ENGN UMR0159 Engineering Library GREAT LAKES RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Contribution No. 2 A Dynamic Height Method for the Determination of Currents in Deep Lakes John C. Ayers Contribution,.No.\,3:Simplified Computations for the'Dynamic Height Method of Current Determination in Lakes John C. Ayers and Roger Bachmann ~' ANN ARBOR, MICHIGAN 1957

Reprinted from LIMNOLOGY AND OCEANOGRAPHY Vol. II, No. 2, April, 1957 Printed in UT.S.A. Simplified Computations for the Dynamic Height Method of Current Determination in Lakes' INTRODUCTION perature and zero hydrostatic pressure A dynamic height method for the deter- (105 &t,o), then applies a compression cormination of current directions and veloci- rection (Aa,,,p) at observed temperature and ties in deep lakes has been developed by in situ pressure to obtain the anomaly at Ayers (1956). This method involves very temperature and pressure. The new speconsiderable amounts of arithmetical cal- cific volume anomaly table (Table 1) was culation and hence contains numerous prepared by graphing the zero-pressure possibilities for arithmetical error. Suc- portions of Ayers' (loc. cit., Table 2) origicessful reduction of the amount of computa- nal anomaly table and reading anomaly tion required would achieve both economy values (105 t,,o) at each 0.1~C. The new of time and reduced chances for error. table of coefficients of compression (Table This paper presents a successful simplified 2) was prepared by graphing and reading computation routine for the freshwater at each 0.1C Ayers' (loc. cit., Table 1) dynamic height method. compressibility coefficients. As in the origThe simplified computation routine con- inal paper, the coefficient of compressibility sists of a more convenient rearrangement of (in cm3/cm3/atmosphere) is numerically the original computation method and in no equal to compression in cm3/atmosphere way changes the theoretical considerations (Ct) when applied to a volume of one cubic upon which that method was based. The centimeter. The pressure effects are reader is referred to Ayers (1956) for the handled, as in the old method, by the full development of theory and the original decibar system: pressure in atmospheres, method. p = depth in meters +.- 10 meters/atmosFormal thanks are tendered to the Uni- phere. As before, 105 Ct X p = Aatp and versity of Michigan Biological Station, 105 t,' - Aat,p = 105 t,p, the required where much of this work was carried out. anomaly at temperature and pressure. As in the old method, the resultant THE SIMPLIFIED METHOD anomaly (105 t.8p) is required for the surThe new method differs from the original face temperature, for each subsurface in two ways: 1) the expansive effects of inflection of the temperature curve, and temperature and the compressive effects of for the temperature at the reference level. pressure are independently determined and COMPARISONS OF THE NEW AND applied, and 2) new tables directly readable OLD METHtODS to 0.10C have been prepared. Independent application of the thermal and pressure As indicated above, the required resultant effects eliminates the laborious double anomaly for any subsurface temperature interpolations commonly required in the and pressure can be obtained (in the new original method. method) by two table readings, one diviThe new method first determines the sion, one multiplication, and one subtracspecific volume anomaly at observed tem- tion-a total of five operations. Under the old method obtaining a single resultant 1 Contribution No. 3 from the Great Lakes Research Institute, University of Michigan, and anomaly at temperature and pressure inContribution No. 883 from the Woods Hole volved a total of thirteen operations as Oceanographic Institution. follows: 155

156 NOTES AND COMMENT TABLE 1. Specific volume anomaly (1056 t,o) ~C 0.1.2.3.4.5.6.7.8.9 0 13.0 12.4 11.5 10.8 10.2 9.6 9.0 8.4 7.9 7.4 1 7.0 6.6 6.2 5.7 5.3 4.9 4.6 4.2 3.8 3.5 2 3.0 2.8 2.6 2.3 2.0 1.8 1.6 1.4 1.2 1.1 3 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 1.0 1.1 1.2 1.3 1.5 1.7 1.9 2.1 2.4 2.7 6 3.0 3.3 3.7 4.1 4.5 4.9 5.3 5.7 6.1 6.5 7 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 8 12.0 12.6 13.2 13.9 14.6 15.3 16.0 16.7 17.4 18.3 9 19.0 19.7 20.4 21.2 22.0 22.8 23.6 24.4 25.3 26.1 10 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 11 37.0 38.0 39.1 40.2 41.3 42.5 43.6 44.7 45.8 46.9 12 48.0 49.1 50.2 51.4 52.6 53.8 55.0 56.2 57.4 58.7 13 60.0 61.3 62.6 63.9 65.2 66.5 67.8 69.1 70.4 71.7 14 73.0 74.4 75.8 77.2 78.6 80.0 81.4 82.8 84.2 85.6 15 87.0 88.6 90.2 91.8 93.4 95.0 96.6 98.2 99.8 101.4 16 103.0 104.7 106.4 108.1 109.8 111.5 113.2 114.9 116.6 118.3 17 120.0 121.8 123.6 125.4 127.2 129.0 130.8 132.6 134.4 136.2 18 138.0 139.9 141.8 143.7 145.6 147.5 149.4 151.3 153.2 155.1 19 157.0 159.0 161.0 163.0 165.0 167.0 169.0 171.0 173.0 175.0 20 177.0 179.1 181.2 183.3 185.4 187.5 189.6 191.7 193.8 195.9 21 198.0 200.3 202.6 204.9 207.2 209.5 211.8 214.1 216.4 218.7 22 221.0 223.3 225.6 227.9 230.2 232.5 234.8 237.1 239.4 241.7 23 244.0 246.4 248.8 251.2 253.6 256.0 258.4 260.8 263.2 265.6 24 268.0 270.6 273.2 275.8 278.4 281.0 283.6 286.2 288.8 291.4 TABLE 2. Coefficients of compression (105 Ct) ~C 0.1.2.3.4.5.6.7.8.9 0 5.250 5.243 5.236 5.230 5.224 5.219 5.214 5.209 5.204 5.200 1.195.191.187.183.179.176.173.170.167.163 2.160.157.154.151.148.145.142.139.136.133 3.130.128.125.122.120.118.115.112.110.108 4.105.103.100.098.095.092.089.087.085.082 5.080.078.076.074.072.070.068.066.064.062 6.060.058.057.055.053.051.049.047.045.044 7.043.041.039.038.036.034.033.031.029.028 8.027.025.024.023.022.020.019.018.016.014 9.013.012.011.010.008.007.006.004.003.001 10 5.000 4.998 4.997 4.996 4.994 4.993 4.992 4.990 4.989 4.987 11 4.985.984.983.982.981.980.979.978.977.976 12.975.974.973.971.970.969.968.967.966.965 13.964.962.961.960.959.958.957.956.955.954 14.953.953.952.951.950.949.948.947.946.945 15.944.943.942.941.941.940.939.938.938.937 16.936.936.935.934.933.932.932.931.931.930 17.929.929.928.928.927.926.925.924.923.923 18.922.921.921.920.919.919.918.918.917.916 19.916.915.914.914.913.913.912.912.911.911 20.910.910.909.909.908.908.907.907.906.906 21.905.905.904.904.903.903.903.903.902.902 22.902.902.901.901.901.901.900.900.899.899 23.898.898.898.897.897.896.896.895.894.894 24.893 a. Reading anomalies at temperatures b. Interpolating to anomaly at observed adjacent to the observed temperature and temperature and less than observed presat pressure less than observed pressure sure (1 subtraction, 1 multipli(2 table readings) cation, and 1 addition)

NOTES AND COMMENT 157 c. Reading anomalies at temperatures stations taken in Lake Michigan on 28 June adjacent to observed temperature and at 1955 have been worked out by each method. pressure greater than observed pressure While there were minor variations in the (2 table readings) insignificant terminal decimal places, the d. Interpolating to anomaly at observed final dynamic heights rounded off to the temperature and greater than observed same values by either method. There is, pressure (3 operations as in b above) then, no loss of accuracy resulting from the e. Interpolating between anomalies at simpler new method. The total labor (with observed temperature (items b and d above) calculator) required for the two determinafor anomaly at observed temperature and tions of this topography was 31 man-hours pressure (3 operations as in b above). by the original method and 13.5 manThe new computation reduces to less than hours by the new. half the number of operations (and the time, labor, and chance for error) involved in REFERENCES obtaining each individual resultant specific AYERS, J. C. 1956. Adynamic height method for volume anomaly. the determination of currents in deep lakes. Limnol. & Oceanogr., 1: 150-161. Subsequent computations leading to the Limnol. & Oceanogr. 1: final dynamic height at each station are JOHN C. AYERS carried out as in the original paper and are Great Lakes Research Institute and Departno less time-consuming and laborious than ment of Zoology, University of Michigan; before. They do benefit, however, by the and Woods Hole Oceanographic Institution lesser chance for error inherent in the new ROGER BACHMANN method. Idaho Cooperative Wildlife Research Unit, As a comparison of the accuracy of the University of Idaho, two methods, the dynamic heights of 56 Moscow, Idaho

Reprinted from LIMNOLOGY AND OCEANOGRAPHY Vol. 1, No. 3, July, 1956 Printed in U.S.A. A Dynamic Height Method for the Determination of Currents in Deep Lakes' JOHN C. AYERS Department of Conservation, Cornell University and Woods Hole Oceanographic Institution ABSTRACT An adaptation of the oceanographer's dynamic height method to freshwater conditions, the method develops directly from the Smithsonian Tables of water density as a function of temperature and from Amagat's coefficients of compressibility. Pressure effects are handled by use of the decibar system. The calculated dynamic heights are plotted and contoured. Current directions are obtained by application of the geostrophic principle; current velocities are computed from the slopes of the surface topography. A table of specific volume anomalies, necessary for the height computation, has been developed. Sources of potential error are reviewed and assessed. The method has been applied to three synoptic surveys of Lake Huron, and the results obtained are both internally consistent and in good agreement with the behaviors of other parameters. When used with proper caution the method appears to be a promising technique for the study of certain circulation phenomena of large deep lakes. INTRODUCTION wind-caused mixing; the latter was used in It is well known that freshwater is Lake Huron (Table 3). On the basis of slightly compressed by pressure and ex- our present knowledge, the method appears panded by temperatures above and below to be applicable in lakes where the pre4~ Centigrade. For some decades the ponderance of depths is more than three oceanographer has made use of similar times the mean depth to the thermocline. phenomena to calculate current direction It is strongly urged that currents deterand velocity from density data, and from mined by this method be checked against the reciprocal of density-specific volume. currents obtained by other means; lack of It appears that with certain modifications agreement may indicate that significant the oceanographer's dynamic height method bottom currents are present and that of current determination can be applied to bottom stress must be considered. The at least the deeper lakes. dynamic height method may be unsuccessThe degree to which this application ful during periods of spring and fall turnmay be limited by the shallowness of lakes over. is not yet clear. The method depends upon the selection of a subsurface reference level at which currents are virtually absent. The dynamic (or geopotential) height It is imperative that the reference level be method deals with the calculated lengths below the depth of the turbulent mixing of water columns which are considered to and return flow that result from wind ac- be standing upon the horizontal reference tion. It appears from the work of Morti- level. In each component segment of the mer (1951, 1954) that turbulent mixing water column at each station the resultant and return may, under ordinary winds, effect, on length, of compression due to attain two or three times the mean depth hydrostatic pressure of overlying water of the thermocline. The depth to which and expansion due to temperatures above summer warming penetrates may also be or below 4~ C is determined. Summation used as a measure of the over-all depth of of the resultant effects in all the component 1 Contribution No. 2 from the Great Lakes Re- segments gives the column's length change search Institute and Contribution No. 823 from resulting from the expansions and/or comthe Woods Hole Oceanographic Institution. pressions of its parts. Addition of this 150

DYNAMIC HEIGHT METHOD FOR LAKE CURRENTS 151 length change to the theoretical length of TABLE 1. Compressibility coefficients and comthe column under standard conditions gives pression in unit volume (Ct) at temperatures bethe calculated height of the column top tween 0 and 30 degrees Centigrade, based on Amagat the calculated height of the column top (1893) above the reference level under the prevailing conditions of pressure and tempera- 0C Compressibility coeffi- Compression in 1 cm3 cient* (cmt/cmp/atm) (Ct)* (cmc/atm) ture. The differing heights of the column X 10-6 X 10-6 tops at the various stations produce a 0 52.50 52.50 "topography" of the water surface. From 1 51.95 51.95 this dynamic or geopotential topography the 2 51.60 51.60 3 51.30 51.30 current directions and velocities are ob- 4 51.05 51.05 tained. 5 50.80 50.80 The freshwater method's primary length 6 50.60 50.60 unit is the centimeter. Its use allows un- 7 50.43 50.43 8 50.27 50.27 usual convenience in operating with specific 9 50.13 50.13 volume. 10 50.00 50.00 The effects of temperature on volume 11 49.85 49.85 per unit mass have been obtained from the 12 49.75 49.75 Smithsonian Tables as quoted in the Hand- 13 49.64 49.64 14 49.53 49.53 book of Chemistry and Physics (1949, Table 15 49.44 49.44 of relative density and volume of water). 16 49.35 49.35 The volume per unit mass is the specific 17 49.30 49.30 volume. 18 49.22 49.22 19 49.15 49.15 Volume/unit mass = specific volume 20 49.10 49.10 = at,0 (at observed temperature (1) 22 49.02 49.02 and zero hydrostatic pressure) 23 48.98 48.98 24 48.93 48.93 The coefficients of compressibility used 25 48.90 48.90 are those of Amagat (1893)2 as given in the 26 48.88 48.88 Handbook of Chemistry and Physics (op. cit., 27 48.85 48.85 28 48.82 48.82 Table of compressibility of liquids). 29 48.81 48.81 Amagat's coefficients for 1 to 25 atmos- 30 48.80 48.80 pheres pressure at 0, 10, and 20~ C were graphed, and the graph extrapolated to * Second decimal places only approximate. 30~ C; from the graph the coefficient at each degree was obtained (Table 1). therefore essentially numerically equal to Amagat's unit "contraction in unit the pressure in decibars. Also volume per atmosphere" (cm3/cm3/atm) when multiplied by the number of volumes (2) -atmospheres pressure = p involved becomes compression per atmosphere (cm3/atm); when one specific volume Compression at observed temperature, is used the compression is numerically the Ct, (Table 1) multiplied by the in situ pressame as the coefficient and has the unit sure at depth gives compression in situ cm3/gm/atm. ~~~cm3/~gm/atm.. Ct X p = compression in situ = Aatp. (3) Since one atmosphere of hydrostatic pressure = 1 bar = 10 decibars = the It is assumed that a lake has constant hydrostatic pressure of ten meters of water, surface area during the short period when and one meter of water produces one data are being collected, i.e., the lake decibar of pressure, the depth in meters is consists of a finite number of water columns each of constant 1 cm2 cross section. Vol2 The author is aware of Beyer's (1954) conclusion that Amagat's coefficients are slightly tooy h large. The use of Amagat's coefficients may in- consisting entirely of changes in length, troduce a small systematic error. and each component specific volume of

152 JOHN C. AYERS TABLE 2. Specific volume anomaly (105t,,p)* (Specific volume length anomaly when applied to columns of 1 cm2 cross section) Depth Pressure Temperature, ~C feet meters atmospheres 0 1 2 3 4 5 6 7 0 0 0 13.0 7.0 3.0 1.0 0 1.0 3.0 7.0 5 0.5 10.4 4.4 0.4 -1.6 -2.5 -1.5 0.5 4.5 10 1.0 7.8 1.8 -2.2 -4.1 -5.1 -4.1 -2.1 2.0 15 1.5 5.1 -0.8 -4.7 -6.7 -7.7 -6.6 -4.6 -0.6 50 15.2 1.52 5.0 -0.9 -4.8 -6.8 -7.8 -6.7 -4.7 -0.7 25 2.5 -0.1 -6.0 -9.9 -11.8 -12.8 -11.7 -9.7 -5.6 100 30.5 3.05 -3.0 -8.8 -12.7 -14.6 -15.6 -14.5 -12.4 -8.4 35 3.5 -5.4 -11.2 -15.1 -17.0 -17.9 -16.8 -14.7 -10.7 150 45.7 4.57 -11.0 -16.7 -20.6 — 22.4 -23.3 -22.2 -20.1 -16.1 50 5.0 -13.3 -19.0 -22.8 -24.7 -25.5 -24.4 -22.3 -18.2 100 10. -39.5 -45.0 -48.6 -50.3 -51.1 -49.8 -47.6 -43.4 150 15. -65.8 -70.9 -74.4 -76.0 -76.6 -75.2 -72.9 -68.7 200 20. - 92.0 -96.9 - 100.2 - 101.6 -102.1 - 100.6 - 98.2 -93.9 250 25. -118.3 -122.9 -126.0 -127.3 -127.6 -126.0 -123.5 -119.1 Depth Pressure, Temperature,'C feet meters atmospheres 7 8 9 10 11 12 13 14 0 0 0 7.0 12.0 19.0 27.0 37.0 48.0 60.0 73.0 5 0.5 4.5 9.5 16.5 24.5 34.5 45.5 57.5 70.5 10 1.0 2.0 7.0 14.0 22.0 32.0 43.0 55.0 68.0 15 1.5 -0.6 4.5 11.5 19.5 29.5 40.5 52.6 65.6 50 15.2 1.52 -0.7 4.4 11.4 19.4 29.4 40.4 52.5 65.5 25 2.5 -5.6 -0.6 6.5 14.5 24.5 35.6 47.6 60.6 100 30.5 3.05 -8.4 -3.3 3.7 11.8 21.8 32.8 44.9 57.9 35 3.5 — 10.7 -5.6 1.5 9.5 19.6 30.6 42.6 55.7 150 45.7 4.57 -16.1 -11.0 -3.9 4.2 14.2 25.3 37.3 50.4 50 5.0 -18.2 -13.1 -6.1 2.0 12.1 23.1 35.2 48.2 100 10. -43.4 -38.3 -31.1 -23.0 -12.9 -1.8 10.4 23.5 150 15. -68.7 -63.4 -56.2 -48.0 -37.8 -26.6 -14.5 -1.3 200 20. -93.9 -88.5 -81.3 -73.0 -62.7 -51.5 -39.3 -26.1 250 25. -119.1 -113.7 -106.3 -98.0 -87.6 -76.4 -64.1 -50.8 Depth Pressure, Temperature,'C feet meters atmospheres 14 15 16 17 18 19 20 21 0 0 0 73.0 87.0 103.0 120.0 138.0 157.0 177.0 198.0 5 0.5 70.5 84.5 100.5 117.5 135.5 154.5 174.5 195.6 10 1.0 68.0 82.1 98.1 115.1 133.1 152.1 172.1 193.1 15 1.5 65.6 79.6 95.6 112.6 130.6 149.6 169.6 190.6 50 15.2 1.52 65.5 79.5 95.5 112.5 130.5 149.5 169.5 190.5 25 2.5 60.6 74.6 90.7 107.7 125.7 144.7 164.7 185.7 100 30.5 3.05 57.9 71.9 88.0 105.0 123.0 142.0 162.0 183.0 35 3.5 55.7 69.7 85.7 102.7 120.8 139.8 159.8 180.8 150 45.7 4.57 50.4 64.4 80.5 97.5 115.5 134.5 154.6 175.6 50 5.0 48.2 62.3 78.3 95.4 113.4 132.4 152.5 173.5 100 10. 23.5 37.6 53.7 70.7 88.8 107.9 127.9 149.0 150 15. -1.3 12.8 29.0 46.1 64.2 83.3 103.4 124.4 200 20. -26.1 — 11.9 4.3 21.4 39.6 58.7 78.8 99.9 250 25. -50.8 -36.6 -20.4 -3.3 15.0 34.1 54.3 75.4 Depth Pressure, Temperature,'C feet meters atmospheres 21 22 23 24 25 26 27 28 0 0 0 198.0 221.0 244.0 268.0 294.0 320.0 347.0 375.0 5 0.5 195.6 218.6 241.6 265.6 291.6 317.6 344.6 372.6 10 1.0 193.1 216.1 239.1 263.1 289.1 315.1 342.1 370.1 15 1.5 190.6 213.7 236.7 260.7 286.7 312.7 339.7 367.7 50 15.2 1.52 190.5 213.6 236.6 260.6 286.6 312.6 339.6 367.6 25 2.5 185.7 208.7 231.8 255.8 281.8 307.8 334.8 362.8 100 30.5 3.05 183.0 206.1 229.1 253.1 279.1 305.1 332.1 360.1 35 3.5 180.8 203.8 226.9 250.9 276.9 302.9 329.9 357.9 150 45.7 4.57 175.6 198.6 221.6 245.6 271.7 297.7 324.7 352.7 50 5.0 173.5 196.5 219.5 243.5 269.6 295.6 322.6 350.6 100 10. 149.0 172.0 195.0 219.1 245.1 271.1 298.2 326.2 150 15. 124.4 147.5 170.5 194.6 220.7 246.7 273.7 301.8 200 20. 99.9 123.0 146.0 170.1 196.2 222.2 249.3 277.4 250 25. 75.4 98.5 121.6 145.7 171.8 197.8 224.9 253.0 * Decimals are only approximate.

DYNAMIC HEIGHT METHOD FOR LAKE CURRENTS 153 each column may be considered to contain which the profile is broken down into its unit mass in a'cube' of 1 cm2 cross section essentially linear portions and the temperaand variable length. Thus compression ture and depth (pressure) of each inflection effects may be subtracted from the specific point recorded. Enough inflection points volume at temperature to obtain the re- should be read to reproduce the profile sultant length of the specific volume'cube' well when these points on a graph are at in situ temperature and pressure connected by straight lines. The required specific volume length atg, - at,, A anomaly for the surface temperature is specific volume length in situ (4) found by interpolation between the proper atmp. values of Table 2. Similarly the anomaly To reduce the physical size of the num- for each inflection point is found, by single bers it is convenient to work with the interpolation when the depth (pressure) specific volume length anomaly which, falls on a tabulated pressure level, or by further, is multiplied temporarily by 105. double interpolation when observed temperature and pressure both fall between ats, - 1.000000 tabulated values. = specific volume length anomaly (5) The anomaly is determined for the sur-at:.,,p face, for each inflection point, and for the The computations leading to the tabu- reference level. The anomalies are averlated (Table 2) specific volume length aged by successive pairs and each average anomaly of water at 150 C and 25 meters is multiplied by the depth interval (in depth will serve as an example of the way centimeters) covered by that pair to give the table was constructed. the calculated length anomaly, AD, of that segment of the water column. Each AD a at 15~, 0 hydrostatic pressure ( ) is divided by 105 to return it to actual at, = 1.00087 cm3/gm value; the AD's are summed cumulatively from zero at the surface down to the referp = 25 meters 10 meters/atm 2.5 atm (Eq. 2) ence level; and the cumulated sum at the reference level is converted to meters (see Table 3). The cumulated length anomaly -X 10-236Cm3/gm/at3/gm (Eq. 3) added to the reference level's depth under standard conditions gives the calculated a5,2.5 = (1.00087 cm3/gm + 1 cm2/gm) length of the existing unit-cross-section - (123.6 X 10-6cm3/gm + 1 cm2/gm) water column. The final calculated length = 1.00087 cm - 123.6 of the water column is its dynamic height X 10-6 cm = 1.000746 cm (Eq. 4) or geopotential height. In Table 3 the reference level was the 60-decibar surface = 1.000746 cm (Eq. 5) whose depth under standard conditions would be 60 meters. 105615,2.6 = 74.6 cm At stations where the depth is less than (see t = 15~, p = 2.5 atm in Table 2) that of the chosen reference level it is permissible to substitute the lower parts of The specific volume length anomalies a near-by station of adequate depth, as in (1066t,,) for each degree between 0 and 28~ C Figure 4. When this is done the isobaric and at several pressure levels are given in surfaces become horizontal (slopeless and Table 2. This table is of minimal size so currentless) between the identical parts of far as numbers of pressure levels are con- the two stations. This device was sugcerned. The user is advised to expand it gested by Nansen and introduced by by adding other pressure levels in the depth Helland-Hansen (1934). It is apparently (pressure) range which pertains to his lake. satisfactory in places where bottom curComputation begins with a fairly de- rents are virtually absent, but may be tailed reading of the temperature profile, in seriously in error in places where currents

154 JOHN C. AYERS TABLE 3. Dynamic height computation, Station 40, survey Synoptic II, 27 July 1954 Depth Tempera- 105,pat Depth Cumulative cm ture C temp. & Average interval Product AD, cm umulative ~~~~depth cinterv~AD m meters 0 18.6 149.4 0 141.5 520 73580 0.74 520 17.9 133.6 0.74 99.4 520 51688 0.52 1040 13.8 65.2 1.26 49.0 480 23520 0.24 1520 11.3 32.7 1.50 9.2 1530 14076 0.14 3050 5.1 -14.3 1.64 - 16.7 760 - 12692 -0.13 3810 4.4 -19.0 1.51 -21.0 760 - 15960 -0.16 4570 4.3 -23.0 1.35 -26.7 1430 -38181 -0.38 6000 4.3 -30.3 0.97 0.97 0.010 add reference level depth at standard conditions 60.000 meters Dynamic height in dynamic meters 60.010 along the bottom are significant. This Since the distance between contours is device should be used only with great inversely proportional to the velocity, it is caution at stations so shallow that major convenient to determine the velocities of parts of the water column require substitu- the fastest current, the slowest current, tion of values. and of a few intermediate ones, then to Current directions are obtained by plot- construct a trumpet-shaped graph (see ting the dynamic height for each station Figs. 1, 2, and 3) from which velocity may and drawing height contours of the surface be read by transferring the intercontour topography at regular intervals. Current- distance with dividers. direction arrowheads may be drawn on the contour lines by application of the geo- APPLICATIONS OF THE METHOD strophic principle: the currents are situated The dynamic height method has been on the slopes of the topography, flow parallel used to determine the surface current patto the contours, and flow in such direction terns and velocities in Lake Huron at three that the topographic'high' is on the right different times in the summer of 1954. At (in the Northern Hemisphere) when the the end of June the major part of the lake observer looks in the direction of flow. was still in essentially its spring condition Current velocity between each pair of — with no thermocline yet formed. At stations can be obtained from: v = 10 this time the current pattern (Fig. 1) was id/0.000145 sin c (Sverdrup et al. 1942: almost identical to that deduced by Har392) where id is the slope of the surface rington (1895) from drift bottles released between the stations and sin 4 is the nat- by lake steamers throughout the entire ural trigonometric function of the latitude shipping season, approximately late March of the mid-point between the station pair. to early December. It was also in agreeIf the slope is expressed in meters of height ment with the results of Millar (1952) who difference per meter of horizontal distance confirmed Harrington's current pattern by between the stations, the velocity will be studies of intake-water temperatures on in meters per second; if the slope is in lake steamers throughout the shipping cm/m, velocity will be in centimeters per season. For these reasons it is believed second. LaFond (1951: 98) gives a similar that the pattern observed in June approximethod of calculating current velocity mates the "fundamental" winter-to-winter from the same parameters. circulation pattern of the lake.

_46!.000 OX 5999`-~ Q ~ ss.194 0..~~~~ GEOPOTENTlAL TOPOGRAPHY OF THE LAKE SURFACE 60-000 ~~~~~~~~~(DYNAMIC METERS) &6O ~~~~~~RELATIVE TO THE 60-DECIBAR SURFACE FO i SYNOPTIC I la 30. *D 28,29 JUNE 954 DEPSTAT1ONS.OD SHALLOW STATIONS 6QOOO. ~......30-FAtHOM CONTOUR -440 U PROBABLE UPWELLING. S PROBABLE SINKING` 0. NS z MILES PER HOUR-3 FIG.1.The retoLa0 5 n t t the 6r FI.1 hecretso ae uo n 82 un 94 in eaintote6 eibrsrae

46' ql." - 9... 3... _ %.025 _j u'... 60.005r 60 2JY 4 )60. 0 2~)^5\}60.0 1 601 5...... 0FTO OT 60. 00 z.1 60.00 I I I I 45 1 2.015'. o JI~s PER..l,/g 2oF T.. LAKE SURFACE e 50. 1 8 4 -8 >8 FIG. 2. Th curentsofLkeHron n27July1954 l GEOPOTENTIAL 6sra (DYNAMIC METERS)..,~ j.. OF THE LAKE SRAEC RELATIVE TO THE 60-OECIBAR SURFACE....... ~~~~SYNOPTIC IE.... E) "';........ f'::~~~ 27 JULY 1954.0.06005 ~ DEEP STATIONS, E) SHALLO STATIONS 60.0 60.01! ~~~~~~~~~~~~~~............ 30-FATHOM CONTOUR \-P U PROBABLE UPWELLING, S PROBABLE SINKING 6.020 60.025.. 60.020' 6025 i ~~ 0 0020 8860.02' 60-deci..2.3 FIG. 2. The currents of Lake Huron on 27 July 1954, in relation to the 60-decibar surface.

o~~~~~~~~~~~~~~~~~~ 0.0~~~~. Z0.005: 600' IjGQQ3O 00~~~~~~~~~~~~~~~~~ "~0'0\s 1 60. 020" III It'''.- 60.005 ~ ~ ~ ~ ~ 6002 6. CT 60.01........... 60.020 10.1.2.33. 60.0 ~ ~ ~ 0:1 MILS PER HOURF 60. RLATIVETO THE60-DECSAR SU RF A 85 84 83~~~~~~~~~~~~~~~ 82~~~~~~~ 81 80~5 UUS 15 FIG.. 3.2 The currents of Lake Huron on 25 August 1954, in relation to the60decibar surface. This is apparently a non-equilibrium condition. E.1 60.020 6003 U sd~~~~~o ~ Uz 60.066004 0635'4 60.036 SRELATNETO THEMILES PER HOUR C 85! MO 83 820 81".80* FI..Th crens fLae uono 2 Agst194 i rltin oth 0-ecbr urae.Thsisapaenl ano-qulirumcodtin

158 JOHN C. AYERS 7-r/. ~024-.z.~~~~~~~~..028 I~~~~~~~~~~~~~~ 04.0193 I ],016.012.o12-I I I I AID D o15. 2,W ADD.008.0150 E\D.0154 001'O ~~~~.0146 i I.00 I.oI I-~ I I I 0~ ~ ~ ~~~~~~~~01,0041 I AD AD.0074 I00 I0/ 133 I -- 15.2 I I I~ I ~./ D=15.3o 008 I o Ik02 461 30.5 I 0~I' I~ I.00o4- I O I'~'D AD - I.0027 -.027 008 I I 0 02-2 ~t.00 0 0 2 9-.0 027 O4~, —, ~ ~~004-.. Ao5D D5,3-: A tA 5..j.o087 —. j,- 0035,00 36. L.o0 I0-037 1AD08 D = 1 5.3M I Io FIG. 4. The distribution of dynamic heights and current velocities in the Oscoda, Michigan-Southampton, Ontario section on 27 July 1954. Current velocities in meters/second are shown on the decibar surfaces. The AD values, in dynamic meters, in the between-station intercontour blocks are averages applying to those blocks. Interception of the isobaric surfaces by the bottom is indicated at each end. D values on the right are the intercontour intervals, in meters, under standard conditions. The insert shows the location of the section.

DYNAMIC HEIGHT METHOD FOR LAKE CURRENTS 159 In the interval from late June to late parently responded rather readily to the July a strong thermocline developed over wind shift. the whole lake. The survey of 27 July Prior to the survey of 25 August 1954 took place after winds from quarters more (Fig. 3) the winds were from more nearly northerly than usual, and the current pat- normal westerly quarters, and the current tern (Fig. 2) appeared to be a wind-in- pattern appeared to indicate a partial duced distortion of the fundamental return toward the June condition. The pattern; the warm light surface water ap- fact that the outflows from Lakes Michigan TABLE 4. Computation of water transport through the Oscoda-Southampton section, 27 July 1954 Data for second and fourth columns are from Figure 4. Intersurface Isobaric surface Intersurface Intersurface area Average velocity Transport Transport sum pair, decibars depth (D + AD), m2 m/sec* m3/sec** m3/sec** dynamic m* Transport between stations 29 and 30, 23979 meters apart 0-15.2 15.2193 364943 0.0054 1970 north 15.2-30.5 -- 0 0 30.5-45.7 -- 0 0 45.7-60.0 0 0 1970 north Transport between stations 30 and 31, 21243 meters apart 0-15.2 15.2193 323304 0.0016 517 north 15.2-30.5 15.3074 325175 0.0039 1268 north 30.5-45.7 15.1973 322836 0.0011 355 north 45.7-60.0 15.2965 324944 0.0013 422 north 2562 north Transport between stations 31 and 32, 26393 meters apart 0-15.2 15.2169 401620 0.0225 9036 south 15.2-30.5 15.3058 403966 0.0095 3838 south 30.5-45.7 15.1973 401102 0.0009 361 south 45.7-60.0 15.2964 403718 0.0017 686 south 13921 south Transport between stations 32 and 39, 22048 meters apart 0-15.2 15.2150 335460 0.0145 4864 south 15.2-30.5 15.3023 337385 0.0068 2294 south 30.5-45.7 15.1971 335066 0.0021 704 south 45.7-60.0 15.2963 337253 0.0022 742 south 8604 south Transport between stations 39 and 40, 17703 meters apart 0-15.2 15.2146 269344 0.0058 1562 north 15.2-30.5 15.3014 270881 0.0030 813 north 30.5-45.7 15.1971 269034 0.0027 726 north 45.7-60.0 15.2962 270789 0.0027 731 north 3832 north Transport between stations 40 and 41A, 44096 meters apart 0-15.2 15.2154 670938 0.0009 604 north 15.2-30.5 - - 0 0 30.5-45.7 -0 0 45.7-60.0 - - 0 0 604 north Net transport, m3/sec 13557 south * Last decimal place only approximate. ** Last two places only approximate.

160 JOHN C. AYERS and Superior, in flowing through Lake the last week of July 1954 the outflow of the Huron, would have to cross dynamic height St. Clair River was 216,000 ft3/second contours in both the upper and lower ends (U. S. Lake Survey, Detroit, personal of the lake indicates that this survey covered communication). The order-of-magnitude a transient non-equilibrium condition in agreement between these figures is probthe lake. ably all that could be expected, for there The surface current directions of all are areas between the section ends and the three surveys were in excellent agreement shores, and below the reference level, where with the movements of drift bottles re- no transport figures could be obtained; in leased on the days of the surveys. Average addition, the Helland-Hansen substitucurrent velocities obtained by the dynamic tions in the bottoms of the end stations height method in June and July were not force the supposition of no currents or significantly different, statistically, from transports there. those of drift bottles released on those surveys. From the August survey, drift POTENTIAL ERRORS OF THE bottle returns were too few to permit METHOD velocity comparison. Sverdrup et al. (op. cit., pp. 393-4) review As a further check on the method, a the sources of potential error in the dynamic water transport estimate has been made height method, and every user of the for the 27 July section from Oscoda, Michi- method should be familiar with them. gan, to Southampton, Ontario (Fig. 4). If They include errors stemming from the the method is correct, summation of the neglect of frictional effects and errors due products of average velocity and area from to the slope of the reference surface. Curthe component parts of the section should rents obtained from dynamic computations give a rough measure of the volume of are relative to any currents at the reference water passing through the section. This level. In considering these relative curvolume was compared with the outflow of rents as absolute currents an error is inLake Huron via the St. Clair River. The troduced if the reference surface slopes distribution of dynamic heights across the (has currents). Errors from these sources section, average dynamic height increments appear to be small in the ocean, and (in in between-station blocks, and velocities the first approximation) may be considered between station pairs are shown in Figure small in lakes that are wide in comparison 4; the computations are summarized in to the inertia circle, whose radius is: r - Table 4. v/2w sin' (where v is maximum surface Current velocity was calculated at each velocity and 2w sin' is the double angular of the 0, 15.2, 30.5, 45.7, and 60.0 decibar velocity of the earth in that latitude). surfaces between each pair of stations in The presence of internal seiches or inthe section. An average velocity was ob- ternal waves during the period of observatained for each intercontour area between tion may be a source of potential error, but each pair of stations; each intercontour the works of Mortimer (1951, 1954, 1955) average dynamic height increment, AD, indicate that in large deep lakes the periods between each pair of stations was added to of such oscillations are long compared to the corresponding standard-condition depth the ordinary work-day. In the first apinterval, D, and multiplied by the distance proximation, internal seiches of large deep between stations to give the area of each lakes may be considered to cause little error intercontour between-station block; the in one-day synoptic surveys such as were average velocity in each block was multi- used on Lake Huron. Their effect on the plied by the block area; and the resulting accuracy of the method in smaller shallower volume transports were summed alge- lakes remains to be investigated. braically. The resulting estimate of trans- The most probable source of significant port through the section was about 13,500 error in applications of the dynamic height m3/second or about 473,000 ft3/second. In method to lakes appears to be the direct

DYNAMIC HEIGHT METHOD FOR LAKE CURRENTS 161 effect of the wind. Pronounced changes behaviors of other parameters (Ayers et al. in wind direction and/or velocity produce 1956). changes in the distribution of density, and significant accelerations (not taken into REFERENCES account by the method) may be present AMAGAT, E. H. 1893. Memoires sur l'elasticitd until equilibrium is regained. Prolonged et la dilation des fluides jusqu'aux tres hautes relatively constant winds may remove pressions. Ann. Chim. (Phys.), 29: 505-574. surface water from the up-wind shore and AYERS, J. C., D. C. CHANDLER, G. H. LAUFF, AND D. V. ANDERSON. 1956. The currents and pile it againstw the down-wind shore, im- water masses of Lake Huron. Great Lakes parting an'artificial' slope to the surface. Research Institute, University of Michigan, The lee-shore sinking and windward-shore Ann Arbor. (in press) upwelling produced by such winds may be BEYER, R. T. 1954. Formulae for sound veexpected to tilt subsurface isobaric surfaces locity in sea water. J. Mar. Res., 13: 113-121. HANDBOOK OF CHEMISTRY AND PHYSICS (C. D. in the opposite direction from the surface Hodgman, Ed.) 1949. 31st Ed. Chemical slope-further increasing the depth of Rubber Publ. Co., Cleveland, Ohio. xviii + warm surface water down-wind and de- 2737 pp. creasing it up-wind. Further studies on HARRINGTON, M. W. 1895. Surface currents of the Great Lakes, as deduced from the movethe effect of the wind, and upon the degree the Great Lakes, as deduced from the movethe eect o he, and pon te eee ments of bottle papers during the seasons of to which shallowness of lakes limits the 1892, 1893, and 1894. U. S. Weather Bureau, use of the method, are needed. Bulletin B (rev. ed.). HELLAND-HANSEN, BJ. 1934. The Sognefjord CONCLUSIONS section. pp 257-274 in James Johnstone Memorial Volume. Univ. Press, Liverpool. Although many potential sources of error 348 pp. are to be considered in applications of the LAFOND, E. C. 1951. Processing oceanographic dnmcsof them data. H. O. Publ. No. 614, U. S. Navy Hydynamic height method, most of them drog. Off., Washington, D. C. vi + 114 pp. appear to lead to relatively small errors and MILLAR, F. G. 1952. Surface temperatures of nearly all may be minimized by careful the Great Lakes. J. Fish. Res. Bd. Canada, planning and by choice of days for observa- 9: 329-376. tion. Used with proper caution, the dy- MORTIMER, C. H. 1951. Water movements in stratified lakes, deduced from observations in namic height method appears to be a Windermere and model experiments. Union promising technique for the investigation Gdodes. et Geophys. int., Assoc. int. d'Hyof certain circulation phenomena, in large drol. scientif., Assemblee g6n. de Bruxelles lakes at least. The method appears to 1951, 3: 336-349. for des t lermia. Tstion —. 1954. Models of the flow-pattern in be more satisfactory for determnatons lakes. Weather, 9:177-184. of surface current patterns and velocities -. 1955. Some effects of the earth's rotathan for subsurface water transport com- tion on water movements in stratified lakes. putations. Application of the method in Proc. Int. Assoc. Limnol., 12: 66-77. Lake Huron has produced logical and SVERDRUP, H. U., M. W. JOHNSON, AND R. H. FLEMING. 1942. The oceans, their physics, reasonable results which are both internally chemistry, and general biology. Prenticeconsistent and in good agreement with the Hall, New York. x + 1087 pp.

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