# gsw_internal_energy_second_derivatives

    second derivatives of internal energy (75-term equation)

## 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(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 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                                [ 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(SA,CT,p)

    u_SA_SA =

       1.0e-04 *

      -0.819630879789472
      -0.815991440934607
      -0.829998881000640
      -0.850454751660842
      -0.275191185099451
       0.935521724807209

    u_SA_CT =

       1.0e-03 *

      -0.131647989755567
      -0.131253260179302
      -0.143764305018288
      -0.175089006292492
       0.044242679479252
       0.582715895326373

    u_CT_CT =

      -0.000723349499915
      -0.000720061216325
      -0.000745410281876
      -0.000733890626293
       0.000470364582417
       0.003405297338392

    u_SA_P =

       1.0e-07 *

      -0.002844346969485
      -0.008509204835375
      -0.019897886928086
      -0.042812697313995
      -0.102544848040703
      -0.171670276420447

    u_CT_P =

       1.0e-07 *

      -0.001722190998963
      -0.005214908835889
      -0.015155996668111
      -0.048975118096772
      -0.129113528684486
      -0.230925961569104

## 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.

    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