We present three-dimensional direct numerical simulations and an analytic model of reflection-driven magnetohydrodynamic (MHD) turbulence in the solar wind. Our simulations describe transverse, non-compressive MHD fluctuations within a narrow magnetic flux tube that extends from the photosphere, through the chromosphere and corona and out to a heliocentric distance
$r$
of 21 solar radii
$(R_{\odot })$
. We launch outward-propagating ‘
$\boldsymbol{z}^{+}$
fluctuations’ into the simulation domain by imposing a randomly evolving photospheric velocity field. As these fluctuations propagate away from the Sun, they undergo partial reflection, producing inward-propagating ‘
$\boldsymbol{z}^{-}$
fluctuations’. Counter-propagating fluctuations subsequently interact, causing fluctuation energy to cascade to small scales and dissipate. Our analytic model incorporates dynamic alignment, allows for strongly or weakly turbulent nonlinear interactions and divides the
$\boldsymbol{z}^{+}$
fluctuations into two populations with different characteristic radial correlation lengths. The inertial-range power spectra of
$\boldsymbol{z}^{+}$
and
$\boldsymbol{z}^{-}$
fluctuations in our simulations evolve toward a
$k_{\bot }^{-3/2}$
scaling at
$r>10R_{\odot }$
, where
$k_{\bot }$
is the wave-vector component perpendicular to the background magnetic field. In two of our simulations, the
$\boldsymbol{z}^{+}$
power spectra are much flatter between the coronal base and
$r\simeq 4R_{\odot }$
. We argue that these spectral scalings are caused by: (i) high-pass filtering in the upper chromosphere; (ii) the anomalous coherence of inertial-range
$\boldsymbol{z}^{-}$
fluctuations in a reference frame propagating outwards with the
$\boldsymbol{z}^{+}$
fluctuations; and (iii) the change in the sign of the radial derivative of the Alfvén speed at
$r=r_{\text{m}}\simeq 1.7R_{\odot }$
, which disrupts this anomalous coherence between
$r=r_{\text{m}}$
and
$r\simeq 2r_{\text{m}}$
. At
$r>1.3R_{\odot }$
, the turbulent heating rate in our simulations is comparable to the turbulent heating rate in a previously developed solar-wind model that agreed with a number of observational constraints, consistent with the hypothesis that MHD turbulence accounts for much of the heating of the fast solar wind.
Analytic derivation of the inertial range of compressible turbulence
