According to this website,
A Birkeland current is a set of currents that flow along geomagnetic field lines connecting the Earth’s magnetosphere to the Earth's high latitude ionosphere.
These then are driven by solar wind in the Earth's magnetosphere.
Birkeland currents are caused by the movement of plasma perpendicular to the magnetic field. They can also be created in the laboratory by multi-terawatt pulsed power generators. The term Birkeland current also commonly refers to electric currents in other planet's ionospheres that follow magnetic field lines. The term can also be used to refer to any field aligned electric current in space.
So it seems (though I'll be looking into it more) that they can only form on specific magnetic field lines.
Fun fact: small-scale variations in these currents produce the Aurora borealis and australis once the currents reach the upper atmosphere. Not related at all, but cool.
Hope this helps!
Another website that has more information is this one.