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tokens: region (black dashed line). For the viscous simulation on the other hand, we can see that the Poynting flux leaving the coronal shell region (solid black line) is substantially smaller than the Poynting flux entering the region (green dashed line). This difference is of the same order as the internal energy increase in the shell region (not shown here). The Poynting flux entering the shell region through the near TR-coronal boundary (green lines) differs in the ideal and viscous simulation due to the different mixture of propagating and standing waves in this region. For the ideal simulation, the standing wave component dominates, which reduces the amount of Poynting flux through the near boundary, as the standing regime creates a node in v z (see e.g. Prokopyszyn et al. 2019 ). Comparing with coronal energy requirements, the difference between the average Poynting flux entering and leaving through the TR-coronal boundaries of the shell region for the viscous simulation is of the order of 1.5−2 Jm −2 s −1 , several orders of magnitude too low, even for the Quiet Sun (∼3 × 10 2 Jm −2 s −1 , Withbroe Noyes 1977 ). Fig. 9. Time integrated Poynting flux , with S the boundaries of the left shell, for the viscous (solid lines) and ideal simulation (dashed lines). The green lines represent the lower TR-coronal boundary and the black lines the upper boundary. 4. Discussion and conclusions In this paper, we use 2.5D numerical simulations to investigate whether a pre-existing density profile is modified by the evaporative upflows following (viscous) heating by phase mixing of Alfvén waves. The Alfvén waves are driven through an additional forcing term in the momentum equation. The simulations account for the effects of gravitational stratification, thermal conduction, and optically thin radiation. A fully resolved atmospheric model (including the chromosphere as a mass reservoir) is achieved with relatively modest resolution by artificially broadening the TR using the approach proposed by Lionello et al. (2009) and Mikić et al. (2013) . To include a transverse density profile (perpendicular to the background magnetic field) in the initial setup, a heating function which varies in the cross-field direction is imposed. Throughout the simulations (running for a total time of more than 5000 s), a complex combination of Alfvén waves ( v z ) and longitudinal perturbations ( v y ) is present in the domain. The longitudinal perturbations are a combination of long-period oscillations resulting from the ongoing evolution due to the imposed cross-field heating profile, the ponderomotive effects associated with the driven Alfvén waves, and the evaporative upflows resulting from heating occurring in the coronal part of the shell regions of the loop due to phase mixing of the Alfvén waves. By comparing with the results from the ideal simulations, we are able to distinguish the evaporative upflows present in the viscous simulations from the other (ideal) longitudinal