Form Drag on Ocean Flows

Form drag is a central process linking rough topography to ocean mixing processes. The history of the measurement of form drag in the ocean and atmosphere is reviewed. The basic equations governing form drag and its role in energy conversion are derived for a Boussinesq fluid, focusing on the case of tidal flow. Observational and numerical results are shown for stratified, tidal flow near a headland in Puget Sound, Washington. There the net form drag far exceeds the frictional drag. Speculative comments are made concerning when and where form drag is likely to be important. Introduction and a Brief History We all understand form drag (sometimes called "pressure drag"). It is simply the spatial integral of bottom pressure times bottom slope (Baines, 1995, p.13; Kundu and Cohen, 2002, p. 338; Gill, 1982, p. 145). It is also often quantified higher in the water column as the vertical Reynolds stress due to waves, as in Gill. Similarly, the rate of doing work by the form drag is intuitively obvious. It is just the form drag times the speed of the interior flow. However, there are times when these simple definitions can be troublesome. For example, in a container of motionless fluid shaped like a pint of beer, the boundary pressure times the boundary slope (in the vertical direction) has a non-zero integral. Yet we intuitively know that there is no real form drag in this case, so is there a problem with our definition? Similarly, when the flow is tidal, or it is difficult to define an undisturbed interior flow, what is the rate of doing work by the form drag? When is it appropriate to calculate form drag on an isolated feature, when the feature is not an isolated Gaussian bump, but is instead part of a deeply-incised continental slope? Further, how is it that form drag can be part of an energy equation when the boundary is stationary, and hence can do no work? Questions such as these are motivation for including here a rigorous derivation of the form drag, and its role in energy conversion. First studied in aeronautics, form drag is ubiquitous in geophysical flows such as airflow over mountains. Smith (1978) measured the form drag on a small mountain using a number of microbarographs at ground stations. He discusses in detail the problems facing this seemingly direct approach. The main source of error is that one does not know the absolute height of the instrument to great enough accuracy (1 m in his case!). To get around this he reasoned that if you could just find a time with very little wind in the record, then you could use that to determine absolute pressures for each record. Atmospheric scientists have refined this approach, and have also used aircraft measurements to try to determine vertical momentum fluxes in sections above mountains. Many of these are reviewed in Davies and Phillips (1985). Hafner and Smith (1985) used a 2-D array of microbarographs in the Alps to get form drag vectors, instead of just looking at it as a process acting normal to a ridge crest. The physical processes which give rise to form drag can be either the generation of internal waves, or flow separation. The drag exerted can be substantial, ranging from about equal to the frictional drag up to 10 times that value for the Southern Alps of New Zealand. Since the flows which generate form drag are typically of much smaller scale than those resolved by atmospheric numerical models, there has been considerable work on the parameterizing of form drag, as exemplified by Lott and Miller (1997). They find significant improvements in predicted flow result for flow over mountains. A nice review of both the physics and the forecasting implications is given by Schär (2002). In contrast to atmospheric studies, estimates of form drag generated by oceanic obstacles are extremely rare, because it is difficult to make highly spatially resolved bottom pressure measurements