def enthalpy_second_derivatives(SA, CT, p):
r"""Calculates the following three second-order derivatives of specific
enthalpy (h),
(1) h_SA_SA, second-order derivative with respect to Absolute Salinity
at constant CT & p.
(2) h_SA_CT, second-order derivative with respect to SA & CT at
constant p.
(3) h_CT_CT, second-order derivative with respect to CT at constant SA
and p.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
CT : array_like
Conservative Temperature [:math:`^\circ` C (TEOS-10)]
p : array_like
pressure [dbar]
Returns
-------
h_SA_SA : array_like
The second derivative of specific enthalpy with respect to
Absolute Salinity at constant CT & p. [J/(kg (g/kg)^2)]
h_SA_CT : array_like
The second derivative of specific enthalpy with respect to SA and
CT at constant p. [J/(kg K(g/kg))]
h_CT_CT : array_like
The second derivative of specific enthalpy with respect to CT at
constant SA and p. [J/(kg K^2)]
References
----------
.. [1] 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. See Eqns. (A.11.18), (A.11.15) and (A.11.12.)
.. [2] McDougall T.J., P.M. Barker, R. Feistel and D.R. Jackett, 2011: A
computationally efficient 48-term expression for the density of seawater in
terms of Conservative Temperature, and related properties of seawater. To
be submitted to Ocean Science Discussions.
Modifications:
2011-03-29. Trevor McDougall.
"""
# NOTE: The Matlab version 3.0 mentions that this function is unchanged,
# but that's not true!
pt0 = pt_from_CT(SA, CT)
abs_pt0 = Kelvin + pt0
t = pt_from_t(SA, pt0, 0, p)
temp_ratio = (Kelvin + t) / abs_pt0
rec_gTT_pt0 = 1 / gibbs(n0, n2, n0, SA, pt0, 0)
rec_gTT_t = 1 / gibbs(n0, n2, n0, SA, t, p)
gST_pt0 = gibbs(n1, n1, n0, SA, pt0, 0)
gST_t = gibbs(n1, n1, n0, SA, t, p)
gS_pt0 = gibbs(n1, n0, n0, SA, pt0, 0)
part = ((temp_ratio * gST_pt0 * rec_gTT_pt0 - gST_t * rec_gTT_t) /
(abs_pt0))
factor = gS_pt0 / cp0
# h_CT_CT is naturally well-behaved as SA approaches zero.
def enthalpy_derivative_CT_CT(SA, CT, p):
return (cp0 ** 2 * ((temp_ratio * rec_gTT_pt0 - rec_gTT_t) /
(abs_pt0 * abs_pt0)))
# h_SA_SA has a singularity at SA = 0, and blows up as SA approaches zero.
def enthalpy_derivative_SA_SA(SA, CT, p):
SA[SA < 1e-100] = 1e-100 # NOTE: Here is the changes from 2.0 to 3.0.
h_CT_CT = enthalpy_derivative_CT_CT(SA, CT, p)
return (gibbs(n2, n0, n0, SA, t, p) -
temp_ratio * gibbs(n2, n0, n0, SA, pt0, 0) +
temp_ratio * gST_pt0 ** 2 * rec_gTT_pt0 -
gST_t ** 2 * rec_gTT_t - 2.0 * gS_pt0 * part +
factor ** 2 * h_CT_CT)
"""h_SA_CT should not blow up as SA approaches zero. The following lines of
code ensure that the h_SA_CT output of this function does not blow up in
this limit. That is, when SA < 1e-100 g/kg, we force the h_SA_CT output to
be the same as if SA = 1e-100 g/kg."""
def enthalpy_derivative_SA_CT(SA, CT, p):
h_CT_CT = enthalpy_derivative_CT_CT(SA, CT, p)
return cp0 * part - factor * h_CT_CT
return (enthalpy_derivative_SA_SA(SA, CT, p),
enthalpy_derivative_SA_CT(SA, CT, p),
enthalpy_derivative_CT_CT(SA, CT, p))
def enthalpy_first_derivatives(SA, CT, p):
r"""Calculates the following three derivatives of specific enthalpy (h)
(1) h_SA, the derivative with respect to Absolute Salinity at
constant CT and p, and
(2) h_CT, derivative with respect to CT at constant SA and p.
(3) h_P, derivative with respect to pressure (in Pa) at constant SA and CT.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
CT : array_like
Conservative Temperature [:math:`^\circ` C (TEOS-10)]
p : array_like
pressure [dbar]
Returns
-------
h_SA : array_like
The first derivative of specific enthalpy with respect to Absolute
Salinity at constant CT and p. [J/(kg (g/kg))] i.e. [J/g]
h_CT : array_like
The first derivative of specific enthalpy with respect to CT at
constant SA and p. [J/(kg K)]
h_P : array_like
The first partial derivative of specific enthalpy with respect to
pressure (in Pa) at fixed SA and CT. Note that h_P is specific
volume (1/rho.)
See Also
--------
TODO
Notes
-----
TODO
References
----------
.. [1] 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. See Eqns. (A.11.18), (A.11.15) and (A.11.12.)
Modifications:
2010-09-24. Trevor McDougall.
"""
# FIXME: The gsw 3.0 has the gibbs derivatives "copy-and-pasted" here
# instead of the calls to the library! Why?
pt0 = pt_from_CT(SA, CT)
t = pt_from_t(SA, pt0, 0, p)
temp_ratio = (Kelvin + t) / (Kelvin + pt0)
def enthalpy_derivative_SA(SA, CT, p):
return (gibbs(n1, n0, n0, SA, t, p) -
temp_ratio * gibbs(n1, n0, n0, SA, pt0, 0))
def enthalpy_derivative_CT(SA, CT, p):
return cp0 * temp_ratio
def enthalpy_derivative_p(SA, CT, p):
return gibbs(n0, n0, n1, SA, t, p)
return (enthalpy_derivative_SA(SA, CT, p),
enthalpy_derivative_CT(SA, CT, p),
enthalpy_derivative_p(SA, CT, p),)