A new instability driven by the combined effect of wind stress and rotation in a sheared liquid layer

TL;DR

This work investigates the linear stability of a wind-stress–driven shear flow in a rotating system, modeling oceanic conditions with a rotating lower boundary. Using Chebyshev spectral collocation and longwave asymptotics, the authors uncover new longwave instability modes that emerge when the rotational Reynolds number $Re_\u03a9$ is nonzero, with the most unstable mode being a spanwise longwave disturbance ($k=0$, $m>0$). The leading-order spanwise phase speed $c_0$ depends only on $Re_\u03a9$, and the full three-dimensional longwave analysis agrees well with numerical results, confirming the physical reality of the instability. The findings illuminate how Earth’s rotation can interact with surface wind shear to modify ocean dynamics, potentially affecting large-scale wave and boundary-layer processes.

Abstract

We examine the linear stability of a shear flow driven by wind stress at the free surface and rotation at the lower boundary, mimicking oceanic flows influenced by surface winds and rotation of Earth. The linearised eigenvalue problem is solved using the Chebyshev spectral collocation method and a longwave asymptotic analysis. Our results reveal new longwave instability modes that emerge for non-zero rotational Reynolds numbers. It is observed that the most unstable mode, characterised by the lowest critical parameters, corresponds to longwave spanwise disturbances with vanishing streamwise wavenumber. The asymptotic analysis, which shows excellent agreement with numerical results, analytically confirms the existence of this instability. Thus, the present study demonstrates the hitherto unreported combined influence of wind stress and rotation of Earth on ocean dynamics.

Figures (12)

• Figure 1: Schematic of the flow configuration used to investigate the effect of rotation on a wind-stress-driven shear flow ($\tau$). The lower wall is located at $z = 0$, and the free surface at $z = d$. In dimensionless form, the free surface is expressed as $z = 1 + h(x, y, t)$, where $h(x, y, t)$ denotes the perturbation superimposed on the base state. The lower plane rotates with a constant angular velocity $\Omega$, and the analysis is carried out in a reference frame that rotates with the Earth.
• Figure 2: Comparison of the neutral stability curve obtained from the present solver with that of smith1982instability. The remaining parameters are set as ${\it Re}_{\Omega} = 0$, $m=0$ and $S = 0$. The critical Reynolds number and the critical wavenumber are found to be ${\it Re}_c = 34.2$ and $k_c = 2.43$, respectively, which are in excellent agreement with the values reported by smith1982instability.
• Figure 3: (a) The variation of the temporal growth rate of instability $\omega_i$ with $k$ for $m=0.01$, and (b) the variation of $\omega_i$ with $m$ for $k=0$, at different values of $Re_\Omega$. The rest of the parameters are $Re=0.1$ and $S=0.01$.
• Figure 4: Neutral stability curves in the $Re_\Omega - k$ plane for different values of $m$, delineating the stable and unstable regions. Panel (b) presents a magnified view of the bottom-left corner of panel (a). The rest of the parameters are $Re=0.1$ and $S=0.01$.
• Figure 5: Variation of (a) $\omega_i$ and (b) $\omega_r$ with $Re_\Omega$ for different values of $Re$. Here, $S = 0.01$. We set $k = 0.01$ and $m = 0$ to examine the streamwise longwave instability.