i know for a fact that the alpine type S alpine subs are dual voice coil. The jaycar amp you have is a 2 channel. when you bridge 2 channels in an amplifier, you effectively half the resistance. normally in single channel mode you can run a 4 ohm speaker at 4 ohms, and a 2 ohm speaker at 2 ohms. bridging the amp will change this. as a result the least resistance you can run on a bridged amplifier (unless specified) is 4 ohms. as this will run the amp in 2 ohm mode.
I am supprised that you didn't kill the amplifier with the alpine sub as the system would have been running in 1 ohm, probably giving an effective power output of anywhere between 800 and 1000w rms. and the amp would have been able to cook bacon and eggs .
depending on how you want to go, you could run 2 dual voice coil subs (one of each channel) in 2 ohm mode, which will give 500 wrms or (starting to get more technical now) run each woofer in 2 ohm (voice coils in paralell) and then run both woofers in series (this will give an effective resistance of 4 ohms) meaning that you will get your 500w and probably not do any damage.
However as everyone has said before, more power is better than less, but too much is just as bad as not enough. if that makes sense.
just to remember, when working with equal resistances i.e. 4 ohm and 4 ohm, in series + > - > + > - it will give a resistance of 8 ohms so just add the 2 voice coil impedecnes together. when in parallell + > + and - > - it will give 2 ohms, just divide the impedences by 2
hope this helps