Effect of depth discontinuity on interfacial stability of tangential-velocity discontinuity in shallow-water flow

It is well known that an interface of tangential velocity discontinuity is necessarily unstable, regardless of the velocity difference's strength, so-called the Kelvin-Helmholtz instability (KHI). However, the instability is suppressed for shallow water flows if the Froude number, defined by the ratio of the velocity difference to the gravity wave's speed, is sufficiently large. In this investigation, we examine the effect of the depth difference of two fluid layers on the KHI. The depth difference enhances instability. The dispersion equation is obtained in a sextic polynomial, the interface is thus linearly stable if the dispersion equation has enough six real roots. Given the Froude number M₁ in the instability range, the growth rate sensitively depends on the depth ratio r=H₁/H₂ and increases monotonically with the depth ratio difference from unity. The critical value of the Froude number for stabilization varies with the depth ratio and attains the minimum value 8 for equal depth. This behavior is verified by asymptotic and numerical analysis. Numerically, we illustrate that if the depth ratio r=0.5, the Froude number is equal or greater than 33≈5.745 to stabilize the interface and it is 4.08 for r=2.