Homework 2
Homework 2¶
Local atmospheric pressure changes are balanced by changes in the local ocean surface height through what is called the inverted barometer effect, whereby increased local atmospheric pressure is balanced by a local depression in the ocean surface, while reduced atmospheric pressure is balanced by a rise in the ocean surface.
Assuming constant hydrostatic pressure at \(100~\mathrm{m}\) depth in the ocean, how much do you expect the ocean surface to rise for a \(1~\mathrm{mb}\) (i.e., \(100~\mathrm{Pa}\) or \(100~\mathrm{N\,m}^{-2}\)) decrease in atmospheric pressure?
The strongest hurricanes can have atmospheric pressures at their cores as low as \(920~\mathrm{mb}\). Assuming that atmospheric pressure is about \(1020~\mathrm{mb}\) on a typical day, about how much would the inverted barometer effect contribute to storm surge where the eye of such a hurricane makes landfall?
Use your knowledge of hydrostatic balance to estimate the percentage of an iceberg’s volume that protrudes above the surface. The density of ice is about \(900~\mathrm{kg\,m}^{-3}\). Assume that hydrostatic pressure does not change horizontally from beneath the iceberg to ice-free areas surrounding the iceberg.
Hint: The shape of the iceberg does not matter, but you can assume a nice cube shape if it helps set up your calculation.
In the previous exercise, the ice floats with its top above the surrounding sea level due to its lower density compared to the surrounding water. Similarly, a warm-core, anticyclonic eddy is a blob of low density water that floats above higher density water with its top above the surrounding sea level. Later in the course we will learn about the horizontal force balance that maintains the shape of the eddy and the high surface height at its center. For now, consider a simple model of an anticyclonic eddy near Hawaii composed of two layers. The upper layer is warm with a uniform density of \(1023~\mathrm{kg\,m}^{-3}\), the lower layer is cold with a uniform density of \(1027~\mathrm{kg\,m}^{-3}\), and the depth of the interface between the layers changes across the eddy. Suppose the surface height increases from the periphery of the eddy to a height of \(0.2~\mathrm{m}\) at the center of the eddy. Also suppose that in the lower layer, the pressure at the center of the eddy is identical to the pressure at the periphery.
Make a sketch showing the sea surface height and the interface between the layers changing across the eddy. You may need to adjust your sketch of the layer interface after answering part (b), but it is always good to start with a sketch of what you know.
Remember that we defined our model of the eddy such that the pressure in the lower layer does not change from the center of the eddy to the periphery. Using hydrostatic balance, pick a single depth in the lower layer, and equate the pressures at that depth between the center and periphery, taking care to account for the elevated surface height at the center. Calculate how much lower or higher the layer interface is in the center compared to the periphery. (You can approximate it; no need for a calculator.)
Illustrate and explain why the answer does not depend on the average thickness of the upper layer.
Answer the following questions regarding centrifugal force. Note that the magnitude of Earth’s angular velocity, \(\Omega = |\vec{\Omega}|\approx 7.3\times10^{-5}~\mathrm{rad\,s}^{-1}\).
What is the centrifugal force per unit mass on a particle fixed to the Earth at the equator? Which direction is the force?
How does the magnitude of the centrifugal force per unit mass at the equator compare to gravitational acceleration?
What is the value of the centrifugal force per unit mass on a particle fixed to the Earth at \(45^\circ\mathrm{N}\)?
What is the deviation angle of a plumb line at \(45^\circ\mathrm{N}\) from the direction of the gravitational force toward the Earth’s center of mass?
For the following cases, draw a vector diagram showing the relationship between two times Earth’s rotation, \(2\vec{\Omega}\), the velocity of the fluid particle, \(\vec{u}\), and the Coriolis force per unit mass on the particle, \(-2\vec{\Omega}\times\vec{u}\). Then make the approximation of a local Cartesian plane such that \(2\vec{\Omega}=f\hat{k}\) and make a new vector diagram in each case.
A fluid particle traveling westward at \(45^\circ\mathrm{S}\).
A fluid particle traveling southward at \(45^\circ\mathrm{N}\).
Suppose you kick a soccer ball in the air directly to the east at a latitude of \(45^\circ\mathrm{N}\). If the ball travels \(60~\mathrm{m}\) and you assume it is has a constant horizontal speed of \(20~\mathrm{m\,s}^{-1}\), then …
Calculate the Coriolis force per unit mass on the ball. Which direction is the Coriolis force on the ball?
Calculate the displacement of the ball in the direction of the Coriolis force after travelling \(60~\mathrm{m}\). Remember that distance, \(d\), is related to acceleration (or force per unit mass), \(a\), and time, \(t\), by \(d=\frac{1}{2}at^2\).
Answer the following:
Which two terms in the horizontal momentum equations are involved in Ekman dynamics?
Is wind stress applied to the entire Ekman layer?
Why does the depth-integrated Ekman transport per unit width only depend on wind stress and the Coriolis parameter?
Answer the following:
A typical tradewind speed is about \(7~\mathrm{m\,s}^{-1}\). Estimate the magnitude of the wind stress.
If the wind is coming from \(30^\circ\) north of east, what is the zonal component of this stress?
What is the corresponding meridional Ekman volume transport per unit width around the latitude of Hawaiʻi?
If this transport per unit width were uniformly distributed within a mixed layer \(40~\mathrm{m}\) thick, what meridional velocity component would that imply?
If the wind were uniform over a zonal span of about \(100^\circ\), what meridional volume transport, in Sverdrups, would these tradewinds yield? Note: \(1~\mathrm{Sverdrup~(Sv)}\,=\,10^{6}~\mathrm{m}^3\,\mathrm{s}^{-1}\).
If the wind speed over that zonal span is reduced to \(3.5~\mathrm{m\,s}^{-1}\), by what factor would the volume transport differ from what you just calculated? Note: You should not have to redo the entire set of calculations.
You have two meteorological moorings on the International Dateline, one at \(20^\circ\mathrm{N}\) and one at \(30^\circ\mathrm{N}\). You observe that the wind vector is directed to the southwest at \(20^\circ\mathrm{N}\) with a wind speed of \(7~\mathrm{m\,s}^{-1}\). At \(30^\circ\mathrm{N}\) there is no wind at all.
What is the magnitude and direction of the Ekman transport at each mooring?
What are the meridional and zonal components of the Ekman transports at each mooring?
What can you say about the horizontal divergence of the Ekman transport?