Sensitivity of External Magnetic Field on the Change in Cross-section of a Toroidal Current

This study validates numerical predictions regarding the sensitivity of toroidal magnetic fields to cross-sectional area changes and the existence of an angle of invariance by analyzing measured magnetic field data from the Aditya Upgrade tokamak plasma.

The Gist

Imagine a giant, glowing donut made of super-hot electricity (plasma) floating inside a fusion machine called a tokamak. This "electric donut" carries a massive amount of current, which creates a magnetic field around it. Scientists need to measure this magnetic field very precisely to understand how the donut is behaving and to keep it stable.

This paper is about a specific question: What happens to the magnetic field outside the donut if the donut gets slightly fatter or thinner?

Here is the breakdown of their discovery, explained simply:

1. The "Goldilocks" Spot (The Angle of Invariance)

The researchers discovered something surprising. If you stand at different spots around the donut and watch it change size (get fatter or thinner), the magnetic field you measure behaves in two completely opposite ways:

• The "Inboard" Side (The Inner Curve): Imagine standing on the inside curve of the donut. If the donut gets fatter (its cross-section grows), the magnetic field you feel actually gets weaker.
• The "Outboard" Side (The Outer Curve): Now, imagine standing on the outside curve. If the donut gets fatter, the magnetic field you feel gets stronger.

It's like a see-saw. On one side, the field goes down when the donut grows; on the other side, it goes up.

But, there is a special spot in between these two sides where nothing changes. No matter if the donut gets fatter or thinner, the magnetic field at this specific angle stays exactly the same. The scientists call this the "Angle of Invariance." It's like a "Goldilocks" zone where the magnetic field is immune to the donut's size changes.

2. The Computer Prediction vs. Real Life

Before doing any experiments, the team used computer simulations (like a video game physics engine) to predict this behavior. They calculated exactly where this "magic angle" should be based on the size of the machine.

They predicted that for their specific machine (Aditya Upgrade), this special angle should be around 62 degrees.

3. The Real-World Experiment

To prove their computer model was right, they went to the actual fusion machine in India. They couldn't easily change the size of the plasma donut directly, so they used a clever trick:

• The Trick: They watched the plasma donut move up and down slightly. By looking at pairs of magnetic sensors placed symmetrically (one on the left, one on the right) and averaging their readings, they could mathematically simulate what the field would look like if the donut had changed its size while staying perfectly centered.
• The Result: They measured the magnetic field at 16 different spots around the donut during many plasma shots.

4. The Conclusion

The real-world data matched the computer predictions perfectly.

• They confirmed that the magnetic field drops on the inside and rises on the outside when the plasma gets bigger.
• They confirmed that the "magic angle" where the field doesn't change lies between 56 and 78 degrees.
• This range perfectly includes their computer-predicted value of 62.3 degrees.

In a nutshell: The paper proves that the magnetic field around a fusion plasma donut is very sensitive to the donut's thickness, but in a predictable way. There is a specific "sweet spot" angle where the field ignores the size changes entirely. This helps scientists understand the "shape" of the magnetic environment better, which is crucial for keeping these fusion machines running safely.

Technical Summary

Technical Summary: Sensitivity of External Magnetic Field on the Change in Cross-section of a Toroidal Current

Problem Statement
In toroidal fusion devices, such as tokamaks, the magnetic field topology is critical for plasma confinement and diagnosis. While the magnetic field ($B$) generated by a toroidal current column with Mega-Ampere (MA) magnitude and a cross-sectional area of tens of centimeters is well-understood in the far field, its behavior in the close vicinity of the current column is complex. It cannot be accurately modeled as a simple filamentary or surface current. Previous numerical studies using tools like POISSON/SUPERFISH predicted that the sensitivity of the external magnetic field to changes in the current column's cross-sectional area ($a_0$) depends heavily on the angular location of the measurement point. Specifically, these studies suggested the existence of an "angle of invariance" ($\theta_I$) where the magnetic field remains constant despite changes in the cross-section. However, these numerical predictions required experimental validation using real plasma data.

Methodology
This study validates the numerical predictions using experimental data from the Aditya Upgrade (Aditya-U) tokamak. The methodology involves three primary components:

1. Numerical Modeling: The authors employed a filamentary approach to calculate the theoretical magnetic field. The toroidal current column was modeled as $N$ co-axial filaments with a current density profile defined by $J(r) = J_0 (1 - r^2/a_0^2)^\gamma$. This method was cross-verified against established solvers (POISSON/SUPERFISH, FEMM), showing agreement within 3% uncertainty.
2. Experimental Setup: Magnetic field measurements were conducted using a Mirnov probe garland consisting of 16 probes installed on a single poloidal plane with $22.5^\circ$ angular separation. Data was collected from Ohmic plasma discharges.
3. Data Processing and Validation Technique: To isolate the effect of cross-sectional area changes while keeping the current column center fixed (a condition difficult to achieve directly in tokamaks), the authors utilized a specific data processing technique. By analyzing plasma discharges where the plasma column underwent vertical shifts ($dY = \epsilon$) but negligible horizontal shifts ($dX \approx 0$), they calculated the magnetic field for a reduced minor radius ($a_0 - \epsilon$) centered at the geometric origin. This was achieved by algebraically averaging the magnetic field readings from symmetric pairs of probes located at $\pm \theta$. This averaging technique was theoretically validated to introduce a maximum uncertainty of only 0.9%, regardless of current density profiling.

Key Results
The study successfully correlated experimental measurements with numerical predictions regarding the sensitivity of $B$ to changes in $a_0$:

• Opposing Trends: The experimental data confirmed that as the minor radius ($a_0$) increases, the magnetic field magnitude decreases at inboard locations (angles $\theta < \theta_I$) and increases at outboard locations (angles $\theta > \theta_I$).
• Angle of Invariance ($\theta_I$): The analysis identified a specific angular range where the magnetic field is insensitive to changes in the cross-section. Experimental data from fifty plasma discharges placed $\theta_I$ between $56.25^\circ$ and $78.75^\circ$.
• Theoretical Agreement: For the Aditya-U geometry (Major radius $R_0 = 75$ cm, probe distance $\rho = 27.5$ cm), the theoretical calculation using the derived linear relation $\theta_I = [1.297 - 0.571 (\rho/R_0)]$ radians yielded a value of $62.32^\circ$. This theoretical value falls squarely within the experimentally observed range, confirming the numerical model.
• Profile Independence: The study demonstrated that the invariance phenomenon holds true regardless of whether the current density profile is uniform or peaked, consistent with Ampere's law which dictates that the external field depends on the total enclosed current rather than the internal distribution.

Significance and Claims
The paper claims to provide the first experimental validation of the dependency of the external magnetic field on the cross-sectional area of a toroidal current channel. The significance of this work lies in:

1. Diagnostic Precision: It highlights that minute changes in the cross-section of MA-order currents can induce significant variations in the local magnetic field (tens of Gauss in medium-sized tokamaks). Understanding this sensitivity is crucial for the precise calibration and interpretation of magnetic diagnostics installed near the plasma column.
2. Validation of Numerical Models: The successful alignment of experimental data with the "angle of invariance" predicted by numerical approaches validates the use of these models for understanding magnetic topology in the vicinity of toroidal currents.
3. Methodological Contribution: The paper establishes a robust experimental technique using symmetric probe averaging to simulate a fixed-center, variable-radius scenario, overcoming the practical limitations of controlling plasma position in tokamak experiments.

The authors conclude that the topology of the external magnetic field is severely impacted by the cross-sectional geometry of the torus, and this relationship is now empirically verified, offering a more complete understanding of magnetic diagnostics in fusion machines.