SIO 210 Problem Set 3 November 4, 2011 Due Nov. 14, 2011 1. At 20 N, both the ocean and the atmosphere carry approximately 2 PW of heat poleward, for a total of about 4 PW (see figure). If (at this latitude) both transport heat through meridional overturn: (a) What is the atmospheric circulation that would carry its heat poleward at 20 N? (name it) Hadley Circulation (b) Calculate the volume and mass transport of the overturn for the ocean that would carry 2 PW, assuming that the temperature difference between the cold and warm parts is 15 C. (This is a tremendous simplification, combining the Atlantic and Pacific together.) Heat transport = ρc p V(ΔT). Therefore, using standard values for density and specific heat: V = (2 PW)/( ρc p ΔT) = (2 x 10 15 W)/[(1025 kg/m 3 )(3850 J/kg C)(15 C)] = 33.8x10 6 m 3 /sec = 33 Sv If we write this in terms of kg/sec, using the density of seawater, we get about 33x10 9 kg/sec = 33 Sv (c) Calculate the volume and mass transport of the overturn for the atmosphere that would carry 2 PW, assuming the temperature difference is 60 C. (Assume that the density of air is 1 kg/m 3, and that its specific heat is 1000 J/kg K.) Same equation V = (2 PW)/( ρc p ΔT) = (2 x 10 15 W)/[(1kg/m 3 )(1000J/kg C)(60 C)] = 33.3x10 9 m 3 /sec If we write this in terms of kg/sec, using the density of air, this becomes almost exactly the same as for the ocean: 33x10 9 kg/sec = 33 Sv
(d) Sketch the two overturns in the vertical on the diagram, assuming the atmospheric transport is within the troposphere. (won t put this on the plot because it requires doing some work with illustrator) $%-&./0!'!( &1!$2(!"#$%&'( )*+,( 2. This is a map of the annual mean wind stress for the Indian Ocean. (a) Mark the location of the westerly winds. (south of about 40 S) Mark the location of the trade winds in the southern hemisphere. (between the equator and about 30 S) (b) Mark the location where the winds are dominated by the Asian monsoon in the Northern Hemisphere. (in the Arabian Sea and Bay of Bengal) Is the Indian Ocean s annual mean (shown here) dominated by the Southwest Monsoon or the Northeast Monsoon? Annual mean pattern looks like the SW Monsoon, which just means that these winds are stronger than the NE monsoon
(c) For each of these three wind regimes (westerlies, trades and monsoonal), indicate the direction of Ekman transport on the map. Arrows to the right of prevailing winds in NH and to the left of prevailing winds in the SH (d) In the Southern Hemisphere, between 20 S and about 50 S, is there Ekman downwelling or Ekman upwelling? downwelling How do you know? Ekman transport is convergent (to the left of the wind), resulting in downwelling (e) Which direction should the general circulation flow between 20 S and 50 S, where you indicated the Ekman up or downwelling? Northward, resulting from downwelling. (Then the eastward flow is on the south side of this gyre and the westward flow is on the north wide of this gyre, connecting to a western boundary current along Africa.) What is the dynamical reason for this direction of flow? Potential vorticity conservation: squashing due to Ekman pumping creates equatorward flow towards lower planetary vorticity (f) Suppose the magnitude of the Ekman downwelling or upwelling at 30 S, in the center of this region, is w E = 1 x 10-6 m/sec
If this velocity is applied for 1 year, how much would a water column be stretched or squashed in that year? (in meters) Simply calculate the distance traveled by a particle at this speed for one year: D = 1 x 10-6 m/sec x (3.15x10 7 sec/yr) = 31.5 m (g) Using potential vorticity conservation, estimate the change in latitude that would result from this stretching or squashing. Assume that the stretching affects only the top 500 m layer of the ocean. Assume that relative vorticity is negligible. Find the meridional velocity using the Sverdrup balance equation: β(v*h) = f w E At 30 S, the Coriolis parameter f = 2Ωsin(latitude) = 1.4x10-4 /sec sin(30 ) = 0.7x10-4 /sec and the β parameter (which I did not define in class) is β = 2Ω cos(latitude)/(radius of Earth) = 1.4x10-4 /sec cos(30 )/(6371x10 3 m) = 1.9x10-11 /(m sec) Using H = 500 m, v = f w E /(βh) = (0. 7x10-4 /sec)( 1 x 10-6 m/sec)/[1.9x10-11 /(m sec) * 500 m] = 0.007 m/sec In one year, the distance traveled would be: 233 km (The speed and distance seem to be off by a factor of 10.) (h) What speed does this translate to in cm/sec or m/sec? (already answered) (i) The Indian Ocean is about 5,000 km wide at this latitude. If this same north-south velocity applies at all longitudes across the Indian Ocean, what is the total meridional transport, again assuming a layer that is 500 m thick? Transport comes directly from the above equation, but taking the velocity 0.007 m/sec * 500 m * 5x10 6 m = 17.5 Sv which is about right. 3. Match the kind of wave to its speed a. capillary waves, b. long swell, c. tsunami in open ocean, d. sound in water, e. seismic waves in solid earth, few tens of m/sec - swell 1500 m/sec sound in water few kilometers/s - seismic waves 200 m/sec - tsunami few cm/sec to few tens of cm/sec capillary (ii) The speed of propagation of ocean surface gravity waves would decrease if gravity a. increased, b. decreased, (Explain your answer.) Phase and group speeds depend linearly on gravity the larger gravity is, the stronger the restoring force which speeds the waves up. It would also decrease if the density of the water were a. increased, b. decreased to a value only somewhat larger than the density of air. (Explain your answer.) If the density difference is weak, the restoring force is weak, and the
waves slow down. (This would be similar to internal waves, which ride the much weaker stratification within the ocean.) (iii) The phenomenon of refraction is important both in gravity waves that enter shallow water on the way to the beach and in sound waves that travel vertically in the ocean. In the ocean, sound waves originating at the depth of the sound speed minimum (the SOFAR channel) are generally refracted a. back towards the center of the SOFAR channel, b. up or down away from the center of the SOFAR channel. (Explain your answer.) This is a wave guide that traps waves: waves that originate in the SOFAR channel have lower speeds and when they move up or down into regions of higher sound speed, they cannot propagate, so they turn back to the sound speed minimum.