  We model the solar horizontal velocity power spectrum at scales larger than
granulation using a two-component approximation to the mass continuity
equation. The model takes four times the density scale height as the integral
(driving) scale of the vertical motions at each depth. Scales larger than this
decay with height from the deeper layers. Those smaller are assumed to follow a
Kolomogorov turbulent cascade, with the total power in the vertical convective
motions matching that required to transport the solar luminosity in a mixing
length formulation. These model components are validated using large scale
radiative hydrodynamic simulations. We reach two primary conclusions: 1. The
model predicts significantly more power at low wavenumbers than is observed in
the solar photospheric horizontal velocity spectrum. 2. Ionization plays a
minor role in shaping the observed solar velocity spectrum by reducing
convective amplitudes in the regions of partial helium ionization. The excess
low wavenumber power is also seen in the fully nonlinear three-dimensional
radiative hydrodynamic simulations employing a realistic equation of state.
This adds to other recent evidence suggesting that the amplitudes of large
scale convective motions in the Sun are significantly lower than expected.
Employing the same feature tracking algorithm used with observational data on
the simulation output, we show that the observed low wavenumber power can be
reproduced in hydrodynamic models if the amplitudes of large scale modes in the
deep layers are artificially reduced. Since the large scale modes have reduced
amplitudes, modes on the scale of supergranulation and smaller remain important
to convective heat flux even in the deep layers, suggesting that small scale
convective correlations are maintained through the bulk of the solar convection
zone.
