Five ways to solve the Yoshida jet problem of wind-driven equatorial flow

The Yoshida jet is a prototype for wind-driven flow in the tropical ocean. We review the Hermite solution discovered sixty years ago, and show how series acceleration through the Hutton-Moore or Euler schemes is necessary to obtain useful accuracy. An explicit solution in terms of Bessel functions is given here for the first time. It is also shown that much simpler analytic solutions, approximate but accurate, are given by low order rational Chebyshev series and Two-Point Padé approximants. Numerically, the Fourier sine domain truncation method combined with the change of coordinate y ∈ sinh(L t) gives quasi-geometric convergence for most problems. When applied to the Yoshida jet, however, the sine/sinh rate of convergence is awful compared to that of the rational Chebyshev spectral method because wind-driven tropical flows decay very slowly as O(1/y). The log-weakened-geometric rate of convergence of the sine-sinh method, realized for functions that decay exponentially fast for large |y|, is replaced by root-exponential convergence with total error proportional to exp(−qN) for some positive constant q. Except for the explicit Bessel solutions, similar considerations should apply to tropical wind-driven flows in general.