TECHNICAL REPORT AN ANALYTICAL METHOD OF ICE POTENTIAL CALCULATION ALLEN L. BROWN Applied Oceanography Branch Division of Oceanography SEPTEMBER 1954 U. S. NAVY HYDROGRAPHIC OFFICE WASHINGTON, D. C. ABSTRACT es Techniques for computing the ice potential as developed by Zubov and Defant are tedious and laborious. In this report these techniques are examined analytically and a simplified rapid method of computation is developed. This new method enables the long-range ice forecaster to limit the detailed analysis of oceanographic data to those locations where ice formation is indicated. FOREWORD The increasing importance of defense installations in northern areas has increased greatly the responsibilities of the U. S. Navy in supplying bases in Arctic waters, where sea ice is often an oper- The Hydrographic Office is charged with the respon— ating obstacle. sibility of developing and testing techniques for observing and fore- casting sea ice conditions. Standardized techniques for observing, charting, and reporting sea ice are now in operational use by the Navy, as described in publications issued by the Hydrographic Office. Heretofore, techniques for forecasting the formation, growth, and movement of sea ice have not been published by this Office. This publication describes a method of long-range forecasting of ice for- Since this technique is still in the develop- mation and growth. mental stage, the Hydrographic Office welcomes comments as to its operational value. -_ HO ew CEE CP: B. COCHRAN ‘-“ Captain, U. S. Navy Hydrographer I mn NO DISTRIBUTION LIST CNO (Op-03, 03D3, 31, 316, 32, 33, 332, Oh, 05, 533, 55) BUAER (2) BUSHIPS (2) BUDOCKS (2) ONR (Code 100, 102, 410, 416, 420, 430, 461, 466) NOL (2) NEL (2) NRL (2) COMOPDEVFOR (2) COMSTS (2) CODTMB (2) AROWA SUPNAVACAD (2) NAVWARCOL (2) NAVPOSTGRADSCOL, Monterey (2) COMDT COGARD (ITIP) (2) USC&GS (2) CG USAF (AFOOP) CGAWS (2) CGNEAC (2) USAF CAMBRSCHLAB (2) USWB (2) CIA (2) BEB (2) SIPRE (2) ASTIA (2) ARTRANSCORP CE (2) INTLHYDROBU, Monaco (2) ARCRSCHLAB, COL, Alaska ARCINSTNA (2) WHOL (2) SIO (2) UNIV WASH (2) TEXAS A&M (2) CBI (2) iv CONTENTS Page Foreword e e e e e e e e e e e ° e e ° oe e e e e e e e e e 414 Dilsusaloruslen IWS 6565 60000000000 0000000 iv Figures eye (ee. 6) 0. '@ ete e> Hor 6s” 36, 6) e e OF=.67, (6). 6) @" 10: (6: 6 e Vv Tables e e eo e oe e es oe e e e e e e e e e e e e e e e e e e Vv A oe Int rodu ct fon e e e e e e e e & e e e oe e e e e e e e e 1 B. Derivation of the Ice Potential for Salinity 2 2h. 70 ©/o0 e e e e e e e e e e e e e e e e ° e e e AL C. Derivatioh of the Ice Potential for Salinity SANTO MO Oh eee Doe ea ee ke PRS Lh De Conclusions e eo e e e e e e e e e e e °° e ° e e e e e 7 Bibliography e e e e e e ° e e e e e e e e e e e e e e e e 8 Appendix I. Derivation of a Forma for the Sensible Heat Loss Qp(h) Associated with Convection to h as a Result Git ISS Rormneilo 6 65 4 4.6 5 0.0.0 4.50.06 0.0 0-5 i] II. Freezing Temperatures and Densities for Sea Water, Vralge ay IMAG 6 GG. ol Ono: OO) DOO sO BO en soe Gn 10 III. Exact Solution of the Ice Potential Equation . . © 13 FIGURES Figure 1 ae Determination Graph for S 221.70 °/o0, 85 3 Figure 2 Ice Determination Graph for S <2h4.70 °/o09, k= .85 6 TABLES Table 1 Temperature and Density of Freezing for Sea Water of Higher Salinities ale Table 2 Freezing and Maximum Densities with Associated Temperature for Sea Water of Lower Salinities 12 He a MOP CGC a ertia | A. INTRODUCTION The Zubov—Defant ice potential computation technique, although not difficult, is quite tedious and in essence graphical or tabular. Further more, the results of the computation may indicate that ice formation is very improbable. Clearly there is need for a technique that will indicate with considerably less labor both possible ice formation and the quantity of ice. In addition the technique should be placed on a simple analytic basis to eliminate the need for several large-scale density nomographs or bulky tables, B. DERIVATION OF THE ICE POTENTIAL FOR SALINITY > 24.70 ©/o0 When the salinity of sea water equals or exceeds 24.70 °/o0, the maximum density occurs at the temperature of freezing, i, = — 0.0985 Cl. -OoONok2 Grn (1) where Ele= CSI 020310) 7-805: (2) It can be shown that the results of thermohaline convection, which are obtained by mixing of infinitesimal layers of sea water from the sur- face to a depth h, are identical with those obtained when mixing is con- sidered for the entire layer from the surface to h.* The density of freezing Of (h) , for a column of mixed water of vy F) depth h and with mean salinity S o.h. Where ? ? 4 h Sah = Tea S (2) aedizgeeny (3) ? is so nearly linear (i.e. the error is less than+0.01) that we can express it as o, (h) = % K (So hd» Soh = 0.8104 S, ,—0.1600. (,) Yhen considering a graphic representation of the freezing density (Fig. 1), it is quite apparent that ice must be formed to attain thermo- haline mixing of a column of water of depth h, mean salinity So h? i Cars and a density at h, OF i [T¢h),S(h) ) if Th is greater than “f(h). Le oT The graphical equivalent of this statement is that the point ( So,h: h) ? *See Appendix I plots above the line (.000) in Figure 1 which indicates the density of freezing. In the case of the first point (7%, : 5 So, h, ) shown in the figure, it is evident that the column of! water(9, h, oan be given a uniform density ( Th, )by temperature che iges alone ‘and that no freezing results, However, for the column of water(o , ho) with coordinates (Shes Sa, ho ) temperature changes alone can take it only to the line of the density of freezing. In order to obtain the uniform density (Ones a change in the mean salinity (A So, ho) is needed, From the geometry it is clear that ASoh=("h— 7) ton a,; (5) but % = Soh cota, +b, ; (6) therefore, ASo,h =(7h—5)) tan a,- Soh (7) A mean salinity change of ASoh has (Defant, 1949) an ice equivalent of hAS fe) é nih = *PL)So he (8) Pw whe re Pi /Pw (the ratio of the density of ice to that of sea water) is taken to be 0.9, and where k is the proportional part of the salt re- leased from the sea water that is frozen (k averages about 0.85). Equation 8 becomes lon i (hy n(ty [tf b= btane, — 5 | (9) ‘ Soih or in numerical units 137.14( [+ 9:1600 ) So, h where 1; is expressed in centimeters, h in meters, So, h in 0o/o9, and Th in density units (P—1) x 10°. (bh) =h (4) [ =m], (10) *See Appendix III G8 = '°%OLb2 2S YOS HdVYD NOILVNIWNSL3G 3011 3YNdIS (°%) ALINITVS ZE 9E Se ve €€ ce TE O€ 6¢ 8c Le 9¢ Gc ve (7+) ALISN3G C. DERIVATION OF THE ICE POTENTIAL FOR SALINITY < 24.70 °/oo When S < 24.70 °/o0, the temperature of maximum density is higher than the freezing temperature, so it is possible for convection to reach a depth,z equal h,under two sets of conditions for some values of %(h). The following represent all the mutually exclusive conditions in ane water column (0,h) that have a bearing on convection: 0, (h) > Oma (Sop) (11) (ies (h)Z omay (SoH) (12) Toh® T¢ (S, h) 9 O% (h)= 0% (Sp), (13) di So sh) , (5, ,)4-ene mae (ere h)< FAN ) 0.8075 S, ,— 0.087, (21) G8 = ‘°%OL' 25S YOS HdVYO NOIWNINYSLSG 391 (°%) ALINIIVS ve ce og 8T 91 vas er Ot @ AYNdI4 (4) ALISNAG there is implied in order to achieve convective mixing to a depth h, an ice equivalent + h)+0.087 a 130.72| (22) * 1,(h)=h tet. 88 { So,h where the numerical units of h, 9% (t), a. h and 1;(h) are defined as they were for (10). : D. CONCLUSIONS This new approach to the ice potential computation technique has several advantages: (a) it is simpler and speedier than the ee method; (b) one can see immediately, by plotting the points (Th, Sys h) whether freezing or freezing with subsequent melting occurs as a roetit of thermohaline convection; and (c) it is easy to estimate the depth of mixing for which ereerere initially occurs, When the curve, generated by the points (7h, S of h)» remains parallel to a constant percentage line (constant 1;/h)*’or it turns back toward the density of the freezing line, tremendous quantitites of thermal energy are involved for addi- tional increases in the ice thickness. One can decide what depths of mixing are of interest (usually that of the initial freeze and/or that for some specified ice thickness) and compute the associated Q'S (sensible heat losses) alone by the resultant formula proved in APPENDIX I. * See Appendix IIT * For plotting lines of constant s/o on either "Ice Determination ae the following relation is obvious: oy = cot a[(.9k)(1,/h) +1] Sy, +b (23) ie BIBLIOGRAPHY DEFANT, A. 1949. "Convection and Ice Potential In the Polar Shelf Seas." Geografiska Annaler, Heft XXXI, pp. 25-35 SVERDRUP, H. Aey JOHNSON, M.W., AND FLEMING, R.H. The Oceans, Their Physics, Chemistry, and General Biology. New York: Prentice = Hall, 1942. ZUBOV, N. N. 1938. Morskie Vody i L'dy (Marine Water and Ice). Moskva Gidrometeoizdat, 453 pp. 1938 APPENDIX I DERIVATION OF A FORMULA FOR THE SENSIBLE HEAT LOSS Qp(h) ASSOCIATED WITH CONVECTION TO h AS A RESULT OF ICE FORMATION Since 1;(h) is in general small in comparison with h, Qr(h) is very nearly h —_— — Qe (n)=f" Cw (2) Pu (Z) {T(Z)— Ty( So,n+ASon)} d2 where cy(z) and P wh2) are respectively the. specific heat and density of the sea water at the depth z. Now the product cyw(z) Pw(z) is relatively constant for all sea waters and AS, yp, as a resujt of the initial assumption, must be. small in comparison with So,he Hence, ove has Qr-(h) CwPw h[To,h - Ty (So,h)]. APPENDIX IT FREEZING TEMPERATURES AND DENSITIES FOR SEA WATER Tables 1 and 2 demonstrate the linear relationship between salinity and density of freezing and between salinity and maximm density, respectively. The temperatures of freezing were computed by equation 1. The densities of freezing were determined from the tables in H. 0. Pub. No. 615 using salinity and temperature of freezing as the arguments. The maximum densities were obtained by inspecting the same tables for maximum values at the indicated salinity. The temperatures of maximum density are computed by the approximate formula by Sverdrup. T (Spay) = 400°C. - 0.215 S. It will be noted in Table 2, that a range of temperatures of maximum density actually satisfies the maximum density value to three decimal places in %,. However, the numerical accuracy that has been maintained in Tables 1 and 2 is greater than the optimum accuracy in actual practice. 10 TABLE 1 TEMPERATURE AND DENSITY OF FREEZING FOR SEA WATER OF HIGHER SALINITIES Salinity Temperature of Freezing Density of Freezing S °/oo Te(S) (°C) (Tg, 8) A 2h, -1.30 19.290 808 25 -1.35 20.098 808 26 -1.41 20.906 809 27 ~1.46 21.715 809 28 -1.52 22052h 809 29 ~1.57 23 333 810 30 -1.63 2h. 143 810 Sit -1.68 2h,.953 811 32 -1.74 25.764 812 33 -1.80 26.576 812 3h -1.85 27.388 B1h, 35 -1.91 28.202 813 36 -1.97 29.015 aE TABLE! 2 FREEZING AND MAXIMUM DENSITIES WITH THE ASSOCIATED TEMPERATURES FOR SEA WATER OF LOWER SALINITIES Temperature Freezing Maximum Temperature Practical! Salinity of Freezing Density Density of Maximum Range of Ss °/oo0 ° o Density o ® / Te (°C) Cp max romance) 76 mex) (°C) A A (@) -0.00 -0,132 0.000 4.00 Be Th 4.23 1 -0.06 0.717 849 0.839 839 3.78 Blobs} BloSyh ~ 2 -0.11 1.528 811 1.639 800 3.57 3.28 3.83 3 -0.17 20337 809 2.439 800 3.36 3.02 3.67 4 -0.22 3.147 810 3.240 801 3.14 3.04 3.23 5 -0.27 3.956 809 4.040 800 2.92 2075 3.09 6 -0. 33 4.76h 808 4.840 800 2071 2.49 2.94 7 -0.38 52573 807 5.6h0 800 20h9 2621 2019 8 -0.43 6.380 807 6.441 801 2.28 2.27 862230 9 -0.49 7.188 808 7.2h1 800 2.06 ol Bo BL 10 -0.54 7.995 807 8.042 801 1.85 Sh dS wal. -0.59 8.802 807 8.842 &00 1.64 1.30 1.99 12 -0.65 9.609 807 9.644 802 1.42 1.33 1.54 13 -0, 70 10.416 807 10.445 801 1.20 0.97 1.45 Wh, -0.75 11.222 806 11.247 802 0.99 0.83 dol} 15 -0.81 12.029 807 12.049 802 0.78 0.58 0.97 16 -0.86 12.835 806 12.851 802 0.56 0.24 0.87 a7, -0.92 13.642 807 13.654 803 0.34 0.0h 0.64 18 -0.97 14.448 806 14.458 804 0.13 =0.01 0.26 19 -1.02 15-255 807 15.262 804 -0.08 0.19 0.00 20 -1.08 16.061 806 16.066 804 -0.30 -0.57 -0.05 Pole =-1.13 16.868 807 16.871 805 -0.51 -0.80 -0.26 22 -1.19 17.675 807 17.677 806 -0.73 ~-O.94 -0.56 23 -1.2) 18.483 808 18.483 806 -0.94 1.24 -0.69 2h, -1.30 19.290 807 19.290 807 -1.16 -1.47 -0.90 25 -1.35 20.098 808 20.098 808 -1.38 -1.62 -1.18 12 APPENDIX IIT EXACT SOLUTION OF THE ICE POTENTIAL EQUATION Defant and Zubov have both shown that for the formation of 14 centimeters of ice, when convection has reached a depth of h centi- meters, the change in the mean salinity of the column of water (o0,h), must be (FL Ss Pw) ks (3.1) where k is the proportional part of the salt released from the sea water that is frozen. If 1]; 1s small in comparison with h, a first approximation to 1), lo, is P Ae. Saiyan lo=h (24 Sop (3.2) Kai, Denote r = ome (+2) : (3-8) Then r is an index of the pergentage error in the first approximation to lj. 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