Secondary instability of Ekman layer rolls

Summary

We study the non-linear stages of the instability of the neutrally-stratified Ekman flow. We find that two-dimensional equilibrated rolls exist, and are reached very closely, although not exactly, after a small random initial perturbation to the Ekman flow has evolved and reached nonlinear saturation. We perform in turn a linear stability analysis of these equilibrated rolls and find that they are subject to an instability of the hyperbolic type. Finally, we investigate the influence of the latitude and of the direction of the geostrophic wind on these stability properties.

1 Introduction

The Ekman spiral flow is an exact solution of the Navier-Stokes equations in the presence of rotation and a rigid boundary. It is useful as a prototype flow for idealized dynamical studies of the planetary boundary layer (PBL). In such studies, the Reynolds number is understood as a turbulent Reynolds number. Values about 500 are considered typical of the PBL. Lilly established that the neutrally-stratified Ekman flow is subject to an inflexion point instability (Lilly, 1966). The rolls often observed in the neutral planetary boundary layer are usually interpreted as the outcome of this instability. However important characteristics of its non-linear development remain unclear and are investigated in this work :

(i) Does the instability saturate, and how ?

(ii) How stable or unstable are the equilibrated rolls with respect to three-dimensional perturbations ?

(iii) How sensitive is the secondary instability to latitude ?

2. Numerical methods

The non-divergent velocity field is completely described by the along-roll velocity and vorticity. These two scalar dynamical fields are decomposed on a set of basis functions that satisfy the appropriate boundary conditions (Spalart et al, 1989 ; Coleman and Spalart, 1990 ; Foster, 2005). This spatial discretization is spectrally accurate. For temporal integrations, we use a third-order Adams-Bashforth scheme with implicit treatment of viscosity. When searching for exactly equilibrated rolls (see below), each Newton iteration involves the resolution of a linear system with several thousands of unknowns. Addressing the linear stability of these saturated rolls is an eigenproblem of comparably large size. Direct methods are inappropriate for such large problems and we successfully use matrix-free, iterative methods to solve them.

3. Nonlinear saturation of the primary instability

Due to nonlinear interactions with itself and the basic flow after the initial stage where the linear approximation is valid, a linearly unstable perturbation may either saturate or directly produce chaos. The convective instability for instance is known to saturate into convective rolls but contradictory conclusions concerning the Ekman flow can be found in the literature. Coleman (1990) runs direct numerical simulations at Re=400 and finds no equilibrated rolls. Foster (2005) performs high-order amplitude expansions at Re=500 and finds that they admit a steady state.

We run a 2D DNS similar to Coleman's at Re=500. The instability grows with a time scale T of about 100 L/U where L is the boundary layer depth and U the geostrophic wind. After a few T, nonlinear interactions become dominant and drive the flow to a travelling quasi-equilibrium, presenting slow oscillations with a period of several tens of T. These oscillations decay but even more slowly. Using a Newton's method, we search for an exactly equilibrated flow close to this quasi-equilibrium. Very small corrections are enough for us to find an exactly equilibrated flow. Hence the equilibrated rolls exist, and are reached very closely, although not exactly, after a small random initial perturbation to the Ekman flow has evolved. Interestingly, while the observations mention contra-rotationg rolls, these equilibrated rolls are co-rotating, as one would expect from the saturation of the instability of a parallel shear flow.

4. Secondary instability of saturated rolls

The equilibrated rolls form a two-dimensional flow, independent on the along-roll coordinates. Such columnar vortices are subject to different families of secondary instability. The Kelvin-Helmholtz rolls that result from the saturation of a free-shear instability are known to suffer in neutral or stable stratification from elliptic and hyperbolic instabilities (Peltier 2003). Unlike Kelvin-Helmholtz rolls, saturated Ekman rolls have along-roll velocity, like swirling jets, which are subject to other types of instabilities. The presence of a rigid boundary could affect the stability properties as well. Using matrix-free Krylov methods, we search for three-dimensional modes of instability. We find that Ekman rolls suffer from a secondary instability of hyperbolic type.

5. Dependence on latitude

The horizontal component of the Coriolis vector and the direction of the geostrophic wind are known to influence the domain of primary instability. We investigate their influence on the nonlinear saturation and on the secondary instability as well.