What are the ways to perform air traffic control on long distance flights?
Direct link vs satellite


  There are three major ways of sending control messages to an aircraft:
  Direct radio link [when] in line of sight, satellite, ground stations,
  e.g., tracon.


The way the question starts brings confusion so the many comments. There are two categories of radio links involved in voice/data communication: Between a ground station and the aircraft and via a relay, e.g. a satellite.


Satellites are merely used today for air traffic control, it's a thing of the future (2 to 3 years for tracking the aircraft using ADS-B). However satellites are sometimes used for aircraft communication with the airline technical staff and/or the manufacturers. However the information exchanged here is not ATC related at all: It can be technical, e.g. live ACARS sent to airline technical staff or to the engine manufacturer, to prepare maintenance at the destination airport. It can also be internal exchange with the airline operations. Satellite communication is also used for passenger services like telephone and Internet access aboard the aircraft.
So ATC uses the other kind of links, the direct ones, which means a transmitter on the ground. However these "ground stations" can serve different purposes because ATC is organized so that ATC facilities are specialized.


Simple ATC organization

There are ATCOs on the site who may add details to this answer, but I'll draw the mainlines. In the US, ATC operational facilities are divided into 4 categories:


ATC Tower. Controllers are located in the airfield tower and they control aircraft on the ground (e.g. at their gates, or on the taxiways), aircraft on the runways (landing or taking off), aircraft approaching the airfield to land, aircraft leaving the airfield and climbing to their cruise altitude. The tower controllers have a good sight on all those aircraft.
Tracon (terminal radar approach control). Controllers maybe located in the tower or at other locations. They are in charge of taking over the aircraft that just took off and are now departing. They continue the control that was started by their colleagues of the Tower. They also take care of the descending aircraft before they are handed over to their colleagues of the Tower for landing. As you see, tracon actually takes aircraft from cruise altitude to near the landing runway, and from near the takeoff runway to cruise altitude.
Next, there is the enroute ATC center for controlling aircraft in cruise, between the departing tracon and the arrival tracon. This facility is also known as air route traffic control center (ARTCC). The enroute controllers can be located anywhere, they don't need to see the sky, and usually they don't see it at all, they work in a dark room using only displays and microphones. There are a small number of centers in the US to control the whole US airspace.


This split is shown on this figure:

  

that is extracted from this question.


The last type of ATC facility is the oceanic ATC which controls an ocean area, because the procedures and tools used are a bit particular. Oceans are not belonging to a particular country, however all countries have agreed to delegate the control to a particular country for a particular area. In the US, for the East coast, New-York oceanic center will control an area, Canada will control another, UK and France will manage the other side of the pond. These centers control areas named OCA, for oceanic control area. See this map for North Atlantic. When the aircraft progress along its route, it is handed over to the next area controller.


This was the first part: We have clarified who are the controllers and ATC facilities, where they are and what phase of flight they control. Now let's see the radio equipment they use.

Radio links

First let's clarify between radio and radar.


Radar is used to determine the position of the aircraft. This can be done everywhere, except over oceans. The reason is radar has a limited range, and there are no radar stations in the ocean (at least not for civil aviation).
Radar stations are located at selected locations, on hills and other high places, to increase their range. some radar are located on airfields too, but that's not a requirement. Actually, all information collected by all radar stations are gathered and distributed to all control centers as a synthetic picture.
Radio is used to exchange voice messages and data. As I wrote, all radio links have a ground station at the center. This ground station is usually at the same location than the controllers. So some ATC radio equipment is at the airfield, some at the enroute centers, etc. For OCA, the equipment can be remote from the controllers, but this is just a technical detail.


Aviation radio for ATC messages can use two types of frequencies:


VHF (very high frequency) which works mostly in line of sight, so the range is something around 50 km to 200 km depending on propagation conditions and location. VHF is used when the aircraft is flying over lands. Controllers at the towers and tracons use VHF, as they deal with aircraft that are either under they eyes (tower) or at a moderate distance from the airfield (tracon). Enroute use them too as they control the part of the sky around the transmitter.
HF (high frequency) works on long distance, like between continents. HF is used by OCA controllers, enroute ATC can also use them. The particularity of HF links is they work somehow erratically, the links can be broken at any time when an aircraft flies over an ocean. The reason is HF waves are reflected by the high ionized atmosphere and the ground, but the degree of ionization varies with time. OCA facilities have procedures to deal with that.
A special word on ATC communication in OCA: Satellites are not used by ATC today. In OCA, ATC communicates with aircraft using HF transmitters located on the ground. When the distance or propagation conditions make the HF communication impossible, ATC (or the aircraft crew) will ask another aircraft to relay the message. Aircraft equipped with satellite phone can also try to contact the OCA center using telephone. If communication is still impossible, then the crew will continue as already cleared by ATC, and after they will continue as indicated in the flight plan which was filed with ATC before the flight. ATC will clear enough airspace around the assumed position of the aircraft to prevent collisions. This is not very different than when there is a loss of communication over land. See section Communications Failure After Entering NAT Region at the end of this document (NAT is the standard North Atlantic route system often used to cross the Atlantic Ocean).



  For an intercontinental flight, how does the control know which ground
  station to use (since it wouldn't make sense to connect all ground
  stations to each airport) for control messages?


I hope you have got a better understanding, and know now this is not the case. Each ATC facility (tower, tracon, enroute center) uses different radio stations and a particular tower controls only the aircraft at this airport, using the local transmitter. The key is that ATC works with pre-agreed procedures to hand off control to the next facility when the aircraft leaves a given control area. An ATC unit controls aircraft in its area, and uses a ground radio station nearby the aircraft. OCA ATC indeed works with aircraft that are remote, and sometimes very far.


  Are the control links (e.g., control-to-ground-station-to-aircraft) to
  communicate with an aircraft predetermined before take-off?


Yes they are, not on a paper, but by the procedures. When the aircraft starts its engines, the crew is in contact with the "ground controller" who is a controller in the tower, specialized in controlling aircraft which aren't on the runways. When the aircraft is ready for takeoff, the ground controller hands over to someone else in the tower who manages the takeoffs (this is "the tower controller"), then after take off the tower hands over to the nearby tracon, then to enroute (there will be several successive enroute centers), etc.

The current controller hands over to the next controller on the aircraft route, when the aircraft is leaving their area of responsibility, this is part of the ATC organization itself.

"Plates" used by crew indicate how to reach a given control facility. For example, on this chart, the frequency of the Denver ARTCC is mentioned explicitly.


Source

This is in case you are lost because this information is normally given by the previous controller prior to handing you over to Denver enroute center:

And here is how to contact the different ATC units at Paris - Charles De Gaulle:


Source



More: ATC Facilities