SIO 210 Problem Set 3 ANSWER KEY November 15, 2010 Due Nov. 22, 2010 (Monday) 1. Waves. Very long swells generated by a distant storm are observed in the open ocean. Their period is T seconds. (a) What is their wavelength L in terms of their period? These are deep water waves, so the relation between wavelength and period is derived from relation between phase speed and wavelength: c = L/T in general for all waves. For deep water waves,

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c = gL /2π So gL/2π = c2 = L2/T2 and therefore L = g T2/2π (b) Evaluate the result numerically, assuming T=20 s and g = 9.8 m/s2. Using a calculator, L = (9.8 m/s2)(20s)(20s)/ 2π = 623.89 m (c) What is the phase speed c of these waves (evaluate numerically)? c = L/T = 623.89m/20s = 31.19 m/s (d) As these waves pass by, water parcels at the surface carry out circular orbits (forward at the crest, then downward, backward at the trough, then upward). If the waveheight (from crest to trough) of these waves is H, what is the diameter D of the orbit? D = H (e) What is the average speed s with which water particles traverse the orbit? The orbit length is 2πR = πD = πH. The speed is therefore πH/T. (f) Numerically evaluate the result for s for waves of period T=20 s and height H=10 m. s = π (10 m)/(20 sec) = 1.57 m/sec. 2. Tides. (a) Suppose the earth rotates about its axis once every 36 h rather than once every 24 h, but otherwise no detail of the astronomy (distance to the moon, radius of Earth) or of the shape of the ocean floor or of the coasts is changed. What would the dominant period of the tide be? Explain. It would be ½ the rotation rate, therefore 18 hours. (b) Would the spatial distribution and/or amplitude of the tide generating force change? Explain. No, there would be no change since the tide generating force, which is F = gmM/r2 would not change (no change in moon or Earth mass or separation distance). (c) Would the spatial distribution and/or amplitude of the actual ocean tide and/or of ocean tidal currents change? To answer fully, you may need to think back in the course about dynamical matters that were discussed in connection with ocean

currents. Explain. Myrl was looking for a clearly expressed answer here, as the answers can vary. One answer is that yes, the spatial distribution and amplitude of the tide would change. That is to say, the dynamic tide would change. An easy way to see this is to note that the spatial pattern and amplitude of tides of different frequencies look very different from one another. 3. Geostrophy. Pacific Ocean. The accompanying figure (next page) has a potential temperature section and a neutral density section from south to north in the subtropical North Pacific. (A small map showing its location is also shown.) a) Why is potential temperature more useful to plot than temperature? Because it removes the effect of compressibility. Because potential temperature is a conservative property while temperature is not, it is a better tracer of flow. b) What is neutral density? Why is it even more useful than potential temperature for looking at geostrophic currents? Why do its contours resemble potential temperature? Neutral density is density calculated with a continuously varying reference pressure. It is computed empirically from the mean distribution of temperature and salinity, and is accessed via a lookup table. It is more useful than potential temperature because density in general is more useful, since the full density is required for calculating the hydrostatic pressure differences and hence geostrophic velocity shear. Its contours resemble potential temperature because in most places density is more strongly related to temperature than to salinity. c) Assume that the circulation represented by this section is in geostrophic balance. Assume that the flow is very weak at the maximum depth of the section and increases upward. Which direction is the flow in the bracketed regions labeled A and B? (A) ___out of the page______ and (B) ______into the page______ Indicate this direction on the plot. (not shown in answer key) d) What are the names of the upper ocean currents in the regions labeled A and B? A – Kuroshio Extension. (Some of you answered North Pacific Current, which is adequate, but this section is very close to Japan, where the current is actually called the Kuroshio.) B - North Equatorial Current. e) Using your result from c), sketch a smoothed version of the sea surface height above the plot. Do this for the whole section. Make this SSH look like a smooth mirror of the thermocline depth, as estimated by eye from the isotherm distribution. It should rise steeply over the Kuroshio, then fall gently towards the south to a low around 7N, then rise again, and the fall again towards the equator. f) Explain in words how the distribution of isotherms/isoneutrals in bracketed area A can lead to a geostrophic velocity difference between the bottom of the section and the top of the section.

The brackets enclose the Kuroshio. Just south of the Kuroshio, there is a pool of warm water and deeper thermocline; north of the Kuroshio the thermocline is shallower and there is no warm pool of water (this isn’t the “Pacific warm pool”, rather it is the STMW). Close to the sea surface, there is more mass above a given depth south of the Kuroshio than north of the Kuroshio, because of the sea surface height distribution. Therefore pressure is higher south of the Kuroshio than north. As we go down in the water column, the denser water north of the Kuroshio begins to compensate for this difference in pressure across the Kuroshio; the pgf weakens and therefore the Kuroshio velocity weakens. This effect is larger and larger with depth, and the velocity keeps weakening. Another way to say this is that the density gradient across the Kuroshio creates a geostrophic velocity shear, with highest eastward velocity at the sea surface decreasing continuously to weaker velocity with increasing depth. g) Just to the left (south) of the current labeled A, there is a thickening of the isotherms and isoneutrals. What is the water mass name we give to this layer?_____North Pacific Subtropical Mode Water_

(See WOCE Pacific Hydrographic Programme Atlas for origin of plots and other plots, including station map: http://www-pord.ucsd.edu/whp_atlas/pacific_index.html)

4. Equatorial circulation This is the upper ocean, tropical portion of the same section as on the previous page. Tickmarks at the bottom are distance and not the latitudes. (Look on previous page for latitude locations)

(a) Label the approximate location of the North Equatorial Countercurrent. Explain again in terms of isotherm distribution. The NECC is an eastward flow. It is therefore identified by isotherms that tilt upwards towards the north. We see this at stations 32 to about 38 (stations marked at the top). (b) Indicate above the plot the regions where you expect to find Ekman pumping and Ekman suction. Pumping (downwelling): north of station 38 or so; between the equator and station 32 or so. Suction (upwelling): in the NECC band, and just on the equator due to divergent Ekman transport under the trade winds. (c) Label the approximate location of the Equatorial Undercurrent. Explain in terms of the isotherm distribution. (It might not be precisely on the equator in this synoptic section.) At the spread in isotherms centered around 120 m, and at about station 15. The spreading isotherms indicate upwelling above this, which is due to Ekman suction (part b), and indicate geostrophic shear on either side of the equator, noting that

geostrophy is pretty good within ¼ degree of the equator. The spread thus creates eastward flow in the EUC relative to much weaker westward surface flow in the SEC. The downward spread of isotherms below the EUC reduces its eastward flow to zero and then reverses it to westward (the EIC). (d) Indicate the direction of the current below the Equatorial Undercurrent. Explain in terms of isotherm distribution, working downwards from the EUC. Westward flow of the EIC. See answer to c.