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* strain_rate_distribution lets selection for distribution of tidal strain rate.
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* If 'constant', the tidal strain rate is fixed to 'Constant tidal strain rate'.
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* If 'latitudinal variation', 'Maximum tidal strain rate' and 'Minimum tidal strain rate' are used.
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* Then, the equation follows as (maximum_tidal_strain_rate - minimum_tidal_strain_rate)*cos(theta/2)+(maximum_tidal_strain_rate+minimum_tidal_strain_rate)/2.
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* This is shown in Fig.3 of Nimmo et al. (2007) (https://doi.org/10.1016/j.icarus.2007.04.021).
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*/
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double tidal_frequency;
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double elastic_shear_modulus;
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double constant_tidal_strain_rate;
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/**
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* Specify the distribution of time-averaged tidal strain rate.
* time-averaged strain rate at certain location = local_strain_rate
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* tidal frequency = tidal_frequency
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* elastic shear modulus = elastic_shear_modulus
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* If 'Use latitudinal variation of strain rate' is true, local_strain_rate is calculated
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* with one period cosine function having maximum tidal strain rate at poles and minimum tidal strain rate at equator.
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* This variation of tidal strain rate follows as (maximum_tidal_strain_rate - minimum_tidal_strain_rate)*cos(theta/2)+(maximum_tidal_strain_rate+minimum_tidal_strain_rate)/2
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* where theta is polar angle from spherical coordinates.
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* If false, constant_tidal_strain_rate is used.
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*/
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for (unsignedint q=0; q<heating_model_outputs.heating_source_terms.size(); ++q)
# This parameter file tests the tidal heating plugin for a case where the
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# viscosity is constant and the tidal heating is dependent on latitude.
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#
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# The equation implemented in this heating model is from Tobie et al. (2003) (https://doi.org/10.1029/2003JE002099),
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# which is defined as:
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# H= 2*(viscosity)*(time-averaged tidal strain rate)^2/(1+((viscosity)*(tidal frequency)/(shear modulus))^2)), where
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# viscosity is the viscosity derived from the material model at every point (constant in this test - 1e14 Pa s)
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# The latitudinal variation of (time-averaged tidal strain rate) is simplified with cosine function between maximum tidal strain rate and minimum tidal strain rate.
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# The variation can be found in Fig.3 of Nimmo et al. (2007) (https://doi.org/10.1016/j.icarus.2007.04.021).
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#
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# The governing equation in the model is simplified as (density)*(specific heat capacity)*dT/dt=H, as unit of H is W/m^3.
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# As expected, temperature increases with time as the convective and conductive processes are not active.
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#
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# Analytical values are 2.85946709e+02 K and 1.65676318e+02 K at the pole and equator, respectively.
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# These values align with ASPECT's results within numerical accuracy. ASPECT's results are 2.86027243e+02 K and 1.65661824e+02 K at poles and equators, respectively.
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set Dimension = 3
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set Use years in output instead of seconds = true
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set End time = 1e6
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set Output directory = tidal_heating_latitude
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set Maximum first time step = 1e5
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set CFL number = 0.8
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set Maximum time step = 1e5
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set Pressure normalization = surface
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set Surface pressure = 0
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subsection Geometry model
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set Model name = spherical shell
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subsection Spherical shell
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set Outer radius = 1560800
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set Inner radius = 1460800
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set Opening angle = 360
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end
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end
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subsection Initial temperature model
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set Model name = function
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subsection Function
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set Coordinate system = spherical
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set Variable names = r, phi,theta
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set Function expression = 100
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end
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end
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subsection Boundary velocity model
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set Zero velocity boundary indicators = top, bottom
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end
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subsection Gravity model
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set Model name = radial constant
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subsection Radial constant
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set Magnitude = 0 #1.3
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end
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end
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subsection Material model
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set Model name = simpler
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subsection Simpler model
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set Reference density = 917
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set Reference specific heat = 2110
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set Reference temperature = 100
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set Thermal conductivity = 0 #1.93
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set Thermal expansion coefficient = 0 #1.6e-4
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set Viscosity = 1e14
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end
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end
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subsection Heating model
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set List of model names = tidal heating
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subsection Tidal heating
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set Tidal frequency = 2.048e-5
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set Elastic shear modulus = 3.3e9
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set Custom distribution of tidal strain rate = latitudinal variation
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set Maximum tidal strain rate = 2.81e-10
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set Minimum tidal strain rate = 1.67e-10
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end
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end
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subsection Formulation
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set Formulation = Boussinesq approximation
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end
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subsection Mesh refinement
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set Initial global refinement = 1
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set Initial adaptive refinement = 0
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set Time steps between mesh refinement = 0
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end
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subsection Postprocess
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set List of postprocessors = velocity statistics, temperature statistics, visualization, basic statistics, \
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pressure statistics, material statistics, heating statistics
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subsection Visualization
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set Time between graphical output = 1e5
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set Output format = vtu
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set List of output variables = material properties, strain rate, shear stress, stress, nonadiabatic pressure, heating
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