source: inundation/pyvolution/smf.py @ 1995

#
# slide_tsunami function
#

"""This function returns a callable object representing an initial water
   displacement generated by a submarine sediment slide.

Using input parameters:

Required
 length  downslope slide length
 depth   water depth to slide centre of mass
 slope   bathymetric slope

Optional
 x0      x origin (0)
 y0      y origin (0)
 alpha   angular orientation of slide in xy plane (0)
 w       slide width (0.25*b)
 T       slide thickness (0.01*b)
 g       acceleration due to gravity (9.8)
 gamma   specific density of sediments (1.85)
 Cm      added mass coefficient (1)
 Cd      drag coefficient (1)
 Cn      friction coefficient (0)
 psi     (0)
 dx      offset of second Gaussian (0.2*width of first Gaussian)
 kappa   multiplier for sech^2 function (3.0)
 kappad  multiplier for second Gaussian function (0.8)
 zsmall  an amount near to zero (0.01)

The following parameters are calculated:

 a0      initial acceleration
 ut      theoretical terminal velocity
 s0      charactistic distance of motion
 t0      characteristic time of motion
 w       initial wavelength of tsunami
 a2D     2D initial amplitude of tsunami
 a3D     3D initial amplitude of tsunami

The returned object is a callable double Gaussian function that represents
the initial water displacement generated by a submarine sediment slide.

Adrian Hitchman
Geoscience Australia, June 2005
"""

def slide_tsunami(length, depth, slope, width=None, thickness=None, \
                  x0=0.0, y0=0.0, alpha=0.0, \
                  gravity=9.8, gamma=1.85, \
                  massco=1, dragco=1, frictionco=0, psi=0, \
                  dx=None, kappa=3.0, kappad=0.8, zsmall=0.01):

    from math import sin, tan, radians, pi, sqrt, exp

    #if width not provided, set to typical value
    if width is None:
        width = 0.25 * length

    #if thickness not provided, set to typical value
    if thickness is None:
        thickness = 0.01 * length

    #calculate some parameters of the slide

    sint = sin(radians(slope))
    tant = tan(radians(slope))
    tanp = tan(radians(psi))

    a0 = gravity * sint * ((gamma-1)/(gamma+massco)) * (1-(tanp/tant))
    ut = sqrt((gravity*depth) * (length*sint/depth) \
                    * (pi*(gamma-1)/(2*dragco)) * (1-(tanp/tant)))
    s0 = ut**2 / a0
    t0 = ut / a0

    #calculate some parameters of the water displacement produced by the slide

    w = t0 * sqrt(gravity*depth)
    a2D = s0 * (0.0574 - (0.0431*sint)) \
             * (thickness/length) \
             * ((length*sint/depth)**1.25) \
             * (1 - exp(-2.2*(gamma-1)))
    a3D = a2D / (1 + (15.5*sqrt(depth/(length*sint))))

    #a few temporary print statements
    print '\nThe slide ...'
    print '\tLength: ', length
    print '\tDepth: ', depth
    print '\tSlope: ', slope
    print '\tWidth: ', width
    print '\tThickness: ', thickness
    print '\tx0: ', x0
    print '\ty0: ', y0
    print '\tAlpha: ', alpha
    print '\tAcceleration: ', a0
    print '\tTerminal velocity: ', ut
    print '\tChar time: ', t0
    print '\tChar distance: ', s0
    print '\nThe tsunami ...'
    print '\tWavelength: ', w
    print '\t2D amplitude: ', a2D
    print '\t3D amplitude: ', a3D

    #keep an eye on some of the assumptions built into the maths

    if ((slope < 5) or (slope > 30)):
        print 'WARNING: slope out of range (5 - 30 degrees) ', slope
    if ((depth/length < 0.06) or (depth/length > 1.5)):
        print  'WARNING: d/b out of range (0.06 - 1.5) ', depth/length
    if ((thickness/length < 0.008) or (thickness/length > 0.2)):
        print 'WARNING: T/b out of range (0.008 - 0.2) ', thickness/length
    if ((gamma < 1.46) or (gamma > 2.93)):
        print 'WARNING: gamma out of range (1.46 - 2.93) ', gamma

    return Double_gaussian(a3D=a3D, wavelength=w, width=width, \
                           x0=x0, y0=y0, alpha=alpha, \
                           dx=dx, kappa=kappa, kappad=kappad, zsmall=zsmall)

#
# slump_tsunami function
#

"""This function returns a callable object representing an initial water
   displacement generated by a submarine sediment slump.

Using input parameters:

Required
 length  downslope slump length
 depth   water depth to slump centre of mass
 slope   bathymetric slope

Optional
 x0      x origin (0)
 y0      y origin (0)
 alpha   angular orientation of slide in xy plane (0)
 w       slump width (1.0*b)
 T       slump thickness (0.1*b)
 R       slump radius of curvature (b^2/(8*T))
 del_phi slump angular displacement (0.48)
 g       acceleration due to gravity (9.8)
 gamma   specific density of sediments (1.85)
 Cm      added mass coefficient (1)
 Cd      drag coefficient (1)
 Cn      friction coefficient (0)
 dx      offset of second Gaussian (0.2*width of first Gaussian)
 kappa   multiplier for sech^2 function (3.0)
 kappad  multiplier for second Gaussian function (0.8)
 zsmall  an amount near to zero (0.01)

The following parameters are calculated:

 a0      initial acceleration
 um      maximum velocity
 s0      charactistic distance of motion
 t0      characteristic time of motion
 w       initial wavelength of tsunami
 a2D     2D initial amplitude of tsunami
 a3D     3D initial amplitude of tsunami

The returned object is a callable double Gaussian function that represents
the initial water displacement generated by a submarine sediment slump.

Adrian Hitchman
Geoscience Australia, June 2005
"""

def slump_tsunami(length, depth, slope, width=None, thickness=None, \
                  radius=None, dphi=0.48, x0=0.0, y0=0.0, alpha=0.0, \
                  gravity=9.8, gamma=1.85, \
                  massco=1, dragco=1, frictionco=0, \
                  dx=None, kappa=3.0, kappad=0.8, zsmall=0.01):

    from math import sin, radians, sqrt

    #if width not provided, set to typical value
    if width is None:
        width = length

    #if thickness not provided, set to typical value
    if thickness is None:
        thickness = 0.1 * length

    #if radius not provided, set to typical value
    if radius is None:
        radius = length**2 / (8.0 * thickness)

    #calculate some parameters of the slump

    sint = sin(radians(slope))

    s0 = radius * dphi / 2
    t0 = sqrt((radius*(gamma+massco)) / (gravity*(gamma-1)))
    a0 = s0 / t0**2
    um = s0 / t0

    #calculate some parameters of the water displacement produced by the slump

    w = t0 * sqrt(gravity*depth)
    a2D = s0 * (0.131/sint) \
             * (thickness/length) \
             * (length*sint/depth)**1.25 \
             * (length/radius)**0.63 * dphi**0.39 \
             * (1.47 - (0.35*(gamma-1))) * (gamma-1)
    a3D = a2D / (1 + (2.06*sqrt(depth/length)))

    #a few temporary print statements
    print '\nThe slump ...'
    print '\tLength: ', length
    print '\tDepth: ', depth
    print '\tSlope: ', slope
    print '\tWidth: ', width
    print '\tThickness: ', thickness
    print '\tRadius: ', radius
    print '\tDphi: ', dphi
    print '\tx0: ', x0
    print '\ty0: ', y0
    print '\tAlpha: ', alpha
    print '\tAcceleration: ', a0
    print '\tMaximum velocity: ', um
    print '\tChar time: ', t0
    print '\tChar distance: ', s0
    print '\nThe tsunami ...'
    print '\tWavelength: ', w
    print '\t2D amplitude: ', a2D
    print '\t3D amplitude: ', a3D

    #keep an eye on some of the assumptions built into the maths

    if ((slope < 10) or (slope > 30)):
        print 'WARNING: slope out of range (10 - 30 degrees) ', slope
    if ((depth/length < 0.34) or (depth/length > 0.5)):
        print  'WARNING: d/b out of range (0.34 - 0.5) ', depth/length
    if ((thickness/length < 0.10) or (thickness/length > 0.15)):
        print 'WARNING: T/b out of range (0.10 - 0.15) ', thickness/length
    if ((radius/length < 1.0) or (radius/length > 2.0)):
        print 'WARNING: R/b out of range (1 - 2) ', radius/length
    if ((dphi < 0.10) or (dphi > 0.52)):
        print 'WARNING: del_phi out of range (0.10 - 0.52) ', dphi
    if ((gamma < 1.46) or (gamma > 2.93)):
        print 'WARNING: gamma out of range (1.46 - 2.93) ', gamma

    return Double_gaussian(a3D=a3D, wavelength=w, width=width, \
                           x0=x0, y0=y0, alpha=alpha, \
                           dx=dx, kappa=kappa, kappad=kappad, zsmall=zsmall)

#
# Double_gaussian class
#

"""This is a callable class representing the initial water displacment 
   generated by a submarine mass failure (smf -> sediment slide or slump).

Using input parameters:

Required
 w       initial wavelength of tsunami
 a3D     3D initial amplitude of tsunami
 width   width of smf

Optional
 x0      x origin of smf
 y0      y origin of smf
 alpha   angular orientation of smf in xy plane (0)
 dx      offset of second Gaussian (0.2*width of first Gaussian)
 kappa   multiplier for sech^2 function (3.0)
 kappad  multiplier for second Gaussian function (0.8)
 zsmall  an amount near to zero (0.01)

Adrian Hitchman
Geoscience Australia, June 2005
"""

class Double_gaussian:

    def __init__(self, a3D, wavelength, width, x0, y0, alpha, \
                 dx, kappa, kappad, zsmall):
        self.a3D = a3D
        self.wavelength = wavelength
        self.width = width
        self.x0 = x0
        self.y0 = y0
        self.alpha = alpha
        self.kappa = kappa
        self.kappad = kappad

        if dx is None:
            self.determineDX(zsmall=zsmall)
        else:
            self.dx = dx

    def __call__(self, x, y):
        """Make Double_gaussian a callable object.

        If called as a function, this object returns z values representing
        the initial 3D distribution of water heights at the points (x,y)
        produced by a submarine mass failure.
        """

        from math import sin, cos, radians, exp, cosh
        from Numeric import zeros, Float

        #ensure vectors x and y have the same length
        N = len(x)
        assert N == len(y)

        am = self.a3D
        wa = self.wavelength
        wi = self.width
        x0 = self.x0
        y0 = self.y0
        alpha = self.alpha
        dx = self.dx
        kappa = self.kappa
        kappad = self.kappad

        #double Gaussian calculation assumes water displacement is oriented
        #E-W, so, for displacement at some angle alpha clockwise from the E-W
        #direction, rotate (x,y) coordinates anti-clockwise by alpha

        cosa = cos(radians(alpha))
        sina = sin(radians(alpha))

        xr = ((x-x0) * cosa - (y-y0) * sina) + x0
        yr = ((x-x0) * sina + (y-y0) * cosa) + y0

        z = zeros(N, Float)

        for i in range(N):
            try:
                z[i] =  -am / ((cosh(kappa*(yr[i]-y0)/(wi+wa)))**2) \
                            * (exp(-((xr[i]-x0)/wa)**2) - \
                                kappad*exp(-((xr[i]-dx-x0)/wa)**2))
            except OverflowError:
                pass

        return z

    def determineDX(self, zsmall):
        """Determine a suitable offset for the second Gaussian function.

        A suitable offset for the second Gaussian function is taken to
        be some fraction of the 'width' of the first Gaussian function.

        The 'width' of the first Gaussian is obtained from the range of
        the x coordinates over which the function takes values from
        'near zero', through 1, and back to 'near zero'.

        The parameter zsmall passed to this function specifies how much
        'near zero' is.
        """

        from math import sqrt, log, e

        a = self.a3D
        c = self.wavelength

        self.dx = 2.0 * (c * sqrt(-log((zsmall/a),e))) / 5.0