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gsw_CT_from_enthalpy_exact.m
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function CT = gsw_CT_from_enthalpy_exact(SA,h,p)
% gsw_CT_from_enthalpy_exact Conservative Temperature from specific
% enthalpy of seawater
%==========================================================================
%
% USAGE:
% CT = gsw_CT_from_enthalpy_exact(SA,h,p)
%
% DESCRIPTION:
% Calculates the Conservative Temperature of seawater, given the Absolute
% Salinity, SA, specific enthalpy, h, and pressure p. The specific
% enthalpy input is calculated from the full Gibbs function of seawater,
% gsw_enthalpy_t_exact.
%
% Note that this function uses the full Gibbs function. There is an
% alternative to calling this function, namely
% gsw_CT_from_enthalpy(SA,h,p), which uses the computationally
% efficient 75-term expression for specific volume in terms of SA, CT
% and p (Roquet et al., 2015).
%
% INPUT:
% SA = Absolute Salinity [ g/kg ]
% h = specific enthalpy [ J/kg ]
% p = sea pressure [ dbar ]
% ( i.e. absolute pressure - 10.1325 dbar )
%
% SA & h need to have the same dimensions.
% p may have dimensions 1x1 or Mx1 or 1xN or MxN, where SA & h are MxN.
%
% OUTPUT:
% CT = Conservative Temperature ( ITS-90) [ deg C ]
%
% 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
%
% McDougall, T.J., 2003: Potential enthalpy: A conservative oceanic
% variable for evaluating heat content and heat fluxes. Journal of
% Physical Oceanography, 33, 945-963.
%
% 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.
%
% Roquet, F., G. Madec, T.J. McDougall, P.M. Barker, 2015: Accurate
% polynomial expressions for the density and specifc volume of seawater
% using the TEOS-10 standard. Ocean Modelling.
%
% The software is available from http://www.TEOS-10.org
%
%==========================================================================
%--------------------------------------------------------------------------
% Check variables and resize if necessary
%--------------------------------------------------------------------------
if ~(nargin == 3)
error('gsw_CT_from_enthalpy_exact: requires three inputs')
end
[ms,ns] = size(SA);
[mh,nh] = size(h);
[mp,np] = size(p);
if (mh ~= ms | nh ~= ns)
error('gsw_CT_from_enthalpy_exact: SA and h must have same dimensions')
end
if (mp == 1) & (np == 1) % p scalar - fill to size of SA
p = p*ones(ms,ns);
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_from_enthalpy_exact: Inputs array dimensions do not agree; check p')
end %if
if ms == 1
SA = SA.';
h = h.';
p = p.';
transposed = 1;
else
transposed = 0;
end
%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------
SA(SA < 0) = 0; % This line ensures that SA is non-negative
CT_freezing = gsw_CT_freezing(SA,p,0); % This is the CT freezing temperature
h_below_freeze = gsw_cp0; %This allows for water to be approx. 1C below the freezing temperature
h_freezing = gsw_enthalpy_CT_exact(SA,CT_freezing,p);
if any(h(:) < (h_freezing(:) - h_below_freeze)) %The input, seawater enthalpy h, is less than the enthalpy at the freezing temperature, i.e. the water is frozen
[I] = find(h < h_freezing);
SA(I) = NaN;
end
h_40 = gsw_enthalpy_CT_exact(SA,40*ones(size(SA)),p);
if any(h(:) > h_40(:)) % The input, seawater enthalpy h, is greater than the enthalpy when CT is 40 C
[I] = find(h > h_40);
SA(I) = NaN;
end
CT = CT_freezing + (40 - CT_freezing).*(h - h_freezing)./(h_40 - h_freezing); % First guess of CT
[dummy, h_CT] = gsw_enthalpy_first_derivatives_CT_exact(SA,CT,p);
%--------------------------------------------------------------------------
% Begin the modified Newton-Raphson iterative procedure
%--------------------------------------------------------------------------
CT_old = CT;
f = gsw_enthalpy_CT_exact(SA,CT_old,p) - h;
CT = CT_old - f./h_CT ; % this is half way through the modified Newton's method (McDougall and Wotherspoon, 2013)
CT_mean = 0.5*(CT + CT_old);
[dummy, h_CT] = gsw_enthalpy_first_derivatives_CT_exact(SA,CT_mean,p);
CT = CT_old - f./h_CT ; % this is the end of one full iteration of the modified Newton's method
% After 1 iteration of this modified Newton-Raphson iteration, the error in
% CT is no larger than 5x10^-14 degrees C, which is machine precision for
% this calculation.
if transposed
CT = CT.';
end
end