and results will be given in terms of the radial length $a_\rho = \sqrt{\hbar/m\omega_\rho}$.

The solutions for each component $\psi_\alpha({\boldsymbol\rho})$ (where $\alpha=\{1,2\}$) are obtained by solving the coupled GPEs [@Esry1997; @Pu1998; @Ho1996], $$\label{Eq:GPE}
\left[H_{\rm sp}({\boldsymbol\rho}) + \frac{\hbar^2}{m}\sum_\beta g_{\alpha\beta}n_\beta({\boldsymbol\rho})-\mu_\alpha \right]\psi_\alpha({\boldsymbol\rho}) = \mathbf{0},$$ where the condensate densities $n_\alpha = |\psi_\alpha({\boldsymbol\rho})|^2$ are individually unit-normalized, $\mu_\alpha$ are the corresponding chemical potentials, and the single-particle Hamiltonian is $H_{\rm sp}({\boldsymbol\rho}) = -\hbar^2\nabla^2/2m + V({\boldsymbol\rho},z=0)$. The dimensionless interactions are described by $g_{\alpha\beta}=N_\beta\sqrt{8\pi}a_{\alpha\beta}/a_z$, where $N_\beta$ is the number of atoms in component $\beta$. In this work, we restrict our attention to balanced populations, $N_1=N_2\equiv N/2$, and intracomponent interactions, $g_{11}=g_{22}\equiv g$. However, more general mixtures, along with a physical example, are discussed in Sec. \[Sec:ExpConsid\].

Variational Model {#Sec:Var}
-----------------

In addition to numerically solving the GPEs, we derive analytical predictions for the stationary-state solutions, including a prediction for the miscible-to-immiscible phase transition. To achieve this, motivated by the one-dimensional work of [@Navarro2009], we implement a chirped gaussian ansatz (superscript $\mathcal V$) to approximate the wavefunctions, $$\psi^{\mathcal V}_\alpha({\boldsymbol\rho})=A\exp\left[\frac{-(x\mp B)^2-y^2}{2W^2}\right] \exp\left[i(C \pm Dx+Ex^2)\right],$$ where variational parameters describe the amplitude $A$, component separation $2B$, phase $C$, wavenumber $D$, chirp $E$ and width $W$. The phase is a product of the chemical potential 