The Continuity Equation
The conservation-of-mass equation for a plasma fluid, the first moment of the kinetic equation.
Conservation of particles
The continuity equation states that the number of particles of a species changes only through flux across boundaries and sources or sinks. For number density n and fluid velocity v:
dn/dt + div(n v) = SThe divergence term is the net outflow; S represents ionization, recombination, fueling, and, for fusion products, creation. In the absence of sources, particles are strictly conserved.
Where it comes from
Multiplying the Boltzmann or Vlasov equation by 1 and integrating over velocity yields exactly this equation, with n the zeroth moment and n v the first moment (the particle flux). It is the first rung of the fluid moment hierarchy.
In MHD and transport
- In single-fluid MHD it becomes the mass continuity equation for the mass density rho
- In transport codes it is flux-surface-averaged to a one-dimensional radial diffusion equation
- Quasi-neutrality links electron and ion continuity through the ambipolar constraint
How it is solved numerically
Continuity is a hyperbolic conservation law, solved with conservative finite-volume schemes that guarantee particles are neither created nor destroyed numerically. Upwind or flux-limited methods prevent spurious oscillations at steep density fronts, such as the edge pedestal or a fueling pellet.
In fusion modeling, the particle-continuity balance determines fueling requirements: to sustain density against transport losses, external fueling (gas puffing, pellets) must replace what leaves. Predicting the fueling needed to hold the design density of the Hyperion breeder is a continuity-equation calculation coupled to the particle transport model.