Advection Term in the Lorenz 96 Model The Lorenz 96 Model is defined as
$$\frac{dx_i}{dt}=\underbrace{(x_{i+1}-x_{i-2})x_{i-1}}_{advection}-x_i+F$$
with some forcing $F$ and periodic boundary conditions so that $x_{i+N}=x_i$ for some $N$.
I do not have an intuitive explanation why the first term on the RHS of the equation describes some advection process. The advection term can be multiplied out to yield $x_{i+1} x_{i-1}$ and $-x_{i-1} x_{i-2}$. The first term means an increase in $x_i$ in time when $x_{i+1}$ and $x_{i-1}$ have the same sign and the second term means an increase in $x_i$ when $x_{i-1}$ and $x_{i-2}$ have different signs.
I don't understand how this relates to the concept of advection.
 A: The fluid equation's convective term, $(\mathbf{u}\cdot \nabla)\mathbf{u}$, becomes in 1-D:
$$ u\;\partial u/ \partial x,$$
which can be discretized in a variety of ways (see, e.g., 1, 2, 3 and 4) involving quadratic terms and different lattice sites, such as
$$ u_{i}(u_{i+1}-u_{i-1}) \quad \text{ or} \label{dis}\tag{1} $$
$$ u_{i}(u_{i+1}-u_{i}) \text{ for }u_{i}<0,\quad  u_{i}(u_{i}-u_{i-1}) \text{ for } u_{i}>0, $$
among others.
As a toy model, Lorenz 96 offers only a rough approximation of the physics, so if the term is advection-like and preserves energy — and the discretizations above are similar enough (using $u$ for $x$) to L96's $u_{i-1}(u_{i+1}-u_{i-2})$ — that's enough to call it "advection".
As for the intuition, the main differences with respect to the main discretization (\ref{dis}) are i) a shift in the position index, $i \to i-1$, which I guess accounts for a preferential direction of wave propagation in the model, and ii) a one-sided next nearest neighbor interaction (the $u_{i-2}$ dependence) whose role, apart from appearing to make it less local, isn't that clear to me.
