Internal Boundary Layers: Flow over Changes in Surface Roughness and Temperature

Before the 1960s most atmospheric boundary-layer studies dealt with flow over flat homogeneous surfaces, land or water. Heterogeneity was acknowledged but the main concern was to have enough upwind fetch over a uniform surface so that it could be considered locally homogeneous. Monin-Obukhov Similarity Theory was beginning to be appreciated and logarithmic profiles were replacing power laws. Electro-mechanical calculators and slide rules were the main computing tools. Sutton (1953) – Micrometeorology, was a standard text and includes good sections on surface layer profiles following Deacon's (1949) analyses but Monin and Obukhov are not mentioned. Ted Munn's book, Descriptive Micrometeorology (1966) corrected that omission in terms of standard texts. He also talks briefly about Internal Boundary Layers, in the context of Stearn and Malkus's work on flow over heated islands.

In the mid to late 1950s and through 1960s William P. Elliott, Hans Panofsky, Alan Townsend, Frank Bradley, Heinz Lettau and others initiated work on Internal Boundary Layers (IBLs) developing as air flowed over a step change in surface roughness.

On the observational side the studies on the frozen surface of Lake Mendota (near Madison, Wisconsin) by Charles Stearns, John Kutzbach and Heinz Lettau (1961, 1964) are innovative. Their roughness changes were created with 500 bushel baskets in one case (1961) and with the same number of left-over Christmas trees in the other (1964). Frank Bradley's (1968) roughness changes were created with metal spikes laid on top of an airport runway and the natural change from grass to the tarmac runway at Jervis Bay, Australia. There were also wind tunnel studies.

Elliott (1958) provided a relatively simple empirical approach to modelling neutrally stratified IBLs downwind of a step change in roughness, from z₀₁ to z₀₂ (using our notation - different authors tend to use different notations!) and where the upstream surface friction velocity is u_*1 (constant) and downwind of the roughness change it is u_*2(x). Elliott assumes that above the IBL, for z > h(x), the undisturbed velocity profile remains logarithmic U = (u_*1/κ)ln(z/z₀₁), while within the IBL the velocity profile is also assumed logarithmic but with friction velocity u_*2(x) and roughness length z₀₂. Continuity of velocity at z = h(x) and momentum conservation lead to relationships for h(x) and u_*2(x).

Panofsky and Townsend (1964) relaxed the constraint of uniform shear stress (friction velocity) within the IBL, while Taylor (1969) and E.W. Peterson (1969) set up RANS models and solved the equations numerically. Their predicted profiles were smoothed at the top of the IBL but if plotted as U vs ln (z) all show an essentially logarithmic profile within most of the IBL and the same is true of the observations. These, and later, models make boundary layer approximations with regard to pressure gradients and use a variety of RANS closure hypotheses. Some were extended to the Planetary Boundary Layer and to multiple roughness changes (Weng et al, 2010). The flow situation is not ideally suitable for Large Eddy Simulations (LES) or wind tunnel study, although several papers report on wind tunnel work and LES work by Glendenning and Lin (2002), Bou-Zeid et al (2004) and a recent thesis by Sridhar (2019) provide interesting results. The issue as I see it is how to represent the features emerging from down amongst the sub grid roughness elements in a way that LES can add value relative to RANS.

Internal Boundary Layers also arise as a result of changes in thermal conditions at the surface, especially in the classic Thermal IBL situations at coastlines with cold stably stratified air advected over warm surfaces. Pollutant in the stably stratified air can be rapidly mixed down to the ground when the growing TIBL extends up to a polluted layer in a process known as fumigation, (Venkatram, 1976).

Taylor (1970, 1971) added thermal changes to his roughness change RANS calculations and others have done the same. Weng et al (2010) focused on deeper layers and longer fetches in order to compare with JYLEX observations of flow inland from the North Sea over Jutland (Sempreviva et al, 1988). Spatial variations in surface roughness can be combined (linearly) with perturbations caused by topography (Weng and Taylor, 2011) and used for practical applications in models such as MS-Micro or WASP (Dorval et al, 2017).

Progress has certainly been made over the past 100 years but there are still things to do! Modelling can be hard work but is relatively inexpensive while field projects can be costly. In his IBL review, John Garratt (1990) concludes "..... there would seem to be a requirement for an additional comprehensive observational set, perhaps covering a somewhat larger fetch range to encompass the transition between the small-scale problem and that at the mesoscale." There have been some field studies in the almost 30 years since then, including Jegede and Foken (1999) as part of the LINEX project in Germany, Pires et al (2015) at Alcantara, Brazil, and a small project at the Wind Energy Institute on Prince Edward Island, Canada - Miller and Taylor (2016). I would second John's opinion, and maybe look for more than one extra data set. Remote sensing with Doppler lidars can help but for high resolutions turbulence and temperature fields we need basic in situ measurements from multiple masts, UAVs or tethered balloons. A fundamental IBL study could be a fun, basic science project to provide the data needed to validate and compare models and ideas, lets find the support and do it!