  Magnetic fields generated in the Sun's interior by the solar dynamo mechanism
drive solar activity over a range of time-scales. While space-based
observations of the Sun's corona exist only for few decades, direct sunspot
observations exist for a few centuries, solar open flux and cosmic ray flux
variations can be reconstructed through studies of cosmogenic isotopes over
thousands of years. While such reconstructions indicate the presence of extreme
solar activity fluctuations in the past, causal links between millennia scale
dynamo activity, consequent coronal field, solar open flux and cosmic ray
modulation remain elusive. By utilizing a stochastically forced solar dynamo
model we perform long-term simulations to illuminate how the dynamo generated
magnetic fields govern the structure of the solar corona and the state of the
heliosphere -- as indicated by variations in the open flux and cosmic ray
modulation potential. We establish differences in the nature of the large-scale
structuring of the solar corona during grand maximum, minimum, and regular
solar activity phases and simulate how the open flux and cosmic ray modulation
potential varies over time scales encompassing these different phases of solar
activity. We demonstrate that the power spectrum of simulated and reconstructed
solar open flux are consistent with each other. Our study provides the
theoretical basis for interpreting long-term solar cycle variability based on
reconstructions relying on cosmogenic isotopes and connects solar internal
variations to the forcing of the state of the heliosphere.
