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Definition

Active scalar equation

A transport equation whose advecting velocity is determined by the transported scalar.

An active scalar equation couples a

∂tθ+u⋅∇θ=F\partial_t\theta+u\cdot\nabla\theta=F

to a specified u=T[θ]u=\mathcal T[\theta] determining velocity from the scalar. The rule may include solving an elliptic equation, and often imposes div⁡u=0\operatorname{div}u=0. The domain, data, and operator T\mathcal T are part of the model.

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Changing the scalar changes the velocity that transports it. This distinguishes active transport from a passive scalar transported by a separately prescribed velocity. Diffusion or fractional diffusion may also be added.