  We propose a novel one-dimensional model that includes both shock and
turbulence heating and qualify how these processes contribute to heating the
corona and driving the solar wind. Compressible MHD simulations allow us to
automatically consider shock formation and dissipation, while turbulent
dissipation is modeled via a one-point closure based on Alfv\'en wave
turbulence. Numerical simulations were conducted with different photospheric
perpendicular correlation lengths $\lambda_0$, which is a critical parameter of
Alfv\'en wave turbulence, and different root-mean-square photospheric
transverse-wave amplitudes $\delta v_0$. For the various $\lambda_0$, we obtain
a low-temperature chromosphere, high-temperature corona, and supersonic solar
wind. Our analysis shows that turbulence heating is always dominant when
$\lambda_0 \lesssim 1{\rm \ Mm}$. This result does not mean that we can ignore
the compressibility because the analysis indicates that the compressible waves
and their associated density fluctuations enhance the Alfv\'en wave reflection
and therefore the turbulence heating. The density fluctuation and the cross
helicity are strongly affected by $\lambda_0$, while the coronal temperature
and mass loss rate depend weakly on $\lambda_0$.
