Cloud movement, finding temperature, velocity and height from this formula

In summary, the problem involves using equations to find the evolution of a parcel's velocity, temperature and height over time, assuming certain constants and using calculus techniques to solve the equations. For the given constants, the approximate time for the parcel to reach the cloud top is 4.3 minutes.
  • #1
automan13
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1. An unsaturated parcel of air is entrained into a cloud near the region of the cloud top. the updraft velocity U, the ambient lapse rate in ther cloud γ, the entrainment rate E, and the mixing ratio of the condensed water μcan be considered constants along the trajectory of the parcel.
assuming that at t=0 the velocity of the parcel is Up and its acceleration is zero.
Obtain from equations, analytical solutions for the evolution in time of the parcel's velocity, temperature and height.
for simplicity, assume that the term g(Tp-T)/T can be approximated by g(Tp-T)/To where To is a constant average temperature in the cloud. Show that for To=-15℃ and γ=6℃/km, this time equals approximately 4.3 minutes.




2. d/dt(Tp - T) = -E (Lm/Cp)- (P - λc) dz/dt
dUp/dt=(Tp-T)/T g + gm - E (Up - U)

Tp= temperature
Up= velocity
z=height





3. Really bad at math and physics, ANY help at understanding the equations and the problem is really appreciated. Thanks in advance.
 
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  • #2
The equation given in the problem is a differential equation, which is used to describe the change in a physical quantity over time. The equation describes how the temperature difference between the parcel and its surroundings (Tp-T) changes over time, based on the entrainment rate (E), the lapse rate (γ), the mixing ratio of the condensed water (μ), and the updraft velocity (U). The other equation given describes how the parcel's velocity (Up) changes over time, based on the temperature difference between the parcel and its surroundings (Tp-T), the gravitational acceleration (g), the molecular weight of air (m) and the entrainment rate (E). To solve the equations analytically, you need to use calculus techniques to find the solutions for the velocity, temperature and height as functions of time. To do this, you need to assume a constant average temperature (To) and use it in the equation to approximate the term g(Tp-T)/T. Once you have done this, you can then solve the equations for the velocity, temperature and height as functions of time. For the given constants (To=-15℃ and γ=6℃/km), the approximate time for the parcel to reach the cloud top can be calculated by substituting the constants into the equations and solving for the time it takes for the velocity, temperature and height to reach their maximum values. This time should be approximately 4.3 minutes.
 

1. How do clouds move and what affects their movement?

The movement of clouds is primarily influenced by wind patterns and air currents in the atmosphere. These winds can be caused by differences in air pressure, temperature, and the Earth's rotation. Other factors such as topography and geographic features can also impact cloud movement.

2. How can temperature be found using cloud movement?

Clouds can provide valuable information about temperature in the atmosphere. For example, cirrus clouds (high-altitude, wispy clouds) can indicate that the air temperature is below freezing. Additionally, the height of clouds can also give clues about temperature, as colder air tends to sink and warmer air rises.

3. What is the formula for finding the velocity of clouds?

The formula for finding the velocity (speed and direction) of clouds is: velocity = distance ÷ time. This can be measured using tools such as weather radar or satellite imagery, which track the movement of clouds over a specific period of time.

4. How can the height of clouds be determined?

The height of clouds can be estimated using their appearance and type. For example, cumulus clouds (fluffy, low-altitude clouds) typically form at a height of 1-2 kilometers, while cirrus clouds can reach heights of 6-12 kilometers. More accurate measurements can be obtained using specialized instruments such as a ceilometer or lidar.

5. What are some other factors that can be determined from cloud movement?

Cloud movement can also provide information about atmospheric conditions such as humidity and air pressure. Additionally, studying the movement of clouds can help predict weather patterns and identify potential areas for precipitation. It can also give insights into the overall circulation of the atmosphere and how it affects global climate patterns.

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