They gain energy by ramping across a DL and the plasma gets it kinetic energy from the particles impinging on the anode and cathode side of the DL.
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No this is not the source of energy increase in magnetic reconnection. Double layers and magnetic reconnection are two separate phenomena. They generally occur separately.
There are situations in which evidence for both have been observed. e.g. some solar flares (caused by magnetic reconnection) emit beams of electrons that are thought to be accelerated by DL.
In the lab, they do not occur separately.
From what we know to be true about the laboratory proven physics of "reconnection" - the DL plays a role as a causative agent involved in the transfer of energy.
Neglecting the existence of DLs by modeling reconnection events entirely in MHD frameworks does not mean they do not exist in real life.
MHD theory is incapable of accounting for them, which is why they never pop up in discussions of reconnection.
Given that we know localized instabilities will generate regions of varying voltage / potential, we know that there will be DLs formed.