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We rigorously derive from first principles the generic Landau amplitudeequation that describes the primary bifurcation in electrically drivenconvection. Our model accurately represents the experimental system: a weaklyconducting, submicron thick liquid crystal film suspended between concentriccircular electrodes and driven by an applied voltage between its inner andouter edges. We explicitly calculate the coefficient $g$ of the leading cubicnonlinearity and systematically study its dependence on the system'sgeometrical and material parameters. The radius ratio $\alpha$ quantifies thefilm's geometry while a dimensionless group ${\cal P}$, similar to the Prandtlnumber, fixes the ratio of the fluid's electrical and viscous relaxation times.Our calculations show that for fixed $\alpha$, $g$ is a decreasing function of${\cal P}$, as ${\cal P}$ becomes smaller, and is nearly constant for ${\cal P}\gtrsim 1$. As ${\cal P} \to 0$, $g \to \infty$. We find that $g$ is anontrivial and discontinuous function of $\alpha$. We show that thediscontinuities occur at codimension-two points that are accessed by varying$\alpha$.

Codimension-two points in annular electroconvection as a function of aspect ratio