# Form drag and tidal flow over topography

For my PhD thesis, I worked with Parker MacCready at the University of Washington. We collaborated with Jim Moum and Jonathan Nash at Oregon State University on the observational portion of the project.

In estuaries and coastal seas, mixing is an essential process for a healthy ecosystem because it moves nutrients and oxygen vertically within the water column. My research looked at what happens when tidal currents flow back and forth over rough topography such as ridges and headlands creating features like internal waves and eddies and eventual mixing, turbulence and dissipation. Specifically, we measured the form drag in an effort to parameterize these processes.

Our study site was Three Tree Point (TTP) a headland in Puget Sound, WA located about halfway between Seattle and Tacoma. Previous work on the form drag at this site include Pawlak et al. (2003), Edwards et al. (2004), McCabe et al. (2006), and Canals et al. (2009). My thesis included three parts: numerical models and theory of form drag due to tidal flow around idealized headlands, an observational experiment at TTP where seafloor pressure sensors were used to measure the form drag at TTP, and a numerical model of TTP which allowed us to gain further insight into the dynamical processes that create form drag at TTP.

Below, find details about the following parts of my research:

Details about our cruise at Three Tree Point in October 2010 from the Ocean Mixing Group at Oregon State Universtiy.

## An introduction to form drag

Form drag is a force that results from pressure differences across an obstacle in a flow field. In the ocean, form drag occurs when currents flow over and around undersea topography.

Frictional drag and form drag are different. Frictional drag is due to the normal forces that result when a fluid flows over a surface. When the surface is rough there tends to be more frictional drag than when the surface is smooth. Form drag, on the other hand, is due to the normal forces that result when a fluid flows over an object creating areas of high and low pressure. Form drag is calculated by taking the integral of the pressure on the surface times the slope of the object over the entire surface area of the object. In general, streamlined bodies have smaller form drag than bluff bodies.

In the ocean, form drag is associated with eddy generation, internal wave generation and localized mixing.

## Tidal flow around idealized headlands

A full description of this part of the study can be found in a paper published in the Journal of Physical Oceanography (pdf).

This study was motivated by work done at Three Tree Point (TTP), a headland in Puget Sound, WA. TTP is shaped like a ridge on a sloping side wall so some of the water goes around it creating eddies and some of the water goes over it creating internal waves. To isolate the dynamics, we modeled idealized headlands with vertical side walls as seen above.

We found that there are two mechanisms that created form drag. The first part is due simply to the fact that the flow is oscillatory and hence there is a sea surface height tilt that is maximum at slack tide. We refer to this part of the drag as the potential flow drag because it is found by calculating the pressure field due to potential flow around the headlands. Although the potential flow drag can have a large magnitude, it cannot not do tidally averaged work on the flow because its phase is in quadrature with the velocity. The second part of the drag is the separation drag which is due to flow separation on the lee side of the headland. In many cases, its magnitude was much smaller than the potential flow drag, but it accounted for all of the tidally averaged work done on the tidal currents.

Once the potential flow pressure field is removed from the total pressure field, the residual pressure field shows the dynamics that create form drag. There are regions of low pressure associated with local eddies and regions of high pressure associated with flow that is blocked by the eddies and "piles up" on the upstream side of the topography.

## Pressure measurements at TTP

Using an array of seafloor pressure sensors (PPODs), we measured the bottom pressure across Three Tree Point (TTP), a headland with sloping side-walls in Puget Sound, WA. Prior to this study, form drag had not been measured with seafloor pressure sensors.

Form drag was then calculated from a spatial integral of the bottom pressure anomalies.

The form drag varied with the tidal currents. The form drag and power (energy conversion rate from the barotropic tides) were largest during strong flood tides.

Before this study, it was assumed that a bluff body drag law would be the best way to parameterize the form drag at TTP because that is the most common parameterization of form drag. However, we found that a linear wave drag law actually did a much better job. The bluff body drag law drastically underprediced the form drag observed at TTP. A wave drag law takes into consideration stratification and therefore is more accurate.

## Numerical modeling of TTP

To gain further insight into the dynamics that create form drag at TTP, I built a numerical model of the region using output from the MoSSea model to force the currents at the boundaries of the model. The model was made using ROMS. It had 100 m horizontal resolution. Tides were forced at the boundaries.

During maximum flood tide, an eddy is visible just to the south of the tip of TTP as a region of elevated relative vorticity. The pressure is divided into the internal pressure (anomalies due to changes in isopycnal height), external pressure (anomalies due to changes in sea surface height), and dynamic pressure (sum of the internal and external pressures) because these are the parts of the pressure signal that directly contribute to form drag. In the eddy, the sea surface is depressed due to a local balance between the sea surface pressure gradient and the centripetal force of the currents. At the same time, the isopycnals are raised below this sea surface depression. The internal and external pressures tend to counteract each other within the eddy except in regions close to shore where isopycnals cannot move vertically.

This means that even though the sea surface depression within the eddy is quite large, it is counteracted internally leading to a smaller form drag than would be present with the sea surface depression alone.

Internal waves were also created within the model as currents flowed over the ridge-like part of TTP. Internal waves draw down isopycnals on the lee-ward side of the topography leading to a pressure difference that creates form drag.

Overall, we see that at TTP, the internal waves and the eddies do equal amounts of work on the flow. The pressure anomalies from the eddy are larger than those in the internal waves, however, since the external and internal signals work to counteract each other in the eddy, the dynamic pressure anomaly has nearly the same magnitude as the dynamic pressure anomalies from the internal lee waves.

This material is based upon work supported by the National Science Foundation under Grant Nos. OCE-0099058, OCE-0425059 and OCE-0751583. "Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation."