Stability of Horizontal Boundary Layers

Let us now detail the stability properties of an Ekman layer introduced in Part I, page 11. First we will recall how to compute the critical Reynolds number. Then we will describe briefly what happens at larger Reynolds numbers. The first step in the study of the stability of the Ekman layer is to consider the linear stability of a pure Ekman spiral of the form where U∞ is the velocity away from the layer and ζ is the rescaled vertical component ζ = x3/√εν. The corresponding Reynolds number is Let us consider the Navier–Stokes–Coriolis equations, linearized around uE The problem is now to study the (linear) stability of the 0 solution of the system (LNSCε). If u=0 is stable we say that uE is linearly stable, if not we say that it is linearly unstable. Numerical results show that u=0 is stable if and only if Re<Rec where Rec can be evaluated numerically. Up to now there is no mathematical proof of this fact, and it is only possible to prove that 0 is linearly stable for Re<Re1 and unstable for Re>Re2 with Re1<Rec<Re2, Re1 being obtained by energy estimates and Re2 by a perturbative analysis of the case Re=∞. We would like to emphasize that the numerical results are very reliable and can be considered as definitive results, since as we will see below, the stability analysis can be reduced to the study of a system of ordinary differential equations posed on the half-space, with boundary conditions on both ends, a system which can be studied arbitrarily precisely, even on desktop computers (first computations were done in the 1960s by Lilly).

References and Remarks on Rotating Fluids

The problem investigated in this part can be seen as a particular case of the study of the asymptotic behavior (when ε tends to 0) of solutions of systems of the type where Δε is a non-negative operator of order 2 possibly depending on ε, and A is a skew-symmetric operator. This framework contains of course a lot of problems including hyperbolic cases when Δε = 0. Let us notice that, formally, any element of the weak closure of the family (uε)ε>0 belongs to the kernel of A. We can distinguish from the beginning two types of problems depending on the nature of the initial data. The first case, known as the well-prepared case, is the case when the initial data belong to the kernel of A. The second case, known as the ill-prepared case, is the general case. In the well-prepared case, let us mention the pioneer paper by S. Klainerman and A. Majda about the incompressible limit for inviscid fluids. A lot of work has been done in this case. In the more specific case of rotating fluids, let us mention the work by T. Beale and A. Bourgeois and T. Colin and P. Fabrie. In the case of ill-prepared data, the nature of the domain plays a crucial role. The first result in this case was established in 1994 in the pioneering work by S. Schochet for periodic boundary conditions. In the context of general hyperbolic problems, he introduced the key concept of limiting system (see the definition on page 125). In the more specific case of viscous rotating fluids, E. Grenier proved in 1997 in Theorem 6.3, page 125, of this book. At this point, it is impossible not to mention the role of the inspiration played by the papers by J.-L. Joly, G. Métivier and J. Rauch (see for instance and). In spite of the fact that the corresponding theorems have been proved afterwards, the case of the whole space, the purpose of Chapter 5 of this book, appears to be simpler because of the dispersion phenomena.

References and Remarks on the Navier–Stokes Equations

The purpose of this chapter is to give some historical landmarks to the reader. The concept of weak solutions certainly has its origin in mechanics; the article by C. Oseen [100] is referred to in the seminal paper by J. Leray. In that famous article, J. Leray proved the global existence of solutions of (NSν) in the sense of Definition 2.5, page 42, in the case when Ω = R3. The case when Ω is a bounded domain was studied by E. Hopf in. The study of the regularity properties of those weak solutions has been the purpose of a number of works. Among them, we recommend to the reader the fundamental paper of L. Caffarelli, R. Kohn and L. Nirenberg. In two space dimensions, J.-L. Lions and G. Prodi proved in [91] the uniqueness of weak solutions (this corresponds to Theorem 3.2, page 56, of this book). Theorem 3.3, page 58, of this book shows that regularity and uniqueness are two closely related issues. In the case of the whole space R3, theorems of that type have been proved by J. Leray in.

Stability of Navier–Stokes Equations

In this chapter we intend to investigate the stability of the Leray solutions constructed in the previous chapter. It is useful to start by analyzing the linearized version of the Navier–Stokes equations, so the first section of the chapter is devoted to the proof of the well-posedness of the time-dependent Stokes system. The study will be applied in Section 3.2 to the two-dimensional Navier–Stokes equations, and the more delicate case of three space dimensions will be dealt with in Sections 3.3–3.5.

Some Elements of Functional Analysis

Before introducing the concept of Leray’s weak solutions to the incompressible Navier–Stokes equations, classical definitions of Sobolev spaces are required. In particular, when it comes to the analysis of the Stokes operator, suitable functional spaces of incompressible vector fields have to be defined. Several issues regarding the associated dual spaces, embedding properties, and the mathematical way of considering the pressure field are also discussed. Let us first recall the definition of some functional spaces that we shall use throughout this book. In the framework of weak solutions of the Navier– Stokes equations, incompressible vector fields with finite viscous dissipation and the no-slip property on the boundary are considered. Such H1-type spaces of incompressible vector fields, and the corresponding dual spaces, are important ingredients in the analysis of the Stokes operator.

Other Layers

Note that Ω=Ω×[0, 1] is a particular case where the boundary layers are purely horizontal or purely vertical. In the general case of an open domain Ω, the boundaries have various orientations. As long as the tangential plane ∂Ω is not vertical, the boundary layers are of Ekman type, with a size of order where ν is the normal of the tangential plane. When ν.r→0, namely when the tangential plane becomes vertical, Ekman layers become larger and larger, and degenerate for ν.r=0 in another type of boundary layer, called equatorial degeneracy of the Ekman layer. We will now detail this phenomenon in the particular case of a rotating sphere. Mathematically, almost everything is widely open! Let Ω=B(0,R) be a ball. Let θ be the latitude in spherical coordinates. The equatorial degeneracy of the Ekman layer is difficult to study. We will just give the conclusions of the analytical studies of. The Ekman layer is a good approximation of the boundary layer as long as |θ|≫E1/5. For |θ|≪E1/5 the Ekman layer degenerates into a layer of size E2/5. • for |θ|≫(εν)1/5, Ekman layer of size • for |θ| of order (εν)1/5, degeneracy of the Ekman layer into a layer of size (εν)1/5 in depth and (εν)2/5 in latitude. Let us now concentrate on the motion between two concentric rotating spheres, the speed of rotation of the spheres being the same. In this case, Ω=B(0,R) − B(0, r) where 0<r<R. Keeping in mind meteorology, the interesting case arises when R − r ≪ R: the two spheres have almost equal radius. Let us study the fluid at some latitude θ. If θ ≠0, locally, the space between the two spheres can be considered as flat and treated as a domain between two nearby plates. The conclusions of the previous paragraphs can be applied. Two Ekman layers are created, one near the inner sphere and the other one near the outer sphere.

Vertical Layers

From a physical point of view, as well as from a mathematical point of view, horizontal layers (Ekman layers) are now well understood. This is not the casefor vertical layers which are much more complicated, from a physical, analytical and mathematical point of view, and many open questions in all these directions remain open. Let us, in this section, consider a domain Ω with vertical boundaries. Namely, let Ωh be a domain of R2 and let Ω=Ωh × [0, 1]. This domain has two types of boundaries: • horizontal boundaries Ωh × {0} (bottom) and Ωh ×{1} (top) where Ekman layers are designed to enforce Dirichlet boundary conditions; • vertical boundaries ∂Ωh × [0, 1] where again a boundary layer is needed to ensure Dirichlet boundary conditions. These layers, however, are not of Ekman type, since r is now parallel to the boundary. Vertical layers are quite complicated. They in fact split into two sublayers: one of size E1/3 and another of size E1/4 where E =νε denotes the Ekman number. This was discovered and studied analytically by Stewartson and Proudman. Vertical layers can be easily observed in experiments (at least the E1/4 layer, the second one being too thin) but do not seem to be relevant in meteorology or oceanography, where near continents, effects of shores, density stratification, temperature, salinity, or simply topography are overwhelming and completely mistreated by rotating Navier–Stokes equations. In MHD, however, and in particular in the case of rotating concentric spheres, they are much more important. Numerically, they are easily observed, at large Ekman numbers E (small Ekman numbers being much more difficult to obtain). The aim of this section is to provide an introduction to the study of these layers, a study mainly open from a mathematical point of view. First we will derive the equation of the E1/3 layer. Second we will investigate the E1/4 layer and underline its similarity with Prandtl’s equations. In particular, we conjecture that E1/4 is always linearly and nonlinearly unstable. We will not prove this latter fact, which would require careful study of what happens at the corners of the domain, a widely open problem.

Other Systems

The methods developed in this book can be applied to various physical systems. We will not detail all the possible applications and will only quote three systems arising in magnetohydrodynamics (MHD) and meteorology, namely conducting fluids in a strong external “large scale” magnetic field, a classical MHD system with high rotation, and the quasigeostrophic limit. The main theorems of this book can be extended to these situations. The theory of rotating fluids is very close to the theory of conducting fluids in a strong magnetic field. Namely the Lorenz force and the Coriolis force have almost the same form, up to Ohm’s law. The common feature is that these phenomena appear as singular perturbation skew-symmetric operators. The simplest equations in MHD are Navier–Stokes equations coupled with Ohm’s law and the Lorenz force where ∇φ is the electric field, j the current, and e the direction of the imposed magnetic field. In this case ε is called the Hartmann number. In physical situations, like the geodynamo (study of the magnetic field of the Earth), it is really small, of order 10−5–10−10, much smaller than the Rossby number. These equations are the simplest model in geomagnetism and in particular in the geodynamo. As ε→0 the flow tends to become independent of x3. This is not valid near boundaries. For horizontal boundaries, Hartmann layers play the role of Ekman layers and in the layer the velocity is given by The critical Reynolds number for linear instability is very high, of order Rec ∼ 104. The main reason is that there is no inflexion point in the boundary layer profile (10.1.2), therefore it is harder to destabilize than the Ekman layer since the Hartmann profile is linearly stable for the inviscid model associated with (10.1.1). As for Ekman layers, Hartmann layers are stable for Re<Rec and unstable for Re>Rec. There is also something similar to Ekman pumping, which is responsible for friction and energy dissipation. Vertical layers are simpler than for rotating fluids since there is only one layer, of size (εν)1/4. We refer to for physical studies.

Weak Solutions of the Navier–Stokes Equations

The mathematical analysis of the incompressible Stokes and Navier–Stokes equations in a possibly unbounded domain Ω of Rd (d = 2 or 3) is the purpose of this chapter. Notice that no regularity assumptions will be required on the domain Ω. Because of the compactness result stated in Theorem 1.3, page 27, the case of bounded domains will be different (in fact slightly simpler) than the case of general domains. The study of the spectral properties of the Stokes operator previously defined relies on the study of its inverse, which is in fact much easier. We shall restrict ourselves here to the case of the homogeneous Stokes operator which is adapted to the case of a bounded domain.