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gsw_SA_freezing_from_t.m
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gsw_SA_freezing_from_t.m
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function SA_freezing = gsw_SA_freezing_from_t(t,p,saturation_fraction)
% gsw_SA_freezing_from_t Absolute Salinity of seawater at the
% freezing temperature
%==========================================================================
%
% USAGE:
% SA_freezing = gsw_SA_freezing_from_t(t,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 in-situ temperature t 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 set to NaN.
%
% INPUT:
% t = in-situ 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 t is MxN.
%
% OUTPUT:
% SA_freezing = Absolute Salinity of seawater when it freezes, for
% given input values of in situ 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_t: Requires either two or three inputs')
end %if
if ~exist('saturation_fraction','var')
saturation_fraction = 0;
end
if (saturation_fraction < 0 | saturation_fraction > 1)
error('gsw_SA_freezing_from_t: saturation_fraction MUST be between zero and one.')
end
[mt,nt] = size(t);
[mp,np] = size(p);
[map,nap] = size(saturation_fraction);
if (mp == 1) & (np == 1) % p scalar - fill to size of t
p = p*ones(size(t));
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_t: Inputs array dimensions arguments do not agree')
end %if
if (map == 1) & (nap == 1) % saturation_fraction scalar
saturation_fraction = saturation_fraction*ones(size(t)); % fill to size of t
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_t: Inputs array dimensions arguments do not agree')
end %if
if mt == 1
t = t.';
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;
p0 = 2.570124672768757e-1;
p1 = -1.917742353032266e1;
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;
% Set the upper and lower limits where the TEOS-10 Gibbs function is defined.
t(t < -12 | t > 80 | p > 12000) = NaN;
p(t < -12 | t > 80 | p > 12000) = NaN;
SA = -(t + 9e-4*p)./0.06; % Firstly, this is a very rough estimate of SA
% simply to get the saturated CT, CTsat.
SA(SA < 0) = 0; % This line ensures that SA is non-negative.
CT = gsw_CT_from_t(SA,t,p);
% 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));
% This is the inital guess of SA using a purpose-built polynomial in CTsat and p.
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));
t_freezing_zero_SA = gsw_t_freezing(zeros(size(t)),p,saturation_fraction);
% Find t > t_freezing_zero_SA. If this is the case, the input values
% represent seawater that is not frozen (at any positive SA).
[Itw] = find(t > t_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 (t_freezing_zero_SA - t).
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 tfreezing_SA, the derivative of CT_freezing
% with respect to SA at fixed p.
%--------------------------------------------------------------------------
[tfreezing_SA, dummy] = gsw_t_freezing_first_derivatives(SA,p,saturation_fraction);
% Now replace the estimate of SA with the one based on
% (t_freezing_zero_SA - t) when (abs(SA) < SA_cut_off).
if ~isempty(Ico)
SA(Ico) = (t(Ico) - t_freezing_zero_SA(Ico))./tfreezing_SA(Ico);
end
%---------------------------------------------------------------------------
% Begin the modified Newton-Raphson method to find the root of
% f = (t_freezing - t) = 0 for SA.
%---------------------------------------------------------------------------
Number_of_Iterations = 2;
for I_iter = 1:Number_of_Iterations
SA_old = SA;
f = gsw_t_freezing(SA_old,p,saturation_fraction) - t;
SA = SA_old - f./tfreezing_SA;
% This is the half-way point of the modified Newton-Raphson
% method of McDougall and Wotherspoon (2013).
SA_mean = 0.5*(SA + SA_old);
[tfreezing_SA, dummy] = gsw_t_freezing_first_derivatives(SA_mean,p,saturation_fraction);
SA = SA_old - f./tfreezing_SA;
end
%--------------------------------------------------------------------------
% The following lines of code, if implemented, calculate the error of
% this function in terms of in-situ temperature, t.
% After 2 iterations the maximum error in t is 4.2x10^-13 C and in SA it is
% 8x10^-12 g/kg.
%
% SA(SA < 0) = 0;
%
% t_freezing = gsw_t_freezing(SA,p,saturation_fraction);
%
% t_error = abs(t_freezing - t);
% t_error(p > 10000 | SA > 120 | ...
% p + SA.*71.428571428571402 > 13571.42857142857) = NaN;
% if ~isempty(Itw)
% t_error(Itw) = NaN; % If the t input is too warm, then there is
% % no (positive) value of SA that represents frozen seawater.
% end
%-----------------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 t input is too warm, then there is
% no (positive) value of SA that represents frozen seawater.
end
if transposed
SA_freezing = SA_freezing.';
end
end