= Solution
The <Lorenz model of thermosolutal convection> is a <Galerkin method> truncation retaining a roll velocity and the leading temperature and concentration modes. The variable $a$ measures the roll velocity; $b$ is the temperature mode correlated with the vertical motion, while $c$ is the horizontally averaged temperature-profile distortion generated by that motion. The corresponding concentration modes are $d$ and $e$. The symbol $d$ in this model is a solute amplitude, not the dimensional layer depth. The terms $ab$ and $ad$ change the mean scalar profiles; $a(1-c)$ and $a(1-e)$ express advection of the remaining background gradients.
The positive parameters are the <Prandtl number> $\sigma=\nu/\kappa$, the ratio $\tau=\kappa_s/\kappa$ of <solutal diffusivity> to <thermal diffusivity>, and the geometric vertical-mode decay ratio $\varpi$. In the usual fundamental-roll truncation with time normalized to the thermal roll-mode decay rate, $\varpi=4\pi^2/(k^2+\pi^2)$; treating it as an independently specified positive truncation parameter leaves the arguments below unchanged.
The parameter $r$ is the destabilizing thermal <Rayleigh number> divided by the free-slip critical-mode value, and $r_s>0$ measures stabilizing solutal buoyancy in the same thermal-diffusive normalization. For example, using a positive bottom-to-top stabilizing solute difference $\Delta S$ and density coefficient $\beta_S$,
$$
r=\frac{g\alpha_T\Delta T d_{\rm layer}^3}{\nu\kappa R_0(k)},\qquad r_s=\frac{g\beta_S\Delta S d_{\rm layer}^3}{\nu\kappa R_0(k)},\qquad R_0(k)=\frac{(k^2+\pi^2)^3}{k^2}.
$$
If a solutal <Rayleigh number> is instead defined with $\kappa_s$ in its denominator, the coefficient $r_s$ here is $\tau$ times that normalized number. \b[Thermal buoyancy drives the roll; solutal buoyancy opposes it and diffuses at a different rate.] This is the essential mechanism of <thermosolutal convection>.
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