ADCP processing and the homogeneous flow assumption

This problem is designed to illustrate some of the difficulties that can arise if fundamental assumptions of instrument operation are violated.

The transformation equations developed in class (and shown in the handout) to convert horizontal flow to radial velocities (as measured by an ADCP) are given as:

Here, r₁ and r₂ are the radial velocities along each beam (at a given depth), Q is the angle of the beam relative to the vertical axis, and u and w are the horizontal and vertical velocities of the flow, respectively. The subscripts 1 and 2 refer to the velocities seen in each of the two beams. (In the questions below let u₁ = 50 cm/s.)

a. For a given u₁, w₁ = w₂ = 0.1·u₁ and Q = 20, plot r₁ and r₂ versus u₁ – u₂ letting u₂ range between +u₁ to – u₁.

b. For a given u₁, w₁ = w₂ = 0.1·u₁ and Q, plot u versus u₁ – u₂ letting u₂ range between +u₁ to – u₁. Plot w versus u₁ – u₂ on a separate graph.

import numpy as np

import matplotlib.pyplot as plt

from matplotlib.font_manager import FontProperties

from pylab import *

#Definition of variable

#r1, r2, are the radial velocities along each beam(at a given depth)

#THETA, is the angle of the beam relative to the vertical axis

#u and v are the horizontal and vertical velocities of the flow

Data=[]

THETA=20*np.pi/180

u1=50

w1=w2=0.1*u1

#creating the loop

for u2 in range(u1,-u1,-2):

r1=(-u1*np.sin(THETA))+(w1*np.cos(THETA))

r2=(u2*np.sin(THETA))+(w2*np.cos(THETA))

U=(0.5*(u1+u2))+((0.5*(w1-w2))/np.tan(THETA))

W=(0.5*(w1+w2))+((0.5*(u1-u2))*np.tan(THETA))

ur=u1-u2

mat=[r1,r2,u1,u2,ur,U,W]

Data.append(mat)

Datos=np.array(Data)

r1=Datos[0:,0]

r2=Datos[0:,1]

ur=Datos[0:,4]

U=Datos[0:,5]

W=Datos[0:,6]

subplot(221)

plot(ur, r1, 'b-', ur, r2, 'r-')

ylabel('r1 and r2')

xlabel('u1-u2')

suptitle('ADCP processing', fontsize=16)

legend( ('r1', 'r2'), loc='upper right')

subplot(223)

plot(ur,U,'r-')

xlabel('u1-u2')

ylabel('U')

subplot(224)

plot(ur,W,'r-')

xlabel('u1-u2')

ylabel('W')

show()

a. For a given u₁, w₁ = w₂ = 0.1·u₁,and u₂ = u₁/2, plot u versus Q, letting Q range from 0 to pi/2. Plot w versus Q on a separate graph.

import numpy as np

import matplotlib

import matplotlib.pyplot as plt

from pylab import *

#Definition of variable

Data=[]

u1=50

w1=w2=0.1*u1

u2=u1/2

#create the loop

for T in range(0,60,2):

r1=(-u1*np.sin(T*np.pi/180))+(w1*np.cos(T*np.pi/180))

r2=(u2*np.sin(T*np.pi/180))+(w2*np.cos(T*np.pi/180))

U=(0.5*(u1+u2))+((0.5*(w1-w2))/np.tan(T*np.pi/180))

W=(0.5*(w1+w2))+((0.5*(u1-u2))*np.tan(T*np.pi/180))

ur=u1-u2

theta=T*np.pi/180

mat=[r1,r2,u1,u2,ur,U,W,T,theta]

Data.append(mat)

Datos=np.array(Data)

r1=Datos[0:,0]

r2=Datos[0:,1]

ur=Datos[0:,4]

U=Datos[0:,5]

W=Datos[0:,6]

T=Datos[0:,7]

theta=Datos[0:,8]

plt.plot(T,W,'r')

subplot(121)

plot(T,U, 'b-')

ylabel('U')

xlabel('Theta')

suptitle('ADCP processing', fontsize=16)

subplot(122)

plot(T,W,'r-')

xlabel('Theta')

ylabel('W')

plt.show()

Oceanic influences in hurricanes

There are some evidences that connect this atmospheric event with the ocean. From the beginning, the process for its development requires at least 26°C the temperature at the ocean surface water. Other clearest evidence of the ocean influences is that, some the hurricane tracks are similar to patterns of distribution the surface warm ocean waters.

As a result of the interchange of heat between the ocean surface water and the atmosphere (affected by the thunderstorms occurrence) the hurricanes intensifies gradually its strength, in fact when the hurricanes reaches coastal zones (not ocean influences) the hurricanes reduces its intensity, the same occurs when the temperature of the ocean surface is reduced. In both cases the energy supplied to the hurricanes is reduced.

The oceanic influences is continues through the process of intensification and feed backing for the hurricane. The hurricane winds in the area causes vertical mixing in the oceanic surface increasing the mixed layer and reducing the surface temperature, this interacting process between the oceanic surface and hurricane is studying in numerical models that consider the depth of mixed layer as an essential input for the calculus of process of intensification of hurricane.

Ocean eddies contribute to the hurricanes development. The mixed layer in ocean eddies area is deeper that other places, in terms of heat available it is traduced as a major amount of energy available for the hurricanes development.

For these arguments exposed it is mandatory that hurricanes would not occur without the oceanic influence.