On the minimum electron transport coefficients in tokamaks in the range of low collision frequencies

There are two close empirical scalings, namely, the T-11 and neo-Alcator ones, that provide correct estimates for the energy confinement time in tokamaks in ohmic heating regimes in the linear part of the dependence τ E ( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaaceiGab8NBayaaraWaaSbaaSqaaiaa-vgaaeqaaaaa % !3C86! $$ \bar n_e $$ ) in the range of low values of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaaceiGab8NBayaaraWaaSbaaSqaaiaa-vgaaeqaaaaa % !3C86! $$ \bar n_e $$ and 〈ν * 〉 ≤ 1. The similar character of electron energy confinement in this range, which expands with increasing magnetic field B 0, has stimulated the search for dimensionless parameters and simple physical models that would explain the experimentally observed dependences χ e ∼ 1/n e and τ Ee ∼ % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaaceiGab8NBayaaraWaaSbaaSqaaiaa-vgaaeqaaaaa % !3C86! $$ \bar n_e $$ . In 1987, T. Okhawa showed that the experimental data were satisfactorily described by the formula χe⊥ = (c 2/ω 2 )ν e /qR, in deriving of which the random spatial leap along the radius r on the electron trajectory was assumed to be the same as that in the coefficient of the poloidal field diffusion, while the repetition rate of these leaps was assumed to be ν e /qR. In 2004, J. Callen took into account the decrease in the fraction of transient electrons with increasing toroidal ratio ɛ = r/R and corrected the coefficient c 2/ω 2 in Okhawa equation by the factor σ ‖ Sp /σ ‖ neo . If one takes into account this correction and assumes that the frequency of the stochastic process is equal to the reciprocal of the half-period of rotation of a trapped electron along its banana trajectory, then the resulting expression for χe⊥ will coincide with the T-11 scaling: χ an ∞ ɛ1.75(T e /A i )0.5/(n e qR) at A i = 1. If the same stochastic process also involves ions, it may result in the opening of the orbit of a trapped ion at the distance ∼(c/ω pe )(m i /m e )1/4. In this case, the calculated coefficient of electron and ion diffusion D is close to D an ≈ χ an /2.