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The first wind nebula around a magnetar was recently discovered in X-raysaround Swift~J1834.9$-$0846. We study this magnetar's global energetics and theproperties of its particle wind or outflows. At a distance of $\sim4\;$kpc,Swift~J1834.9$-$0846 is located at the center of the supernova remnant (SNR)W41 whose radius is $\sim 19\;$pc, an order of magnitude larger than that ofthe X-ray nebula ($\sim2\;$pc). The association with SNR W41 suggests a commonage of $\sim5-100\;$kyr, while its spin-down age is $4.9$~kyr. A small natalkick velocity may partly explain why a wind nebula was detected around thismagnetar but not around other magnetars, most of which appear to have largerkick velocities and may have exited their birth SNR. We find that the GeV andTeV source detected by Fermi/LAT and H.E.S.S., respectively, of radius$\sim11\;$pc is most likely of hadronic origin. The dynamics and internalstructure of the nebula are examined analytically to explain the nebula'scurrent properties. Its size may naturally correspond to thediffusion-dominated cooling length of the X-ray emitting $e^+e^-$ pairs. Thismay also account for the spectral softening of the X-ray emission from thenebula's inner to outer parts. Analysis of the X-ray synchrotron nebula impliesthat (i) the nebular magnetic field is $\gtrsim 11\;\mu$G (and likely$\lesssim30\;\mu$G), and (ii) the nebula is not powered predominantly by themagnetar's quiescent spin-down-powered MHD wind, but by other outflows thatcontribute most of its energy. The latter are most likely associated with themagnetar's bursting activity, and possibly dominated by outflows associatedwith its past giant flares. The energy source for the required outflows cannotbe the decay of the magnetar's dipole field alone, and is most likely the decayof its much stronger internal magnetic field.

Learning About the Magnetar Swift J1834.9-0846 from its Wind Nebula