Anomalous Transport in Tokamaks: Why Heat Leaks Faster Than Collisions Allow — Epoche C2
The measurement that has to be explained A magnetically confined plasma is often described as a bucket with a slow leak: the field holds the particles, collisions let them hop across field lines, and better engineering will steadily reduce the loss. The first two statements are true and the third does not follow, because the leak that is measured is not the leak that collisions produce. This review traces the causal chain from that discrepancy to its consequence for reactor design. Units are SI throughout, following Wesson, except that collision frequencies are evaluated with the standard Coulomb-collision expression in its customary practical form, with density in cm$^{-3}$ and temperature in eV. Fix a representative set of core parameters and use them everywhere below: toroidal field $B = 3$ T, ion and electron temperatures $T_i = T_e = 5$ keV, electron density $n_e = 5\times10^{19}$ m$^{-3}$, major radius $R = 3$ m, minor radius $a = 1$ m so that the inverse aspect ratio is $\varepsilon = a/R = 1/3$, safety factor $q = 2$, deuterium fuel, and Coulomb logarithm $\ln\Lambda = 17$. First cause examined: collisions alone Classical transport is a random walk in which the step is the ion gyroradius $\rho_i$ — the radius of the ion's circular orbit about a field line — and the stepping rate is the ion–ion collision frequency $\nu_{ii}$. With ion thermal speed $v_{th,i} = \sqrt{2T_i/m_i} = 6.92\times10^{5}$ m s$^{-1}$ for the deuteron mass $m_i = 3.34\times10^{-27}$ kg, the gyroradius is $\rho_i = m_i v_{th,i}/eB = 4.8\times10^{-3}$ m. The collision frequency at these parameters is $\nu_{ii} = 82$ s$^{-1}$. The classical ion heat diffusivity is then $\chi_i^{cl} \approx \rho_i^2 \nu_{ii} = (4.8\times10^{-3})^2 \times 82 = 1.9\times10^{-3}$ m$^2$ s$^{-1}$. Toroidal geometry makes this worse, and neoclassical theory says by how much. The field is stronger on the inboard side, so a fraction $\sqrt{\varepsilon}$ of the ions is magnetically trapped and traces banana-shaped orbits of width $w_b = q\rho_i/\sqrt{\varepsilon}$. Such an ion needs only to be scattered through a small angle to be detrapped, so its effective collision rate is enhanced to $\nu_{ii}/\varepsilon$. The random walk built from these three factors gives $$\chi_i^{neo} \;\approx\; \sqrt{\varepsilon}\;w_b^2\;\frac{\nu_{ii}}{\varepsilon} \;=\; \varepsilon^{-3/2}q^2\rho_i^2\nu_{ii},$$ which reduces to the classical result when $\varepsilon \to 1$ and $q \to 1$, as it must. Inserting $\varepsilon^{-3/2} = 3^{3/2} = 5.20$ and $q^2 = 4$ gives $\chi_i^{neo} = 20.8 \times 1.9\times10^{-3} = 3.9\times10^{-2}$ m$^2$ s$^{-1}$. Measured core ion diffusivities in this parameter range are of order $1$ m$^2$ s$^{-1}$. The ratio is $1/0.039 = 26$, between one and two orders of magnitude. In the electron channel the gap is larger still, because the electron gyroradius is smaller than the ion's by roughly the square root of the mass ratio. The signature that identifies the real cause A discrepancy of a factor of 26 could in principle be a missing collisional effect. One observation rules that out, and it is the hinge of the whole argument. Neoclassical diffusivity scales as $\chi_i^{neo} \propto \rho_i^2 \nu_{ii} \propto T_i \times n_e T_i^{-3/2} = n_e T_i^{-1/2}$. A hotter plasma should therefore confine better , and adding heating power should improve confinement. Experiment shows the opposite. The empirical energy confinement scaling assembled for the ITER Physics Basis gives $\tau_E \propto P^{-0.69}$, where $P$ is the heating power, so the stored energy behaves as $W = \tau_E P \propto P^{0.31}$. Doubling the heating power multiplies the stored energy by only $2^{0.31} = 1.24$: a 100% increase in power buys a 24% increase in energy content. Loss that grows when you drive it harder is not loss caused by collisions. The mechanism, and why it is self-limiting The cause is turbulence, and the driver is the temperature gradient itself. In a plasma with a temperature that falls outwards, the ion temperature gradient mode — the ion temperature gradient mode — a drift wave amplified because the ion magnetic drift carries hot ions faster than cold ones, so on the outboard side a temperature perturbation feeds itself from the background gradient — becomes linearly unstable once the gradient is steep enough — becomes linearly unstable once the gradient is steep enough. The threshold is conventionally written in terms of the normalised inverse gradient scale length $R/L_{Ti}$, with $L_{Ti} = |T_i/\nabla T_i|$; measurements on JET and elsewhere place the critical value at roughly 4 to 6. The resulting heat flux is well described by $$Q_i \;=\; n_e\, T_i\, \frac{v_{th,i}\rho_i^2}{R^2}\;\chi_s\left(\frac{R}{L_{Ti}} - \frac{R}{L_{Ti}}\bigg|_{crit}\right)\Theta\!\left(\frac{R}{L_{Ti}} - \frac{R}{L_{Ti}}\bigg|_{crit}\right),$$ where $\Theta$ is the step function that switches the flux on above threshold and $\chi_s$ is a large dimensionless stiffness coefficient. The gyro-Bohm prefactor is fixed by dimensions: the only length available to the turbulence is $\rho_i$ and the only speed is $v_{th,i}$, so a mixing-length diffusivity is $\rho_i^2 v_{th,i}/a = 16$ m$^2$ s$^{-1}$ with the parameters above. That is 16 times the measured value, and the discrepancy is informative rather than embarrassing: it is the saturated level far above threshold, and the plasma does not operate far above threshold. This is the causal core. Because $\chi_s$ is large, a small excess of gradient over the critical value carries the entire heat flux. Extra heating power therefore raises the gradient only slightly and raises the loss almost in proportion. The gradient is pinned near marginal stability, the profile shape is nearly fixed, and the core temperature is set by the pedestal temperature at the edge multiplied by an almost constant profile factor. Transport in this regime is called stiff, and stiffness is the reason the confinement scaling degrades with power. The two brakes, bot