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tokens: 2–8 minutes. Figure 2. Same as Figure 1 , but P 0 = 80 s and σ 0 = 30 s. Green dashed lines label the prominent quasi-periodic oscillations in the range of 2–8 minutes. The cone-of-influence is cross-hatched; within it the spectrum should be considered unreliable. 3. NUMERICAL EXPERIMENTS In order to explore how stochastic transients affect the upper atmosphere, we create a set of numerical experiments and investigate how the power spectrum is altered observationally throughout its propagation in a stratified atmosphere. We do not attempt to model realistic transients at photospheric temperature and density, but only demonstrate what response could be observed if this sort of perturbation was triggered in the lower atmosphere. 3.1. Empirical Model of the Atmosphere Figure 3 plots the one-dimensional model of the atmosphere used in this study. Below z = 2.22 , we implemented the mid-age sunspot umbra model of Maltby et al. ( 1986 ); close to the bottom boundary ( z 1.2 ), we set a constant temperature (6471 ). Above z = 2.22 , we adapted the C7 model of Avrett Loeser ( 2008 ); near the upper boundary (), we set a constant temperature of 1.3 . This kind of model of temperature has been used in a number of previous studies, e.g., Heggland et al. ( 2007 ), Botha et al. ( 2011 ), and Snow et al. ( 2015 ). Figure 3. Empirical model of the atmosphere used in this study; the inset enlarges the temperature profile at z 2.5 . In hydrostatic equilibrium, the pressure is obtained by integrating the gravitational force ρg , where p , ρ , and T are the plasma pressure, density, and temperature, respectively, is the gravitational acceleration at the surface of the Sun, k B is the Boltzmann constant, and m p is the mass of a proton. We choose a base density of at z = 0 , so at coronal height the density is of the order of , see Figure 3 . In one-dimensional MHD simulation, the magnetic field decouples from hydrodynamic equilibrium; we used B = 10 , in line with the z -axis, in our simulations. The plasma parameters were normalized by choosing a set of three to be constant: the length L 0 = 1.0 , the density , and the magnetic field strength B 0 = 10 . Then the normalization factor for the velocity is , the time t 0 = 31.7 s, the gravitational constant , and the temperature T 0 = 0.06 . So the variables become dimensionless by using , , , , , , , etc., where symbols with a hat are dimensionless variables. 3.2. Magnetohydrodynamic Simulation We used the AMRVAC package (Keppens et al. 2012 ; Porth et al. 2014 ) to solve the ideal MHD equations with the finite-volume method: where is the fluid velocity, the magnetic field vector, the solar gravity vector, the unit tensor, the total energy density, the total pressure, γ the adiabatic index, and μ 0 the magnetic permeability of free space. The AMRVAC code was configured to solve Equations ( 6 )–( 10 ) in Cartesian coordinates [ x , y , z ]. The physical quantities were assumed to be invariant along the x - and