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gsw_enthalpy_second_derivatives_CT_exact.m
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function [h_SA_SA, h_SA_CT, h_CT_CT] = gsw_enthalpy_second_derivatives_CT_exact(SA,CT,p)
% gsw_enthalpy_second_derivatives_CT_exact second derivatives of enthalpy
% =========================================================================
%
% USAGE:
% [h_SA_SA, h_SA_CT, h_CT_CT] = gsw_enthalpy_second_derivatives_CT_exact(SA,CT,p)
%
% DESCRIPTION:
% Calculates the following three second-order derivatives of specific
% enthalpy (h),
% (1) h_SA_SA, second-order derivative with respect to Absolute Salinity
% at constant CT & p.
% (2) h_SA_CT, second-order derivative with respect to SA & CT at
% constant p.
% (3) h_CT_CT, second-order derivative with respect to CT at constant SA
% and p.
%
% Note that this function uses the full Gibbs function. There is an
% alternative to calling this function, namely
% gsw_enthalpy_second_derivatives(SA,CT,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 ]
% CT = Conservative Temperature (ITS-90) [ deg C ]
% p = sea pressure [ dbar ]
% ( i.e. absolute pressure - 10.1325 dbar )
%
% SA & CT need to have the same dimensions.
% p may have dimensions 1x1 or Mx1 or 1xN or MxN, where SA & CT are MxN.
%
% OUTPUT:
% h_SA_SA = The second derivative of specific enthalpy with respect to
% Absolute Salinity at constant CT & p. [ J/(kg (g/kg)^2) ]
% h_SA_CT = The second derivative of specific enthalpy with respect to
% SA and CT at constant p. [ J/(kg K(g/kg)) ]
% h_CT_CT = The second derivative of specific enthalpy with respect to
% CT at constant SA and p. [ J/(kg K^2) ]
%
% AUTHOR:
% Trevor McDougall and Paul Barker. [ [email protected] ]
%
% 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.
% See Eqns. (18) and (22)
%
% 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.
%
% This software is available from http://www.TEOS-10.org
%
%==========================================================================
%--------------------------------------------------------------------------
% Check variables and resize if necessary
%--------------------------------------------------------------------------
if ~(nargin == 3)
error('gsw_enthalpy_second_derivatives_CT_exact: Requires three inputs')
end %if
if ~(nargout == 3)
error('gsw_enthalpy_second_derivatives_CT_exact: Requires three outputs')
end %if
[ms,ns] = size(SA);
[mt,nt] = size(CT);
[mp,np] = size(p);
if (ms ~= mt | ns ~= nt )
error('gsw_enthalpy_second_derivatives_CT_exact: SA and CT do not have the same dimensions')
end %if
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_enthalpy_second_derivatives_CT_exact: The dimensions of p do not agree')
end %if
if ms == 1
SA = SA.';
CT = CT.';
p = p.';
transposed = 1;
else
transposed = 0;
end
%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------
cp0 = gsw_cp0; % from Eqn. 3.3.3 of IOC et al. (2010).
pr0 = zeros(size(SA));
T0 = gsw_T0;
pt0 = gsw_pt_from_CT(SA,CT);
rec_abs_pt0 = 1./(T0 + pt0);
t = gsw_pt_from_t(SA,pt0,pr0,p);
temp_ratio = (T0 + t).*rec_abs_pt0;
rec_gTT_pt0 = 1./gsw_gibbs(0,2,0,SA,pt0,pr0);
rec_gTT = 1./gsw_gibbs(0,2,0,SA,t,p);
gSAT_pt0 = gsw_gibbs(1,1,0,SA,pt0,pr0);
gSAT = gsw_gibbs(1,1,0,SA,t,p);
gSA_pt0 = gsw_gibbs(1,0,0,SA,pt0,pr0);
gSASA = gsw_gibbs(2,0,0,SA,t,p);
gSASA_pt0 = gsw_gibbs(2,0,0,SA,pt0,pr0);
part_a = temp_ratio.*rec_gTT_pt0 - rec_gTT;
% h_CT_CT is naturally well-behaved as SA approaches zero.
h_CT_CT = cp0.*cp0.*rec_abs_pt0.*rec_abs_pt0.*part_a;
part_b = rec_abs_pt0.*(temp_ratio.*gSAT_pt0.*rec_gTT_pt0 - gSAT.*rec_gTT);
factor = gSA_pt0./cp0;
% h_SA_CT should not blow up as SA approaches zero. The following lines
% of code ensure that the h_SA_CT output of this function does not blow
% up in this limit. That is, when SA < 1e-100 g/kg, we force the h_SA_CT
% output to be the same as if SA = 1e-100 g/kg.
if any(SA < 1e-100)
SA(SA < 1e-100) = 1e-100;
rec_gTT_pt0 = 1./gsw_gibbs(0,2,0,SA,pt0,pr0);
rec_gTT = 1./gsw_gibbs(0,2,0,SA,t,p);
gSAT_pt0 = gsw_gibbs(1,1,0,SA,pt0,pr0);
gSAT = gsw_gibbs(1,1,0,SA,t,p);
gSA_pt0 = gsw_gibbs(1,0,0,SA,pt0,pr0);
part_b = (temp_ratio.*gSAT_pt0.*rec_gTT_pt0 - gSAT.*rec_gTT).*rec_abs_pt0;
factor = gSA_pt0./cp0;
end
h_SA_CT = cp0.*part_b - factor.*h_CT_CT;
% h_SA_SA has a singularity at SA = 0, and blows up as SA approaches zero.
h_SA_SA = gSASA - temp_ratio.*gSASA_pt0 ...
+ temp_ratio.*gSAT_pt0.*gSAT_pt0.*rec_gTT_pt0 ...
- gSAT.*gSAT.*rec_gTT ...
- 2.*gSA_pt0.*h_SA_CT./cp0 - (factor.*factor).*h_CT_CT;
if transposed
h_SA_SA = h_SA_SA.';
h_SA_CT = h_SA_CT.';
h_CT_CT = h_CT_CT.';
end
end