The Energy Equation
The evolution of internal energy or pressure for a plasma fluid, closing the fluid hierarchy with heat flux and sources.
Energy conservation
The energy equation is the third moment of the kinetic equation, governing the pressure or internal energy of a fluid species. In pressure form:
(3/2) dp/dt + (3/2) div(p v) + p div v = -div(q) - Pi:grad v + QThe terms are the change and advection of internal energy, compressional heating (p div v), the divergence of the heat flux q, viscous heating, and the source Q (ohmic, fusion, radiation, exchange with other species). It is where heating and cooling enter the fluid description.
The closure problem
The heat flux q is the next moment, so the energy equation is not closed by itself. A closure relation is needed: the adiabatic assumption sets q = 0 and gives p rho^(-gamma) = constant; collisional closures give Fourier-law conduction q = -kappa grad T (the Braginskii result); collisionless plasmas need kinetic closures.
How it is solved numerically
- Advected with continuity and momentum as a coupled hyperbolic system
- The conduction term is parabolic and treated implicitly because parallel kappa is enormous
- Radiation and fusion sources are evaluated from local n and T each step
In magnetized plasmas the extreme anisotropy of heat conduction (parallel conduction vastly exceeding perpendicular) is the main numerical challenge, requiring field-aligned coordinates or carefully limited fluxes.
Its role in fusion
The energy balance determines the temperature profile and therefore the fusion rate, which depends steeply on temperature. Coupled to power balance, it decides whether a plasma reaches and holds fusion conditions, as in the design-point analysis of the Hyperion breeder.