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gsw_CT_freezing_first_derivatives_poly.m
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gsw_CT_freezing_first_derivatives_poly.m
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function [CTfreezing_SA, CTfreezing_P] = gsw_CT_freezing_first_derivatives_poly(SA,p,saturation_fraction)
% gsw_CT_freezing_first_derivatives_poly first derivatives of
% Conservative Temperature at which seawater freezes (poly)
%==========================================================================
%
% USAGE:
% [CTfreezing_SA, CTfreezing_P] = gsw_CT_freezing_first_derivatives_poly(SA,p,saturation_fraction)
%
% DESCRIPTION:
% Calculates the first derivatives of the Conservative Temperature at
% which seawater freezes, with respect to Absolute Salinity SA and
% pressure P (in Pa) of the comptationally efficient polynomial fit of the
% freezing temperature (McDougall et al., 2014).
%
% 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:
% CTfreezing_SA = the derivative of the Conservative Temperature at
% freezing (ITS-90) with respect to Absolute Salinity at
% fixed pressure [ K/(g/kg) ] i.e. [ K kg/g ]
%
% CTfreezing_P = the derivative of the Conservative Temperature at
% freezing (ITS-90) with respect to pressure (in Pa) at
% fixed Absolute Salinity [ K/Pa ]
%
% AUTHOR:
% Trevor McDougall, 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 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_CT_freezing_first_derivatives_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_CT_freezing_first_derivatives_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_CT_freezing_first_derivatives_poly: 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_CT_freezing_first_derivatives_poly: 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
%--------------------------------------------------------------------------
SA(SA < 0) = 0; % This line ensures that SA is not negative
% 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;
SA_r = SA.*1e-2;
x = sqrt(SA_r);
p_r = p.*1e-4;
% a = 0.014289763856964; % Note that a = 0.502500117621/35.16504.
% b = 0.057000649899720;
% Note that -0.018994561378548 = -a -a*b - 2.4*b/gsw_SSO
% and, 4.632588654871302e-05 = 2*a*b./gsw_SSO
CTfreezing_SA = (c1 + x.*(1.5*c2 + x.*(2*c3 + x.*(2.5*c4 + x.*(3*c5 + 3.5*c6*x)))) ...
+ p_r.*(c10 + x.*(1.5*c11 + x.*(2*c13 + x.*(2.5*c16 + x.*(3*c19 + 3.5*c22*x)))) ...
+ p_r.*(c12 + x.*(1.5*c14 + x.*(2*c17 + 2.5*c20*x)) ...
+ p_r.*(c15 + x.*(1.5*c18 + 2*c21*x))))).*1e-2...
- saturation_fraction.*(1e-3).*(-0.018994561378548 - SA.*4.632588654871302e-05);
CTfreezing_P = (c7 + SA_r.*(c10 + x.*(c11 + x.*(c13 + x.*(c16 + x.*(c19 + c22*x))))) ...
+ p_r.*(2*c8 + SA_r.*(2*c12 + x.*(2*c14 + x.*(2*c17 + 2*c20*x))) ...
+ p_r.*(3*c9 + SA_r.*(3*c15 + x.*(3*c18 + 3*c21*x))))).*1e-8;
% set any values that are out of range to be NaN.
CTfreezing_SA(p > 10000 | SA > 120 | ...
p + SA.*71.428571428571402 > 13571.42857142857) = NaN;
CTfreezing_P(p > 10000 | SA > 120 | ...
p + SA.*71.428571428571402 > 13571.42857142857) = NaN;
if transposed
CTfreezing_SA = CTfreezing_SA.';
CTfreezing_P = CTfreezing_P.';
end
end