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Equation 1 · Part 3 · From Origins to Frontier: A History of Mathematics, Proof, and Scientific Computation

Symbol J

∂ζ∂t+J(ψ,ζ)=0,ζ=∇2ψ,\frac{\partial \zeta}{\partial t} + J(\psi, \zeta) = 0, \qquad \zeta = \nabla^2 \psi,
JJ

What this part means

the Jacobian term expressing advection of vorticity by the flow itself.

Its job in the formula

J is part of the quantity the equation computes from the expression on the right.

Where the article explains it

where ψ\psi is the stream function describing the horizontal flow, ζ\zeta is the relative vorticity, and J(ψ\psi,ζ\zeta) is the Jacobian term expressing advection of vorticity by the flow itself.

The passage around this formula

…the barotropic vorticity equation used in that 1950 work can be written, in simplified form, as ∂ζ∂t+J(ψ,ζ)=0,ζ=∇2ψ\frac{\partial \zeta}{\partial t} + J(\psi, \zeta) = 0, \qquad \zeta = \nabla^2 \psi. where ψ\psi is the stream function describing the horizontal flow, ζ\zeta is the relative vorticity, and J(ψ\psi,ζ\zeta) is the Jacobian term expressing advection of vorticity by the flow itself. This is stated here not to reproduce the numerics of the original run but because the equation exposes exactly what made the problem hard to do by hand and tractable by machine: it is a single nonlinear…

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the article section

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