The Shafranov Shift

The outward displacement of inner flux surfaces relative to the plasma boundary, driven by pressure and internal current.

What It Is

In a toroidal equilibrium the nested flux surfaces are not concentric: the centers of inner surfaces are shifted outward, toward the low-field side, relative to the plasma boundary. This displacement is the Shafranov shift. It is a direct consequence of force balance in a torus, where the plasma pressure and the hoop force of the toroidal current must be held by the poloidal field, and the result is an outboard compression of flux surfaces.

Scaling

The shift grows with poloidal beta and with the internal inductance of the current profile, and it scales with the inverse aspect ratio. Roughly, the normalized shift of the magnetic axis relative to the geometric center increases in proportion to beta_p + l_i/2 times epsilon, where l_i measures current-profile peaking. High-pressure, peaked-current plasmas therefore show large shifts.

Consequences

The outward shift compresses flux surfaces on the outboard side, steepening pressure gradients there and increasing the local magnetic well, which can be stabilizing for ballooning modes at high beta, the origin of second-stability access. It also moves the magnetic axis, changing the effective aspect ratio seen by the core plasma and affecting heating and current-drive deposition.

Relevance

The Shafranov shift is computed self-consistently by any Grad-Shafranov equilibrium solver and must be accounted for in coil design and plasma positioning. Spherical tokamaks such as the Hyperion breeder concept, with low aspect ratio and high beta, exhibit pronounced shifts, which feed into stability and transport analysis at the design stage.