I think the label ‘AC adapter’ here is a bit misleading. This seems clearly a DC adapter (or more strictly AC to DC adapter). Note the label that marks the – / + of the connector.
Another way to tell is to use your multimeter to measure the AC voltage of the output of the transformer. If the transformer outputs DC, the AC voltage reading will be almost zero.
You will need an AC adapter, such as this one:
http://www.homedepot.com/p/Orbit-Transformer-24V-750Ma-57040/202899607#.UfWLPReJBpl
You can also search ‘sprinkler transformer’ on Amazon and there are plenty of choices.