hexagons [e.g., @Palm1960; @Chandrasekhar1961; @Toomre1977] and the decomposition of convective motions into planforms, which are made up from such patterns in the horizontal direction [e.g., @Chandrasekhar1961; @Gough1975]. Due to the continuity equation and under the assumption of a steady flow, the vertical flows and the horizontal divergence show the same pattern close to the surface.

![image](fig2){width="\linewidth"}

We now look at a few possible patterns in detail. The top row in Figure \[figPatterns\] shows simple convective rolls. Such a roll is given by a pattern $X(\bx)$ for which $$\begin{aligned}
X(\bx) = h_k^2 \left( e^{i \bk_0 \cdot \bx} + c.c. \right), \label{eqRolls}\end{aligned}$$ where $\bk_0=(k_{x,0},k_{y,0})$ is a chosen wave vector and we use the notation $+ c.c.$ to indicate that the previous terms are repeated with a complex conjugate. The top row in Figure \[figPatterns\] shows three examples of rolls for three different wave vectors $\bk_0,\bl_0,\bmm_0$, which are kept fixed for the entire section and are shown in the small subpanels of Figure \[figPatterns\] as colored arrows. These wave vectors form a triad, that is $\bk_0+\bl_0=\bmm_0$. The wave vectors chosen here have the same length and are at an angle of 60 degrees (up to a small deviation caused by the discretization).

We 