We now consider wave refraction due to velocity strain and shear associated with vortical mean flows. Such refraction changes the waves' pseudomomentum field and, arguably, the central topic of wave-mean interactions outside simple geometry is how such pseudomomentum changes are related to the leading-order mean-flow response. The same question was satisfactorily answered in simple geometry by the pseudomomentum rule. However, refractive changes in the pseudomomentum do not rely in any essential way on wave dissipation or external forces, and yet they can irreversibly change the total amount of pseudomomentum in the wave field. This makes clear that the usual pseudomomentum rule of simple geometry, which equates such changes to an effective force exerted on the mean flow, must be modified.

As we shall see, the conservation law for the sum of pseudomomentum and GLM impulse is the key for understanding the wave-mean interactions in the presence of refraction. We will illustrate this by a number of examples consisting of wavepackets and confined wavetrains. The most important result is the following: if the concept of an effective mean force makes sense at all, then this force is not exerted at the location of the wavepacket, but at the location of the vortices that induce the straining field. This gives the wave—mean interactions a non-local character that was clearly absent in simple geometry, where the effective mean force was always exerted at the location of the wavepacket.

We study the Boussinesq system, which is the simplest fluid model that admits internal gravity waves. These dispersive waves are ubiquitous in the atmosphere and ocean and they owe their restoring mechanism to the stable stratification in these environments, i.e., to the fact that density decreases with altitude. Internal gravity waves are typically far too small in scale (especially vertical scale) to be resolvable within global numerical models and therefore their dynamics and their interactions with the mean flow must be parametrized (i.e., put in by hand) based on a combination of observations and theory. Consequently, the classic wave-mean interaction theory for internal gravity waves has been extensively developed and this provides a convenient starting point for us.

Boussinesq system and stable stratification

The simplest fluid model that captures the effect of stable internal stratification is the Boussinesq model, which can be derived from the Euler equations and its dissipative counterparts under the assumption of a small density contrast across the fluid, together with ∇ . u = 0. Importantly, the latter constraint filters sound waves and thereby reduces the number of degrees of freedom compared to the full Euler system.

Before writing down the governing equations we note that as a realistic model for atmospheric and oceanic flows the Boussinesq system is mostly limited by its global restriction to small density contrasts across the fluid, and that these limitations are much more severe in the atmosphere than in the ocean.

We now turn to wave—mean interactions involving Rossby waves, the peculiar vorticity waves whose linear dynamics was described briefly in §4.2.2. Unlike acoustic waves or gravity waves, the dynamics of Rossby waves is essentially linked to the layerwise advection of PV, and this gives the mathematical description of Rossby waves and of their interactions with a mean flow a very special character, including the one-way phase propagation of Rossby waves.

The easiest model in which to study this topic is the quasi-geostrophic approximation to the shallow-water equations on a β-plane. However, the results easily generalize to three-dimensional stratified flow.

Quasi-geostrophic dynamics

We have no interest in gravity waves in this chapter and therefore we use the simplest theoretical approximation that filters these waves whilst retaining the balanced flow structure of Rossby waves and shallow-water vortices. This is accomplished by the quasi-geostrophic approximation to the equations, which is essentially a nonlinear extension of the linear balanced mode. These equations use a single dynamical variable, namely the PV.

Overall, the use of PV and of balanced flow systems based on PV advection and PV inversion (such as the quasi-geostrophic system or its many variants) are key concepts in atmosphere ocean fluid dynamics. For instance, balanced models were an essential component of the first successful numerical weather forecasts. Such a direct quantitative use of balanced models is less important today, but the insights that can be gained from studying such reduced dynamical systems remain as valuable as ever.

We now embark on a journey into new theoretical territory: Lagrangian-mean theory based on particle-following averaging. This theory allows a sharper and more succinct description of material advection in fluid dynamics, which greatly simplifies the mean description of material invariants such as scalar tracers or, crucially, of vorticity and potential vorticity. Based on this, Lagrangian-mean theory is superior to Eulerian-mean theory in the description of flow dynamics to do with vorticity and circulation. This will be particularly important for wave–mean interactions outside simple geometry, where long-range mean pressure fields greatly complicate the description of the mean flow based on momentum budgets.

On the downside of Lagrangian-mean theory we need to count the increased structural complexity of the theory, the requirement to evolve particle displacements alongside with the usual flow variables, and the potential breakdown of the particle-following flow map under averaging in the case of large-amplitude waves. Thus whether Eulerian or Lagrangian averaging is more efficient can depend on the problem at hand. As before, we take the view that Eulerian and Lagrangian concepts are complementary to each other in the optimal description of all aspects of fluid motion, and this is certainly true in wave–mean interaction theory as well.

We begin with a general discussion of Lagrangian averaging and of small-amplitude Stokes corrections and then give a comprehensive introduction to the so-called generalized Lagrangian-mean theory (GLM), which formally is not restricted to small-amplitude waves.

Geometric wave theory is the natural extension of WKB theory to situations in which the still layer depth H (and therefore the wave speed) is a slowly varying function of both x and y, and possibly even of t, although we will not consider that case here. In fact, even for constant H geometric wave theory is useful because it allows the computation of the structure of normal modes in bounded domains with irregular shapes, i.e., shapes for which there is no simple explicit expression for the normal modes.

The basic assumption of geometric wave theory is that there is a scale separation between the rapidly varying phase of the wavetrain on the one hand, and the slowly varying layer depth and wavetrain parameters such as amplitude and wavenumber on the other. Of course, in bounded domains the domain size must also be large compared to the wavelength. This basic assumption leads to a flexible and generic asymptotic procedure for solving for the wave field. Eventually, with the inclusion of dispersive effects, geometric wave theory becomes the ray-tracing method, which is the Swiss army knife for computing the asymptotic behaviour of small-scale waves in many fields of physics, including GFD.

A peculiarity of the progression from one-dimensional WKB theory to twodimensional geometric wave theory and finally to dispersive ray tracing is that the structure of the theory becomes easier, not harder,asits generality increases.

Vascular endothelial cells are cultured on the inside of a 10-cm long hollow tube that has aninternal diameter of 3 mm. Culture medium flows through the tube at Q= 1 ml/s. The cells produce a cytokine, EDGF, at a rate nEDGF (production rate per cell area) that depends on the local wall shear stressaccording to nEDGF = kτwall, wherek is an unknown constant with units of ng/dyne per s. The flow in the tubeis not fully developed, such that the shear stress is known to vary with axial position according toτwall = τ0(1 –βx), where β= 0.02 cm−1, τ0= 19 dyne/cm2, and x is the distance from the tubeentrance. Under steady conditions a sample of medium is taken from the outlet of the tube, and theconcentration of EDGF is measured to be 35 ng/ml in this sample. What isk?

Flow occurs through a layer of epithelial cells that line the airways of the lung due to avariety of factors, including a pressure difference across the epithelial layer(ΔP = P0) and, in the case of transientcompression, to a change in the separation between the two cell membranes, w2, as a function of time. We consider these cases sequentially below. Note that the depthof the intercellular space into the paper is L, and the transition in cellseparation from w1 to w2 occurs over alength δ much smaller than H1 andH2. (See the figure overleaf.)