# gsw_internal_energy_second_derivatives_CT_exact

    second derivatives of internal energy

## Contents

- USAGE:
- DESCRIPTION:
- INPUT:
- OUTPUT:
- EXAMPLE:
- AUTHOR:
- VERSION NUMBER:
- REFERENCES:

## USAGE:

    [u_SA_SA, u_SA_CT, u_CT_CT, u_SA_P, u_CT_P] = ...
                   gsw_internal_energy_second_derivatives_CT_exact(SA,CT,p)

## DESCRIPTION:

    Calculates the following five second-order derivatives of
      internal energy,
     (1) u_SA_SA, second order derivative with respect to Absolute Salinity
         at constant CT & p.
     (2) u_SA_CT, second order derivative with respect to SA & CT at
         constant p.
     (3) u_CT_CT, second order derivative with respect to CT at constant
         SA & p.
     (4) u_SA_P, second-order derivative with respect to SA & P at
         constant CT.
     (5) u_CT_P, second-order derivative with respect to CT & P at
         constant SA

    Note that this function uses the full Gibbs function.  There is an
    alternative to calling this function, namely
    gsw_internal_energy_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                                [ 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:

    u_SA_SA  =  The second derivative of internal energy with respect to
                Absolute Salinity at constant CT & p.   [ (J/kg)(g/kg)^-2 ]
    u_SA_CT  =  The second derivative of internal energy with respect to
                SA & CT at constant p.              [ (J/kg)(g/kg)^-1 K^-1]
    u_CT_CT  =  The second derivative of internal energy with respect to
                CT at constant SA and p.                    [ (J/kg) K^-2 ]
    u_SA_P   =  The second derivative of internal energy with respect to
                SA & P at constant CT.            [ (J/kg)(g/kg)^-1 Pa^-1 ]
    u_CT_P   =  The second derivative of internal energy with respect to
                CT & P at constant SA.                [ (J/kg) K^-1 Pa^-1 ]

## EXAMPLE:

    SA = [34.7118; 34.8915; 35.0256; 34.8472; 34.7366; 34.7324;]
    CT = [28.7856; 28.4329; 22.8103; 10.2600;  6.8863;  4.4036;]
    p =  [     10;      50;     125;     250;     600;    1000;]

    [u_SA_SA, u_SA_CT, u_CT_CT, u_SA_P, u_CT_P] = ...
                           gsw_internal_energy_second_derivatives_CT_exact(SA,CT,p)

    u_SA_SA =

       1.0e-04 *

      -0.083825343496748
      -0.083416792635045
      -0.084948480848605
      -0.085554093346234
      -0.014989876934701
       0.151822855467358

    u_SA_CT =

       1.0e-03 *

      -0.131960094412962
      -0.131426154575457
      -0.143661243353954
      -0.176542994620612
       0.038873103109403
       0.578487406867507

    u_CT_CT =

      -0.000723400187344
      -0.000719998992295
      -0.000745814518760
      -0.000736301199729
       0.000456106079104
       0.003383686798365

    u_SA_P =

       1.0e-07 *

      -0.002355222222595
      -0.007035025555756
      -0.016468311540362
      -0.035407474852142
      -0.084808691842265
      -0.141942939648594

    u_CT_P =

       1.0e-07 *

      -0.001718583707015
      -0.005204387424044
      -0.015207379618322
      -0.049042313140502
      -0.129085760730138
      -0.230817132562563

## AUTHOR:

    Trevor McDougall and Paul Barker.          [ help@teos-10.org ]

## VERSION NUMBER:

    3.05 (16th February, 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 the TEOS-10 web site.

    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