Effective population size when the population sizes varies from season to season
This is derived from studying how heterozygosity changes over time. The standard equation for change in heterozygosity ($H$) with constant population size ($N$) is:

$H_t = \left(1 - \frac{1}{2N}\right)^tH_0$

When $N$ varies between generations you use the product of this formula:

$H_t = \left(1 - \frac{1}{2N_0}\right)\left(1 - \frac{1}{2N_1}\right)...\left(1 - \frac{1}{2N_{t-1}}\right)H_0 = \prod_{i=0}^{t-1}\left(1 - \frac{1}{2N_{i}}\right)H_0$

To get the overall $N_e$ you need to find the population size that gives the corresponding decrease in heterozygocity over t generations i.e:

$\left(1 - \frac{1}{2N_{e}}\right)^t = \prod_{i=0}^{t-1}\left(1 - \frac{1}{2N_{i}}\right)$

Rearranging gives:

$N_e = \frac{1}{2\left[1-\left[\prod_{i=0}^{t-1}\left(1 -\frac{1}{2N_{i}}\right)\right]^{1/t}\right]}$

This expression can be approximated by the harmonic mean, which is easy to verify with some toy data (you get small deviations if yearly population sizes are very small). The explanation given above can also be found in Hedrick (2009, p217ff), along with some nice examples that integrates the effects on effective population size from several factors.