1. Self-consistent mean field forces in two-fluid models of turbulent plasmas C. C. Hegna University of Wisconsin Madison, WI CMSO Meeting Madison, WI August 6, 2004

2. Theses The properties of turbulent plasmas are described using the two-fluid equations. Global constraints are derived for the fluctuation induced mean field forces that act on the ion and electron fluids. Relationship between relaxation of parallel momentum flows and parallel currents CCH, Self-consistent mean-field forces in turbulent plasmas: current and momentum relaxation, Physics of Plasmas 5, 2257 (1998); 3480 (1998). --- RFP physics was largely the motivation

3. In resistive MHD dynamo theory, a mean field force is identified Fluctuations affects mean field dynamics in resistive MHD through a dynamo electric field Write all quantities as mean field and fluctuations The bracket <> notation denotes either an ensemble average or an average over the small spatial scales or fast time scales of the fluctuations Mean field Ohms Law

4. Global conservation laws have motivated local forms for the mean field force of resistive MHD In resistive MHD, fluctuations do not dissipate helicity, but do dissipate energy. (Boozer, J. Plasma Physics, 1986; Bhattacharjee and Hameiri, PRL 1986; Phys. Fluids 1987; Strauss, Phys. Fluids 1985). These condititions are used to motivate a local form for the mean-field force in toroidal confinement devices --- fluctuations generate an additional electron viscosity or hyper-resitivity, not a dynamo. K 2 is a profile dependent positive function satisfying boundary conditions. Consistent with the Taylor state, F gets large, ---> J || /B = constant

5. Two-fluid equations can be written in a concise form The exact two-fluid momentum balance equations –These equations can be written more concisely with the identification of the canonical momentum. Momentum balance equations Plasma flow for each species

6. Fluctuations induce mean field forces on both the ion and electron species. For simplicity, we consider a cylindrical plasmas with all the usual boundary conditions. Quantities are split into equilibrium and fluctuating quantities, Nonlinearities produce fluctuation induced mean field forces (actually forces per unit charge) Note, the first term contains both the MHD and Hall dynamo terms. For the electrons, v e =u - J/ne + O(m e /m i ).

7. Three global properties of the mean-field forces can be shown Mean field momentum balance equations Three global constraints can be shown. The last condition can also be written using F ||M = F ||i -F ||e, F ||O = (m i F |||e +m e F ||i )/(m i +m e ) In two-fluid theory, fluctuations do not dissipate generalized helicity, but do dissipate energy.

8. A number of assumptions are used to prove the three global constraints Simplifying assumptions used in the constraint derivations: –Fluctuation amplitudes are small compared to the mean magnetic field – The equilibrium quantities evolve on a slow diffusive time scale Viscosities and radial mean flow are ordered with resistivity. Parallel heat flux is ordered small to be consistent with the neglect of heat flux, –The viscous force is dissipative for both species –All other equilibrium flows are ordered small - probably not a crucial assumption,may be generalized to equilibrium with flow –Ion and electron skin depths are small. With ~ 1, s ~ s

9. The implied local forms suggest a coupling between current and flow evolution The parallel components of the turbulent mean field force –Can rewrite these equations as Those that appear in Ohms law And the total momentum balance Coefficients are spatially dependent functions that vanish on the boundary and satisfy k e 2 > 0, k i 2 > 0, (L e + L i ) 2 < k e 2 k i 2 /4 e 2 > 0, i 2 > 0, ( e + i ) 2 < e 2 i 2 /4

10. Summary Using a modest number of assumptions, three global constraints are derived for turbulence induced mean field forces in two-fluid models of plasmas. These constrains imply functional forms for the parallel mean-field forces in the Ohms law and the total momentum balance equations suggesting the fluctuations relax the plasma to states with field aligned current and bulk plasma momentum. Applications to flow profile evolution during discrete dynamo events on MST?