Homework 4

Homework 4

Solutions

  1. In class, we talked about “planetary” vorticity and “relative” vorticity. Explain what each term means physically, and specify how each is computed.

  2. Potential vorticity is \((\zeta + f)/H\), where \(\zeta\) is the relative vorticity, \(f\) is the Coriolis parameter (or planetary vorticity), and \(H\) is the thickness of a water column. In class, we discussed how potential vorticity is conserved in the absence of torques (i.e., away from the boundaries in the geostrophic interior). Using conservation of potential vorticity and assuming we are in the northern hemisphere, and noting that there will be more than one possibility in each case, state what could happen if we do each of the following to a water column:

    1. Increase the relative vorticity

    2. Move the column to a higher latitude

    3. Shrink the water column

  3. Explain (briefly!) to one of your non-PO oceanography colleagues how a simple consideration of angular momentum and mass conservation for the ocean leads to the robust conclusion that western boundary currents should be strong and directed poleward in the subtropical gyres.

  4. Imagine that the pattern of wind stress over the North Atlantic stays the same, but the magnitude of wind stress doubles. What changes would you expect to see in the large-scale ocean circulation?

  5. For the gyre-in-a-tank demonstration from class, explain how the sloping bottom of the tank is an analogue for background vorticity field in the real ocean related to the change in \(f\) with latitude.

  6. Consider an idealized ocean basin that is \(1\times 10^4~\mathrm{km}\) wide. Over this basin, the wind blows in a simple wind pattern. The wind is purely zonal, and the zonal component of the wind speed, \(U_w^x\), increases linearly with latitude from a speed of \(−7\,\mathrm{m\,s}^{−1}\) at \(20^\circ\mathrm{N}\) to \(+7\,\mathrm{m\,s}^{−1}\) at \(30^\circ\mathrm{N}\). Note that \(f\) and \(\beta\) are both functions of latitude, although it is fine to approximate \(\beta\) as a constant over this latitude range.

    1. What is the total Ekman volume transport in Sverdrups (not transport per unit width) across the entire ocean basin at \(20^\circ\mathrm{N}\), \(25^\circ\mathrm{N}\), and \(30^\circ\mathrm{N}\)?

    2. Is this transport convergent or divergent? Does this imply downward or upward Ekman pumping? Is your answer consistent with what you expect from the wind-stress curl? Why?

    3. Based on (b) only, which direction do you expect the Sverdrup transport to be? Why?

    4. Using Sverdrup balance, estimate the total Sverdrup volume transport across the entire ocean basin at \(20^\circ\mathrm{N}\), \(25^\circ\mathrm{N}\), and \(30^\circ\mathrm{N}\)?

    5. If the depth of the ocean is \(1000\,\mathrm{m}\), what is the average velocity associated with the Sverdrup transport over the basin interior at \(20^\circ\mathrm{N}\), \(25^\circ\mathrm{N}\), and \(30^\circ\mathrm{N}\)?

    6. Using answers from (a) and (d) what is the geostrophic volume transport in the basin interior at \(20^\circ\mathrm{N}\), \(25^\circ\mathrm{N}\), and \(30^\circ\mathrm{N}\)?

    7. What is the direction and magnitude of volume transport in the western boundary current at \(20^\circ\mathrm{N}\), \(25^\circ\mathrm{N}\), and \(30^\circ\mathrm{N}\)?

    8. If the depth of the ocean is \(1000\,\mathrm{m}\) and the western boundary current is \(100\,\mathrm{km}\) wide, what is the average velocity in the western boundary current at \(20^\circ\mathrm{N}\), \(25^\circ\mathrm{N}\), and \(30^\circ\mathrm{N}\)?

  7. Compare/contrast North Atlantic Deep Water and Antarctic Bottom Water. Explain how the formation process accounts for the water property differences in these two water masses.

  8. Ten statements are given below that are appropriate for North Atlantic Deep Water (NADW), Antarctic Bottom Water (AABW), or Common Water (CW). For each one, pick the most appropriate water mass for that statement. If more than one seems appropriate, pick the one that you think is most appropriate.

    1. Relatively warm and saline

    2. Very “old” water

    3. High in oxygen and low in nutrients

    4. Largest volume in Worthington’s seawater census

    5. Sea ice production is very important in generation

    6. Formed in the Weddell Sea

    7. Relatively homogeneous

    8. Not formed at the sea surface

    9. Cooling by winds very important in generation

    10. Formed near Greenland

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