Flux pumping and bifurcated relaxations of helical core in 3D magnetohydrodynamic modelling of ASDEX Upgrade plasmas

Using the JOREK code to simulate ASDEX Upgrade plasmas, this study investigates the self-regulating flux pumping mechanism driven by a pressure-gradient instability, successfully reproducing experimental core profiles and mapping the bifurcated transitions between helical, sawtooth, and island states across varying dissipation and beta parameters.

The Gist

Imagine a giant, donut-shaped oven called a tokamak. Inside this oven, we try to cook a soup of super-hot gas (plasma) to create clean, limitless energy. The secret to keeping this soup hot and stable is a magnetic "cage" that holds it in place.

However, this magnetic cage has a nasty habit of getting sick. Every now and then, the center of the soup gets too crowded with electric current, causing a sudden, violent "hiccup" called a sawtooth crash. This hiccup dumps the heat out of the center, ruining the cooking process. For decades, scientists have tried to stop these hiccups.

The Discovery: The "Self-Healing" Soup

Recently, scientists at the ASDEX Upgrade (a real tokamak in Germany) discovered a special way to cook. They found a "hybrid" recipe where the soup enters a sawtooth-free state. Instead of crashing, the soup seems to "pump" itself.

Think of it like a self-stirring pot. Normally, if you put a heavy ingredient in the middle of a pot, it sinks and clumps up. But in this special state, the soup has a magical internal mechanism that constantly shuffles the heavy ingredients (electric current) from the center to the edges. This keeps the center light and stable, preventing the "hiccup" (sawtooth crash) from ever happening.

The Study: Simulating the Magic

The authors of this paper used a super-computer program called JOREK to simulate this process. They wanted to understand how this self-stirring works and, more importantly, what conditions are needed to keep it going.

Here is what they found, broken down into simple concepts:

1. The "Dynamo" Engine

The magic "stirring" is driven by something called a dynamo. Imagine a tiny, invisible fan inside the soup. This fan is powered by a specific wobble in the magnetic field (called an m/n = 1/1 instability).

• How it works: This wobble creates an electric field that pushes the electric current away from the center.
• The Result: The current stays flat and spread out, like a pancake, rather than piling up in a peak. As long as this fan keeps spinning, the soup stays stable.

2. The "Friction" Problem (Dissipation)

The researchers realized that this system is very sensitive to friction (or "dissipation").

• Low Friction (The Sweet Spot): If the soup is very "slippery" (low viscosity and resistivity), the magnetic fan spins perfectly. The current gets redistributed smoothly, and the plasma stays in the happy, stable Flux Pumping state.
• Medium Friction: If you add a little too much friction, the fan starts to stutter. The soup begins to hiccup again, but in a rhythmic way. This is the Sawtooth state (the bad hiccup).
• High Friction: If the soup is too thick and sticky, the fan stops working entirely. The current piles up in the center, and the system collapses into a messy, unstable state.

The Analogy: Think of it like a skateboarder on a ramp.

• Flux Pumping: The ramp is perfectly smooth (low friction). The skater glides effortlessly, maintaining a perfect loop.
• Sawtooth: The ramp is a bit bumpy. The skater wobbles and falls, then gets back up, wobbles, and falls again (the cycle of hiccups).
• High Friction: The ramp is covered in sand. The skater can't move at all and just sinks into the sand.

3. The "Beta" Factor (Pressure vs. Magnetism)

The study also looked at how "pressurized" the soup is compared to the strength of the magnetic cage (this is called Beta).

• They found that you need high pressure (a very hot, energetic soup) to keep the magnetic fan spinning strong enough to fight the friction. If the soup isn't hot enough, the fan dies, and the hiccups return.

4. The "Tipping Point"

The most exciting finding is that this "perfect state" is fragile.

• In the computer simulations, they found that even if the soup looks stable, hidden "vibrations" (higher-frequency waves) can sneak in.
• If these vibrations get too strong, they jam the magnetic fan. The system suddenly flips from the stable "Flux Pumping" mode back into the chaotic "Sawtooth" mode. It's like a tightrope walker who looks steady until a sudden gust of wind knocks them off.

Why Does This Matter?

This research is a roadmap for building future fusion power plants (like ITER and DEMO).

• The Goal: We need these power plants to run for hours or days without stopping. The "sawtooth hiccups" are a major obstacle to doing that.
• The Solution: This paper tells us exactly how "slippery" and "hot" the plasma needs to be to achieve that self-stirring, hiccup-free state.
• The Future: The authors are now building a "surrogate model" (a fast, simplified calculator) based on these findings. This will help engineers quickly check if a specific recipe for a fusion reactor will work before they even turn on the real machine.

In a Nutshell

This paper is about learning how to keep a fusion reactor's magnetic cage from collapsing. They discovered that under the right conditions (very low friction and high heat), the plasma creates its own internal "stirring mechanism" that keeps it stable. However, this mechanism is delicate; if the conditions aren't perfect, the system crashes back into chaos. By mapping out these conditions, we are one step closer to building a machine that can provide clean energy forever.

Technical Summary

1. Problem Statement

Controlling sawtooth oscillations is critical for the stable, long-pulse operation of next-generation tokamaks like ITER and DEMO. The "flux pumping" phenomenon, observed in ASDEX Upgrade (AUG) hybrid scenario experiments, offers a promising self-regulating solution. Flux pumping is characterized by a sawtooth-free, quiescent state where the central safety factor ($q_0$) is clamped near unity, and toroidal current density is anomalously redistributed radially.

However, the operational parameter space for achieving flux pumping remains uncertain, particularly regarding the influence of system dissipation (viscosity and resistivity) and plasma beta ($\beta$). Previous Magnetohydrodynamic (MHD) models often operated at lower Hartmann numbers (higher dissipation) than experimental conditions or relied on reduced models that could not fully capture the nonlinear dynamics. There is a need for quantitative, full 3D MHD modeling at realistic, low-dissipation parameters to define the stability boundaries and understand the transition mechanisms between flux pumping and sawtooth instabilities.

2. Methodology

The study utilizes the JOREK code, a nonlinear, full MHD simulation tool, to model AUG discharge #36663.

• Model Physics: A two-temperature (ions and electrons), single-fluid, full MHD model is employed. Improvements over previous work include the introduction of equipartition terms in temperature equations and the application of isotropic resistivity with appropriate poloidal current sources.
• Base Case: Simulations are initialized with realistic AUG parameters: $q_0 \approx 1.0$, normalized beta $\beta_N \approx 3.0$, plasma current $I_p \approx 0.8$ MA, and experimental resistivity/viscosity values.
• Parameter Scans:
  • Dissipation Scan: The Hartmann number ($H$) and Magnetic Prandtl number ($P$) are varied by scaling kinematic viscosity ($\nu$) and resistivity ($\eta$). $H$ acts as a proxy for system dissipation (low $H$ = high dissipation).
  • Beta Scan: Plasma beta ($\beta_N$ and $\beta_p$) is scaled down to investigate the threshold for sustaining flux pumping.
  • Long-term Simulations: Simulations run for thousands of normalized time units to observe bifurcations and long-term stability.
• Analysis: The study analyzes the evolution of the safety factor ($q_0$), current density profiles, dynamo electric fields, and mode structures (Poincaré plots, radial mode structures).

3. Key Contributions

• Quantitative Reproduction: Successfully reproduces the experimental "clamped" current density and flat $q$-profile ($q_0 \approx 1$) in AUG flux pumping, validating the dynamo effect mechanism at realistic, low-dissipation parameters ($H \approx 7.9 \times 10^7$).
• Bifurcation Mapping: Systematically maps the bifurcation of core plasma states across a wide range of Hartmann numbers and plasma betas, identifying four distinct regimes.
• Mechanism Elucidation: Clarifies the role of the $m/n=1/1$ quasi-interchange-like instability in generating the dynamo electric field and identifies the destabilizing role of higher harmonics ($n \ge 2$) in the breakdown of flux pumping.
• Operational Window Estimation: Derives a theoretical operating window for flux pumping in terms of electron density ($N_e$) and temperature ($T_e$) by linking MHD parameters to ITG and Finn viscosity models.

4. Key Results

A. Validation of Flux Pumping Mechanism

The 3D simulations confirm that the flux pumping state is sustained by a dynamo electric field generated primarily by the $m/n=1/1$ MHD instability.

• The dynamo field is negative in the core ($\rho_p < 0.2$), increasing current diffusion, and positive in the outer core ($0.2 < \rho_p < 0.4$), increasing current density.
• This redistribution flattens the current density profile, keeping $q_0 \approx 1$ and preventing the internal kink instability that triggers sawteeth.
• The results quantitatively match experimental IDE reconstructions, including the current density deficit ($\delta J_\phi \approx -2$ MA/m$^2$).

B. Bifurcated Plasma States vs. Hartmann Number ($H$)

Scanning system dissipation reveals four distinct plasma states:

1. Flux Pumping (FP): Occurs at very low dissipation (High $H \gtrsim 10^6$). Characterized by a stable, flat current profile and $q_0 \approx 1$.
   • Instability: At marginal $H$ ($\sim 10^6$), flux pumping is fragile. The emergence of $n \ge 2$ harmonics reduces the net dynamo amplitude, leading to high-frequency oscillations and eventual transition to sawteeth.
2. Sawtooth (ST): Occurs at moderate dissipation ($10^5 \lesssim H \lesssim 10^7$). Characterized by periodic $q_0$ oscillations (kink cycling) driven by intermittent dynamo activity.
3. Single Crash (SC): Occurs at higher dissipation ($10^4 \lesssim H \lesssim 10^6$). A giant sawtooth crash occurs once, followed by a quasi-stationary state with a resistive $m/n=1/1$ magnetic island and $q_0 < 1$ (but $>0.7$).
4. Quasi-Stationary Island (QS): Occurs at very high dissipation (Low $H \lesssim 10^4$). The dynamo is suppressed (noise level), leading to a peaked current density and $q_0 \approx 0.7$, similar to 2D diffusion models without dynamo effects.

C. Beta Thresholds and Transition Dynamics

• Flux pumping requires a minimum plasma beta ($\beta_p$) to drive the pressure-gradient instability.
• Discrepancy: The single-fluid model predicts flux pumping at lower $\beta$ ($\beta_N \approx 2$) than observed in experiments (where sawteeth appear at $\beta_N \approx 2$). This suggests that two-fluid effects and kinetic energetic particle physics (e.g., fishbones) are necessary to accurately predict the experimental threshold.
• Long-term Evolution: In marginal cases, flux pumping can transition to small sawteeth over long timescales due to the destabilization of higher-order harmonics ($n \ge 2$).

D. Estimated Operational Window

By correlating the critical Hartmann number ($H_c \approx 3 \times 10^7$) and beta threshold with ITG and Finn viscosity models, the authors estimate the flux pumping window in the $N_e - T_e$ plane:

• High Temperature: Required to maintain low Spitzer resistivity.
• Moderate Density: High density increases dissipation (lowering $H$), pushing the system out of the flux pumping regime.
• Conclusion: Flux pumping is likely accessible in high-temperature, moderate-density regimes.

5. Significance

• Theoretical Advancement: This work bridges the gap between reduced MHD models (valid at high dissipation) and full experimental conditions (low dissipation), confirming that flux pumping is a robust solution even at experimentally relevant Hartmann numbers, provided the plasma beta is sufficient.
• Predictive Capability: The identification of the "fragility" of flux pumping against $n \ge 2$ modes provides a crucial diagnostic for experimentalists: maintaining a pure $n=1$ spectrum is essential for stability.
• Future Device Design: The study highlights the limitations of single-fluid models and underscores the necessity of incorporating two-fluid and kinetic effects to precisely define the operational window for ITER and DEMO.
• Surrogate Modeling: The findings support the development of fast surrogate models (flight simulators) to efficiently predict flux pumping accessibility, reducing the computational cost of full 3D nonlinear simulations for scenario planning.