Many large newspapers in California are bemoaning the very low turnout in special U.S. House and legislative elections recently. Some newspapers are editorializing in favor of eliminating special elections for the legislature, and advocating that the Constitution be changed to let the Governor appoint legislators to fill vacant seats. See this Los Angeles Times editorial, and this Santa Rosa Press Democrat editorial. The newspapers are correct that voter turnout in recent special elections has been low. Ever since the top-two rules were in force, starting in 2011, the median voter turnout in California special legislative and U.S. House elections has been 13.84%. The average has been 15.80%. There have been 19 special elections under top-two rules.
If one looks at special elections for U.S. House and legislature in the period just before top-two rules went into effect, one notes that the median turnout was 20.27% and the average turnout was 21.93%. That covers the special elections held in 2009 and 2010, and also the special election held January 4, 2011, because it was held under the old rules.
Thus, whether one uses the median turnout, or the average turnout, turnout is approximately one-third lower under the top-two rules than in the corresponding period of time before it went into effect. One reason for lower turnout is that top-two makes it impossible for minor party candidates to get into the run-off, if there is one. Under the old rules for special elections, which were blanket primary elections, the top vote-getter from each party went into the run-off, if there was one (and there usually was a run-off). This encouraged minor party members to run, because they could campaign for a longer period of time. Also, under the old rules, minor party members only needed 150 signatures to avoid paying the filing fee, but now they need either 3,000 or 1,500, depending on whether they are running for U.S. House/State Senate, or Assembly.