Equipartition and Transport in Two-dimensional Electrostatic Turbulence

Turbulent equipartition is investigated for the nonlinear evolution of pressure driven flute modes in a plasma in an inhomogeneous magnetic field. Numerical solutions of the model equations on a bounded domain with sources and sinks show that the turbulent fluctuations give rise to up-gradient transport, a "pinch flux", of heat or particles. The averaged equilibrium density and temperature profiles approach the profiles n ∼ B and T ∼ B predicted by turbulent equipartition. Recently, a new approach has been suggested for predicting the quasi-steady profiles in tokamak plasmas [1, 2, 3, 4]. It is based on the existence of Lagrangian invariants in the presence of turbulence. The basic assumption is that the turbulent mixing causes the equipartition of these invariants over the accessible phase space, a state denoted Turbulent EquiPartition (TEP) [1]. Since the Lagrangian invariants depend on the magnetic field B, a homogeneous distribution of these invariants implies that if B is inhomogeneous, so are the density and the temperature. Therefore, the fluxes that drive the plasma towards TEP may be up-gradient. In a two-dimensional plasma model the corresponding mechanism is easily understood. If the magnetic field B = ẑB(x, y) is inhomogeneous, the E × B-drift vE = (ẑ × ∇φ)/B is compressible, and the relation ∇ · (BvE) = 0 implies that n/B is a Lagrangian invariant. Another Lagrangian invariant is given by the specific entropy T /n. This gives the TEP profiles n ∼ B and T ∼ B. If the diamagnetic drift vd = −(ẑ ×∇p)/(neB) is also taken into account, these quantities are no longer exact invariants. Here we present simulations of TEP with self-consistently generated electrostatic turbulence. The density and temperature profiles are allowed to develop self-consistently under the influence of external heating. When the pressure peaking exceeds the instability threshold p′ > (B5/3)′, thermal energy can be released by displacing hot and dense fluid parcels to regions with weaker magnetic field, where they expand adiabatically, and the turbulence sets in. The turbulent fluxes are observed to include pinch fluxes. A basic requirement on our model is that it describes the fluid drifts accurately in the presence of an inhomogeneous magnetic field B = ẑB(x, y). It must also correctly describe the adiabatic compression and heating of a fluid parcel that is displaced into a region of larger B. (These requirements are not met in the commonly used models where the Rayleigh-Taylor Instability (RTI) is caused by an "artificial gravity".) Our model is based on the continuity equation for the electron density and the Braginskii transport equation for the electron temperature accounting for both the E × B and the diamagnetic drifts (cf. Ref. [5]), together with the ion vorticity equation for cold ions. The system of equations is closed by assuming quasi-neutrality. Assuming that the density n, temperature T and the inhomogeneous magnetic field B deviate only slightly from constant reference levelsN , T and B: n = N (1 + ñ(x, y, t)), T = T (1 + T̃ (x, y, t)), B = B(1 + B̃(x, y)), (1) we obtain the model equations for the small quantities ñ, T̃ and B̃: ∂n ∂t + {φ, n−B}+ {n+ T,B} = 0, (2) ∂T ∂t + { φ, T − 2 3 B } + { 2 3 n+ 7 3 T,B } = 0 , (3) ∂∇2φ ∂t + { φ,∇φ } + {n+ T,B} = 0. (4) For convenience we have dropped the tilde. The potential is normalized by T /e, the time by ω−1 ci = mi/(eB), and the space variables by ρ = (T /mi)/ωci. Note that Eqs. (2)-(4) are scale invariant with respect to multiplying the dependent variables and B with a constant and dividing t by the same constant. Equations (2)-(3) possess the Lagrangian invariants l± = ± √ 5/2(n − B) + 3T/2 − n, corresponding to the invariants L± derived in Ref. [5]. They are advected by the velocities v± = ẑ × ∇[φ − n − (1 ± (5/2))T ], which are neither fluid nor guiding center velocities, but rather Riemann-like characteristics. Thus, the TEP profiles can be expected to be given by a spatially homogeneous distribution of l±, which implies n−B = const., T − 2B/3 = const. , (5) These are the same as the TEP profiles that would result ifn/B andT /nwere exact Lagrangian invariants using the expansion in Eqs. 1. Equations (2)-(4) conserve the energy-like integral E = ∫ [1 2 (∇φ) + (n+ T )B ] dx dy. (6) The first term is the kinetic energy, while the second term has the form of potential energy. It represents that part of the thermal energy which can be converted to kinetic energy when fluid parcels are displaced to a region with weaker magnetic field. In order to investigate the linear stability we consider a slab model where the equilibrium gradients are in the x-direction. We linearize Eqs. (2)-(4) around the background profiles n0(x), T0(x) andB(x) and assume a waveform exp(ikr− iωt) in the local approximation. The dispersion relation reads ck [ c + 10 3 cB′ + 5 3 (B′)2 ] + cB(n0 + T ′ 0 − 5 3 B′) + 5 3 (B)(n0 −B′) = 0 (7) where we have introduced c = ω/ky and the prime denotes differentiation with respect to x. The long wavelength solution is c ≈ −B(n0 + T ′ 0 − 3B)/k. This is recognized as a special case of the RTI. It is unstable for B(n0 + T ′ 0 − 5 3 B′) > 0. (8) Assuming that the magnetic field is decreasing with increasing x, i.e., B′ < 0, the instability condition becomes (n0 + T0 − 5B/3)′ < 0. Solutions of Eq. (7) shows that there is a finite wavenumber cut-off corresponding to k ≈ ρ−1 (in dimensional units) for the RTI in this model, contrary to models for RTI where the magnetic field inhomogeneity is represented by an "artificial gravity". The instability sets in if the pressure profile n0 + T0 is more peaked than 5B/3. We observe that the TEP-profiles are marginally stable.