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gsw_t_freezing_poly.m
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function t_freezing = gsw_t_freezing_poly(SA,p,saturation_fraction)
% gsw_t_freezing_poly in-situ temperature at which seawater freezes
% (poly)
%==========================================================================
%
% USAGE:
% t_freezing = gsw_t_freezing_poly(SA,p,saturation_fraction)
%
% DESCRIPTION:
% Calculates the in-situ temperature at which seawater freezes from a
% comptationally efficient polynomial.
%
% INPUT:
% SA = Absolute Salinity [ g/kg ]
% 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 SA is MxN.
%
% OUTPUT:
% t_freezing = in-situ temperature at which seawater freezes. [ deg C ]
% (ITS-90)
%
% AUTHOR:
% Trevor McDougall, Paul Barker and Rainer Feistal [ 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 sections 3.33 and 3.34 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.
%
% The software is available from http://www.TEOS-10.org
%
%==========================================================================
%--------------------------------------------------------------------------
% Check variables and resize if necessary
%--------------------------------------------------------------------------
if ~(nargin == 2 | nargin == 3)
error('gsw_t_freezing_poly: 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_t_freezing_poly: saturation_fraction MUST be between zero and one.')
end
[ms,ns] = size(SA);
[mp,np] = size(p);
[msf,nsf] = size(saturation_fraction);
if (mp == 1) & (np == 1) % p scalar - fill to size of SA
p = p*ones(size(SA));
elseif (ns == np) & (mp == 1) % p is row vector,
p = p(ones(1,ms), :); % copy down each column.
elseif (ms == mp) & (np == 1) % p is column vector,
p = p(:,ones(1,ns)); % copy across each row.
elseif (ns == mp) & (np == 1) % p is a transposed row vector,
p = p.'; % transposed then
p = p(ones(1,ms), :); % copy down each column.
elseif (ms == mp) & (ns == np)
% ok
else
error('gsw_t_freezing_poly: Inputs array dimensions arguments do not agree')
end
if (msf == 1) & (nsf == 1) % saturation_fraction scalar
saturation_fraction = saturation_fraction*ones(size(SA)); % fill to size of SA
elseif (ns == nsf) & (msf == 1) % saturation_fraction is row vector,
saturation_fraction = saturation_fraction(ones(1,ms), :); % copy down each column.
elseif (ms == msf) & (nsf == 1) % saturation_fraction is column vector,
saturation_fraction = saturation_fraction(:,ones(1,ns)); % copy across each row.
elseif (ns == msf) & (nsf == 1) % saturation_fraction is a transposed row vector,
saturation_fraction = saturation_fraction.'; % transposed then
saturation_fraction = saturation_fraction(ones(1,ms), :); % copy down each column.
elseif (ms == msf) & (ns == nsf)
% ok
else
error('gsw_t_freezing_poly: Inputs array dimensions arguments do not agree')
end
if ms == 1
SA = SA.';
p = p.';
saturation_fraction = saturation_fraction.';
transposed = 1;
else
transposed = 0;
end
%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------
SA(SA < 0) = 0; % This line ensure that SA is non-negative.
CT_freezing = gsw_CT_freezing_poly(SA,p,saturation_fraction);
t_freezing = gsw_t_from_CT(SA,CT_freezing,p);
%--------------------------------------------------------------------------
% This function, gsw_t_freezing_poly, calculates the in-situ freezing
% temperature, t_freezing, of seawater by first evaluating a polynomial of
% the Conservative Temperature at which seawater freezes, CT_freezing,
% using the GSW function gsw_CT_freezing_poly. The in-situ freezing temperature
% is then calculated using the GSW function gsw_t_from_CT. However, if one
% wanted to compute the in-situ freezing temperature directly from a single
% polynomial expression without first calculating the Conservative
% Temperature at the freezing point, the following lines of code achieve
% this. The error of the following fit is similar to that of the present
% function, gsw_t_freezing_poly, and ranges between -8e-4 K and 3e-4 K when
% compared with the in-situ freezing temperature evaluated by Newton-
% Raphson iteration of the equality of the chemical potentials of water in
% seawater and in ice. (Note that the in-situ freezing temperature can be
% found by this exact method using the function gsw_t_freezing).
%
% c0 = 0.002519;
%
% c1 = -5.946302841607319;
% c2 = 4.136051661346983;
% c3 = -1.115150523403847e1;
% c4 = 1.476878746184548e1;
% c5 = -1.088873263630961e1;
% c6 = 2.961018839640730;
%
% c7 = -7.433320943962606;
% c8 = -1.561578562479883;
% c9 = 4.073774363480365e-2;
%
% c10 = 1.158414435887717e-2;
% c11 = -4.122639292422863e-1;
% c12 = -1.123186915628260e-1;
% c13 = 5.715012685553502e-1;
% c14 = 2.021682115652684e-1;
% c15 = 4.140574258089767e-2;
% c16 = -6.034228641903586e-1;
% c17 = -1.205825928146808e-2;
% c18 = -2.812172968619369e-1;
% c19 = 1.877244474023750e-2;
% c20 = -1.204395563789007e-1;
% c21 = 2.349147739749606e-1;
% c22 = 2.748444541144219e-3;
%
% SA_r = SA.*1e-2;
% x = sqrt(SA_r);
% p_r = p.*1e-4;
%
% t_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)));
%
% Adjust for the effects of dissolved air
% t_freezing = t_freezing - saturation_fraction.*(1e-3).*(2.4 - SA./70.33008);
%
%---------------This is the end of the alternative code--------------------
t_freezing(p > 10000 | SA > 120 | ...
p + SA.*71.428571428571402 > 13571.42857142857) = NaN;
if transposed
t_freezing = t_freezing.';
end
end