FIGURE 3. Electron Fermi acceleration in squashed flux bubbles.

Shown are the results of two-dimensional simulations of electron acceleration in isolated, squashed flux bubbles, where Fermi acceleration can be studied without the interference from the parallel electric fields associated with magnetic reconnection. The release of magnetic energy as the bubbles contract and become round is essentially the same as in the contraction of magnetic islands during magnetic reconnection. a, The initial magnetic field lines in a squashed magnetic bubble superimposed over the uniform, isotropic electron temperature T_e0 = 0.1. The bubble is initially in force balance in the y direction (the variation in the density enabling the plasma pressure to balance the magnetic pressure) but not in the x direction. Initially , where n₀ and B₀ are the maximum values of the density and magnetic field, and B_z is zero. Realistic values of the electron mass (m_i/1,836) and velocity of light (c = 100c_A) are required so the electrons can undergo many Fermi reflections during the contraction of the bubble. b, The parallel electron temperature T_e and magnetic field lines at late time. Note the increase in the parallel temperature within the bubble and the near circular shape of the final magnetic field lines. The change in the perpendicular temperature is small. Sixty percent of the released magnetic energy is transferred to electrons. c, T_e and the magnetic field lines at late time shown from a simulation that is identical to that shown in a and b but with a larger initial electron pressure (_e0 = 1.1). In this case, the back pressure from the accelerated electrons prevents the full contraction of the bubble, consistent with the discussion following equation (2).