-
Notifications
You must be signed in to change notification settings - Fork 0
/
gsw_SA_freezing_from_CT_poly.m
306 lines (271 loc) · 12.1 KB
/
gsw_SA_freezing_from_CT_poly.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
function SA_freezing = gsw_SA_freezing_from_CT_poly(CT,p,saturation_fraction)
% gsw_SA_freezing_from_CT_poly Absolute Salinity of seawater at
% the freezing point (poly)
%==========================================================================
%
% USAGE:
% SA_freezing = gsw_SA_freezing_from_CT_poly(CT,p,saturation_fraction)
%
% DESCRIPTION:
% Calculates the Absolute Salinity of seawater at the freezing temperature.
% That is, the output is the Absolute Salinity of seawater, with the
% fraction saturation_fraction of dissolved air, that is in equilibrium
% with ice at Conservative Temperature CT and pressure p. If the input
% values are such that there is no positive value of Absolute Salinity for
% which seawater is frozen, the output, SA_freezing, is put equal to NaN.
%
% INPUT:
% CT = Conservative Temperature (ITS-90) [ deg C ]
% p = sea pressure [ dbar ]
% ( i.e. absolute pressure - 10.1325 dbar )
%
% OPTIONAL:
% saturation_fraction = the saturation fraction of dissolved air in
% seawater
% (i.e., saturation_fraction must be between 0 and 1, and the default
% is 0, air free)
%
% p & saturation_fraction (if provided) may have dimensions 1x1 or Mx1 or
% 1xN or MxN, where CT is MxN.
%
% OUTPUT:
% SA_freezing = Absolute Salinity of seawater when it freezes, for
% given input values of Conservative Temperature
% pressure and air saturation fraction. [ g/kg ]
%
% AUTHOR:
% Trevor McDougall and Paul Barker [ help@teos-10.org ]
%
% VERSION NUMBER: 3.05 (27th January 2015)
%
% REFERENCES:
% IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of
% seawater - 2010: Calculation and use of thermodynamic properties.
% Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
% UNESCO (English), 196 pp. Available from http://www.TEOS-10.org.
% See section 3.33 of this TEOS-10 Manual.
%
% McDougall, T.J., P.M. Barker, R. Feistel and B.K. Galton-Fenzi, 2014:
% Melting of Ice and Sea Ice into Seawater and Frazil Ice Formation.
% Journal of Physical Oceanography, 44, 1751-1775.
%
% McDougall T. J. and S. J. Wotherspoon, 2013: A simple modification of
% Newton's method to achieve convergence of order 1 + sqrt(2). Applied
% Mathematics Letters, 29, 20-25.
%
% The software is available from http://www.TEOS-10.org
%
%==========================================================================
%--------------------------------------------------------------------------
% Check variables and resize if necessary
%--------------------------------------------------------------------------
if ~(nargin == 2 | nargin == 3)
error('gsw_SA_freezing_from_CT_poly: Requires either two or three inputs')
end
if ~exist('saturation_fraction','var')
saturation_fraction = 0;
end
if (saturation_fraction < 0 | saturation_fraction > 1)
error('gsw_SA_freezing_from_CT_poly: saturation_fraction MUST be between zero and one.')
end
[mt,nt] = size(CT);
[mp,np] = size(p);
[map,nap] = size(saturation_fraction);
if (mp == 1) & (np == 1) % p scalar - fill to size of CT
p = p*ones(size(CT));
elseif (nt == np) & (mp == 1) % p is row vector,
p = p(ones(1,mt), :); % copy down each column.
elseif (mt == mp) & (np == 1) % p is column vector,
p = p(:,ones(1,nt)); % copy across each row.
elseif (nt == mp) & (np == 1) % p is a transposed row vector,
p = p.'; % transposed then
p = p(ones(1,mt), :); % copy down each column.
elseif (mt == mp) & (nt == np)
% ok
else
error('gsw_SA_freezing_from_CT_poly: Inputs array dimensions arguments do not agree')
end %if
if (map == 1) & (nap == 1) % saturation_fraction scalar
saturation_fraction = saturation_fraction*ones(size(CT)); % fill to size of CT
elseif (nt == nap) & (map == 1) % saturation_fraction is row vector,
saturation_fraction = saturation_fraction(ones(1,mt), :); % copy down each column.
elseif (mt == map) & (nap == 1) % saturation_fraction is column vector,
saturation_fraction = saturation_fraction(:,ones(1,nt)); % copy across each row.
elseif (nt == map) & (nap == 1) % saturation_fraction is a transposed row vector,
saturation_fraction = saturation_fraction.'; % transposed then
saturation_fraction = saturation_fraction(ones(1,mt), :); % copy down each column.
elseif (mt == map) & (nt == nap)
% ok
else
error('gsw_SA_freezing_from_CT_poly: Inputs array dimensions arguments do not agree')
end %if
if mt == 1
CT = CT.';
p = p.';
saturation_fraction = saturation_fraction.';
transposed = 1;
else
transposed = 0;
end
%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------
aa = 0.014289763856964; % Note that aa = 0.502500117621/35.16504.
b = 0.057000649899720;
c0 = 0.017947064327968736;
%
c1 = -6.076099099929818;
c2 = 4.883198653547851;
c3 = -11.88081601230542;
c4 = 13.34658511480257;
c5 = -8.722761043208607;
c6 = 2.082038908808201;
%
c7 = -7.389420998107497;
c8 = -2.110913185058476;
c9 = 0.2295491578006229;
%
c10 = -0.9891538123307282;
c11 = -0.08987150128406496;
c12 = 0.3831132432071728;
c13 = 1.054318231187074;
c14 = 1.065556599652796;
c15 = -0.7997496801694032;
c16 = 0.3850133554097069;
c17 = -2.078616693017569;
c18 = 0.8756340772729538;
c19 = -2.079022768390933;
c20 = 1.596435439942262;
c21 = 0.1338002171109174;
c22 = 1.242891021876471;
p0 = 2.570124672768757e-1;
p1 = -1.917742353032266e+1;
p2 = -1.413382858617969e-2;
p3 = -5.427484830917552e-1;
p4 = -4.126621135193472e-4;
p5 = -4.176407833276121e-7;
p6 = 4.688217641883641e-5;
p7 = -3.039808885885726e-8;
p8 = -4.990118091261456e-11;
p9 = -9.733920711119464e-9;
p10 = -7.723324202726337e-12;
p11 = 7.121854166249257e-16;
p12 = 1.256474634100811e-12;
p13 = 2.105103897918125e-15;
p14 = 8.663811778227171e-19;
p_r = p.*1e-4;
% Form the first estimate of SA_freezing from a polynomial in CT and p_r.
SA = -(CT + 9*p_r)./0.06; % A rough estimate to get the saturated CT.
SA(SA < 0) = 0;
% CTsat is the estimated value of CT if the seawater were saturated with
% dissolved air, recognizing that it actually has the air fraction
% saturation_fraction; see McDougall et al. (2014).
CTsat = CT ...
- (1-saturation_fraction).*(1e-3).*(2.4-aa.*SA).*(1+b.*(1-SA./35.16504));
SA = p0 + p.*(p2 + p4*CTsat + p.*(p5 + CTsat.*(p7 + p9*CTsat) ...
+ p.*(p8 + CTsat.*(p10 + p12*CTsat) + p.*(p11 + p13*CTsat + p14*p)))) ...
+ CTsat.*(p1 + CTsat.*(p3 + p6*p));
CT_freezing_zero_SA = c0 + p_r.*(c7 + p_r.*(c8 + c9.*p_r)) ...
- saturation_fraction.*(2.4e-3).*(1 + b);
% Find CT > CT_freezing_zero_SA. If this is the case, the input values
% represent seawater that is not frozen (at any positive SA).
[Itw] = find(CT > CT_freezing_zero_SA); % Itw stands for "I_too_warm"
if ~isempty(Itw)
SA(Itw) = NaN;
end
% Find -SA_cut_off < SA < SA_cut_off, replace the above estimate of SA
% with one based on (CT_freezing_zero_SA - CT).
SA_cut_off = 2.5; % This is the band of SA within +- 2.5 g/kg of SA = 0,
% which we treat differently in calculating the initial
% values of both SA and dCT_dSA.
[Ico] = find(abs(SA) < SA_cut_off);
SA(SA < 0 & SA >= -SA_cut_off) = 0;
% Find SA < -SA_cut_off, set them to NaN.
SA(SA < -SA_cut_off) = NaN;
%--------------------------------------------------------------------------
% Form the first estimate of dCT_dSA, the derivative of CT with respect
% to SA at fixed p.
%--------------------------------------------------------------------------
SA_r = 0.01*SA;
x = sqrt(SA_r);
dCT_dSA_part = 2*c1 + x.*(3*c2 + x.*(4*c3 + x.*(5*c4 + x.*(6*c5 + 7*c6.*x))))...
+ p_r.*(2*c10 + p_r.*(2*c12 + p_r.*(2*c15 + 4*c21.*x.*x))...
+ x.*x.*(4*c13 + 4*c17.*p_r + 6*c19.*x.*x)...
+ x.*(3*c11 + 3*p_r.*(c14 + c18.*p_r)...
+ x.*x.*(5*c16 + 5*c20.*p_r + 7*c22.*x.*x)));
dCT_dSA = 0.5*0.01*dCT_dSA_part ...
- saturation_fraction.*(1e-3).*(-aa.*(1 + b.*(1-SA./35.16504)) ...
- b.*(2.4 - aa.*SA)./35.16504);
% Now replace the estimate of SA with the one based on
% (CT_freezing_zero_SA - CT) when (abs(SA) < SA_cut_off).
if ~isempty(Ico)
SA(Ico) = (CT(Ico) - CT_freezing_zero_SA(Ico))./dCT_dSA(Ico);
end
%--------------------------------------------------------------------------
% Begin the modified Newton-Raphson method to solve the root of
% CT_freezing = CT for SA.
%--------------------------------------------------------------------------
Number_of_Iterations = 2;
for I_iter = 1:Number_of_Iterations
%--------------------------------------------------------------------------
% CT_freezing temperature function evaluation (the forward function
% evaluation), being the same as gsw_CT_freezing_poly(SA,p,saturation_fraction).
%--------------------------------------------------------------------------
SA_r = 0.01*SA;
x = sqrt(SA_r);
SA_old = SA;
CT_freezing = c0 + SA_r.*(c1 + x.*(c2 + x.*(c3 + x.*(c4 + x.*(c5 + c6.*x))))) ...
+ p_r.*(c7 + p_r.*(c8 + c9.*p_r)) + SA_r.*p_r.*(c10 + p_r.*(c12 + p_r.*(c15 ...
+ c21.*SA_r)) + SA_r.*(c13 + c17.*p_r + c19.*SA_r) + x.*(c11 + p_r.*(c14 ...
+ c18.*p_r) + SA_r.*(c16 + c20.*p_r + c22.*SA_r))) ...
- saturation_fraction.*(1e-3).*(2.4 - aa.*SA).*(1 + b.*(1 - SA./35.16504));
SA = SA_old - (CT_freezing - CT)./dCT_dSA;
% This is the half-way point of the modified Newton-Raphson
% method of McDougall and Wotherspoon (2013).
SA_r = 0.5*0.01*(SA + SA_old); % This is now the mean value of SA and SA_old.
x = sqrt(SA_r);
dCT_dSA_part = 2*c1 + x.*(3*c2 + x.*(4*c3 + x.*(5*c4 + x.*(6*c5 + 7*c6.*x))))...
+ p_r.*(2*c10 + p_r.*(2*c12 + p_r.*(2*c15 + 4*c21.*x.*x))...
+ x.*x.*(4*c13 + 4*c17.*p_r + 6*c19.*x.*x) + x.*(3*c11 + 3*p_r.*(c14 ...
+ c18.*p_r) + x.*x.*(5*c16 + 5*c20.*p_r + 7*c22.*x.*x)));
dCT_dSA = 0.5*0.01*dCT_dSA_part ...
- saturation_fraction.*(1e-3).*(-aa.*(1 + b.*(1-SA./35.16504)) ...
- b.*(2.4 - aa.*SA)./35.16504);
SA = SA_old - (CT_freezing - CT)./dCT_dSA;
% This is the end of one full iteration of the modified
% Newton-Raphson method of McDougall and Wotherspoon (2013).
end
%--------------------------------------------------------------------------
% The following lines of code, if implemented, calculates the error of
% this function in terms of Conservative Temperature, CT_error.
% With Number_of_Iterations = 1, the maximum error in CT is 2x10^-7 C.
% With Number_of_Iterations = 2, the maximum error in CT is 7x10^-15 C,
% which is the machine precision of the computer.
% Number_of_Iterations = 2 is what we recommend.
%
% SA_r = 0.01*SA;
% x = sqrt(SA_r);
% CT_freezing = c0 ...
% + SA_r.*(c1 + x.*(c2 + x.*(c3 + x.*(c4 + x.*(c5 + c6.*x))))) ...
% + p_r.*(c7 + p_r.*(c8 + c9.*p_r)) ...
% + SA_r.*p_r.*(c10 + p_r.*(c12 + p_r.*(c15 + c21.*SA_r)) + SA_r.*(c13 + c17.*p_r + c19.*SA_r) ...
% + x.*(c11 + p_r.*(c14 + c18.*p_r) + SA_r.*(c16 + c20.*p_r + c22.*SA_r))) ...
% - saturation_fraction.*(1e-3).*(2.4 - aa.*SA).*(1 + b.*(1 - SA./35.16504));
%
% CT_error = abs(CT_freezing - CT);
%
% CT_error(p > 10000 | SA > 120 | ...
% p + SA.*71.428571428571402 > 13571.42857142857) = NaN;
%
%--------------------This is the end of the error calculation--------------
SA_freezing = SA;
%Find SA that are out of range, set them to NaN.
SA_freezing(p > 10000 | SA > 120 | ...
p + SA.*71.428571428571402 > 13571.42857142857) = NaN;
if ~isempty(Itw)
SA_freezing(Itw) = NaN; % If the CT input is too warm, then there is
end % no (positive) value of SA that represents frozen seawater.
if transposed
SA_freezing = SA_freezing.';
end
end