Kinetically driven plasma waves are fundamental for a description of the thermodynamical properties of the Earth's magnetosheath. The most commonly observed ion-scale instabilities are generated by temperature anisotropy of the ions, such as the mirror and proton cyclotron instabilities. We investigate here the spatial resolution dependence of the mirror and proton cyclotron instabilities in a global hybrid-Vlasov simulation using the Vlasiator model; we do this in order to find optimal resolutions and help future global hybrid-Vlasov simulations to save resources when investigating those instabilities in the magnetosheath. We compare the proton velocity distribution functions, power spectra and growth rates of the instabilities in a set of simulations with three different spatial resolutions but otherwise identical set-up. We find that the proton cyclotron instability is absent at the lowest resolution and that only the mirror instability remains, which leads to an increased temperature anisotropy in the simulation. We conclude that the proton cyclotron instability, its saturation and the reduction of the anisotropy to marginal levels are resolved at the highest spatial resolution. A further increase in resolution does not lead to a better description of the instability to an extent that would justify this increase at the cost of numerical resources in future simulations. We also find that spatial resolutions between 1.32 and 2.64 times the inertial length in the solar wind present acceptable limits for the resolution within which the velocity distribution functions resulting from the proton cyclotron instability are still bi-Maxwellian and reach marginal stability levels. Our results allow us to determine a range of spatial resolutions suitable for the modelling of the proton cyclotron and mirror instabilities and should be taken into consideration regarding the optimal grid spacing for the modelling of these two instabilities, within available computational resources.

中文翻译：

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全局混合-Vlasov模拟中磁石场波的分辨率依赖性

动力学驱动的等离子波是描述地球磁热的热力学性质的基础。最常见的离子级不稳定性是由离子的温度各向异性（例如镜面和质子回旋加速器不稳定性）产生的。我们在这里使用Vlasiator模型在全球混合Vlasov模拟中研究反射镜和质子回旋加速器不稳定性的空间分辨率依赖性；我们这样做是为了找到最佳分辨率，并帮助未来的全球混合-Vlasov模拟在调查磁石场中的不稳定性时节省资源。我们在一组具有三种不同空间分辨率但设置相同的模拟中比较了质子速度分布函数，功率谱和不稳定性的增长率。我们发现质子回旋加速器的不稳定性在最低分辨率下是不存在的，只有镜面不稳定性得以保留，这导致了模拟中温度各向异性的增加。我们得出的结论是，质子回旋加速器的不稳定性，饱和度以及各向异性向边缘水平的减小都可以在最高的空间分辨率下得到解决。分辨率的进一步提高并不能更好地描述这种不稳定性，以至于在以后的模拟中以数值资源为代价证明这种增加是合理的。我们还发现空间分辨率在1.32和2之间。太阳风惯性长度的64倍为分辨率提供了可接受的极限，在该极限内，质子回旋加速器的不稳定性所导致的速度分布函数仍是双麦克斯韦式的，并达到了边际稳定性水平。我们的结果使我们能够确定适用于质子回旋加速器和镜面不稳定性建模的空间分辨率范围，并应考虑在可用的计算资源范围内为这两个不稳定性建模的最佳网格间距。