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tokens: the ion density in the “strong gradient” simulation at , the LHDI fluctuations are highlighted in the unstable plane (yellow) and in a cut that is perpendicular to the ambient magnetic field. The initialization of the simulations ensures pressure balance by means of the Vlasov equilibrium proposed in Alpers (1969) , Pu et al. (1981) , which shapes a plasma boundary with density and magnetic field asymmetries, uniform temperature, with no electric field and no velocity nor magnetic field shear. Hereafter, we refer to the side I (resp. side II ) of this boundary as the high (resp. low) density side. The expressions for the initialization profiles are: Here, T = T i + T e is the uniform temperature, PBC is the constant of pressure balance (set to have the plasma beta β I = 10/3 and β II = 1/12), and w is a constant that defines the width of the layer – set to 0.98 (resp. 0.5) in the “strong” (resp. “weak”) gradient case. We use a density and magnetic field asymmetry of n I / n II = 10 and B I / B II = 0.5. The shape of these initialization profiles is shown in the left panel of Fig. 1 . In the following, all quantities are normalized to ion quantities in side I : the ion gyrofrequency, ; the ion skin depth, ; and the Alfvén speed . The numerical box dimensions are: L x = L y = 2 π and L z = 20 π , with a number of cells, namely, N x = N y = 288 and N z = 1472. The box is elongated (by a factor of in the magnetic field direction in order to reliably include the narrow cone angle over which unstable LHDI modes grow. The timestep used in the simulations is d t = 3.4 × 10 −4 to satisfy the CFL stability condition. The simulations are ran for several tens of the ion gyroperiods until electron acceleration saturates. We use 100 macro-particles per cell and a second order spline interpolation for the macro-particles. We use a reduced ion-to-electron mass ratio m i / m e = 100 to make simulations computationally feasible while maintaining a sufficient scale separation between ions and electrons. The implications of using such a reduced mass ratio are discussed in Sect. 4 . We use an ion-to-electron temperature ratio T i / T e = 1, and a plasma-to-cyclotron frequency ratio of . The magnitude of the density asymmetry used in these simulations encompasses the typical parameters observed in small planetary magnetospheres such as Mercury ( Gershman et al. 2015 ). Two different simulations are investigated using two different layer widths: (i) a steeper boundary case (with the inverse gradient scale length ε n = 1, defined in Eq. ( 7 )), hereafter called a “strong gradient” simulation; and (ii) a smoother boundary case (with ε n = 0.5), hereafter called the “weak gradient” simulation. These parameters are chosen to ensure that the amplitude of the LHDI fluctuations is well above the PIC noise level and also to saturate through the two different mechanisms introduced in Sect. 2.1 : ion trapping or current relaxation. 3. Evolution of LHDI and associated