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gsw_enthalpy_second_derivatives.m
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function [h_SA_SA, h_SA_CT, h_CT_CT] = gsw_enthalpy_second_derivatives(SA,CT,p)
% gsw_enthalpy_second_derivatives second derivatives of enthalpy
% (75-term equation)
% =========================================================================
%
% USAGE:
% [h_SA_SA, h_SA_CT, h_CT_CT] = gsw_enthalpy_second_derivatives(SA,CT,p)
%
% DESCRIPTION:
% Calculates the following three second-order derivatives of specific
% enthalpy (h),using the computationally-efficient expression for
% specific volume in terms of SA, CT and p (Roquet et al., 2015).
% (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 the 75-term equation has been fitted in a restricted range of
% parameter space, and is most accurate inside the "oceanographic funnel"
% described in McDougall et al. (2003). The GSW library function
% "gsw_infunnel(SA,CT,p)" is avaialble to be used if one wants to test if
% some of one's data lies outside this "funnel".
%
% 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)
%
% McDougall, T.J., D.R. Jackett, D.G. Wright and R. Feistel, 2003:
% Accurate and computationally efficient algorithms for potential
% temperature and density of seawater. J. Atmosph. Ocean. Tech., 20,
% pp. 730-741.
%
% 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: Requires three inputs')
end %if
if ~(nargout == 3)
error('gsw_enthalpy_second_derivatives: 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: 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: 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
%--------------------------------------------------------------------------
SA(SA<0) = 0;
%db2Pa = 1e4; % factor to convert from dbar to Pa
%cp0 = 3991.86795711963; % from Eqn. (3.3.3) of IOC et al. (2010).
sfac = 0.0248826675584615; % sfac = 1/(40*(35.16504/35)).
offset = 5.971840214030754e-1; % offset = deltaS*sfac.
x2 = sfac.*SA;
xs = sqrt(x2 + offset);
ys = CT.*0.025;
z = p.*1e-4;
% h001 = 1.0769995862e-3;
% h002 = -3.0399571905e-5;
% h003 = 3.3285389740e-6;
% h004 = -2.8273403593e-7;
% h005 = 2.1062306160e-8;
% h006 = -2.1078768810e-9;
% h007 = 2.8019291329e-10;
% h011 = -1.5649734675e-5;
% h012 = 9.2528827145e-6;
% h013 = -3.9121289103e-7;
% h014 = -9.1317516383e-8;
% h015 = 6.2908199804e-8;
h021 = 2.7762106484e-5;
h022 = -5.8583034265e-6;
h023 = 7.1016762467e-7;
h024 = 7.1739762898e-8;
h031 = -1.6521159259e-5;
h032 = 3.9639828087e-6;
h033 = -1.5377513346e-7;
h042 = -1.7051093741e-6;
h043 = -2.1117638838e-8;
h041 = 6.9111322702e-6;
h051 = -8.0539615540e-7;
h052 = 2.5368383407e-7;
h061 = 2.0543094268e-7;
h101 = -3.1038981976e-4;
h102 = 1.21312343735e-5;
h103 = -1.9494810995e-7;
h104 = 9.0775471288e-8;
h105 = -2.2294250846e-8;
h111 = 3.5009599764e-5;
h112 = -4.7838544078e-6;
h113 = -1.8566384852e-6;
h114 = -6.8239240593e-8;
h121 = -3.7435842344e-5;
h122 = -1.18391541805e-7;
h123 = 1.3045795693e-7;
h131 = 2.4141479483e-5;
h132 = -1.72793868275e-6;
h133 = 2.5872962697e-9;
h141 = -8.7595873154e-6;
h142 = 6.4783588915e-7;
h151 = -3.3052758900e-7;
% h201 = 6.6928067038e-4;
% h202 = -1.7396230487e-5;
% h203 = -1.6040750532e-6;
% h204 = 4.1865759450e-9;
h211 = -4.3592678561e-5;
h212 = 5.5504173825e-6;
h213 = 1.8206916278e-6;
h221 = 3.5907822760e-5;
h222 = 1.46416731475e-6;
h223 = -2.1910368022e-7;
h231 = -1.4353633048e-5;
h232 = 1.5827653039e-7;
h241 = 4.3703680598e-6;
h301 = -8.5047933937e-4;
h302 = 1.87353886525e-5;
h303 = 1.6421035666e-6;
h311 = 3.4532461828e-5;
h312 = -4.9223558922e-6;
h313 = -4.5147285423e-7;
h321 = -1.8698584187e-5;
h322 = -2.4413069600e-7;
h331 = 2.2863324556e-6;
h401 = 5.8086069943e-4;
h402 = -8.6611093060e-6;
h403 = -5.9373249090e-7;
h411 = -1.1959409788e-5;
h421 = 3.8595339244e-6;
h412 = 1.2954612630e-6;
h501 = -2.1092370507e-4;
h502 = 1.54637136265e-6;
h511 = 1.3864594581e-6;
h601 = 3.1932457305e-5;
xs2 = xs.^2;
dynamic_h_SA_SA_part = z.*(-h101 + xs2.*(3.*h301 + xs.*(8.*h401 + xs.*(15.*h501 + 24.*h601.*xs))) ...
+ ys.*(- h111 + xs2.*(3.*h311 + xs.*(8.*h411 + 15.*h511.*xs)) + ys.*(-h121 ...
+ xs2.*(3.*h321 + 8.*h421.*xs) + ys.*(-h131 + 3.*h331.*xs2 + ys.*(-h141 ...
-h151.*ys)))) + z.*(-h102 + xs2.*(3.*h302 + xs.*(8.*h402 + 15.*h502.*xs)) ...
+ ys.*(-h112 + xs2.*(3.*h312 + 8.*h412.*xs) + ys.*(-h122 + 3.*h322.*xs2 ...
+ ys.*(-h132 - h142.*ys ))) + z.*(xs2.*(8.*h403.*xs + 3.*h313.*ys) ...
+ z.*(-h103 + 3.*h303.*xs2 + ys.*(-h113 + ys.*(-h123 - h133.*ys)) ...
+ z.*(-h104 - h114.*ys - h105.*z)))));
h_SA_SA = 1e8*0.25.*sfac.*sfac.*dynamic_h_SA_SA_part./xs.^3;
dynamic_h_SA_CT_part = z.*(h111 + xs.*(2*h211 + xs.*(3*h311 + xs.*(4*h411 + 5*h511.*xs))) ...
+ ys.*(2*h121 + xs.*(4*h221 + xs.*(6*h321 + 8*h421*xs)) + ys.*(3*h131 ...
+ xs.*(6*h231 + 9*h331.*xs) + ys.*(4*h141 + 8*h241.*xs + 5*h151.*ys )))...
+z.*(h112 + xs.*(2*h212 + xs.*(3*h312 + 4*h412.*xs)) + ys.*(2*h122 ...
+ xs.*(4*h222 + 6*h322.*xs) + ys.*(3*h132 + 6*h232.*xs + 4*h142.*ys)) ...
+ z.*(h113 + xs.*(2*h213 + 3*h313.*xs) + ys.*(2*h123 + 4*h223.*xs ...
+ 3*h133.*ys) + h114*z)));
h_SA_CT = 1e8*0.025.*0.5.*sfac.*dynamic_h_SA_CT_part./xs;
dynamic_h_CT_CT_part = z.*(2.*h021 + xs.*(2.*h121 + xs.*(2.*h221 + xs.*(2.*h321 ...
+ 2.*h421.*xs))) + ys.*(6.*h031 + xs.*(6.*h131 + xs.*(6.*h231 + 6.*h331.*xs)) ...
+ ys.*(12.*h041 + xs.*(12.*h141 + 12.*h241.*xs) + ys.*(20.*h051 + 20.*h151.*xs ...
+ 30.*h061.*ys))) + z.*(2.*h022 + xs.*(2.*h122 + xs.*(2.*h222 + 2.*h322.*xs)) ...
+ ys.*(6.*h032 + xs.*(6.*h132 + 6.*h232.*xs) + ys.*(12.*h042 + 12.*h142.*xs ...
+ 20.*h052.*ys)) + z.*(2.*h023 + xs.*(2.*h123 + 2.*h223.*xs) + ys.*(6.*h133.*xs ...
+ 6.*h033 + 12.*h043.*ys) + 2.*h024.*z)));
h_CT_CT = 1e8.*6.25e-4.*dynamic_h_CT_CT_part;
if transposed
h_SA_SA = h_SA_SA.';
h_SA_CT = h_SA_CT.';
h_CT_CT = h_CT_CT.';
end
end