Water properties
================

Seawater properties (especially water density) are important in hydrology, e.g., for salt intrusion. The implemented water properties are based on *Algorithms for computation of fundamental properties of seawater. Unesco technical papers in marine science 44. Unesco 1983* (https://unesdoc.unesco.org/90e8e061-800c-4c51-b08d-3529216aaee4).


Pure water density
------------------

The density of water at zero relative pressure (one atmosphere or 101325 Pa) and zero salinity as a function of temperature can be calculated according to different methods. According to Craig (eq. 14):

.. math:: 

	\rho_w (t) =  a_0 + a_1t + a_2 t^2 + a_3t^3 + a_4 a^4 + a_5 t^5

where:

    | t = temperature in °C
    | :math:`a_0` = 999.842594
    | :math:`a_1` =   6.793952E-2
    | :math:`a_2` =  -9.095290E-3
    | :math:`a_3` =   1.001685E-4
    | :math:`a_4` =  -1.120083E-6
    | :math:`a_5` =   6.536332E-9

.. code::

    print(pure_water_density_craig(20))

Output::

    998.2063193824


Pure water density according to *Tanaka, M., et. al; Recommended table for the density of water between 0°C and 40°C based on recent experimental reports, Metrologia, 2001, 38, 301-309* (http://www.personal.psu.edu/users/m/r/mrh318/water-properties/Tanaka-etal-M-2001.pdf) by:

.. math:: 

	\rho_w (t) =  a_5 \left[ 1 - \frac{(t + a_1)^2(t + a_2)}{a_3(t + a_4)} \right]

where

    | t = temperature in °C
    | :math:`a_1` =  -3.983035
    | :math:`a_2` = 301.797
    | :math:`a_3` =   522528.9
    | :math:`a_4` =  69.34881
    | :math:`a_5` = 999.974950

.. code::

    print(pure_water_density(20))

Output::

    998.2067455596167

The following figures show the pure water density and the difference between the methods after Craig and Tanaka et al:

.. image:: images/water_density_pure.png
   :scale: 50 %

.. image:: images/water_density_pure_diff.png
   :scale: 50 %


Water density depencency on salinity
------------------------------------

The density of water at zero relative pressure (one atmosphere or 101325 Pa) as a function of temperature and salinity can be calculated according to (eq. 13):

.. math::

    \rho (S, t) = \rho_w + (b_0 + b_1t + b_2t^2 + b_3t^3 + b_4t^4)S + (c_0 + c_1t + c_2t^2)S^{\frac{3}{2}} + d_0S^2

where

    | :math:`b0` = 8.24493E-1
    | :math:`b1` = -4.0899E-3
    | :math:`b2` = 7.6438E-5
    | :math:`b3` = -8.2467E-7
    | :math:`b4` = 5.3875E-9
    | :math:`c0` = -5.72466E-3
    | :math:`c1` = 1.0227E-4
    | :math:`c2` = -1.6546E-6
    | :math:`d0` = 4.8314E-4


.. image:: images/water_density_salinity.png
   :scale: 50 %


Water density
-------------

Water density is calculated by (eq. 7):

.. math:: 

	\rho (S, t, p) =  \frac{\rho (S, t)}{1 - \frac{p}{K(S, t, p)}}

where

.. math:: 

    K(S, t, p)


.. for a given temperature T in the range 0<T<40◦C, salinity S in the range 0<S<42 PSU and pressurepis determined from the equation ofstate (UNESCO 1981):


Secant bulk modulus of pure water
---------------------------------

.. math::

    K_w(t) = 19652.21 + (148.4206 + (-2.327105 + (1.360477E-2 + -5.155288E-5 * t) * t) * t) * t

Secant bulk modulus at zero pressure
------------------------------------

.. math::

    K(S, t) = K_w(t) + (54.6746 + (-0.603459 + (1.09987E-2 + (-6.1670E-5 * t) * t) * t) * t) * s + (7.944E-2 + 1.6483E-2 * t + -5.3009E-4 * t * t) * (s^1.5)

Secant bulk modulus of pure water
---------------------------------

.. math::

    K(S, t, p) = \frac{p}{(V - V_0)/V_0}

	K(S, t, p) = K(S, t, p) = 


Water conductivity and salinity
-------------------------------

Conductivity is usually expressed in micro- or millisiemens per centimeter (:math:`\mu S/cm` or :math:`mS/cm`). Typical drinking water has a conductivity between 0.05 and 0.50 mS/cm

Salinity is the total concentration of salts in water, usually expressed in g/kg. Seawater has a salinity of ca. 35 g/kg