What parameter is used to choose the direction of the flux in a magnetic circuit?
Sometimes it's easier to think about magnetic circuits like electrical circuits, this way you can reuse your vast knowledge of how currents behave.

Here is an electronic model of the magnetic circuit.







simulate this circuit &ndash; Schematic created using CircuitLab

In your image, the current flowing in the coil that gives rise to the magnetic flux \$\Phi_1\$ and the current in the coil that gives rise to the flux \$\Phi_2\$ goes in the same counter-clockwise direction. This means that you will have a flux flowing in the same direction. Both of the flux's will flow upwards and join constructively at the top of the T-junction. The flux at \$\Phi_3\$ has the current in its coil flowing in the clockwise direction, so it's forcing the flux to flow along the right hand side downwards. 

Magnetic flux behaves very much like current does in electronics (in an ideal world), the sum of all flux entering a node is = 0. Just like KCL (Kirchhoff's Current Law). With high reluctance, the flux will be low, just like having high resistance will make the current low in an electrical circuit. If you want a proper magnetic circuit then you need to make proper low permeability path's for the flux to flow through (like iron, or cobalt for an example), just like you would in an electrical circuit to make, say a lamp light up. You wouldn't be able to light up a 60 W bulb with a 1 kΩ resistor in series with it. 

Here's some magnetic parameters and electric parameters that behave in a very similar way.


\$U = R×I\$, Electromagnetic force (measured in volts) = Resistance × Current
\$ℱ = R×\Phi\$, Magnetomotive force = Reluctance × Magnetic flux


Coils with current flowing through them will actually give you a MMF (Magnetomotive force), so it should be voltage sources in the schematic above, but I have depicted it as current sources because what we're interested in is the flux, which behaves like a current. In the same way you can have electrical norton and thevenin circuits, the same way you can have magnetic norton and thevenin circuits. 

As a charged capacitor is to a voltage source, current flowing through a coil is to a MMF source. A capacitor with a very large capacitance behaves a little bit like a regular battery, in case you are unfamiliar with how charged capacitors behave. 

In order to answer your question, black on white, it is the current flowing through the coils that decide which direction the flux will flow in. Imagine a typical screw seen from above, looking down at its head. The tip is just penetrating a piece of wood. If you got a coil around this screw  with current flowing in clockwise direction, then the flux will flow downwards through the center of the screw. If you would use a screwdriver and rotate the screw clockwise then it will start penetrating the wood even further. So having current flowing clockwise around the screw will result in a flux that flows in the same direction as if you would physically rotate the screw clockwise. This is also known as the right-hand rule, but meh, I always forget which is which. Screws work better.



With everything said, I hope that the schematic I've shown above can convince you why the flux will flow in the direction it's flowing in.