What to write in email to potential PhD advisor? I would like to apply for following PhD studies in the Nordic countries, particularly in the field of Computer Science. The problem that I found is that it says in the admission procedure that "before applying one should get a Professor or Researcher that agrees to supervise the applicant". 

I have checked up the personal webpages of the Professors and I have found their topics of interests, publications and so on. I like a lot of the topics in which they make research, but I do not really know how to address an email to them asking if they would agree to supervise me. Particularly I have the following doubts:


Should I make a small research article putting my ideas about one research interest of them? The problem here is that I believe I would not have the enough time to go through a lot of papers made on their reseach groups. What can I do if do not have the enough background? Time is running fast for submitting the application and I do not believe I would have enough time to study in deep their research.
Should I send my CV and ask to a specific Professor that I have interest in following PhD studies under their supervision, and if it would be possible if he or she to send me some open question or open research task so that they could see that I will manage to do it well on his or her research group?


I will transcribe what the ad specifically says:


  
  You must do your best to find a suitable supervisor at the department who agrees to supervise you. Please see the research pages
  of the department, and especially the descriptions of the research
  groups.
  You must agree on a study plan, a research plan and a financing plan together with your (found) supervisor,  and submit an application
  to the steering committee of the department's doctoral programme.
  


also about the supervisor it says:


  A student who wants to become a PhD student should start the
  application process by finding a supervisor at the department, who
  works in the field of interest of the student and who is willing to
  supervise the student. If the student is unsure of which supervisor to
  contact, he or she may first contact the research coordinator of the
  department who will steer the student in the right direction.
  Generally, a supervisor is a professor and/or principal investigator
  employed by the department.


For what I read it seems, at least to me, that one should really have in mind a good research proposal before applying, am I wrong? or only with an email will suffice to contact the potential supervisor?

What should I do in this situation?

Thanks
If you have a specific potential advisor at a university, start by sending a brief (and polite, of course) email. Ask whether she would be willing to take a new student and say a few words about your background (what you did in your bachelor/master, where and with whom). Thus you can show that your background is compatible with the advisor's research. If you manage to add a few words about how you liked something specific about the advisor's recent work, that will certainly be a big plus. End by thanking the supervisor for her time and say that you are ready to provide more info if necessary. The advisor can then help you get through the formalities if she sees a potential in you.