Context. The heating and structure of the solar chromosphere depends on the underlying magnetic field, among other parameters. The lowest magnetic flux of the solar atmosphere is found in the quiet Sun internetwork and is thought to be provided by the small-scale dynamo (SSD) process.

Aims. Our aim is to understand the chromospheric structure and dynamics in a simulation with purely SSD generated magnetic fields.

Methods. We performed a 3D radiation-magnetohydrodynamic (rMHD) simulation of the solar atmosphere, including the necessary physics to simulate the solar chromosphere. No magnetic field was imposed beyond that generated by an SSD process. We analysed the magnetic field in the chromosphere, and the resulting energy balance.

Results. Plasma at chromospheric temperatures reaches high into the atmosphere, with small, transient regions reaching coronal temperatures. An average Poynting flux of 5 × 10⁶ erg cm⁻³ s⁻¹ is found at the base of the chromosphere. The magnetic field in the chromosphere falls off more slowly with height than predicted by a potential field extrapolation from the radial component of the photospheric field. Starting in the middle chromosphere, the magnetic energy density is an order of magnitude higher than the kinetic energy density and, in the upper chromosphere, is also higher than the thermal energy density. Nonetheless, even in the high chromosphere, the plasma-β in shock fronts and low-field regions can locally reach values above unity.

Conclusions. The interactions between shocks and the magnetic field are essential to understanding the dynamics of the internetwork chromosphere. The SSD generated magnetic fields are strong enough to dominate the energy balance in the mid to upper chromosphere. The energy flux into the chromosphere is 8.16 × 10⁶ erg cm⁻² s⁻¹, higher than the canonical values required to heat the quiet Sun chromosphere and corona. Possibly due to the limited box size, the simulation is unable to maintain a million-degree corona.