Pinch (magnetic fusion)

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A Pinch is a machine used to study plasmas, specifically fusion plasmas. A pinch must statisfy the equilibrium ideal magnetohydrodynamics (MHD) equations, a series of equations that describe the motion of magnetized fluids. The equilibrium version of the ideal MHD equations describe magnetic configurations that lead to a steady state. The plasma contained in the first machines to reach ideal MHD equilibrium would constrict rapidly, thus giving rise to the term pinch. Plasmas not in MHD equilibrium over Alfvénic time scales (typically ~1 μs in laboratory plasmas) will rapidly collapse over 2-3 Alfvénic times.

Pinches are generally divided into three categories, depending on the number of symmetric dimensions assumed when solving the ideal MHD equations.

The first step in designing a new fusion device is ensuring that it satisfies MHD equilibrium.

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[edit] One Dimensional configurations

There are three analytic one dimensional configurations generally studied in plasma physics. These are the θ-pinch, the Z-pinch, and the Screw Pinch. All of the classic one dimensional pinches are cylindrically shaped. Symmetry is assumed in the axial (z) direction and in the azimuthal (θ) direction. It is traditional to name a one-dimensional pinch after the direction in which the current travels.

[edit] The θ-pinch

A sketch of the θ-Pinch Equilibrium. The z directed magnetic field (shown in purple) corresponds to a θ directed plasma current (shown in yellow).
A sketch of the θ-Pinch Equilibrium. The z directed magnetic field (shown in purple) corresponds to a θ directed plasma current (shown in yellow).

The θ-pinch has a magnetic field traveling in the z direction. Using Ampère's law (discarding the displacement term)

\nabla \times \vec{B} = \mu_0 \vec{J}

\vec{B} = B_{z}(r)\hat{z}

\mu_{0} \vec{J} = \frac{d}{d \theta}B_z \hat{r} - \frac{d}{dr}B_z \hat{\theta}

Since B is only a function of r we can simplify this to

\mu_0 \vec{J} = -\frac{d}{dr}B_z \hat{\theta}

So J points in the θ direction. θ-pinches tend to be resistant to plasma instabilities. This is due in part to the frozen in flux theorem, which is beyond the scope of this article.

[edit] The Z-Pinch

A sketch of the z-Pinch Equilibrium. A -θ directed magnetic field (shown in purple) corresponds to a z directed plasma current (shown in yellow).
A sketch of the z-Pinch Equilibrium. A -θ directed magnetic field (shown in purple) corresponds to a z directed plasma current (shown in yellow).

The Z-Pinch has a magnetic field in the θ direction. Again, by electrostatic Ampere's Law

\nabla \times \vec{B} = \mu_0 \vec{J}

\vec{B} = B_{\theta}(r)\hat{\theta}

\mu_{0} \vec{J} = \frac{d}{dr}B_{\theta} \hat{z} - \frac{d}{dz}B_{\theta} \hat{r} = \frac{d}{dr}B_{\theta} \hat{z}

So J points in the z direction. Since particles in a plasma basically follow magnetic field lines, Z-pinches lead them around in circles. Therefore, they tend to have excellent confinement properties.

[edit] The Screw Pinch

The Screw pinch is an effort to combine the stability aspects of the θ-pinch and the confinement aspects of the Z-pinch. Referring once again to Ampere's Law

\nabla \times \vec{B} = \mu_0 \vec{J}

But this time, the B field has a θ component and a z component

\vec{B} = B_{\theta}(r)\hat{\theta} + B_z (r) \hat{z}

\mu_{0} \vec{J} = \frac{1}{r}\frac{d}{dr}(r B_{\theta}) \hat{z} - \frac{d}{dr}B_{z} \hat{\theta}

So this time J has a component in the z direction and a component in the θ direction.

[edit] Two Dimensional Equilibria

A toroidal coordinate system in common use in plasma physics. The red arrow indicates the poloidal direction (θ) and the blue arrow indicates the toroidal direction (φ)
A toroidal coordinate system in common use in plasma physics. The red arrow indicates the poloidal direction (θ) and the blue arrow indicates the toroidal direction (φ)

A common problem with one-dimensional equilibria based machines is end losses. As mentioned above, most of the motion of particles in a plasma is directed along the magnetic field. With the θ-pinch and the screw-pinch, this leads particles to the end of the machine very quickly (as the particles are typically moving quite fast). Additionally, the Z-pinch has major stability problems. Though particles can be reflected to some extent with magnetic mirrors, even these allow many particles to pass. The most common method of mitigating this effect is to bend the cylander around into a torus. Unfortunately this breaks θ symmetry, as paths on the inner portion (inboard side) of the torus are shorter than similar paths on the outer portion (outboard side). Thus, a new theory is needed. This gives rise to the famous Grad-Shafranov equation.

The one dimensional equilibria provide the inspiration for some of the toroidal configurations. An example of this is the ZETA device at Culham England (which also operated as a Reversed Field Pinch). The most well recognized of these devices is the toriodal version of the screw pinch, the Tokamak.

Numerical solutions to the Grad-Shafranov equation have also yielded some equilibria, most notably that of the Reversed Field Pinch.

[edit] Three Dimensional Equilibria

There does not exist a coherent analytical theory for three-dimensional equilibria. The general approach to finding three dimensional equilibria is to solve the vacuum ideal MHD equations. Numerical solutions have yielded designs for stellarators. Some machines take advantage of simplification techniques such as helical symmetry (for example University of Wisconsin's Helically Symmetric eXperiment).

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