Constitutive equations. More...

Functions/Subroutines
subroutine	initialize_models

subroutine	allocate_phases_parameters
	Initialization of relaxation flags. More...

subroutine	eos
	Equation of state. More...

subroutine	sound_speeds (C_mix, mach)
	Local sound speeds. More...

subroutine	eval_densities
	Phases densities. More...

subroutine	solve_cubic (a, b, c, y1, y2, y3)
	Solution of a cubic. More...

subroutine	f_xdis_eq
	Equilibrium Dissolved gas. More...

subroutine	f_beta_eq
	Equilibrium Crystal content. More...

subroutine	f_growth_rate
	This subroutine compute the growth rates for the different crystal phases. More...

subroutine	f_nucleation_rate
	Lithostatic pressure. More...

subroutine	lithostatic_pressure
	Lithostatic pressure. More...

subroutine	hydrostatic_pressure
	Hydrostatic pressure. More...

subroutine	press_relax_term (pressure_relaxation)
	Pressure relaxation term. More...

subroutine	vel_relax_term (velocity_relaxation)
	Velocity relaxation term. More...

subroutine	f_permkc
	Magma permeability. More...

subroutine	mixture_viscosity
	Mixture viscosity. More...

subroutine	f_viscliq
	Magma viscosity. More...

subroutine	f_viscmelt
	Melt viscosity. More...

subroutine	f_theta
	Crystal relative viscosity. More...

subroutine	f_bubbles
	Exsolved gas relative viscosity. More...

subroutine	f_alfa (xtot, xmax, r_beta, r_rho_md, r_rho_2, r_alfa_2)
	Bottom exsolved gas. More...

subroutine	f_alfa3 (r_p_2, xtot, r_beta, r_rho_md, r_rho_g, r_alfa_g)
	Bottom exsolved gas. More...

Variables
real *8	cv_2
	exsolved gas specific heat capacity at constant volume More...

real *8	cv_m
	melt specific heat capacity at constant volume More...

real *8, dimension(:), allocatable	cv_c
	crystals specific heat capacity at constant volume More...

real *8, dimension(:), allocatable	cv_d
	dissolved gas heat capacity at constant volume More...

real *8, dimension(:), allocatable	cv_g
	exsolved gas heat capacity at constant volume More...

real *8	c0_m
	melt sound speed at atmospheric conditions More...

real *8, dimension(:), allocatable	c0_c
	crystals sound speed at atm conditions More...

real *8, dimension(:), allocatable	c0_d
	dissolved gas sound speed at atm conditions More...

real *8, dimension(:), allocatable	c0_g
	exsolved gas sound speed at atm conditions More...

real *8	gamma_2
	exsolved gas adiabatic exponent More...

real *8	gamma_m
	melt adiabatic exponent More...

real *8, dimension(:), allocatable	gamma_c
	crystals adiabatic exponent More...

real *8, dimension(:), allocatable	gamma_d
	dissolved gas adiabatic exponent More...

real *8, dimension(:), allocatable	gamma_g
	exsolved gas adiabatic exponent More...

real *8	rho0_2
	exsolved gas reference density More...

real *8	rho0_m
	melt reference density More...

real *8, dimension(:), allocatable	rho0_c
	crystals reference density More...

real *8, dimension(:), allocatable	rho0_d
	dissolved gas reference density More...

real *8, dimension(:), allocatable	rho0_g
	exsolved gas reference density More...

real *8	t0_2
	exsolved gas reference temperature More...

real *8	t0_m
	melt reference temperature More...

real *8, dimension(:), allocatable	t0_c
	crystals gas reference temperature More...

real *8, dimension(:), allocatable	t0_d
	dissolved gas reference temperature More...

real *8, dimension(:), allocatable	t0_g
	exsolved gas reference temperature More...

real *8	p0_m
	melt reference pressure More...

real *8, dimension(:), allocatable	p0_c
	crystals reference pressure More...

real *8, dimension(:), allocatable	p0_d
	dissolved gas reference pressure More...

real *8, dimension(:), allocatable	p0_g
	exsolved gas reference pressure More...

real *8	bar_e_2
	exsolved gas formation energy More...

real *8	bar_e_m
	melt formation energy More...

real *8, dimension(:), allocatable	bar_e_c
	crystals formation energy More...

real *8, dimension(:), allocatable	bar_e_d
	dissolved gas formation energy More...

real *8, dimension(:), allocatable	bar_e_g
	exsolved gas formation energy More...

real *8	bar_p_m
	melt cohesion pressure More...

real *8, dimension(:), allocatable	bar_p_c
	crystals cohesion pressure More...

real *8, dimension(:), allocatable	bar_p_d
	dissolved gas cohesion pressure More...

real *8, dimension(:), allocatable	bar_p_g
	exsolved gas cohesion pressure More...

real *8	s0_m
	melt reference entropy More...

real *8, dimension(:), allocatable	s0_c
	crystals reference entropy More...

real *8, dimension(:), allocatable	s0_d
	dissolved gas reference entropy More...

real *8, dimension(:), allocatable	s0_g
	exsolved gas reference entropy More...

real *8, dimension(:), allocatable	pc_g
	critical gas pressure More...

real *8, dimension(:), allocatable	tc_g
	critical gas temperature More...

real *8, dimension(:), allocatable	a_g
	parameter for the VDW EOS More...

real *8, dimension(:), allocatable	b_g
	parameter for the VDW EOS More...

character *20	gas_law
	equation of state for gas More...

complex *16	cv_1
	dis.gas+melt+crystals specific heat capacity at constant volume More...

complex *16	e_mix
	total internal energy More...

complex *16	e_1
	local specific internal energy of the melt-crystals phase More...

complex *16	e_2
	local specific internal energy of the exsolved gas More...

complex *16	e_m
	local specific internal energy of the melt More...

complex *16, dimension(:), allocatable	e_c
	local specific internal energy of the crystals More...

complex *16, dimension(:), allocatable	e_d
	local specific internal energy of the dissolved gas More...

complex *16, dimension(:), allocatable	e_g
	local specific internal energy of the exsolved gas More...

complex *16	p_1
	local pressure of the melt-crystals phase More...

complex *16	p_2
	local pressure of the exsolved gas More...

complex *16	s_1
	local specific entropy of the melt-crystals phase More...

complex *16	s_2
	local specific entropy of the exsolved gas More...

complex *16	s_m
	local specific entropy of the melt More...

complex *16, dimension(:), allocatable	s_c
	local specific entropy of the crystals More...

complex *16, dimension(:), allocatable	s_d
	local specific entropy of the dissolved gas More...

complex *16, dimension(:), allocatable	s_g
	local specific entropy of the exsolved gas More...

complex *16	mu_1
	free Gibbs energy of the melt-crystals phase More...

complex *16	mu_2
	free Gibbs energy of the exsolved gas More...

complex *16	mu_m
	free Gibbs energy of the melt More...

complex *16, dimension(:), allocatable	mu_c
	free Gibbs energy of the crystals More...

complex *16, dimension(:), allocatable	mu_d
	free Gibbs energy of the dissolved gas More...

complex *16, dimension(:), allocatable	mu_g
	free Gibbs energy of the exsolved gas More...

complex *16	rho_1
	dis_gas+melt+crystals phase local density More...

complex *16	rho_2
	exsolved gas local density More...

complex *16	rho_m
	melt local density More...

complex *16, dimension(:), allocatable	rho_c
	crystals local density More...

complex *16, dimension(:), allocatable	rho_d
	dissolved gas local density More...

complex *16, dimension(:), allocatable	rho_g
	exsolved gas local density More...

complex *16	rho_md
	dis_gas+melt local density More...

complex *16, dimension(:), allocatable	rhob_c
	crystals bulk density More...

complex *16	rhob_m
	melt bulk density More...

complex *16	alfa_1
	dis_gas+melt+crystals phase local volume fraction More...

complex *16	alfa_2
	total exsolved gas local volume fraction More...

complex *16, dimension(:), allocatable	alfa_g
	exsolved gas phases local volume fraction More...

complex *16	alfarho_2
	bulk density of the exsolved gas More...

complex *16	alfa_m_1
	melt volume fraction in phase 1 More...

complex *16, dimension(:), allocatable	alfa_d_1
	dissolved gas volume fractions in phase 1 More...

complex *16, dimension(:), allocatable	alfa_g_2
	exsolved gas volume fractions in phase 2 More...

complex *16, dimension(:), allocatable	beta
	crystal volume fraction in the melt-crystals phase More...

complex *16, dimension(:), allocatable	beta_eq
	equil. cry. volume fraction in the melt-crystals phase More...

complex *16, dimension(:,:), allocatable	mom_cry
	moments of the crystal referred to the melt-crystals phase More...

complex *16, dimension(:), allocatable	growth_rate
	growth rate for the crystals More...

complex *16, dimension(:), allocatable	nucleation_rate
	nulceation rate for the crystals More...

complex *16	u_1
	melt-crystals phase local velocity More...

complex *16	u_2
	exsolved gas local velocity More...

complex *16	x_1
	melt-crystals phase local mass fraction More...

complex *16	x_2
	exsolved gas local mass fraction More...

complex *16, dimension(:), allocatable	x_c
	crystals mass fraction (with respect to the mixture) More...

complex *16	x_m
	melt mass fraction (with respect to the mixture) More...

complex *16, dimension(:), allocatable	x_d
	dissolved gas mass fraction (with respect to the mixture) More...

complex *16, dimension(:), allocatable	x_g
	exsolved gas mass fraction (with respect to the mixture) More...

complex *16, dimension(:), allocatable	x_d_md
	dissolved gas mass fraction in the melt+dis.gas phase More...

complex *16, dimension(:), allocatable	x_d_md_eq
	equil. dis. gas mass fraction in the melt+dis.gas phase More...

complex *16, dimension(:), allocatable	x_c_1
	cristal mass fractions in phase 1 More...

complex *16	x_m_1
	melt mass fraction in phase 1 More...

complex *16	x_md_1
	melt+dis.gas mass fraction in phase 1 More...

complex *16, dimension(:), allocatable	x_d_1
	dissolved gas mass fractions in phase 1 More...

complex *16, dimension(:), allocatable	x_g_2
	exsolved gas mass fractions in phase 2 More...

complex *16	t
	mixture local temperature More...

complex *16	rho_mix
	mixture local density More...

complex *16	u_mix
	mixture velocity More...

complex *16	cv_mix
	mixture specific heat capacity at constant volume More...

complex *16	visc_mix
	mixture viscosity More...

complex *16	s
	mixture entropy More...

complex *16	c_1
	first phase local sound speed More...

complex *16	c_2
	second phase local sound speed More...

integer	n_drag_models

character(len=30), dimension(20)	available_drag_models

character *30	drag_funct_model
	drag function model More...

complex *16	drag_funct
	drag function for the relative velocity: More...

real *8	drag_funct_coeff
	coefficient for the drag function for the relative velocity: More...

real *8	bubble_number_density
	bubble number density More...

real *8	log10_bubble_number_density

real *8	tortuosity_factor
	tortuosity factor More...

real *8	throat_bubble_ratio
	throat bubble ratio More...

real *8	friction_coefficient

real *8	c_d

real *8	r_a

character *20	p_relax_model
	pressure relaxation model More...

complex *16	tau_p
	pressure relaxation rate More...

real *8	tau_p_coeff
	pressure relaxation coefficient: More...

real *8, dimension(:), allocatable	tau_c
	crystallization parameter: More...

real *8, dimension(:), allocatable	beta0
	chamber (equilibrium) crystal volume fraction More...

real *8, dimension(:), allocatable	beta_max
	maximum crystal volume fraction More...

character *20	crystallization_model
	Model for the equilibrium crystal volume fraction: . More...

real *8	frag_eff
	index of fragmentation in the interval [0;1] More...

integer	fragmentation_model
	Parameter to choose the fragmentation model: . More...

real *8	frag_thr
	Threshold for the fragmentation. More...

real *8	tau_frag_exp
	Fragmentation exponent. More...

complex *16	tau_frag
	fragmentation rate More...

real *8	tau_frag_coeff
	fragmentation coefficient: More...

real *8, dimension(:), allocatable	tau_d
	exsolution parameter: More...

character *20	exsol_model
	String for exsolution model: . More...

real *8, dimension(:), allocatable	solub
	Solubility parameter for the Henry's law. More...

real *8, dimension(:), allocatable	solub_exp

real *8, dimension(:), allocatable	x_ex_dis_in

real *8	grav
	gravitational acceleration More...

real *8	k_cr
	country rock permeability More...

real *8	rho_cr
	contry rock density More...

real *8	visc_2
	gas viscosity More...

complex *16	visc_melt
	melt viscosity More...

complex *16	visc_1
	melt+crystal viscosity More...

complex *16	visc_rel_bubbles
	relative viscosity due to bubbles More...

integer	n_theta_models

character(len=30), dimension(20)	available_theta_models

character *30	theta_model
	Parameter to choose the model for the influence of crystal on the mixture: 'Lejeune_and_Richet1995' 'Dingwell1993' 'Melnik_and_Sparks1999' 'Costa2005' 'Melnik_and_Sparks2005' 'Vona_et_al2011' 'Vona_et_al2011_mod' 'Vona_et_al2013_eq19' 'Vona_et_al2013_eq20' 'Vona_et_al2013_eq21' 'Fixed_value' More...

complex *16	theta
	Relative viscosity of the crystals. More...

real *8	theta_fixed
	Fixed value for the relative viscosity of the crystals. More...

real *8	c1
	Coefficients for the relative viscosity models. More...

real *8	c2

real *8	c3

real *8	p_lith
	Lithostatic pressure. More...

real *8	p_hydro
	Hydrostatic pressure. More...

real *8	zeta_lith
	Elevation above the bottom for the evaluation of the lithostatic pressure. More...

integer	n_bubble_models

character(len=30), dimension(20)	available_bubble_models

character *20	bubbles_model
	Parameter to choose the model for the influence of the bubbles on the mixture: . More...

logical	explosive
	Flag to choose the eruptive style: . More...

complex *16	permkc
	Magma permeability. More...

complex *16	inv_permkc

real *8	alfa_switch

real *8	a_2nd

real *8	b_2nd

real *8	perm0

integer	n_visc_melt_models

character(len=30), dimension(20)	available_visc_melt_models

character *30	visc_melt_model
	Parameter to select the melt viscosity (bubbles and crystal-free) model: . More...

real *8, dimension(12)	wt_init

real *8	rho_w
	Water density. More...

real *8	xa

real *8	xb

real *8	xc

Detailed Description

Constitutive equations.

Author: Mattia de' Michieli Vitturi This module contains the constitutive equations for the closure of the system.

Date: 04/02/2017

Function/Subroutine Documentation

subroutine constitutive::allocate_phases_parameters ( )

Initialization of relaxation flags.

Author: Mattia de' Michieli Vitturi This subroutine set the number and the flags of the non-hyperbolic terms.

Date: 07/09/2012

Definition at line 479 of file constitutive.f90.

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subroutine constitutive::eos ( )

Equation of state.

Author: Mattia de' Michieli Vitturi This subroutine evaluates, from the mixture temperature T, the phase densities $\rho_i$ and the mass fractions, the values of $e_i, p_i, s_i, \mu_i$ for the two phases.

Date: 15/08/2011

Definition at line 573 of file constitutive.f90.

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subroutine constitutive::eval_densities ( )

Phases densities.

Author: Mattia de' Michieli Vitturi This subroutine evaluates the densities of the two phases given the pressures p1 and p2, the temperature T and the mass fraction $x_{d,md}$. $\rho_i = \frac{p_i + \bar{p}_i}{c_{v,i}(\gamma_i-1)T}$

Date: 15/08/2011

Definition at line 729 of file constitutive.f90.

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subroutine constitutive::f_alfa	(	real*8, dimension(n_gas), intent(in)	xtot,
		real*8, dimension(n_gas), intent(in)	xmax,
		real*8, dimension(n_cry), intent(in)	r_beta,
		real*8, intent(in)	r_rho_md,
		real*8, intent(in)	r_rho_2,
		real*8, dimension(n_gas), intent(out)	r_alfa_2
	)

Bottom exsolved gas.

Author: Giuseppe La Spina This subrotine evaluates the exsolved gas volumetric fraction given the the total gas mass fraction and the dissolved gass mass fraction:
$\alpha_2= \frac{x_2(1-\beta)\rho_{md}}{(1-x_{tot})\rho_2 + x_2(1-\beta)\rho_{md}}$ ,
where $x_2=x_{tot}-x_{d,md}$ is the mass fraction of the exsolved gas with respect to the crystal-free magma.

Date: 13/03/12

Parameters

[in]	xtot	total gas mass fraction
[in]	xmax	dissolved gas mass fraction
[in]	r_beta	crystals volume fraction
[in]	r_rho_md	dis.gas+melt density
[in]	r_rho_2	exsolved gas density
[out]	r_alfa_2	exsolved gas volume fraction

Date: 02/12/12

Definition at line 2060 of file constitutive.f90.

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subroutine constitutive::f_alfa3	(	real*8, intent(in)	r_p_2,
		real*8, dimension(n_gas), intent(in)	xtot,
		real*8, dimension(n_cry), intent(in)	r_beta,
		real*8, intent(in)	r_rho_md,
		real*8, dimension(n_gas), intent(in)	r_rho_g,
		real*8, dimension(n_gas), intent(out)	r_alfa_g
	)

Bottom exsolved gas.

Author: Mattia de' Michieli Vitturi This subrotine evaluates the exsolved gas volumetric fraction given the the total gas mass fraction and the dissolved gass mass fraction:
$\alpha_2= \frac{x_2(1-\beta)\rho_{md}}{(1-x_{tot})\rho_2 + x_2(1-\beta)\rho_{md}}$ ,
where $x_2=x_{tot}-x_{d,md}$ is the mass fraction of the exsolved gas with respect to the crystal-free magma.

Date: 13/03/12

Parameters

[in]	r_p_2	gas pressure
[in]	xtot	total gas mass fraction
[in]	r_beta	crystals volume fraction
[in]	r_rho_md	dis.gas+melt density
[in]	r_rho_g	exsolved gas density
[out]	r_alfa_g	exsolved gas volume fraction

Date: 02/12/12

Definition at line 2103 of file constitutive.f90.

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subroutine constitutive::f_beta_eq ( )

Equilibrium Crystal content.

Author: Mattia de' Michieli Vitturi This subrotine evaluates the equilibrium crystal volume fraction as a function of the pressure:
$\beta = \min\left( beta_{max} , 0.45 + 0.55 (0.58815 (\frac{p}{1.D6})^{-0.5226} \right)$ ,

Parameters

[in]	press	liquid pressure
[in]	temp	magmatic mixture temperature
[in]	x_d	dissolved gas mass fraction
[in]	rho_c	density of crystal components
[in]	rho_1	density of phase 1
[out]	beta_eq	equilibrium crystal volume fraction

Date: 08/10/2012

Definition at line 1000 of file constitutive.f90.

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subroutine constitutive::f_bubbles ( )

Exsolved gas relative viscosity.

Author

Mattia de' Michieli Vitturi This subrotine evaluates the relative viscosity due to the influence of exsolved gas on the mixture viscosity according to the model specified in the input file with the parameter bubbles_model:

bubbles_model = Costa 2007 =>
bubbles_model = Einstein =>
bubbles_model = Quane-Russel =>
bubbles_model = Eilers =>
bubbles_model = Sibree =>

Date: 02/10/2012

Definition at line 1937 of file constitutive.f90.

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subroutine constitutive::f_growth_rate ( )

This subroutine compute the growth rates for the different crystal phases.

Author: Mattia de' Michieli Vitturi

Date: 01/03/2017

Definition at line 1047 of file constitutive.f90.

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subroutine constitutive::f_nucleation_rate ( )

Lithostatic pressure.

Author: Mattia de' Michieli Vitturi This subroutine compute the growth rates for the different crystal phases

Date: 01/03/2017

Definition at line 1072 of file constitutive.f90.

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subroutine constitutive::f_permkc ( )

Magma permeability.

Author: Mattia de' Michieli Vitturi This subroutine evaluates the permeability by means of the relationship from Klug and Cashman, 1996:
$perm_{kc}= perm_0*10.0^{-10.2(100*\alpha)^{\frac{0.014}{\alpha}}}$
for $\alpha\geq\overline{alpha}$ For $\alpha<\overline{alpha}$ a quadratic extrapolation is applied.

Date: 02/10/2012

Definition at line 1426 of file constitutive.f90.

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subroutine constitutive::f_theta ( )

Crystal relative viscosity.

Author

Mattia de' Michieli Vitturi This subrotine evaluates the relative viscosity $\theta$ due to the influence of crystal on the mixture viscosity according to the model specified in the input file with the parameter theta_model:

theta_model = 1 => $\theta = \left( 1 - \frac{\beta}{0.7}\right)^{-3,4}$ (Einstein-Roscoe with constants from Lejeune & Richter '95);
theta_model = 2 => $\theta = \left( 1+0.75\frac{\beta/0.84}{1-\beta/0.84} \right)^2$ (Dingwell et al '93);
theta_model = 3 => $\theta = \theta_0$ Melnik & Sparks '99
theta_model = 4 => $\theta = \left[ 1 - c_1 erf\left(0.5 * \sqrt\pi (1+\frac{c_2}{(1-\beta)^{c_3}}) \right)\right]^\frac{-2.5}{c_1}$ (Costa)
theta_model = 5 => $\theta = c_1 \cdot 10 ^{arctan(c_2(\beta-c_3))+\frac{\pi}{2}}$ (Melnik & Sparks '05).

Date: 02/10/2012

Definition at line 1758 of file constitutive.f90.

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subroutine constitutive::f_viscliq ( )

Magma viscosity.

Author: Mattia de' Michieli Vitturi This subroutine evaluates the viscosity of the (bubble-free) magma using the viscosity of the melt phase (bubbles and crystal-free magma) and the relative viscosity $1\theta$ due to the presence of the crsytals.

Date: 20/02/2009

Definition at line 1497 of file constitutive.f90.

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subroutine constitutive::f_viscmelt ( )

Melt viscosity.

Author: Mattia de' Michieli Vitturi This subrotine evaluates the melt viscosity (bubbles and crystal-free) using an empirical relationship of melt viscosity according to water concentration and temperature of melt (Equation 7 in Hess and Dingwell '96):
where w is the concentration of H2O in weight percent.

Date: 13/11/08

Definition at line 1526 of file constitutive.f90.

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subroutine constitutive::f_xdis_eq ( )

Equilibrium Dissolved gas.

Author: Mattia de' Michieli Vitturi This subrotine evaluates the equilibrium dissolved gas mass fraction by means of the Henry's law:
$x_{d,md}^{eq} = s\sqrt P\ (kg/m^3)$
where is a constant. If the pressure is smaller than nucleation pressure then $x_{d,md}^{eq} = x_{tot}$ (total mass gas fraction at the inlet).

Parameters

[in]	pres	liquid pressure
[in]	temp	temperature
[in]	xtot	total gas fractions in the mixture
[in]	alfa_g_2	exsolved gas volume fractions in phase 2
[in]	x_g	gas mass fractions
[in]	rho_md	dis_gas+melt density
[out]	x_d_md_eq	equilibrium dissolved gas

Date: 08/10/12

Definition at line 862 of file constitutive.f90.

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subroutine constitutive::hydrostatic_pressure ( )

Hydrostatic pressure.

Author: Mattia de' Michieli Vitturi This subrotine evaluates the hydrostatic pressure at depth zeta

Date: 13/03/12

Definition at line 1118 of file constitutive.f90.

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subroutine constitutive::initialize_models ( )

Definition at line 415 of file constitutive.f90.

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subroutine constitutive::lithostatic_pressure ( )

Lithostatic pressure.

Author: Mattia de' Michieli Vitturi This subrotine evaluates the lithostatic pressure at depth zeta

Date: 13/03/12

Definition at line 1097 of file constitutive.f90.

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subroutine constitutive::mixture_viscosity ( )

Mixture viscosity.

This subrotine evaluates the viscosity of the mixture (melt+crystals+gas) below the fragmentation level

Date: 11/03/2013

Definition at line 1473 of file constitutive.f90.

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subroutine constitutive::press_relax_term ( complex*16, intent(out) pressure_relaxation )

Pressure relaxation term.

Author: Mattia de' Michieli Vitturi This subrotine evaluates the term for the relaxation of the pressures.

Date: 08/10/2012

Parameters

[out] pressure_relaxation relaxation term

Definition at line 1140 of file constitutive.f90.

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subroutine constitutive::solve_cubic	(	complex*16, intent(in)	a,
		complex*16, intent(in)	b,
		complex*16, intent(in)	c,
		complex*16, intent(out)	y1,
		complex*16, intent(out)	y2,
		complex*16, intent(out)	y3
	)

Solution of a cubic.

This subroutine solves a cubic polynomial in the form y^3+ay^2+by+c=0. When the coefficients are real, the solutions found are real.

Parameters

[in]	a	polynomial coefficient
[in]	b	polynomial coefficient
[in]	c	polynomial coefficient
[out]	y1	solution
[out]	y2	solution
[out]	y3	solution

Date: 20/06/2013

Definition at line 796 of file constitutive.f90.

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subroutine constitutive::sound_speeds	(	real*8, intent(out)	C_mix,
		real*8, intent(out)	mach
	)

Local sound speeds.

Author: Mattia de' Michieli Vitturi This subroutine evaluates the local sound speed of the different phases as:
$C_i = \sqrt \frac{\partial p_i}{\partial \rho_i}$ and the mixture sound speed as:

Sound speeds of the phases

Compressibilites of the phases

Definition at line 647 of file constitutive.f90.

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subroutine constitutive::vel_relax_term ( complex*16, intent(out) velocity_relaxation )

Velocity relaxation term.

Author: Mattia de' Michieli Vitturi This subrotine evaluates the drag function for the relaxation of the velocities

Date: 08/10/2012

Parameters

[out] velocity_relaxation relaxation term

Definition at line 1203 of file constitutive.f90.

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Variable Documentation

real*8 constitutive::a_2nd

Definition at line 387 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::a_g

parameter for the VDW EOS

Definition at line 79 of file constitutive.f90.

complex*16 constitutive::alfa_1

dis_gas+melt+crystals phase local volume fraction

Definition at line 129 of file constitutive.f90.

complex*16 constitutive::alfa_2

total exsolved gas local volume fraction

Definition at line 130 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::alfa_d_1

dissolved gas volume fractions in phase 1

Definition at line 136 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::alfa_g

exsolved gas phases local volume fraction

Definition at line 131 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::alfa_g_2

exsolved gas volume fractions in phase 2

Definition at line 137 of file constitutive.f90.

complex*16 constitutive::alfa_m_1

melt volume fraction in phase 1

Definition at line 135 of file constitutive.f90.

real*8 constitutive::alfa_switch

Definition at line 386 of file constitutive.f90.

complex*16 constitutive::alfarho_2

bulk density of the exsolved gas

Definition at line 133 of file constitutive.f90.

character (len=30), dimension(20) constitutive::available_bubble_models

Definition at line 362 of file constitutive.f90.

character (len=30), dimension(20) constitutive::available_drag_models

Definition at line 181 of file constitutive.f90.

character (len=30), dimension(20) constitutive::available_theta_models

Definition at line 325 of file constitutive.f90.

character (len=30), dimension(20) constitutive::available_visc_melt_models

Definition at line 391 of file constitutive.f90.

real*8 constitutive::b_2nd

Definition at line 387 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::b_g

parameter for the VDW EOS

Definition at line 80 of file constitutive.f90.

real*8 constitutive::bar_e_2

exsolved gas formation energy

Definition at line 59 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::bar_e_c

crystals formation energy

Definition at line 61 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::bar_e_d

dissolved gas formation energy

Definition at line 62 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::bar_e_g

exsolved gas formation energy

Definition at line 63 of file constitutive.f90.

real*8 constitutive::bar_e_m

melt formation energy

Definition at line 60 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::bar_p_c

crystals cohesion pressure

Definition at line 66 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::bar_p_d

dissolved gas cohesion pressure

Definition at line 67 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::bar_p_g

exsolved gas cohesion pressure

Definition at line 68 of file constitutive.f90.

real*8 constitutive::bar_p_m

melt cohesion pressure

Definition at line 65 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::beta

crystal volume fraction in the melt-crystals phase

Definition at line 140 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::beta0

chamber (equilibrium) crystal volume fraction

Definition at line 247 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::beta_eq

equil. cry. volume fraction in the melt-crystals phase

Definition at line 141 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::beta_max

maximum crystal volume fraction

Definition at line 250 of file constitutive.f90.

real*8 constitutive::bubble_number_density

bubble number density

Definition at line 206 of file constitutive.f90.

character*20 constitutive::bubbles_model

Parameter to choose the model for the influence of the bubbles on the mixture:
.

'Einstein'
'Quane-Russel -> For Campi-Flegrei we use 0.63 as reported in the 2004 Report (Task 2.2)
'Eilers' -> Mader et al. 2013
'Sibree' -> Mader et al. 2013
'Taylor' -> Mader et al. 2013
'Mackenzie' -> Mader et al. 2013
'DucampRaj' -> Mader et al. 2013
'BagdassarovDingwell' -> Mader et al. 2013
'Rahaman' -> Mader et al. 2013

Definition at line 375 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::c0_c

crystals sound speed at atm conditions

Definition at line 31 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::c0_d

dissolved gas sound speed at atm conditions

Definition at line 32 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::c0_g

exsolved gas sound speed at atm conditions

Definition at line 33 of file constitutive.f90.

real*8 constitutive::c0_m

melt sound speed at atmospheric conditions

Definition at line 30 of file constitutive.f90.

real*8 constitutive::c1

Coefficients for the relative viscosity models.

Definition at line 349 of file constitutive.f90.

real*8 constitutive::c2

Definition at line 349 of file constitutive.f90.

real*8 constitutive::c3

Definition at line 349 of file constitutive.f90.

complex*16 constitutive::c_1

first phase local sound speed

Definition at line 176 of file constitutive.f90.

complex*16 constitutive::c_2

second phase local sound speed

Definition at line 177 of file constitutive.f90.

real*8 constitutive::c_d

Definition at line 219 of file constitutive.f90.

character*20 constitutive::crystallization_model

Model for the equilibrium crystal volume fraction:
.

'Vitturi2010' => Eq. 4 of de' Michieli Vitturi et al. 2010;

Definition at line 255 of file constitutive.f90.

complex*16 constitutive::cv_1

dis.gas+melt+crystals specific heat capacity at constant volume

Definition at line 90 of file constitutive.f90.

real*8 constitutive::cv_2

exsolved gas specific heat capacity at constant volume

Definition at line 23 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::cv_c

crystals specific heat capacity at constant volume

Definition at line 25 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::cv_d

dissolved gas heat capacity at constant volume

Definition at line 26 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::cv_g

exsolved gas heat capacity at constant volume

Definition at line 27 of file constitutive.f90.

real*8 constitutive::cv_m

melt specific heat capacity at constant volume

Definition at line 24 of file constitutive.f90.

complex*16 constitutive::cv_mix

mixture specific heat capacity at constant volume

Definition at line 171 of file constitutive.f90.

complex*16 constitutive::drag_funct

drag function for the relative velocity:

Definition at line 194 of file constitutive.f90.

real*8 constitutive::drag_funct_coeff

coefficient for the drag function for the relative velocity:

drag_funct_coeff > 0 => positive drag (finite rate velocity relaxation)
drag_funct_coeff $\rightarrow +\infty$ => instantaneous relaxation
drag_funct_coeff = 0 => no velocity relaxation

Definition at line 201 of file constitutive.f90.

character*30 constitutive::drag_funct_model

drag function model

'constant' => drag_funct = 1
'eval' => drag_funct = f(q)
'darcy' => Darcy formulation from Eq. 16 Degruyter et al. 2012
'forchheimer' => Forcheeimer from Eq. 16 Degruyter et al. 2012
'drag' => Stokes law for rising bubbles
'single_velocity' => Single velocity model (u1=u2)

Definition at line 191 of file constitutive.f90.

complex*16 constitutive::e_1

local specific internal energy of the melt-crystals phase

Definition at line 94 of file constitutive.f90.

complex*16 constitutive::e_2

local specific internal energy of the exsolved gas

Definition at line 95 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::e_c

local specific internal energy of the crystals

Definition at line 97 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::e_d

local specific internal energy of the dissolved gas

Definition at line 98 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::e_g

local specific internal energy of the exsolved gas

Definition at line 99 of file constitutive.f90.

complex*16 constitutive::e_m

local specific internal energy of the melt

Definition at line 96 of file constitutive.f90.

complex*16 constitutive::e_mix

total internal energy

Definition at line 92 of file constitutive.f90.

logical constitutive::explosive

Flag to choose the eruptive style:
.

explosive = .TRUE. => explosive eruption
explosive = .FALSE. => effusive eruption

Definition at line 380 of file constitutive.f90.

character*20 constitutive::exsol_model

String for exsolution model:
.

'Henry' => Henry's law;
'Zhang' => Zhang model (Zhang, JVGR '99).

Definition at line 293 of file constitutive.f90.

real*8 constitutive::frag_eff

index of fragmentation in the interval [0;1]

Definition at line 258 of file constitutive.f90.

real*8 constitutive::frag_thr

Threshold for the fragmentation.

Definition at line 268 of file constitutive.f90.

integer constitutive::fragmentation_model

Parameter to choose the fragmentation model:
.

fragmentation_model = 1 => volume fraction
fragmentation_model = 2 => overpressure
fragmentation_model = 3 => strain rate

Definition at line 265 of file constitutive.f90.

real*8 constitutive::friction_coefficient

Definition at line 216 of file constitutive.f90.

real*8 constitutive::gamma_2

exsolved gas adiabatic exponent

Definition at line 35 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::gamma_c

crystals adiabatic exponent

Definition at line 37 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::gamma_d

dissolved gas adiabatic exponent

Definition at line 38 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::gamma_g

exsolved gas adiabatic exponent

Definition at line 39 of file constitutive.f90.

real*8 constitutive::gamma_m

melt adiabatic exponent

Definition at line 36 of file constitutive.f90.

character*20 constitutive::gas_law

equation of state for gas

'ideal' => ideal gas law
'VDW' => Van der Waals equation of state

Definition at line 86 of file constitutive.f90.

real*8 constitutive::grav

gravitational acceleration

Definition at line 303 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::growth_rate

growth rate for the crystals

Definition at line 145 of file constitutive.f90.

complex*16 constitutive::inv_permkc

Definition at line 384 of file constitutive.f90.

real*8 constitutive::k_cr

country rock permeability

Definition at line 306 of file constitutive.f90.

real*8 constitutive::log10_bubble_number_density

Definition at line 207 of file constitutive.f90.

complex*16, dimension(:,:), allocatable constitutive::mom_cry

moments of the crystal referred to the melt-crystals phase

Definition at line 143 of file constitutive.f90.

complex*16 constitutive::mu_1

free Gibbs energy of the melt-crystals phase

Definition at line 111 of file constitutive.f90.

complex*16 constitutive::mu_2

free Gibbs energy of the exsolved gas

Definition at line 112 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::mu_c

free Gibbs energy of the crystals

Definition at line 114 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::mu_d

free Gibbs energy of the dissolved gas

Definition at line 115 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::mu_g

free Gibbs energy of the exsolved gas

Definition at line 116 of file constitutive.f90.

complex*16 constitutive::mu_m

free Gibbs energy of the melt

Definition at line 113 of file constitutive.f90.

integer constitutive::n_bubble_models

Definition at line 360 of file constitutive.f90.

integer constitutive::n_drag_models

Definition at line 179 of file constitutive.f90.

integer constitutive::n_theta_models

Definition at line 323 of file constitutive.f90.

integer constitutive::n_visc_melt_models

Definition at line 389 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::nucleation_rate

nulceation rate for the crystals

Definition at line 146 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::p0_c

crystals reference pressure

Definition at line 55 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::p0_d

dissolved gas reference pressure

Definition at line 56 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::p0_g

exsolved gas reference pressure

Definition at line 57 of file constitutive.f90.

real*8 constitutive::p0_m

melt reference pressure

Definition at line 54 of file constitutive.f90.

complex*16 constitutive::p_1

local pressure of the melt-crystals phase

Definition at line 101 of file constitutive.f90.

complex*16 constitutive::p_2

local pressure of the exsolved gas

Definition at line 102 of file constitutive.f90.

real*8 constitutive::p_hydro

Hydrostatic pressure.

Definition at line 355 of file constitutive.f90.

real*8 constitutive::p_lith

Lithostatic pressure.

Definition at line 352 of file constitutive.f90.

character*20 constitutive::p_relax_model

pressure relaxation model

'constant' => tau_p = 1
'eval' => tau_p = f(q)

Definition at line 228 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::pc_g

critical gas pressure

Definition at line 77 of file constitutive.f90.

real*8 constitutive::perm0

Definition at line 387 of file constitutive.f90.

complex*16 constitutive::permkc

Magma permeability.

Definition at line 383 of file constitutive.f90.

real*8 constitutive::r_a

Definition at line 222 of file constitutive.f90.

real*8 constitutive::rho0_2

exsolved gas reference density

Definition at line 41 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::rho0_c

crystals reference density

Definition at line 43 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::rho0_d

dissolved gas reference density

Definition at line 44 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::rho0_g

exsolved gas reference density

Definition at line 45 of file constitutive.f90.

real*8 constitutive::rho0_m

melt reference density

Definition at line 42 of file constitutive.f90.

complex*16 constitutive::rho_1

dis_gas+melt+crystals phase local density

Definition at line 118 of file constitutive.f90.

complex*16 constitutive::rho_2

exsolved gas local density

Definition at line 119 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::rho_c

crystals local density

Definition at line 121 of file constitutive.f90.

real*8 constitutive::rho_cr

contry rock density

Definition at line 309 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::rho_d

dissolved gas local density

Definition at line 122 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::rho_g

exsolved gas local density

Definition at line 123 of file constitutive.f90.

complex*16 constitutive::rho_m

melt local density

Definition at line 120 of file constitutive.f90.

complex*16 constitutive::rho_md

dis_gas+melt local density

Definition at line 124 of file constitutive.f90.

complex*16 constitutive::rho_mix

mixture local density

Definition at line 169 of file constitutive.f90.

real*8 constitutive::rho_w

Water density.

Definition at line 407 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::rhob_c

crystals bulk density

Definition at line 126 of file constitutive.f90.

complex*16 constitutive::rhob_m

melt bulk density

Definition at line 127 of file constitutive.f90.

complex*16 constitutive::s

mixture entropy

Definition at line 174 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::s0_c

crystals reference entropy

Definition at line 72 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::s0_d

dissolved gas reference entropy

Definition at line 73 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::s0_g

exsolved gas reference entropy

Definition at line 74 of file constitutive.f90.

real*8 constitutive::s0_m

melt reference entropy

Definition at line 71 of file constitutive.f90.

complex*16 constitutive::s_1

local specific entropy of the melt-crystals phase

Definition at line 104 of file constitutive.f90.

complex*16 constitutive::s_2

local specific entropy of the exsolved gas

Definition at line 105 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::s_c

local specific entropy of the crystals

Definition at line 107 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::s_d

local specific entropy of the dissolved gas

Definition at line 108 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::s_g

local specific entropy of the exsolved gas

Definition at line 109 of file constitutive.f90.

complex*16 constitutive::s_m

local specific entropy of the melt

Definition at line 106 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::solub

Solubility parameter for the Henry's law.

Definition at line 296 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::solub_exp

Definition at line 298 of file constitutive.f90.

complex*16 constitutive::t

mixture local temperature

Definition at line 168 of file constitutive.f90.

real*8 constitutive::t0_2

exsolved gas reference temperature

Definition at line 47 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::t0_c

crystals gas reference temperature

Definition at line 49 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::t0_d

dissolved gas reference temperature

Definition at line 50 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::t0_g

exsolved gas reference temperature

Definition at line 51 of file constitutive.f90.

real*8 constitutive::t0_m

melt reference temperature

Definition at line 48 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::tau_c

crystallization parameter:

tau_c = 0 => instantaneous crystallization (equilibrium);
tau_c > 0 => finite rate crystallization;
tau_c = -1 => no crystallization.

Definition at line 244 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::tau_d

exsolution parameter:

tau_d = 0 => instantaneous exsolution (equilibrium);
tau_d > 0 => finite rate exsolution;
tau_d = -1 => no exsolution.

Definition at line 287 of file constitutive.f90.

complex*16 constitutive::tau_frag

fragmentation rate

Definition at line 274 of file constitutive.f90.

real*8 constitutive::tau_frag_coeff

fragmentation coefficient:

tau_frag_coeff = 0 => instantaneous fragmentation;
tau_frag_coeff > 0 => finite rate fragmentation;

Definition at line 280 of file constitutive.f90.

real*8 constitutive::tau_frag_exp

Fragmentation exponent.

Definition at line 271 of file constitutive.f90.

complex*16 constitutive::tau_p

pressure relaxation rate

Definition at line 231 of file constitutive.f90.

real*8 constitutive::tau_p_coeff

pressure relaxation coefficient:

tau_p_coeff = 0 => instantaneous relaxation (sinlge pressure);
tau_p_coeff > 0 => finite rate relaxation;

Definition at line 237 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::tc_g

critical gas temperature

Definition at line 78 of file constitutive.f90.

complex*16 constitutive::theta

Relative viscosity of the crystals.

Definition at line 343 of file constitutive.f90.

real*8 constitutive::theta_fixed

Fixed value for the relative viscosity of the crystals.

Definition at line 346 of file constitutive.f90.

character*30 constitutive::theta_model

Parameter to choose the model for the influence of crystal on the mixture: 'Lejeune_and_Richet1995' 'Dingwell1993' 'Melnik_and_Sparks1999' 'Costa2005' 'Melnik_and_Sparks2005' 'Vona_et_al2011' 'Vona_et_al2011_mod' 'Vona_et_al2013_eq19' 'Vona_et_al2013_eq20' 'Vona_et_al2013_eq21' 'Fixed_value'

Definition at line 340 of file constitutive.f90.

real*8 constitutive::throat_bubble_ratio

throat bubble ratio

Definition at line 213 of file constitutive.f90.

real*8 constitutive::tortuosity_factor

tortuosity factor

Definition at line 210 of file constitutive.f90.

complex*16 constitutive::u_1

melt-crystals phase local velocity

Definition at line 148 of file constitutive.f90.

complex*16 constitutive::u_2

exsolved gas local velocity

Definition at line 149 of file constitutive.f90.

complex*16 constitutive::u_mix

mixture velocity

Definition at line 170 of file constitutive.f90.

complex*16 constitutive::visc_1

melt+crystal viscosity

Definition at line 318 of file constitutive.f90.

real*8 constitutive::visc_2

gas viscosity

Definition at line 312 of file constitutive.f90.

complex*16 constitutive::visc_melt

melt viscosity

Definition at line 315 of file constitutive.f90.

character*30 constitutive::visc_melt_model

Parameter to select the melt viscosity (bubbles and crystal-free) model:
.

'Hess_and_Dingwell1996'
'Romano_et_al2003'
'Giordano_et_al2008'
'Giordano_et_al2009'
'Di_Genova_et_al2013_eqn_3,5'
'Di_Genova_et_al2013_eqn_4,5'

Definition at line 401 of file constitutive.f90.

complex*16 constitutive::visc_mix

mixture viscosity

Definition at line 172 of file constitutive.f90.

complex*16 constitutive::visc_rel_bubbles

relative viscosity due to bubbles

Definition at line 321 of file constitutive.f90.

real*8, dimension(12) constitutive::wt_init

Definition at line 404 of file constitutive.f90.

complex*16 constitutive::x_1

melt-crystals phase local mass fraction

Definition at line 151 of file constitutive.f90.

complex*16 constitutive::x_2

exsolved gas local mass fraction

Definition at line 152 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::x_c

crystals mass fraction (with respect to the mixture)

Definition at line 153 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::x_c_1

cristal mass fractions in phase 1

Definition at line 162 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::x_d

dissolved gas mass fraction (with respect to the mixture)

Definition at line 155 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::x_d_1

dissolved gas mass fractions in phase 1

Definition at line 165 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::x_d_md

dissolved gas mass fraction in the melt+dis.gas phase

Definition at line 158 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::x_d_md_eq

equil. dis. gas mass fraction in the melt+dis.gas phase

Definition at line 159 of file constitutive.f90.

real*8, dimension(:), allocatable constitutive::x_ex_dis_in

Definition at line 300 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::x_g

exsolved gas mass fraction (with respect to the mixture)

Definition at line 156 of file constitutive.f90.

complex*16, dimension(:), allocatable constitutive::x_g_2

exsolved gas mass fractions in phase 2

Definition at line 166 of file constitutive.f90.

complex*16 constitutive::x_m

melt mass fraction (with respect to the mixture)

Definition at line 154 of file constitutive.f90.

complex*16 constitutive::x_m_1

melt mass fraction in phase 1

Definition at line 163 of file constitutive.f90.

complex*16 constitutive::x_md_1

melt+dis.gas mass fraction in phase 1

Definition at line 164 of file constitutive.f90.

real*8 constitutive::xa

Definition at line 409 of file constitutive.f90.

real*8 constitutive::xb

Definition at line 409 of file constitutive.f90.

real*8 constitutive::xc

Definition at line 409 of file constitutive.f90.

real*8 constitutive::zeta_lith

Elevation above the bottom for the evaluation of the lithostatic pressure.

Definition at line 358 of file constitutive.f90.

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