The Momentum Equation
Newton's second law for a plasma fluid element, balancing inertia against pressure and electromagnetic forces.
Force balance for a fluid
The momentum equation is the second moment of the kinetic equation and expresses Newton's second law for a fluid element. For a plasma species it reads:
rho (dv/dt + v.grad v) = -grad p - div(Pi) + n q (E + v x B) + ROn the right: the pressure-gradient force, the viscous stress tensor Pi, the Lorentz force from the electric and magnetic fields, and R the collisional friction with other species. The left side is the inertial acceleration following the fluid.
The MHD reduction
Summing over species and using quasi-neutrality gives the single-fluid MHD momentum equation rho dv/dt = -grad p + J x B, dropping the separate electric force because the net charge is negligible. The J x B force is what magnetic confinement is built on: it balances the plasma pressure gradient in equilibrium.
Equilibrium and flows
- In static equilibrium the inertia vanishes and grad p = J x B, the basis of Grad-Shafranov
- The parallel component gives sound waves and parallel flows along field lines
- The perpendicular component gives the E cross B and diamagnetic drifts
How it is solved numerically
The momentum equation is advanced together with continuity and energy as a coupled hyperbolic system. The J x B coupling to the induction equation makes the set stiff (fast Alfven and magnetosonic waves), so implicit or semi-implicit methods are used for slow-timescale problems. The viscous term, when kept, adds a parabolic (diffusive) part handled implicitly.
Momentum transport also governs plasma rotation, which provides flow shear that suppresses turbulence, a factor in confinement predictions for devices like the Hyperion breeder.