  Recent findings suggest that wall-bounded turbulent flow can take different
statistically stationary turbulent states, with different transport properties,
even for the very same values of the control parameters. What state the system
takes depends on the initial conditions. Here we analyze the multiple states in
large-aspect ratio ($\Gamma$) two-dimensional turbulent Rayleigh--B\'enard flow
with no-slip plates and horizontally periodic boundary conditions as model
system. We determine the number $n$ of convection rolls, their mean aspect
ratios $\Gamma_r = \Gamma /n$, and the corresponding transport properties of
the flow (i.e., the Nusselt number $Nu$), as function of the control parameters
Rayleigh ($Ra$) and Prandtl number. The effective scaling exponent $\beta$ in
$Nu \sim Ra^\beta$ is found to depend on the realized state and thus
$\Gamma_r$, with a larger value for the smaller $\Gamma_r$. By making use of a
generalized Friedrichs inequality, we show that the elliptical instability and
viscous damping determine the $\Gamma_r$-window for the realizable turbulent
states. The theoretical results are in excellent agreement with our numerical
finding $2/3 \le \Gamma_r \le 4/3$, where the lower threshold is approached for
the larger $Ra$. Finally, we show that the theoretical approach to frame
$\Gamma_r$ also works for free-slip boundary conditions.
