Tokamak Shafranov Shift Equilibrium Calculator
Grad-Shafranov MHD Equilibrium: Compute high-beta outward magnetic axis displacement ($\Delta$), flux surface compression, and Shafranov parameter ($\Lambda = \beta_p + l_i/2 - 1$).
Geometry & Equilibrium Parameters
Shafranov Shift & Equilibrium Limit
The Shafranov Shift & Grad-Shafranov Equilibrium
The 2D axisymmetric equilibrium of a magnetized plasma is governed by the Grad-Shafranov equation, balancing the magnetic Lorentz force with plasma pressure: $\mathbf{j} \times \mathbf{B} = \nabla p$.
1. Analytical Formulation
Δ₀ ≈ (a² / 2R₀) · (β_p + l_i/2) · (2 / (1 + κ²))
Λ = β_p + l_i/2 - 1 [Shafranov Parameter]
β_{p,max} ≈ (R₀/a) · ((1+κ²)/2) - l_i/2
2. Physical Consequences
- Flux Compression: Outboard flux surfaces pack closely together, steepening edge gradients.
- Second Stability Regime: At very high $\beta_p$ with strong Shafranov shifts, magnetic well formation can allow entry into the second ballooning stable regime.
Frequently Asked Questions
What is the Shafranov shift in tokamak physics?
The Shafranov shift (named after Soviet physicist Vitaly Shafranov) is the outward horizontal displacement of nested magnetic flux surfaces towards the outboard (low-field) side of a tokamak. It arises because the plasma thermal pressure gradient (\nabla p) pushes the core outward against the toroidal magnetic field, while the poloidal magnetic field lines are shorter on the inboard side, generating an outward hoop force.
How does Shafranov shift relate to the tokamak equilibrium limit?
As plasma pressure (poloidal beta \beta_p) increases, the magnetic axis shifts outward. If \beta_p approaches the equilibrium limit \beta_{p,eq} \sim R_0 / a, the magnetic axis shifts all the way to \Delta \approx a/2, compressing outboard flux surfaces so tightly that an inner magnetic X-point appears on the high-field side, destroying plasma equilibrium.
Does Shafranov shift stabilize or destabilize MHD modes?
In moderate amounts, the Shafranov shift is highly stabilizing! The outward shift increases local magnetic shear and flux tube volume variation (V'' < 0), creating an effective magnetic well on the bad-curvature outboard side that suppresses interchange modes, ballooning modes, and microturbulent transport (ion temperature gradient ITG modes).