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Self Organization in a Diffusion Model of Thin Electric Current Sheets

Student Researcher: Andrew Kercher, Faculty Advisor: Dr. Robert Weigel

Undergraduate Research in Computational Mathematics at George Mason University

Abstract: In space plasmas, thin electric current sheets are observed to form near the surface of the sun, in the solar wind, and within Earth’s magnetotail. The basic physics of this system can be approximated by a diffusion model with a diffusivity that has hysteresis, and switches between low and high, depending on local current amplitude. We show that under constant energy input conditions, this system evolves to a state with characteristics of self-organized criticality.

Source Function: We treat the source as a given function and set it specifically.

This has the effect of steadily increasing the strength of the field reversal and the current sheet that supports it [3].

Energy: The total field energy of the system at any instant is defined as

Although the field strength is being increased by the source function, on average this increase in balanced by the dynamic state that is induced. The system remains close to, but under the critical state. Using the ideas of SOC we are able to begin to draw conclusions about a system in which the dynamics are extremely complicated.

The scalar field 𝐵𝑥 is modeled using one spatial dimension 𝑧, with a variable diffusion coefficient 𝐷(𝑧,𝑡), a source term 𝑆(𝑧,𝑡), and 𝐷𝑚𝑖𝑛 ≪𝐷𝑚𝑎𝑥 . The quantity 𝑄(𝑧,𝑡) has hysteresis, and acts like a physical instability.

𝑄ሺ𝑧,𝑡ሻ= 𝐷𝑚𝑖𝑛 →low state until ฬ𝜕𝐵𝑥𝜕𝑧ฬ> 𝑘 𝑄ሺ𝑧,𝑡ሻ= 𝐷𝑚𝑎𝑥 →high state until ฬ𝜕𝐵𝑥𝜕𝑧ฬ< 𝛽𝑘 0 < 𝛽 < 1

𝜕𝐵𝑥𝜕𝑡 = 𝜕𝜕𝑧 ൬𝐷(𝑧,𝑡) 𝜕𝐵𝑥𝜕𝑧൰+ 𝑆(𝑧)

𝜕𝜕𝑡൫𝐷ሺ𝑧,𝑡ሻ൯= 𝑄൬ฬ𝜕𝐵𝑥𝜕𝑧ฬ൰𝜏 − 𝐷𝜏

𝑄൬ฬ𝜕𝐵𝑥𝜕𝑧ฬ൰= ൜𝐷𝑚𝑖𝑛 Low State 𝐷𝑚𝑎𝑥 High State

−𝐿 ≤ 𝑧 ≤ 𝐿 𝜕𝐵𝑥𝜕𝑧 ሺ−𝐿,𝑡ሻ= 𝜕𝐵𝑥𝜕𝑧 ሺ𝐿,𝑡ሻ= 0

𝑆ሺ𝑧ሻ= 𝑆0 sinቀ𝜋𝑧2𝐿ቁ

Boundary Conditions: The system is evaluated on the spatial interval ൣ �– 𝐿,𝐿൧, where the value of the slope at the bounds is set equal to zero. Thus, there is no transport of 𝐵𝑥 through the boundaries.

To integrate the system, 𝑄 and 𝐷 were initially set equal to 𝐷𝑚𝑖𝑛 . While 𝑄 and 𝐷 were specifically set, the initial values for 𝐵𝑥 seem independent of the systems evolution into a dynamical behavior.

𝐸ሺ𝑡ሻ= න𝑑𝑧𝐵𝑥2

The value 𝑘, which is required to initiate the instability, is greater than the value 𝛽𝑘, which is required to maintain the instability. This evolution of 𝑄 is important for obtaining SOC behavior, along with influencing the progression of the variable diffusion coefficient 𝐷(𝑧,𝑡).

Magnetic field (blue lines)The field and definition of self-organization are still being developed, but

our general features of interest are the development of dynamics and structure that are not imposed by an external agent ([1],[2]).  In this poster we use the concepts and metaphors of SOC to guide our interpretation of modeling results.

Self-organized systems include, but are not limited to:

•Simple rules govern the dynamics. •Thresholds exist within the system. •The threshold is eventually exceeded by the buildup of energy.

The sandpile system depicted in Figure 1 is one of the most simple and well-known examples used to explain features of Self-Organized Criticality. Adding grains of sand slowly increases the energy of the system at a constant rate.

If the local slope exceeds a threshold value, avalanches occur.  The resulting avalanches can occur over a broad range of length and time scales.  In addition, the system evolves to a point where it is, on average, near the critical point.

Self-Organized Criticality

Instability in a Diffusion Model

Model Description

The revised Lu Continuum Model [4]

Summary

References

Long range correlations in space exist, even though the dynamics are governed by local rules.

Figure 1: Sandpile Model

Figure 2: Sandpile Extremes

Systems displaying characteristics associated with SOC dissipate stored energy in avalanches [3]. It is this instability that determines this “fast diffusion”[3]. Once the instability is reached, local slopes cause rapid movement, driving the system back to a stable state.

Analysis

The input of energy by the source term drives the system to the point of criticality. Once the critical point is reached, the system reacts by unloading the energy in avalanches . As time increases, the system will return to a stable state, but steep slopes will be present in many local spatial positions [3].

This internal turbulence keeps the overall system close to but below criticality [3].

When the value of the source function coefficient is increased, the field self-adjusts itself. Although the amplitude decreases, the frequency increases. This self-adjustment allows the total energy of the system to remain unchanged [3].

When 𝑄 is in a low state, 𝑄 = 𝐷𝑚𝑖𝑛 , and it will remain in a low state until the local slope becomes greater then the critical gradient 𝑘, where 𝑘 represents the instability threshold. Moreover, when 𝑄 is in a high state, 𝑄 = 𝐷𝑚𝑎𝑥 , and it will not make a transition into the low state until the local slope becomes less then 𝛽𝑘, where 0 < 𝛽 < 1.

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Figure 6: Total field energy, with all parameters held constant, except the Source Function coefficient, 𝑆0. A) 𝑆0 = 3× 10−4, B) 𝑆0 = 10−4, C) 𝑆0 = 3× 10−3, D) 𝑆0 = 10−3

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Average

Figure 7: Time averaged field strength

Figure 5(A -F): Shows the progression of the system at specific times. The source forces critical gradient to be exceeded. The result is the unloading of energy in avalanches, which balance and overcome the energy input.

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BA C

[1] Per Bak, Chao Tang, Kurt Wiesenfeld. Phys. Rev. Lett. Vol. 59 (1987) 381[2] Henrik Heldtoft Jensen, Kim Christensen and Hans C. Fogedby, Phys. Rev.B, Vol. 40 (1989) 7425[3] A. Klimas et al. Self-organized substorm phenomenon and its relation to localized reconnection in the magnetospheric plasma sheet, J. Geophys. Res., 105(A8), (2000) 18,765-18,780.[4] E. T. Lu. Avalanches in continuum driven dissipative systems, Phys. Rev. Lett., 74(13), (1995) 2511-2514.

A linear diffusion model (Figure 3A), with a constant coefficient (Eq. 1), will be brought to equilibrium with a proportional diffusivity. If an instability threshold exists in nonlinear diffusion (Figure 3B) with a source term (Eq. 2), the nonlinearity causes irregular diffusion which is subject to the limits imposed by the instability.

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Figure 3: A) Linear with source B) Nonlinear with source

(1) 𝜕∅(𝑧,𝑡)𝜕𝑡 = 𝐷𝑚𝑖𝑛 𝜕2∅(𝑧,𝑡)𝜕𝑧2 + 𝑆(𝑧)

(2) 𝜕∅(𝑧,𝑡)𝜕𝑡 = 𝜕𝜕𝑧ቀ𝐷(𝑧,𝑡) 𝜕∅(𝑧,𝑡)𝜕𝑧 ቁ+ 𝑆(𝑧)