Parallel expansion of density bursts G. Chiavassa, H. Bufferand, G. Ciraolo, Ph. Ghendrih, H. Guillard, L. Isoardi, A. Paredes, F. Schwander, E. Serre, P. Tamain M2P2-CNRS laboratory, Ecole Centrale Marseille, IRFM Cea Cadarache, Inria Sophia Antipolis, France

Overview Summary Analytical calculations and numerical computations are performed to describe the parallel expansion of an over-dense region of finite extent. It is shown that the parallel transport of particles generates fronts propagating at supersonic velocities. Time traces of the expanding pulses can be used to estimate the location and size of the initial density burst.

Introduction Evidence of poloidally localized cross-field transport in experiments and theoretical analysis of turbulence transport governs the onset of parallel transport towards equilibrium. When cross-field transport appears in bursts, both for ELM relaxation events and microturbulence, the parallel transport of particles is shown to generate fronts that propagate with supersonic velocities. It is shown that after a short transient the density structure is no longer monotonic and that the two fronts (one co, the other counter the magnetic field) are independent. Furthermore, the time trace of the particle flux at a given location is characterized by a sharp rise followed by a longer time scale relaxation. Comparing the time delay and magnitude of the density burst at two locations allows to estimate the magnitude and the location of the generation of the front.

1-D modeling of parallel transport A minimum system involving only the particle and momentum balances along the magnetic field line is investigated. In the isothermal framework, the sound velocity is constant and can be used to normalize the particle velocity. Length is normalized by the length of the field line and time by the parallel confinement time. The dimensionless equations take the form :

(1)

(3) Therefore the front velocity is always supersonic whatever the initial density Nb, as observed on Figure 1-right.

Expansion of the triangular burst:

Figure 1. Left: Sketch of the density pulse extending from the symmetry point z=0. Dashed area corresponds to the initial density burst. Right: Front velocity CF as a function of the initial density step Nb. Blue curve represent the approximation for small values involved in turbulent transport.

In the following stage of evolution, the triangular burst shifts and expands along the z-axis as observed on Figure 4. The density of the front decreases, which governs in turn a decrease of the front velocity CF from equation (3).

Time evolution of the front is represented on Figure 6. As expected, and accordingly to the front density decrease, the position of the front converges asymptotically towards a sonic displacement.

Analytical solution Using classical shock wave relations [2,3], an analytical expression for the solution of the previous problem can be found in the first instants of the expansion [1]. It can be shown that generically a rarefaction wave preceded by a front will bridge the background plasma to the initial burst as sketched on Figure 1left. The front is a discontinuity between the density NF and the density N0 =1, while the rarefaction wave exhibits an exponential

Transient evolution :

At time t=zb, the rarefactions coming from the left and right discontinuities of the initial burst interact a z=0. The density decreases while the front expands with the velocity CF as represented on Figure 3. After this transient, the front appears to travel in an unperturbed fashion until a triangular shape is reached for Log(N) and M. With our present understanding, it is difficult to determine analytically when such a shape is obtained. This time is refereed as tF0 in the following sections.

We measure : Assuming during this phase the following relation between the efolding and the previous decay rate : we deduce from relation (10) the estimations for the particle content of the initial burst :

instead of

We introduce a reference state with triangular shape for Log(N) extending from z=0 to z=LF0, such that NF0 and CFO are determinated by the initial phase. The extend of the front is set by particle conservation:

Conclusion

Figure 6: Evolution of the triangular front location

Finally, it is also possible to compute the extention and e-folding length (or slope of Log(N)) :

Where MF is the Mach number at the front location related to the front velocity : MF=CF-1/CF. LF(t) is obtained from equation (4). Using relation (5), the time dependance of the front density is governed by: (6) Integration of this equation gives : (7) with being the normalized evolution time.

and

Finally, using data of probe 2 and equation (3), the density of the initial burst is found to be : and its initial extent is :

Modeling the triangular burst evolution:

(4)

leading to

instead of

Figure 4. Evolution of the density during the triangular shaped burst phase of expansion

In [2], a similar triangular front evolution is described. When taking into acount the fact that the shape is here triangular for Log(N), one obtains the relations:

Figure 2: Comparison between exact and numerical density (left) and Mach number (right) for a given time t
(8)

with

On can also estimate the distance of the probes from the origin using equation (9) and the given value for z1+z2=0.6 :

(5)

and is represented by the dashed blue area on Figure 1-left for z>0. At initial time t=0, both the background plasma and the burst are considered at rest with vanishing Mach number.

The initial time for the onset of the triangular pulse is also expressed as a function of the front density :

The density rise is not described but is characterized by the very small relaxation time :

(9)

Numerical simulations Since the solution of system (1) exhibits discontinuities, a robust shock capturing method is necessary. The implemented numerical scheme is based on an approximate Roe solver combined with third order ENO interpolations [2,4]. Time evolution is obtained through second order TVD Runge-Kutta integration. As observed on Figure 2, the numerical solutions obtained with this algorithm match the previous exact ones.

Figure 5: Evolution of the generic front density derived from equation (7)

Given these relations, the front position zF reads :

The right end of the rarefaction moves with velocity CF-1 > 0, while the left end has a negative velocity: -1. As a consequence, this point reaches the symmetry point z=0 at time t=zb. For larger values of t, the analytical solution is not valid and numerical simulations are used to describe the expansion of the front.

(2)

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Figure 7. Left: time traces of density recorded at probes 1 (blue) and 2 (black). Right: highlight of the rise and exponential decay for signal on probe 2. Figure 3. Evolution of the density during the transient between t=zb and the triangular shaped burst.

where N is the density (normalized by the constant value of background plasma, N0) and M is the Mach number. The initial state for the density is described by a constant density extending over a finite length scale and connected to the background plasma by two discontinuities :

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Using this relation we have represented the generic evolution of the front density on Figure 5.

density decay and a linear increase for M. The front density and velocity are characterized by the following equations:

(10)

Characterizing the cross-field transport events The previous analytical expressions allow us to compare the signature of two triangular pluses generated by the initial density bursts. These time traces are represented on figure 7 and correspond to two different measurement points obtained by numerical integration of system (1). The blue curve corresponds to a probe located in the counter direction of the magnetic field line, at z=-z1=-0.1, while the black one corresponds to a probe in the co-direction at z=z2=0.5. The measured signals exhibit a sharp rise followed by an exponential decay highlighted on Figure 7-right for probe 2. Data from point 1 is similar but a rather flat maximum, indicating that z1 is close enough from the location of the initial burst so that the phase of triangular shape has not been reached yet. On figure 7, we first measure the value of the fronts density :

One dimensional analytical work and numerical modeling provide a comprehensive description of the dynamics of the parallel expansion of a density burst. Two fronts propagate in the co and counter magnetic field directions and become independent. Using measurements of the signal at two different locations, the origin, extend and magnitude of the initial burst can be estimated. It is interesting to note that the shape of the recorded density pulse leads to the characteristic ELM form of the particle flux and that the front velocity scales like sound velocity in agreement with experimental measurement for the ELM propagation. The present work could be used as a guideline to analyze the ELM shapes as a signature of the distance over which the parallel pulses have propagated.

References [1] Chiavassa G. et al, Density increment expansion along plasma magnetic field lines, in preparation. [2] Landau L., Lifchitz E., Mécanique des fluides, MIR editions, Moscow 1971, p. 472. [3] Leveque R.J., Numerical Methods for Conservation Laws, Birkhauser, 1992. [4] Shu C.W and Osher S.J., Efficient implementation of essentially nonoscillatory shock-capturing schemes, J.Comput. Phys., 83, 1989.

Acknowledgements This work is supported by ANR ESPOIR contract ANR090035-01.