We consider a compressible laminar boundary layer with uniform pressure when x < x0 and a prescribed large adverse pressure gradient when x > x0. The Illingworth-Stewartson transformation is applied, and the transformed external velocity u1(x) then chosen such that \[ \lambda = -\frac{x_0}{u^2_0}u_1\frac{du_1}{dx}\frac{T_w}{T_s} \] is constant, where Ts is the stagnation temperature.

For large λ, when a thin sublayer exists as the layer reacts to the sharp pressure gradient, inner and outer asymptotic expansions are derived and matched for functions F and S which determine the stream function and the temperature. The equations for F and S are largely uncoupled, in that the first approximation to F is independent of S, the first approximation to S depends only on the first approximation to F, and so on.

The closed equations for the velocity correlation tensor and for the mean-squared displacement of a particle suspended in a stationary homogeneous turbulent flow, with an arbitrary linear law of fluid-particle interaction, are obtained using two assumptions suggested previously for the problem of turbulent self-diffusion: the ‘independence approximation’ and the Gaussian property of the functional distribution of particle velocities. The numerical solution of the derived equations is given for an isotropic system with a model turbulence spectrum. The following characteristics of the particle motion are obtained: (a) the mean kinetic energy, (b) diffusivity, (c) rate of energy dissipation, (d) velocity correlation function, and (e) the correlation function of the relative fluid-particle velocity. The impact of various spectral modes on the characteristics of the particle motion is discussed.

Turbulent transport of fluctuating turbulent energy, turbulent momentum flux, temperature variance, turbulent heat flux, etc. in the upper part of the atmospheric boundary layer is usually dominated by buoyant transport. This transport is responsible for the erosion of the overlying stably stratified region, resulting in progressive thickening of the mixed layer. It is easy to show that a classical gradient transport model for the transport will not work, because it transports energy in the wrong direction. On the other hand, application of the eddy-damped quasi-Gaussian approximation to the equations for the third moments results in a transport model which predicts realistic inversion rise rates and heat-flux profiles. This is also a gradient transport model, but like molecular transport in solutions, a flux of one quantity depends on gradients of all relevant quantities. Transport coefficients are modified by the heat flux, so that the vertical transport is severely reduced near the inversion base. A simple Lagrangian model of transport of an indelible scalar in a stratified flow indicates that the form of the modified transport coefficients results from a marked anisotropic change in the Lagrangian time scale in stratification.