In the Classroom
Using TOPEX Satellite El Niño Altimetry Data to Introduce Thermal Expansion and Heat Capacity Concepts in Chemistry Courses Harvey F. Blanck Department of Chemistry, Austin Peay State University, Clarksville, TN 37044;
[email protected] The U.S./French TOPEX/Poseidon satellite, which was launched in August 1992 into an orbit with an inclination of 66.6° at an altitude of 1336 km with a repeating position cycle of 9.92 days, measures the height of the surface of Earth’s oceans relative to a reference ellipsoid. With appropriate corrections and calibrations the TOPEX altimetry data give an average height over an area 6 to 12 km in diameter accurate to within 3–5 cm, which allows for the detection of large-area, low-profile hills and valleys on the ocean surface. One particularly large bump that occurred during 1997 and persisted into 1998 was in the equatorial Pacific basin and was associated with the El Niño phenomenon during this time period. Although other factors influence the size of bumps, the major contributor is excess thermal energy. Warm water is less dense than cool water and will float somewhat like ice, with a portion above the surface of the cooler surrounding water. The height of the bump can be used to estimate the excess thermal energy in the warmer water. Density and heat capacity are important concepts encountered in most, if not all, introductory chemistry textbooks. Since density, thermal expansion, and heat capacity are used in the calculation of the excess thermal energy in the warmwater bump and since El Niño has been a well-publicized contemporary event, I believe that the use of these calculations will be helpful to both high school and college chemistry teachers. Theory If water is heated, provided no evaporation occurs, the mass remains the same but the temperature and the volume increase. If a vertical column of warm ocean water surrounded by cooler water retained its vertical orientation and did not spread out over the surface of surrounding cooler water, it would float above the surrounding water at a height dependent upon the difference in density between the warm and cool water and the length of the column. The difference in density is caused by the added thermal energy and depends upon the temperature of the warmer and cooler water. A warm-water bump on the ocean may be thought of as a collection of these vertical columns of warm water floating a bit above the surrounding cooler water. In a large-area low-profile warmwater bump, the rate of mixing of warm and cool water is relatively slow, as is the rate at which the warm water will flow toward the surface and flow outward over the cooler surrounding water. One very straightforward way to calculate the excess thermal energy in a warm-water bump is to determine the temperature profile for each of the columns in a collection of columns and compare them to a reference column of cooler water. Columns may extend to a depth of several hundreds
of meters. From the specific heat in units of J g{1 K{1 and the temperature profile of the column, the excess thermal energy for each column may be calculated and then summed over the entire array of columns. While a large amount of temperature data for ocean surfaces and profiles to depths of several hundred meters is available from various sources, including ATLAS buoys in the equatorial Pacific (1), for an El Niño–sized bump, the in situ temperature data (preferably for every 1 degree of latitude and longitude from the surface to 500 m depth) for this method of calculation are not available. What is available is the height of the ocean surface for the entire globe, known to an accuracy of 3–5 cm from the TOPEX altimetry data. When compared to altimetry data for normal seasonal ocean heights, the excess height of a large warm-water bump can be detected. This excess height provides sufficient information to allow an estimate of its excess thermal energy. If the rate of expansion of water per degree were independent of temperature and depth, then the absolute amount of expansion in cm3 from a given amount of thermal energy would be independent of the total amount of water in which it is distributed. For example, if a given amount of excess thermal energy were uniformly distributed in a 30-m column at a temperature 20 °C warmer, the surface height excess would be the same as that generated if the same amount of excess thermal energy were uniformly distributed in a 60-m column at a temperature 10 °C warmer. Although the rate of expansion of seawater does change somewhat with temperature and also with pressure and salinity, the change is not large. Selecting the value at some middle temperature, at a pressure of one bar, and at 35 ppt salinity allows calculation of a reasonably good estimate of the excess thermal energy stored in a warm water column. If ∆η, the excess height at which warm water is floating above cooler reference water is divided by V ′, the rate of volume expansion with temperature in cubic centimeters per gram of water per kelvin, the result is the number of gram– kelvins for a column with a cross section of 1 cm2. If the temperature difference between the warmer and cooler water is ∆T, then dividing the gram-kelvins per cm2 by ∆T gives the mass of the column in grams per square centimeter. As ∆T decreases for any given ∆η, the mass and hence the length of the column increase. Alternatively, Cp, the specific heat in J g{1 K{1, may be used to convert the gram–kelvins per square centimeter to ∆H, the excess thermal energy, in J cm{2. The resulting equation is
∆H =
Cp V′
∆η
JChemEd.chem.wisc.edu • Vol. 76 No. 12 December 1999 • Journal of Chemical Education
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In the Classroom
Notice that the change in temperature that would normally be present in the numerator in a ∆H expression containing Cp is associated with the rate of change of volume with temperature, V ′. An equivalent expression using the density, ρ, in g/cm3 and the coefficient of thermal expansion, α, in K{1 rather than V ′ is given by an equation used by Chambers and Tapley (2):
ρC p ∆H = α ∆η Multiplying Cp by ρ gives the heat capacity in joules per kelvin for each cubic centimeter. The coefficient of thermal expansion is the fractional volume change per kelvin. Data and Calculations Excess height values, ∆η, in millimeters, for 600 cells of 1° latitude by 1° longitude from 4.5° N to 4.5° S and 150° W (210° E) to 90° W (270° E) were provided by NASA’s Jet Propulsion Laboratory (JPL) scientists Akiko Hayashi and Lee-Lueng Fu. These values were derived from TOPEX altimetry data for November 16, 1997, which is approximately when the greatest anomalous behavior for the 1997– 1998 El Niño occurred. Since the values along the equator change very slowly with longitude throughout this selected region, only values for every 10° of longitude are shown in Table 1. The excess height values change more rapidly with a change in latitude. The latitudes shown are for midpoints in each one-degree cell (e.g., 4.5° N is the average for the cell extending from 4° N to 5° N). A JPL color-coded map of the Pacific Ocean derived from the TOPEX data for November 16, 1997, which includes this region, is shown in Figure 1. For orientation purposes, Hawaii (black specks in the green area near the center of the figure) is at 20° N and 155° W, and the Galapagos Islands (black speck in the white area near South America) are on the equator at 90° W. The map and Table 1 show how the sea surface height differs from the height under normal conditions. Normal conditions at each latitude and longitude were obtained from TOPEX data collected in 1993–1996. The values associated with the colors on the map are shown in the color bar. The extreme colors represent the sea level at least 130 mm above or below normal. The largest positive sea surface height anomaly is in white at the equator and is well over 130 mm in much of the area as shown in Table 1.
Figure 1. Sea surface anomaly generated from TOPEX data for the Pacific Ocean on November 16, 1997. The greatest excess height associated with El Niño’s warm water pool is along the eastern equatorial Pacific (white). The western equatorial Pacific has the greatest height deficit (magenta). Large “background” areas such as those in the northern central Pacific have near-normal sea surface heights (green). Reproduced by permission from Jet Propulsion Laboratory, California.
Areas having normal sea surface heights are in green. The large magenta area in the western portion of the equatorial Pacific does not necessarily have lower sea surface heights than green areas, but it is lower than normal because warm equatorial water usually pools in the western Pacific rather than the eastern Pacific. Color-coded maps for other dates as well as a wealth of information on the TOPEX/Poseidon project are available at JPL’s TOPEX Web site (3). V ′ values may be calculated from the rate at which 1/ρ varies with temperature. Densities for seawater at various temperatures and salinities are shown in most oceanography textbooks, but are given in Table 2 for easy reference. Only values for a salinity of 35 parts per thousand, which is a typical Pacific Ocean salinity, are shown, with ordinary water for comparison. Calculated values of 1/ρ are also shown. Values
Table 1. Excess Sea Surface Heights for 18 Latitude and Longitude Cells from 58 S to 58 N and 1508 W to 908 W on November 16, 1997 Longitude
Latitude 4.5° S 3.5° S 2.5° S 1.5° S 0.5° S 0.5° N 1.5° N 2.5° N 3.5° N 4.5° N ∆η /mm
1636
150° W
{40
72
145
194
220
224
197
15 4
118
93
140° W
60
165
248
309
35 0
360
335
290
240
194
130° W
13 4
242
329
391
431
440
41 1
356
296
247
120° W
18 7
286
356
405
43 8
447
425
377
318
263
110° W
21 6
293
359
422
456
451
42 3
37 2
304
249
100° W
274
303
341
393
434
454
441
404
356
305
90° W
230
268
306
332
358
383
384
36 9
346
315
Journal of Chemical Education • Vol. 76 No. 12 December 1999 • JChemEd.chem.wisc.edu
In the Classroom Table 2. Properties of Sea Water and Ordinary Water Temperature/°C
Variable
15
20
25
30
SEA WATER (35 PPT) Density/g cm{3 (1/ρ)/cm g 3
{1
1.02599
1.02478
1.02337
1.02175
0.97467
0.97582
0.97716
0.97871
Cp /J g{1 K {1
3.990
3.993
3.995
3.999
V ′/cm3 g{1 K {1
—
0.000249
0.000289
—
ORDINARY WATER Density/g cm{3
0.99910
0.99820
0.99704
(1/ρ)/cm3 g{1
1.00090
1.00180
1.00297
1.00437
Cp /J g{1 K {1
4.186
4.182
4.180
4.179
V ′/cm3 g{1 K {1
—
0.000207
0.000257
—
0.99565
of V ′ were calculated by dividing the difference between 1/ρ values by the temperature difference. For example, the V ′ for sea water at 20 °C was calculated from the 1/ρ values at 15 °C and 25 °C: V ′ = ∆V/∆T = (0.97716 cm3 g{1 – 0.97467 cm3 g{1)/10 K = 0.000249 cm3 g{1 K{1 Using the values in Table 2 for 25 °C, the excess thermal energy associated with an anomalous excess height of ∆η cm is ∆H = (Cp /V ′) ∆η = (3.995 J g{1 K{1/0.000289 cm3 g{1 K{1) ∆η = (1.38 × 104 J cm{3) ∆η Using the second form of the equation where α = 0.000296 K{1 gives the same numerical result. (Values for α may be found in tables or calculated from V ′ × ρ.) ∆H = (ρCp /α) ∆η = (1.0234 g cm{3 × 3.993 J g{1 K{1/0.000296 K{1) ∆η = (1.38 × 104 J cm{3) ∆η Many of the 1° latitude by 1° longitude cells near the middle of the warm-water bump along the equator are 45 cm higher than normal, in which case, ∆H = (1.38 × 104 J cm{3) (45 cm) = 6.2 × 105 J cm{2 or approximately 620 kJ cm{2. A typical cross section from 5° S to 5° N latitude averages more than 25 cm higher than normal, which is approximately 345 kJ cm{2. At the equator, 1° of latitude and longitude is about 111 km. The 600-cell rectangle is about 1110 km wide and 6660 km long. This is approximately 7.4 × 106 km2, or 7.4 × 1016 cm2; and at 345 kJ cm{2 the anomalous energy within this rectangle of 10° latitude by 60° longitude is a massive 2.5 × 1019 kJ. For comparison, one of the world’s largest coal-fired electrical generator sites, at Cumberland City, Tennessee, produces 1300 megawatts of power from each of two steam turbines by burning 24,000 tons of coal per day. In so doing this huge TVA facility produces 8.2 × 1013 kJ of electrical energy per year. Hence, it would take about 300,000 such facilities one year to produce an amount of electrical energy equivalent to the excess energy stored in this portion of the warm-water ridge in the Pacific Ocean along the equator.
It is instructive to examine how deep the warmer water extends. Normal ocean temperatures decrease with depth until a region of relatively uniform temperature near 0 °C is reached. The decrease is not linear. A typical vertical profile along the Pacific equator during a non–El Niño year is shown in Figure 2.2 of Philander’s book (4 ). The figure shows the water decreasing in temperature slowly from around 28 °C at the surface to 25 °C across 50–150 meters. Of more recent origin are data from ATLAS buoys deployed along and on both sides of the equatorial Pacific. Below the upper layer of warm water the temperature rapidly decreases from 25 to 15 °C in the next 50 to 100 m. This region of rapid decrease with temperature is called the thermocline. The thermocline is followed by a region of much slower decrease in temperature with depth. The temperature eventually declines to near zero. During normal years the top of the thermocline is depressed by warm water to 150 m in the western equatorial Pacific and is shallow or touching the surface in the eastern Pacific against the Americas. During an El Niño year such as 1997–98 the opposite occurs. The thermocline shoals in the western part and deepens in the eastern part. The depth of the warm water can be estimated from the excess surface height. For a 20-cm bump the excess would be (20 cm)/ (0.00029 cm3 g{1 K{1) or 6.9 × 104 g K cm{2. If the average temperature excess were 10 K then the depth of a 1-cm2 warm water column would contain 6.9 × 104 g K/10 K = 6.9 × 103 g. Dividing by the density yields 6.9 × 103 cm3. For a 1-cm2 column the length would be 6.9 × 103 cm or about 70 m. If the temperature excess were 5 K then the depth of the warm water would be 140 m. Depths may also be estimated from the average density of the warm and cool water. The mass in a column of warm water must be equal to the mass in a column of cool water of equal area but ∆η shorter than the warm water column. If the column of warm water of 1-cm2 cross-sectional area extends a distance ∆η + h below the surface, its mass will be (∆η + h) × ρwarm. The cool water column extending to the bottom of the warm water will have a length of h below the normal surface and a mass of h × ρcool. Equating the two masses and solving for h/∆η gives ρwarm/(ρcool – ρwarm). Using the densities from Table 2 for seawater at 30 and 20 °C produces a ratio of 337 to 1. In this case a 20-cm upward displacement of the surface is the result of a warm water column extending 67 m below the surface, while a 10-cm upward displacement of the surface would be from a 34-m column of warm water. The maximum excess height for most of the longitudinal slices of the warm-water ridge (Table 1) is at 0.5° N and the values decrease roughly symmetrically on either side of this maximum. The color-coded map shows a gradual decrease in height of the excess sea surface height both to the north and to the south of the equator, although it is not symmetrical at the extremities. It is instructive to draw what might be a reasonable shape for an ideal warm-water ridge that has a maximum in the center and approaches zero difference at the extremes. One possibility is that the cross section of the warmwater ridge would resemble two Gaussian “error” curves as shown in Figure 2, with the excess sea surface height depicted as a normally oriented Gaussian curve and the subsurface warm region depicted as an inverted Gaussian curve. The
JChemEd.chem.wisc.edu • Vol. 76 No. 12 December 1999 • Journal of Chemical Education
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In the Classroom
Figure 2. General appearance of the cross section of a warm water ridge of excess sea surface height and the subsurface warm water. The subsurface curve is the warm water/thermocline boundary. Because of a large difference between the size of the warm water portion above and below the normal surface, two different scales have been used.
water vapor content must be determined by the satellite to correct the radar altimeter data. Studies of heat storage using direct temperature measurements have been conducted (5), and comparison of TOPEX altimetry data with actual temperature measurements shows them to be in reasonably good agreement (6 ). Low-profile hills and valleys on the ocean are generated or influenced by a variety of factors other than thermal energy. Ocean dynamics are complex indeed. Comparisons of thermal energy (steric effect) and windinduced surface changes have been examined in relation to TOPEX data (7). The calculations of thermal energy excess in warm-water ocean bumps from radar altimetry data alone, while not unreasonable, must be understood to be a simplification for an extremely complex system. The Gaussian model proposed for the cross section of a warm-water ridge requires more study, but it is a useful visual model of the warm-water bump above the normal surface and its subsurface warm-water wedge. I believe students will enjoy these relevant calculations and learn a bit about density, thermal expansion, and heat capacity in the process. I have tried to present sufficient data and detail to allow teachers to pick and choose calculations appropriate to the level of their students. It is evident that dimensional analysis is a distinct advantage in using these equations. I have also tried to include enough descriptive detail of the TOPEX data and El Niño to answer many of the questions students may ask. The Web sites mentioned are very informative with both text and graphics. Acknowledgments
drawing has two scales because the lower portion (which is often referred to as a “V”-shaped warm-water wedge) should be several hundred times larger than the upper warm-water portion, as suggested by the previous calculations. Actual measurements of subsurface temperatures by 70 ATLAS buoys deployed along the equator and 10° either side of the equator through the entire Pacific have been and are being made. The data do show the warm water extending to depths similar to those calculated above and the general wedge shape of the subsurface warm water is apparent, although unfortunately the ATLAS buoy array does not at present extend far enough either north or south of the equator to measure the cross-section extremities. To show that this model is correct will require analysis of much additional data, but the model is easy to draw and is a reasonable visual representation of the way in which the thermocline is depressed by warm water along a warm-water ridge. Discussion Various factors must be taken into account to modify the raw TOPEX radar altimeter data to obtain meaningful information. For example, as mentioned at JPL’s TOPEX Web site, radar propagation speed is altered slightly by variations in water vapor in the atmosphere, and therefore atmospheric
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The data for the cells and the pictorial representation of the relative TOPEX altimetry data of the Pacific basin were very kindly provided by the TOPEX/Poseidon Project conducted by the Jet Propulsion Laboratory of the California Institute of Technology under contract with NASA. I wish to specifically thank JPL scientists Akiko Hayashi and Lee-Lueng Fu for providing data and I especially thank JPL scientist Victor Zlotnicki for helpful comments and suggestions. Several reviewers made helpful suggestions, which were very much appreciated. Literature Cited 1. NOAA’s Pacific Marine Environmental Laboratory (PMEL) Web site; http://www.pmel.noaa.gov and especially http://www.pmel.noaa.gov/ toga-tao/realtime.html (accessed Sep 1999). 2. Chambers, D.P.; Tapley, B. D.; Stewart, R. H. J. Geophys. Res. 1997, 102C, 10525–10533. 3. NASA’s Jet Propulsion Laboratory, TOPEX Web site; http:// topex-www.jpl.nasa.gov (accessed Sep 1999). 4. Philander, G. El Niño, La Niña, and the Southern Oscillation; Academic: New York, 1990; p 63. 5. Yan, X.-H.; Niiler, P. P.; Nadiga, S. K.; Stewart, R. H.; Cayan, D. R. J. Geophys. Res. 1995, 100C, 6899–6926. 6. White, W. B.; Tai, C.-K. J. Geophys. Res. 1995, 100C, 24943– 24954. 7. Stammer, D. J. Geophys. Res. 1997, 102C, 20987–21009.
Journal of Chemical Education • Vol. 76 No. 12 December 1999 • JChemEd.chem.wisc.edu
