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gsw_t_freezing.m
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function t_freezing = gsw_t_freezing(SA,p,saturation_fraction)
% gsw_t_freezing in-situ temperature at which seawater freezes
%==========================================================================
%
% USAGE:
% t_freezing = gsw_t_freezing(SA,p,saturation_fraction)
%
% DESCRIPTION:
% Calculates the in-situ temperature at which seawater freezes. The
% in-situ temperature freezing point is calculated from the exact
% in-situ freezing temperature which is found by a modified Newton-Raphson
% iteration (McDougall and Wotherspoon, 2013) of the equality of the
% chemical potentials of water in seawater and in ice.
%
% An alternative GSW function, gsw_t_freezing_poly, it is based on a
% computationally-efficient polynomial, and is accurate to within -5e-4 K
% and 6e-4 K, when compared with this function.
%
% 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., 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_t_freezing: 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: 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: Inputs array dimensions arguments do not agree')
end %if
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: Inputs array dimensions arguments do not agree')
end %if
if ms == 1
SA = SA.';
p = p.';
saturation_fraction = saturation_fraction.';
transposed = 1;
else
transposed = 0;
end
%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------
% This line ensures that SA is non-negative.
SA(SA < 0) = 0;
% The following code gives a rather accurate polynomial-based expression
% for the freezing temperature, adjsted for the saturation fraction.
% This is the value that is used as the seed for the modified Newton's
% method. The error of the following polynomial ranges between -8e-4 K
% and 3e-4 K when compared with the outout of this function.
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;
% The initial value of t_freezing_exact (for air-free seawater)
tf = 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)));
df_dt = 1000.*gsw_t_deriv_chem_potential_water_t_exact(SA,tf,p) - gsw_gibbs_ice(1,0,tf,p);
% df_dt here is the initial value of the derivative of the function f whose
% zero (f = 0) we are finding (see Eqn. (3.33.2) of IOC et al (2010)).
tf_old = tf;
f = 1000.*gsw_chem_potential_water_t_exact(SA,tf_old,p) - gsw_gibbs_ice(0,0,tf_old,p);
tf = tf_old - f./df_dt ; % this is half way through the modified method (McDougall and Wotherspoon, 2013)
tfm = 0.5.*(tf + tf_old);
df_dt = 1000.*gsw_t_deriv_chem_potential_water_t_exact(SA,tfm,p) - gsw_gibbs_ice(1,0,tfm,p);
tf = tf_old - f./df_dt; % this is the end of one iteration of the modified Newton method
tf_old = tf;
f = 1000.*gsw_chem_potential_water_t_exact(SA,tf_old,p) - gsw_gibbs_ice(0,0,tf_old,p);
tf = tf_old - f./df_dt ; % this is half way through the modified method (McDougall and Wotherspoon, 2013)
% Adjust for the effects of dissolved air
t_freezing = tf - saturation_fraction.*(1e-3).*(2.4 - SA./70.33008);
% If the data is outside the range of applicability of TEOS-10, set the
% output to NaN.
t_freezing(p > 10000 | SA > 120 | ...
p + SA.*71.428571428571402 > 13571.42857142857) = NaN;
if transposed
t_freezing = t_freezing.';
end
% The maximum error is 2x10^-13 degrees C for this code which has one and a
% half iterations of the modified Newton's method. This is machine precision
% for this calculation. This is the maximum error over the whole
% (SA,p) domain with SA varying between 0 and 42 g/kg and p varying
% independently between 0 and 10,000 dbar.
end