Effects of field-aligned flows on standing kink and sausage modes supported by coronal loops

Fundamental standing modes and their overtones play an important role in coronal seismology. We examine how a significant field-aligned flow affects standing modes supported by coronal loops, modeled here as cold magnetic slabs. Of particular interest are the period ratios of the fundamental to its $(n-1)$-th overtone ($P_1/nP_n$) for both kink and sausage modes, and the threshold half-width-to-length ratio for sausage modes. For standing kink modes, the flow significantly reduces $P_1/nP_n$ in general, the effect being particularly strong for larger $n$ and when the density contrast $\rho_0/\rho_e$ between loops and their surroundings is weak. That said, even when $\rho_0/\rho_e$ approaches infinity, this effect is still substantial, reducing the minimal $P_1/nP_n$ by up to 13.7% (24.5%) for $n=2$ ($n=4$) relative to the static case, when the Alfv\'en Mach number $M_A$ reaches 0.8 where $M_A$ measures the loop flow speed in units of the internal Alfv\'en speed. For standing sausage modes, though not negligible, the flow effect in reducing $P_1/nP_n$ is not as strong. However, the threshold half-width-to-length ratio is considerably larger in the flowing case than its static counterpart. For $\rho_0/\rho_e$ in the range $[9, 1024]$ and $M_A$ in the range $[0, 0.5]$, an exhaustive parameter study yields that this threshold is well fitted by our Equation (23) which involves the two parameters in a simple way. This allows one to analytically constrain the combination $(\rho_0/\rho_e, M_A)$ for a loop with known width-to-length ratio when a standing sausage oscillation is identified therein. It also allows one to further examine the idea of partial sausage modes, and the flow is found to reduce significantly the spatial extent where partial modes are allowed.