Pressure, Plate Motion, and Earthquake Size#

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Part I: Pressure variations on and within a terrestrial planet#

Pressure, force per unit area, plays an important role in making a habitable planet. Atmospheric pressure controls the living condition on the planet’s surface while the pressure deep inside the planet dictate which materials at which states exist in the planet. We study pressure on Earth and other planets to understand the dynamic of the planets and find the condition that life may prosper.

Section A: pressure at the surface#

Pressure at the surface mostly comes from the atmosphere layer on top of the rocky planet. The surface pressure dictates whether liquid water can exist on the planet. The variation of the pressure in space and time causes the circulation of air and more importantly, weather.

The air pressure at the Earth’s surface is \(1.013 \times 10^5 \text{ N/m}^2\). This pressure supports the weight of the Earth’s atmosphere. (Weight is the gravitational force: \(\text{weight} = m \times g\).)

TO DO#

Question 1 What is the mass of Earth’s atmosphere? The radius of the Earth: \(6371 \text{ km}\). The gravitational acceleration \(g: 9.8 \text{ m/s}^2\). You assume that weight of the Earth’s atmosphere acts equally on Earth’s surface.

Answer





Question 2 The planets closest to us, Mars and Venus, are in many ways similar to Earth, but their atmospheres are quite different.

Planet

Venus

Mars

Radius

6052 km

3397 km

Gravitational acceleration

9.1 m/s\(^2\)

3.8 m/s\(^2\)

Mass of atmosphere

4.5 x 10\(^{20}\) kg

2.2 x 10\(^{16}\) kg

What is the atmospheric pressure on the surface of Mars and Venus?

Answer





Section B: pressure inside the rocky interior#

While measuring pressure on the surface is easy, measuring deep inside the interior is challenging if possible, because we may not directly access the environment down below. Instead, we determine the pressure from other physical properties including planet’s density which can be determined from seismological observations and astronomical observations.

Dziewonski and Anderson (1980) made the Preliminary Reference Earth Model (PREM) from normal mode observations (mode of the Earth’s vibrations), travel time observations for seismic waves and astronomic and geodetic data. PREM contains physical properties such as density, seismic wave velocities, and other elastic properties at specific depths including major boundaries such as Mohorovičić discontinuity and core-mantle bounary.

../_images/PREM_vp_vs_rho.png

Image: Seismic velocities (\(\alpha\) and \(\beta\)) and density (\(\rho\)) for the Preliminary Reference Earth Model (PREM).

In this problem set, we will use PREM to calculate the mass of the Earth, gravitational acceleration, and pressure at any given depth. We have demonstrated how to calculate the pressure inside a planet \(P(r)\) from the planet’s mass density \(\rho = \rho(r)\) and the pressure at the planet’s surface \(P(R)\).

(1)#\[\begin{equation} P(r) = P(R) + \int_r^R \rho(a) g(a) da \end{equation}\]

The gravitational acceration \(g(r)\) inside a planet can be determined using Newton’s law of gravitation with mass inside the sphere of radius \(r\):

(2)#\[\begin{equation} g(r) = \frac{GM_{\text{inside}}}{r^2} = \frac{G}{r^2} \int_0^r 4\pi a^2 \rho(a) da \end{equation}\]

Note that the density must satisfy the equation

(3)#\[\begin{equation} M_E = \int_0^R 4\pi r^2 \rho(r) dr \end{equation}\]

where \(M_E\) is the planet’s mass and \(R\) is the planet radius. If you wonder about this equation, it is essentially \(\text{mass} = \text{density} \times \text{volume}\) integrated over the entire sphere.

import pint
import pandas as pd
import pint_pandas

def read_mineos_cards(file,header = 3, R = None):
    """
    Read a card deck file of physical properties in mineos format
    
    Input Parameters:
    ----------------
    
    file: mineos card file containing columns with various properties
    
    header: number of lines in the header
    
    R: Mean radius of the planet
    """

    # Get the default unit registry e.g. MKS units
    ureg = pint.get_application_registry()
        
    # set default radius as Earth
    if R is None: R = 6371000.0 * ureg.meter

    names=['radius','rho','vpv','vsv','qkappa','qmu','vph','vsh','eta']
    units =['m','kg/m^3','m/s','m/s','dimensionless','dimensionless','m/s','m/s','dimensionless']
    fields=list(zip(names,units))
    #formats=[np.float for ii in range(len(fields))]
    # modelarr = np.genfromtxt(file,dtype=None,comments='#',skip_header=3,names=fields)
    modelarr = pd.read_csv(file,skiprows=header,comment='#',sep='\s+',names=fields)

    # read the units from last header
    modelarr_ = modelarr.pint.quantify(level=-1)
    
    # Get the depths based on subtracting radius from R
    modelarr_['depth'] = R - modelarr_['radius'].pint.to(R.units)
                            
    return modelarr_
PREM = read_mineos_cards('Files/PREM750_CARDS')

PREM
/home/globalseismology/.conda/envs/fall2022-sef/lib/python3.9/site-packages/pint_pandas/pint_array.py:648: UnitStrippedWarning: The unit of the quantity is stripped when downcasting to ndarray.
  return np.array(qtys, dtype="object", copy=copy)
/home/globalseismology/.conda/envs/fall2022-sef/lib/python3.9/site-packages/pint_pandas/pint_array.py:648: UnitStrippedWarning: The unit of the quantity is stripped when downcasting to ndarray.
  return np.array(qtys, dtype="object", copy=copy)
radius rho vpv vsv qkappa qmu vph vsh eta depth
0 0.0 13088.48 11262.21 3667.8 1327.6 84.6 11262.21 3667.8 1.0 6371000.0
1 6824.0 13088.47 11262.2 3667.79 1327.6 84.6 11262.2 3667.79 1.0 6364176.0
2 13648.0 13088.44 11262.18 3667.78 1327.6 84.6 11262.18 3667.78 1.0 6357352.0
3 20472.0 13088.39 11262.14 3667.75 1327.6 84.6 11262.14 3667.75 1.0 6350528.0
4 27296.0 13088.32 11262.09 3667.72 1327.6 84.6 11262.09 3667.72 1.0 6343704.0
... ... ... ... ... ... ... ... ... ... ...
745 6369800.0 1020.0 1450.0 0.0 57822.5 0.0 1450.0 0.0 1.0 1200.0
746 6370100.0 1020.0 1450.0 0.0 57822.5 0.0 1450.0 0.0 1.0 900.0
747 6370400.0 1020.0 1450.0 0.0 57822.5 0.0 1450.0 0.0 1.0 600.0
748 6370700.0 1020.0 1450.0 0.0 57822.5 0.0 1450.0 0.0 1.0 300.0
749 6371000.0 1020.0 1450.0 0.0 57822.5 0.0 1450.0 0.0 1.0 0.0

750 rows × 10 columns

# grabbing values in 'g/cc' as a pandas series
PREM.rho.pint.to('g/cc')
0      13.088480000000002
1      13.088470000000003
2      13.088440000000004
3      13.088390000000002
4      13.088320000000003
              ...        
745    1.0200000000000002
746    1.0200000000000002
747    1.0200000000000002
748    1.0200000000000002
749    1.0200000000000002
Name: rho, Length: 750, dtype: pint[gram / cubic_centimeter]
# grabbing value in 'g/cc' as a numpy array
PREM.rho.pint.to('g/cc').values.data
array([13.08848, 13.08847, 13.08844, 13.08839, 13.08832, 13.08822,
       13.08811, 13.08798, 13.08783, 13.08766, 13.08746, 13.08725,
       13.08702, 13.08676, 13.08649, 13.0862 , 13.08588, 13.08555,
       13.08519, 13.08482, 13.08442, 13.08401, 13.08357, 13.08311,
       13.08264, 13.08214, 13.08162, 13.08109, 13.08053, 13.07995,
       13.07935, 13.07873, 13.07809, 13.07744, 13.07676, 13.07606,
       13.07534, 13.0746 , 13.07384, 13.07306, 13.07225, 13.07143,
       13.07059, 13.06973, 13.06885, 13.06795, 13.06702, 13.06608,
       13.06512, 13.06413, 13.06313, 13.0621 , 13.06106, 13.06   ,
       13.05891, 13.05781, 13.05668, 13.05553, 13.05437, 13.05318,
       13.05198, 13.05075, 13.0495 , 13.04823, 13.04695, 13.04564,
       13.04431, 13.04296, 13.04159, 13.0402 , 13.03879, 13.03736,
       13.03591, 13.03444, 13.03295, 13.03144, 13.02991, 13.02836,
       13.02679, 13.0252 , 13.02358, 13.02195, 13.0203 , 13.01863,
       13.01693, 13.01522, 13.01349, 13.01173, 13.00996, 13.00816,
       13.00635, 13.00451, 13.00266, 13.00078, 12.99888, 12.99697,
       12.99503, 12.99307, 12.9911 , 12.9891 , 12.98708, 12.98504,
       12.98299, 12.98091, 12.97881, 12.97669, 12.97455, 12.97239,
       12.97021, 12.96801, 12.96579, 12.96355, 12.96129, 12.95901,
       12.9567 , 12.95438, 12.95204, 12.94968, 12.94729, 12.94489,
       12.94247, 12.94002, 12.93756, 12.93508, 12.93257, 12.93005,
       12.9275 , 12.92494, 12.92235, 12.91975, 12.91712, 12.91447,
       12.91181, 12.90912, 12.90641, 12.90368, 12.90094, 12.89817,
       12.89538, 12.89257, 12.88974, 12.88689, 12.88402, 12.88113,
       12.87822, 12.87529, 12.87234, 12.86937, 12.86638, 12.86337,
       12.86034, 12.85729, 12.85421, 12.85112, 12.84801, 12.84488,
       12.84172, 12.83855, 12.83535, 12.83214, 12.82891, 12.82565,
       12.82238, 12.81908, 12.81576, 12.81243, 12.80907, 12.8057 ,
       12.8023 , 12.79888, 12.79545, 12.79199, 12.78851, 12.78501,
       12.78149, 12.77795, 12.7744 , 12.77082, 12.76722, 12.7636 ,
       12.16635, 12.15977, 12.15314, 12.14645, 12.13971, 12.13291,
       12.12605, 12.11914, 12.11218, 12.10515, 12.09807, 12.09093,
       12.08373, 12.07648, 12.06917, 12.06179, 12.05437, 12.04688,
       12.03933, 12.03172, 12.02405, 12.01633, 12.00854, 12.00069,
       11.99278, 11.98481, 11.97678, 11.96868, 11.96053, 11.95231,
       11.94403, 11.93569, 11.92728, 11.91881, 11.91028, 11.90168,
       11.89302, 11.8843 , 11.87551, 11.86666, 11.85774, 11.84875,
       11.8397 , 11.83058, 11.8214 , 11.81215, 11.80284, 11.79346,
       11.784  , 11.77449, 11.7649 , 11.75525, 11.74553, 11.73574,
       11.72588, 11.71595, 11.70595, 11.69589, 11.68575, 11.67554,
       11.66526, 11.65492, 11.6445 , 11.634  , 11.62344, 11.61281,
       11.6021 , 11.59132, 11.58047, 11.56955, 11.55855, 11.54748,
       11.53634, 11.52512, 11.51383, 11.50246, 11.49102, 11.4795 ,
       11.46791, 11.45625, 11.4445 , 11.43269, 11.42079, 11.40882,
       11.39677, 11.38464, 11.37244, 11.36016, 11.3478 , 11.33537,
       11.32285, 11.31026, 11.29758, 11.28483, 11.272  , 11.25909,
       11.2461 , 11.23303, 11.21987, 11.20664, 11.19333, 11.17993,
       11.16645, 11.15289, 11.13925, 11.12553, 11.11172, 11.09783,
       11.08386, 11.0698 , 11.05566, 11.04144, 11.02713, 11.01274,
       10.99826, 10.9837 , 10.96905, 10.95432, 10.9395 , 10.92459,
       10.9096 , 10.89452, 10.87935, 10.8641 , 10.84876, 10.83333,
       10.81781, 10.80221, 10.78651, 10.77073, 10.75486, 10.7389 ,
       10.72285, 10.70671, 10.69048, 10.67416, 10.65775, 10.64124,
       10.62465, 10.60796, 10.59119, 10.57432, 10.55736, 10.5403 ,
       10.52316, 10.50592, 10.48858, 10.47116, 10.45363, 10.43602,
       10.41831, 10.40051, 10.38261, 10.36461, 10.34652, 10.32834,
       10.31005, 10.29168, 10.2732 , 10.25463, 10.23596, 10.2172 ,
       10.19833, 10.17937, 10.16031, 10.14115, 10.1219 , 10.10254,
       10.08309, 10.06353, 10.04388, 10.02412, 10.00427,  9.98432,
        9.96426,  9.9441 ,  9.92384,  9.90348,  5.56645,  5.56175,
        5.55705,  5.55236,  5.54766,  5.54297,  5.53828,  5.53359,
        5.5289 ,  5.52421,  5.51953,  5.51485,  5.51016,  5.50548,
        5.50081,  5.49613,  5.49145,  5.49145,  5.48673,  5.48201,
        5.47729,  5.47257,  5.46785,  5.46313,  5.45842,  5.4537 ,
        5.44899,  5.44427,  5.43956,  5.43485,  5.43013,  5.42542,
        5.42071,  5.416  ,  5.41129,  5.40657,  5.40186,  5.39715,
        5.39244,  5.38773,  5.38302,  5.3783 ,  5.37359,  5.36888,
        5.36417,  5.35945,  5.35474,  5.35002,  5.34531,  5.34059,
        5.33587,  5.33116,  5.32644,  5.32172,  5.317  ,  5.31228,
        5.30755,  5.30283,  5.2981 ,  5.29338,  5.28865,  5.28392,
        5.27919,  5.27446,  5.26972,  5.26498,  5.26025,  5.25551,
        5.25077,  5.24602,  5.24128,  5.23653,  5.23178,  5.22703,
        5.22227,  5.21752,  5.21276,  5.208  ,  5.20323,  5.19847,
        5.1937 ,  5.18893,  5.18415,  5.17938,  5.1746 ,  5.16982,
        5.16503,  5.16024,  5.15545,  5.15065,  5.14586,  5.14106,
        5.13625,  5.13144,  5.12663,  5.12182,  5.117  ,  5.11217,
        5.10735,  5.10252,  5.09769,  5.09285,  5.08801,  5.08316,
        5.07831,  5.07346,  5.0686 ,  5.06374,  5.05887,  5.054  ,
        5.04913,  5.04425,  5.03936,  5.03447,  5.02958,  5.02468,
        5.01978,  5.01487,  5.00996,  5.00504,  5.00012,  4.99519,
        4.99026,  4.98532,  4.98038,  4.97543,  4.97047,  4.96551,
        4.96055,  4.95558,  4.9506 ,  4.94562,  4.94063,  4.93564,
        4.93064,  4.92563,  4.92062,  4.9156 ,  4.91058,  4.90555,
        4.90051,  4.89547,  4.89042,  4.88537,  4.88031,  4.87524,
        4.87016,  4.86508,  4.85999,  4.8549 ,  4.8498 ,  4.84469,
        4.83957,  4.83445,  4.82932,  4.82418,  4.81904,  4.81388,
        4.80873,  4.80356,  4.79839,  4.7932 ,  4.78802,  4.78282,
        4.77761,  4.7724 ,  4.76718,  4.76195,  4.75672,  4.75148,
        4.74622,  4.74096,  4.73569,  4.73042,  4.72513,  4.71984,
        4.71454,  4.70923,  4.70391,  4.69858,  4.69325,  4.6879 ,
        4.68255,  4.67719,  4.67182,  4.66644,  4.66105,  4.65565,
        4.65024,  4.64482,  4.6394 ,  4.63396,  4.62852,  4.62306,
        4.6176 ,  4.61213,  4.60664,  4.60115,  4.59565,  4.59014,
        4.58461,  4.57908,  4.57354,  4.56799,  4.56242,  4.55685,
        4.55127,  4.54568,  4.54007,  4.53446,  4.52883,  4.5232 ,
        4.51755,  4.5119 ,  4.50623,  4.50055,  4.49486,  4.48916,
        4.48345,  4.47773,  4.472  ,  4.46625,  4.4605 ,  4.45473,
        4.44895,  4.44317,  4.44317,  4.4393 ,  4.43543,  4.43156,
        4.42767,  4.42379,  4.4199 ,  4.416  ,  4.4121 ,  4.4082 ,
        4.40428,  4.40037,  4.39644,  4.39252,  4.38858,  4.38465,
        4.3807 ,  3.99214,  3.99097,  3.98981,  3.98864,  3.98748,
        3.98632,  3.98515,  3.98399,  3.98282,  3.98166,  3.98049,
        3.97933,  3.97817,  3.977  ,  3.97584,  3.97584,  3.96744,
        3.95903,  3.95063,  3.94223,  3.93383,  3.92542,  3.91702,
        3.90862,  3.90022,  3.89181,  3.88341,  3.87501,  3.86661,
        3.8582 ,  3.8498 ,  3.8414 ,  3.833  ,  3.82459,  3.81619,
        3.80779,  3.79939,  3.79098,  3.78258,  3.77418,  3.76578,
        3.75737,  3.74897,  3.74057,  3.73217,  3.72376,  3.54326,
        3.53968,  3.53609,  3.53251,  3.52893,  3.52534,  3.52176,
        3.51818,  3.51459,  3.51101,  3.50743,  3.50385,  3.50026,
        3.49668,  3.4931 ,  3.48951,  3.48593,  3.48235,  3.47877,
        3.47518,  3.4716 ,  3.46802,  3.46443,  3.46085,  3.45727,
        3.45368,  3.4501 ,  3.44652,  3.44294,  3.43935,  3.43577,
        3.35949,  3.36016,  3.36082,  3.36148,  3.36214,  3.3628 ,
        3.36346,  3.36413,  3.36479,  3.36545,  3.36611,  3.36677,
        3.36743,  3.36809,  3.36876,  3.36942,  3.37008,  3.37074,
        3.3714 ,  3.37206,  3.37273,  3.37339,  3.37405,  3.37471,
        3.37471,  3.37514,  3.37557,  3.37601,  3.37644,  3.37687,
        3.3773 ,  3.37773,  3.37817,  3.3786 ,  3.37903,  3.37946,
        3.37989,  3.38032,  3.38076,  2.9    ,  2.9    ,  2.9    ,
        2.9    ,  2.9    ,  2.9    ,  2.9    ,  2.9    ,  2.9    ,
        2.9    ,  2.9    ,  2.6    ,  2.6    ,  2.6    ,  2.6    ,
        2.6    ,  2.6    ,  2.6    ,  2.6    ,  2.6    ,  2.6    ,
        2.6    ,  1.02   ,  1.02   ,  1.02   ,  1.02   ,  1.02   ,
        1.02   ,  1.02   ,  1.02   ,  1.02   ,  1.02   ,  1.02   ])

Mini-tutorial#

  • Integration with scipy

SciPy is a collection of mathematical algorithms and convenience functions built on the NumPy extension of Python. SciPy allows us to optimize functions, find roots of functions, interpolate, and integrate. It also provides mathematical and physical constants

Since PREM provided the density at specific depths, and we do not know the analytical form of density, we will use the trapezoidal rule to integrate.

In this mini-tutorial, we will compute a definite integral of \(y = f(x) = 10 - x^2\) from \(x=0\) to \(x=3\). This integral can be evaluated easily the value by hand:

(4)#\[\begin{equation} \int_{-3}^3 (10 - x^2) dx = \left(10x - \frac{x^3}{3}\right) \Big|_{x=-3}^{x=3} = 42 \end{equation}\]

We will use scipy.integrate.trapezoid to integrate the function with given (x,y) using Trapezoidal rule.

Focus on the following lines:

from scipy import integrate

integral = integrate.trapezoid(y_trap, x_trap)

Run the next two cells below and then adjust the slider on the figure. Notice the area approaches 42 when the number of trapezoids increases.

import numpy as np
from scipy import integrate

import plotly.graph_objects as go
from plotly.offline import init_notebook_mode, iplot
init_notebook_mode()