Lorenz model of thermosolutal convection ($a,b,c,d,e$)

A five-mode Galerkin method model retains the roll velocity $a$, its correlated temperature and concentration modes $b,d$, and the mean-profile distortions $c,e$. Its parameters include the Prandtl number $\sigma$, diffusivity ratio $\tau$, thermal forcing $r$, stabilizing solutal forcing $r_s$, and a positive geometric decay ratio $\varpi$. The velocity equation is $\dot{a}=\sigma (-a+rb-r_{s}d)$, while the scalar equations are $\dot{b}=-b+a(1-c)$, $\dot{c}=\varpi (-c+ab)$, $\dot{d}=-\tau d+a(1-e)$ and $\dot{e}=\varpi (-\tau e+ad)$. The solutal forcing is normalized using the thermal diffusion scale, so its steady stabilizing effect is $r_s/\tau$.

Linearization of the Lorenz model of thermosolutal convection at conduction yields $(\lambda +\sigma )(\lambda +1)(\lambda +\tau )-\sigma r(\lambda +\tau )+\sigma r_s(\lambda +1)=0$. A Hopf bifurcation requires the cubic coefficient identity $a_3=a_1{a}_2$ and positive frequency squared. Its threshold is the stated $r_H$, with ${\omega }^2=\sigma r_s(1-\tau )/(\sigma +1)-{\tau }^2>0$. If $r_s={\beta }^2\tau$ and $\delta =\sigma /(1+\sigma )$, then $r_H=1+\tau /\delta +{\beta }^2\tau \delta +O({\tau }^2)$. Comparing with the steady fold of a thermosolutal Lorenz model gives a leading nonnegative difference $\tau (\beta \delta -1{)}^2/\delta$ for $\beta \ge 0$.

A nonzero steady Lorenz model of thermosolutal convection branch obeys $r(x)=(1+x)+r_s\tau (1+x)/({\tau }^2+x)$, with $x=a^2$. For $0<\tau <1$, it has an interior saddle-node bifurcation precisely when $r_s>{\tau }^3/(1-{\tau }^2)$. Differentiating gives $x_{\ast }=-{\tau }^2+\sqrt{r_s\tau (1-{\tau }^2)}$ and the stated minimum. The weaker-versus-stronger threshold distinction matters: a sufficient bound need not be the sharp subcriticality condition. Steady-branch existence alone does not decide oscillatory stability.