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The effective field in this case is vertical i.e. perpendicular to the plane of the liquid.
The lines of force run from N to S both vertically and horizontally. If there were no field, there would be no movement.
There has to be some initial asymmetry to determine the direction of the current flow: if the electrodes and magnetic field were perfectly symmetrical the current would flow equally in both directions and there would be no movement! Once the liquid flow has started, that pretty well determines the current direction depending on whether the positive or negative ions are more mobile.Experiments with bath water (does it spin the other way south of the equator?) have shown that micro-flow persists for hours after the bath is filled. In this case if the bulk of the conductive liquid is flowing, say, clockwise, that will persist when the electrodes are removed, but turbulence at the surfaces may produce local countercurrent eddies that will start the flow in the opposite direction when you move the electrodes.Some old electric clocks had symmetric synchronous motors, and in the days when we had frequent power outages I recall my grandmother's clock sometimes stationary and often running backwards when the power was restored!
If the system is completely symmetrical, nothing will move. It is the principle used in residual current magnetic circuit breakers.
The absence of circumferential current  (the path of circumferential current is indicated by the red arrow in figure 3 of this section) and the axially symmetric layout of the conductors and magnetic materials  are sufficient conditions for total internal confinement of the B field. (Some authors prefer to use the H field). Because of the symmetry, the lines of B flux must form circles of constant intensity centered on the axis of symmetry. The only lines of B flux that encircle any current are those that are inside the toroidal winding. Therefore, from Ampere's circuital law, the intensity of the B field must be zero outside the windings.
I say again, if B= 0 it will work without the magnet.