<p>First of all, actually switching from one department to another in grad school is exceedingly difficult - you often need to apply all over again, and this time they will be prejudiced against you! </p>
<p>Secondly, it is not at all uncommon to have an advisor in a different department. A lot of grad study is interdisciplinary, and some areas can be approached from multiple directions. The advice I was given is that your “home” department should describe at least half of what you are doing, as well as the direction of your research. </p>
<p>For example, if you are looking at computational physics as a physics PhD-candidate you will be focusing on the physics side of things, using the computational science to answer physics questions, and those questions will really be the core of your research while the program is just the tool. Conversely, looking at the same issue as a CompSci candidate you will focus on the process and the algorithms.</p>