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gsw_specvol_second_derivatives_wrt_enthalpy.m
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function [v_SA_SA_wrt_h, v_SA_h, v_h_h] = gsw_specvol_second_derivatives_wrt_enthalpy(SA,CT,p)
% gsw_specvol_second_derivatives_wrt_enthalpy second order derivatives
% of volume specific with respect to enthalpy
% (75-term equation)
% =========================================================================
%
% USAGE:
% [v_SA_SA_wrt_h, v_SA_h, v_h_h] = ...
% gsw_specvol_second_derivatives_wrt_enthalpy(SA,CT,p)
%
% DESCRIPTION:
% Calculates the following three first-order derivatives of specific
% volume (v) with respect to enthalpy,
% (1) v_SA_SA_wrt_h, second-order derivative with respect to Absolute Salinity
% at constant h & p.
% (2) v_SA_h, second-order derivative with respect to SA & h at
% constant p.
% (3) v_h_h, second-order derivative with respect to h at
% constant SA & p.
%
% Note that this function uses the using the computationally-efficient
% 75 term expression for specific volume (Roquet et al., 2015). There is
% an alternative to calling this function, namely
% gsw_specvol_second_derivatives_wrt_enthalpy_CT_exact(SA,CT,p) which uses
% the full Gibbs function (IOC et al., 2010).
%
% This 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. (2010). 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:
% v_SA_SA_wrt_h = The second-order derivative of specific volume with
% respect to Absolute Salinity at constant h & p.
% [ (m^3/kg)(g/kg)^-2 (J/kg)^-1 ]
% v_SA_h = The second-order derivative of specific volume with respect to
% SA and h at constant p. [ (m^3/kg)(g/kg)^-1 (J/kg)^-1 ]
% v_h_h = The second-order derivative with respect to h at
% constant SA & p. [ (m^3/kg)(J/kg)^-2 ]
%
% AUTHOR:
% Trevor McDougall and Paul Barker. [ help@teos-10.org ]
%
% VERSION NUMBER: 3.04 (10th December, 2013)
%
% 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., 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_specvol_second_derivatives_wrt_enthalpy: Requires three inputs')
end %if
if ~(nargout == 3)
error('gsw_specvol_second_derivatives_wrt_enthalpy: Requires three outputs')
end %if
[ms,ns] = size(SA);
[mt,nt] = size(CT);
[mp,np] = size(p);
if (ms ~= mt | ns ~= nt )
error('gsw_specvol_second_derivatives_wrt_enthalpy: SA and CT do not have the same dimensions')
end
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_specvol_second_derivatives_wrt_enthalpy: The dimensions of p do not agree')
end
if ms == 1
SA = SA.';
CT = CT.';
p = p.';
transposed = 1;
else
transposed = 0;
end
%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------
% This line ensures that SA is non-negative.
SA(SA < 0) = 0;
[dummy, v_CT, dummy] = gsw_specvol_first_derivatives(SA,CT,p);
[h_SA, h_CT] = gsw_enthalpy_first_derivatives(SA,CT,p);
[vCT_SA_SA, vCT_SA_CT, vCT_CT_CT, dummy, dummy] = gsw_specvol_second_derivatives(SA,CT,p);
[h_SA_SA, h_SA_CT, h_CT_CT] = gsw_enthalpy_second_derivatives(SA,CT,p);
rec_h_CT = 1./h_CT;
rec_h_CT2 = rec_h_CT.^2;
v_h_h = (vCT_CT_CT.*h_CT - h_CT_CT.*v_CT).*(rec_h_CT2.*rec_h_CT);
v_SA_h = (vCT_SA_CT.*h_CT - v_CT.*h_SA_CT).*rec_h_CT2 - h_SA.*v_h_h;
v_SA_SA_wrt_h = vCT_SA_SA - (h_CT.*(vCT_SA_CT.*h_SA - v_CT.*h_SA_SA) ...
+ v_CT.*h_SA.*h_SA_CT).*rec_h_CT2 - h_SA.*v_SA_h;
if transposed
v_SA_SA_wrt_h = v_SA_SA_wrt_h.';
v_SA_h = v_SA_h.';
v_h_h = v_h_h.';
end
end