Radiation stresses and wave forces#

run initialize/init_3c.ipynb
Packages successfully loaded

We continue on the application of the dispersion relationship in the cross-shore, which was discussed in week 3 and the Wave course. You can use Forward Euler to calculate a derivative, an example in the cell below to calculate dh/dx. Note that the last value of dh_dx remains 0, since there is no forward location. This can be ignored in this notebook.

x = np.arange(0, 10 + 1, 1)
h = -100 + x**2

dh_dx = np.zeros(len(h))  # dh/dx
for i in range(len(dh_dx) - 1):
    dh_dx[i] = (h[i + 1] - h[i]) / (x[i + 1] - x[i])

plt.figure(figsize=(3, 1))
plt.plot(x, h, label="h")
plt.plot(x, dh_dx, label="dh_dx")
plt.legend();
../_images/cd34c4f98d1ed173d6dffe97bdfa870bbc195fcfe913d260536c885f7724e71e.png

1) Wave height (H), radiation stress (Sxx), mean water level by waves (\(\eta\)) for normal incident waves#

In Waves (CIEM3000)you should have programmed a function to compute the wave height (H), radiation stress (Sxx), and mean water level (\(\eta\)) along the cross-shore for normal incident waves. Can you reproduce/copy these functions below? If not, you can load the content in the cell below and use this to answer the upcoming questions. However, we recommend to practice making those functions yourself. As inputs (arguments) you should consider the offshore deep-water wave height (H0), the wave period (T), the offshore water depth (h0) and the bed slope (slope). Below is a setup of the environmental conditions, for a spatial grid that is defined with x_range. The function check_W3_normal_wave_setup will check your answers. You can also ignore RuntimeWarnings, for example “divide by zero encountered in scalar power”.

H0 = 2.0  # The offshore wave height [m]
T = 6  # The wave period [s]
h0 = 20  # The largest water depth
slope = 1 / 100  # The constant bed slope
rho = 1025  # The density of water [kg/m3]
x_range = np.arange(
    0, 2000 + 1, 1
)  # The cross-shore directed grid, the horizontal axis. (stepssize of 1 is required to function properly)


def W3_normal_wave(x_range, H0, T, h0, slope):
    # The environmental conditions
    x = x_range  # the horizontal axis
    zbed = -h0 + slope * x  # bed elevation [m]
    h = -zbed  # still water depth [m]
    h[
        h < 0
    ] = 0  # no negative depths (A depth of 0 can cause warnings, these can be ignored to keep the code straightforward)

    # given conditions
    gamma = 0.8  # wave breaking ratio

    """Complete the code here"""

    return H, n, Ksh, E, Sxx, eta


W3_plot_normal_incident_waves()
/tmp/ipykernel_3694/1359819942.py:66: UserWarning: No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.
  ax.legend(bbox_to_anchor=(-0.02, 1), loc="lower left")
../_images/bccafa827f267c9d382aea7171a828c09d6f6db7b1dc01ae0715d29eb739af3c.png

If you are not able to make the functions, you can set Load_answers = True (rather than False) to get the values for the wave height (H), for n, the shoaling parameter (Ksh), the wave energy (E), the radiant stress (Sxx), and the surface elevation (eta), with mean-sea-level at 0 m. The parameters list the outcomes of the functions

Load_answers = False

if Load_answers == True:
    angle = 0  # The offshore wave angle
    H0 = 1.5  # The offshore wave height [m]
    T = 6  # The wave period [s]
    h0 = 20  # The largest water depth
    slope = 1 / 30  # The constant bed slope
    H = [
        1.5,
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Below is the cross-shore distribution of several wave characteristics shown for the wave conditions described on top. Feel free to study the effects of changing these parameters on the wave characteristics. You can also use these to answer the questions below.

W3_Q1()

2) Wave height (H), radiation stress (Sxx), mean water level by waves (\(\eta\)) for oblique waves#

The influence of refraction should also be considered for waves that approach the shore under an angle. Can you calculate the wave heigth (H), radiation stress (Sxx), and mean water level (eta) for oblique waves by completing the function below? The function W3_plot_oblique_waves(angle1, angle2) will plot these characteristics along the cross-shore for the two given angles. This function will give a runtime warning when the water depth (h) is 0, you may ignore this to keep the code as straightforward as possible.

angle = 30  # The offshore wave angle


def W3_wave_setup(x_range, H0, T, h0, slope, angle):
    # The environmental conditions
    x = x_range  # the horizontal axis
    zbed = -h0 + slope * x  # bed elevation [m]
    h = -zbed  # still water depth [m]
    h[h < 0] = 0  # no negative depths

    # given conditions
    gamma = 0.8  # wave breaking ratio

    """ You can build on the results from normal-indicent waves"""

    ...

    return H, Sxx, eta, c, n  # c and n are useful later on


W3_plot_oblique_waves()
/tmp/ipykernel_3694/1892891519.py:67: UserWarning: No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.
  ax.legend(bbox_to_anchor=(-0.02, 1), loc="lower left")
../_images/4419af04f46f8d6d36cd3b58cec579e2e52ac4e39e8e82ea28dc74f77d2e559a.png

You can use the interactive widget below to assess the influence of changing environmental conditions on the wave characteristics

W3_show_oblique_waves()

3) The shorewards directed force (Fx)#

We dive a bit deeper into the force in the cross-shore direction that drives the wave setup (Fx). Can you calculate this force along the cross-shore for an alongshore uniform coast? For this question, you can consider using a similar approach as the setup is calculated, by using forward euler.

def W3_Fx(x_range, H0, T, h0, slope, angle):
    """It is recommended to use the earlier defined function W3_wave_setup()"""
    H, Sxx, eta, c, n = W3_wave_setup(x_range, H0, T, h0, slope, angle)

    ...

    return Fx


W3_plot_Fx()
W3_show_Fx()

4) Alongshore shear stresses (Syx) and alongshore force (Fy).#

Can you compute the radiation shear stress in y direction (Syx)?

def W3_Syx(x_range, H0, T, h0, slope, angle):
    """It is recommended to use the earlier defined function W3_wave_setup()"""
    H, Sxx, eta, c, n = W3_wave_setup(x_range, H0, T, h0, slope, angle)

    ...

    return Syx


W3_plot_Syx()

And can you compute the alongshore force (Fy) for an alongshore uniform coast?

def W3_Fy(x_range, H0, T, h0, slope, angle):
    """It is recommended to use the earlier defined function(s)"""
    x = x_range
    ...

    return Fy


W3_plot_Fy()
W3_show_Syx_Fy()

5) Longshore current (V)#

Can you calculate the distribution of the longshore current, following expression 5.82 in the book. For simplicity we assume that in this equation h is equal to the still water depth, and the friction factor \(c_f\) is 0.01.

cf = 0.01


def W3_V(x_range, H0, T, h0, slope, angle, cf):
    # The environmental conditions
    x = x_range  # the horizontal axis
    zbed = -h0 + slope * x  # bed elevation [m]
    h = -zbed  # still water depth [m]
    h[h < 0] = 0  # no negative depths

    # With
    gamma = 0.8

    """It is recommended to use the earlier defined function W3_wave_setup()"""
    H, Sxx, eta, c, n = W3_wave_setup(x_range, H0, T, h0, slope, angle)

    ...

    return V


W3_plot_V()
W3_show_V()

6) Wave induced local current, orbital velocity#

Can you calculate the amplitude of the velocity (u0) near the bed through the linear wave theory?

def W3_u0(x_range, H0, T, h0, slope, angle):
    """It is recommended to use the earlier defined function W3_wave_setup()"""
    H, Sxx, eta, c, n = W3_wave_setup(x_range, H0, T, h0, slope, angle)

    L = c * T
    k = 2 * np.pi / L  # The wave number

    snell_constant = np.sin(np.deg2rad(angle)) / c[0]  # apply snell's law
    theta_radians = np.arcsin(snell_constant * c)

    # page 199 of the book
    omega = 2 * np.pi / T
    u0 = 0.5 * omega * H / (np.sin(theta_radians) * k * h)

    return u0


W3_plot_u0()
W3_show_U0()