We investigate the transition from forward to inverse energy cascade in turbulent flows in thin layers, varying the functional form of the forcing and the thickness of the layer. We show that, as the forcing function becomes more three dimensional, the inverse cascade is suppressed and the critical height $h_c$, where the transition occurs, is decreased. We study the dependence of this critical height on a parameter $r$ which measures the dimensionality of the forcing and thus construct a phase-space diagram in the parameter space $r-h$. We discuss the effect of Reynolds number and domain size.

中文翻译：

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强迫维数在薄层湍流能级联中的作用

我们研究了薄层湍流中从正向能量级联到逆向能量级联的转变，改变了强迫作用的功能形式和层的厚度。我们证明，随着强迫函数变得更立体，反向级联被抑制并且临界高度$H_C$，发生过渡的位置会减少。我们研究此临界高度对参数的依赖性$\mathrm{[R}$ 它测量强制的维数，从而在参数空间中构建相空间图 $\mathrm{[R}-H$。我们讨论雷诺数和域大小的影响。