  Eccentric millisecond pulsars (eMSPs) with white dwarf companions exhibit
orbital eccentricities orders of magnitude larger than predicted by turbulent
convection in the white dwarfs' red giant progenitors. The orbital periods of
eMSPs cluster around $P=20-30$ d, remarkably close to the red giant convective
eddy turnover time $t_{\rm eddy}$. We propose that the anomalously large
eccentricities are resonantly driven by convective flows somehow made coherent
when the turnover time matches the tidally locked red giant's spin period,
which is also the tidal forcing period. Numerical simulations of rotating red
giants and magnetic field studies of stars show some evidence for especially
ordered flow patterns when the convective Rossby number $P/t_{\rm eddy}$ is of
order unity. We show that resonant convection boosts eccentricities by a factor
of $(t_{\rm nuc}/P)^{1/2}\approx 3\times 10^3$ over the random-walk values that
characterize conventional MSPs, in good agreement with observations ($t_{\rm
nuc}$ is the giant's nuclear burning time-scale). We also show how variations
in the eddy turnover time arising from red giant metallicity variations can
reproduce the observed effective width of the resonance, $\Delta P/P\approx
0.4$.
