# gsw_internal_energy_second_derivatives_CT_exact

`second derivatives of internal energy`

## 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`