Python Richardson Number Calculation: Temperature, Pressure, and Wind Speed

Richardson number is a dimensionless number that relates the effects of buoyancy and wind shear on turbulent flow. It can be calculated using the following equation:

Ri = g/((Tb/Tz) - 1) * (ΔU/Δz)^2

where,

Ri = Richardson number g = acceleration due to gravity (9.81 m/s^2) Tb = virtual potential temperature at the bottom of the layer (K) Tz = virtual potential temperature at the top of the layer (K) ΔU = wind speed difference between the bottom and top of the layer (m/s) Δz = height difference between the bottom and top of the layer (m)

To calculate Richardson number in Python, you can define a function that takes temperature, pressure, and wind speed as input and returns the Richardson number. Here is an example code:

def richardson_number(temp_bottom, temp_top, pressure, wind_speed_bottom, wind_speed_top, height):
    g = 9.81  # acceleration due to gravity in m/s^2
    R = 287.05  # gas constant for dry air in J/kg K
    cp = 1004.  # specific heat capacity of dry air at constant pressure in J/kg K
    
    # calculate virtual potential temperature at the bottom and top of the layer
    theta_v_bottom = temp_bottom * (1 + 0.61 * (0.622 * (pressure / 100.) / (pressure / 100. - 0.622)))  # virtual potential temperature at the bottom of the layer
    theta_v_top = temp_top * (1 + 0.61 * (0.622 * (pressure / 100.) / (pressure / 100. - 0.622)))  # virtual potential temperature at the top of the layer
    
    # calculate the temperature difference between the bottom and top of the layer
    delta_theta_v = theta_v_top - theta_v_bottom
    
    # calculate the wind speed difference between the bottom and top of the layer
    delta_u = wind_speed_top - wind_speed_bottom
    
    # calculate the Richardson number
    ri = g / (delta_theta_v / height) * (delta_u / height)**2
    
    return ri

Here, we have used the ideal gas law to calculate the density of dry air and the virtual temperature, which is the temperature that a parcel of air would have if it were brought adiabatically to a reference pressure level. We have also used the specific heat capacity of dry air at constant pressure to calculate the virtual potential temperature. Finally, we have used the height difference between the bottom and top of the layer to convert the wind speed difference into a shear term.

To use this function, you can simply call it with the appropriate input values:

temp_bottom = 20.  # temperature at the bottom of the layer in degrees Celsius
temp_top = 10.  # temperature at the top of the layer in degrees Celsius
pressure = 1000.  # pressure in hPa
wind_speed_bottom = 5.  # wind speed at the bottom of the layer in m/s
wind_speed_top = 10.  # wind speed at the top of the layer in m/s
height = 100.  # height difference between the bottom and top of the layer in meters

ri = richardson_number(temp_bottom, temp_top, pressure, wind_speed_bottom, wind_speed_top, height)

print('Richardson number:', ri)

This will print the calculated Richardson number for the given input values.