The simple initial conditions in our experiments notwithstanding, we conjecture that the building blocks and individual processes involved in the dynamo action and equipartition of the aforementioned dynamo might be of general importance in the studies of astrophysical dynamos. It is an observed fact for example that the Sun possesses a structured, large-scale mean magnetic field of strength comparable to equipartition, which goes through a periodic dynamo process. Thus, studying equipartition dynamos such as the one in the present simulations may be an important step towards the understanding of the fast growth and the subsequent equilibration of astrophysical magnetic fields.
References
- Archontis, 2000
- Archontis et.al, 2007
- Cameron and Galloway, 2006a
- Cameron and Galloway, 2006b
- Cowling 1934
- Dorch and Archontis, 2004
- Galloway and Proctor, 1992
- Moffatt 1978
- Parker 1979
