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By Karl E. Gustafson, James A. Sethian

Vortex tools have emerged as a brand new type of strong numerical innovations to investigate and compute vortex movement. This booklet addresses the theoretical, numerical, computational, and actual elements of vortex tools and vortex movement

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Likewise for H. Thus for n = nx the ratio PjQ is specified as a certain function of/ and m, and again for n = n2 as some other function of / and m. 126) and the prescribed ratios PJQi9 P2/Q2. One important point to remember is that, to satisfy the radiation condition, for each homogeneous plane wave of the spectrum that solution for nt must be chosen which ensures that the time-averaged power flux Re JE Λ H* is directed away from the plane z = 0. 124) is negative. The nature of the relationship between the directions of phase propagation and power fiux are liable to be quite complicated, but would have to be understood in any particular problem under investigation.

The Three-dimensional Case The power radiated by an arbitrary current density flowing in the plane z = 0 is now considered. The time-averaged power flux across any plane z = constant ( > 0 , say) is 00 Re \ ff (EXH* - £„//*) dxdy. 12) where it is understood that P and Q each have arguments (/, m), and that P* and g* each have arguments (/', m'). 12) mean that contributions are made to the quadruple integral only when both / = /' and m = rri. 12) is unity; whereas, if I2 + m2 > 1, then ]/l — I2 — m2 is pure imaginary and the entire integrand (apart from the delta functions) with / = /', m = m' is pure imaginary.

I)} Q(Lm) have been associated directly with Hx and Hy, the tangential components of H; that is, in effect, with the components of current density y*, jy. This choice accords with the point of view adopted in the introduction to the theory, and is the most convenient for present purposes. The spectrum function for Hz follows at once from div H = 0, and those for the components of E are then obtained from curl H = /ωε0Ε. 109) — OO oo j,(x, y) = 2 / / P(l, m) eik><-lx+m>° dl dm. 110) — OO These double Fourier transforms can be thought of as repeated single Fourier transforms.

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