You basically have two or three choices for this.
You need to use pulse width modulation, PWM, controller of some sort since the currents are too large for an efficient linear solution that does not need a heat-sink the size of a shoe-box.
The issue with PWM is you are driving a purely resistive load. That means you are switching large amps at large voltages with sharp edges. As we all know that translates into a rather effective EMI transmitter.
As such you have two options.
1. High Frequency PWM with an added inductor.
simulate this circuit – Schematic created using CircuitLab
Using the above technique you can maintain the current, all be it with a ripple, in the coil and wire at a fairly steady rate and the PWM action will be switching voltage and a sustaining current level.
However, since the load wire resistance is very low, the coil needs to be large so it's DC resistance is much less than the wire so you do not lose power by heating the inductor.
2. Low Frequency PWM
The alternative is to use low frequency PWM.
simulate this circuit
Low frequency means drive the wire like it would be driven from the AC line, i.e 100, or 200Hz. In this scenario you are swill switching the full current but now you are doing it at a frequency that is much less problematic from an EMI point of view.
However, it may still be prudent to add a small inductor coil to reduce the current rise time a just a little.
3. A fully integrated current driver.
You could implement a fully integrated current control system with feedback using an LED driver chip like the LT3086 from Linear Technology. That would be a high frequency PWM system but again you are going to need a large inductor, magnetically and physically.
This would be the ultimate design for precise control of current, but may well be overkill for your application.
In all cases it is very important to design your circuit so the MOSFET turns on and off as quickly as it can, especially with a high frequency PWM system. During transition the resistance of the MOSFET changes rapidly, however there is a considerable amount of power lost as heat during that transition. As such you can find the MOSFETS will overheat despite the temperature you think they should be running at. That may mean adding a push-pull pre-driver before the MOSFET to deliver sufficient current to fill and dump the gate capacitances quickly.
Obviously your power supply needs to be capable to supply the full current on your wire, but you also need a significant charge reservoir capacitor close to the driver circuit. Although capable of providing the max current, the power supply itself may not be capable of adapting to such high transitions in current immediately.
Of the three approaches I mention, I personally would opt for the low frequency PWM method with fast MOSFETS. If need be, I would also add some small inductance into the line to limit the rise time and noise spikes a little.
If you are driving this thing from an AC transformer you may want to consider a normal "dimmer" type triac circuit.