  We present a theoretical analysis of some unexplored aspects of relaxed
Bose-Einstein condensate dark matter (BECDM) haloes. This type of ultralight
bosonic scalar field dark matter is a viable alternative to the standard cold
dark matter (CDM) paradigm, as it makes the same large-scale predictions as CDM
and potentially overcomes CDM's small-scale problems via a galaxy-scale de
Broglie wavelength. We simulate BECDM halo formation through mergers, evolved
under the Schr\"odinger-Poisson equations. The formed haloes consist of a
soliton core supported against gravitational collapse by the quantum pressure
tensor and an asymptotic $r^{-3}$ NFW-like profile. We find a fundamental
relation of the core=to-halo mass with the dimensionless invariant $\Xi \equiv
\lvert E \rvert/M^3/(Gm/\hbar)^2$ or $M_{\rm c}/M \simeq 2.6 \Xi^{1/3}$,
linking the soliton to global halo properties. For $r \geq 3.5 \,r_{\rm c}$
core radii, we find equipartition between potential, classical kinetic, and
quantum gradient energies. The haloes also exhibit a conspicuous turbulent
behavior driven by the continuous reconnection of vortex lines due to wave
interference. We analyse the turbulence 1D velocity power spectrum and find a
$k^{-1.1}$ power-law. This suggests the vorticity in BECDM haloes is
homogeneous, similar to thermally-driven counterflow BEC systems from condensed
matter physics, in contrast to a $k^{-5/3}$ Kolmogorov power-law seen in
mechanically-driven quantum systems. The mode where the power spectrum peaks is
approximately the soliton width, implying the soliton-sized granules carry most
of the turbulent energy in BECDM haloes.
