It's hard to say exactly what the cause is without knowing what the site/keywords are, but I'd recommend checking if these are all New Visits and/or Unique Visitors and where they are geographically located. I've seen huge spikes in traffic by the same group of people/computers in foreign countries before who were hitting my site over and over for some unknown reason. Also, check which search engine is sending the traffic, which landing pages they visited first, etc.
And of course, this spike could be legit, especially if your site ranks for keywords related to a particular topic that happened to get a lot of searches yesterday (breaking news, a controversial report debunking commonly held belief, viral video, etc).