Solving for ground temperature using atmospheric layer models and energy budgets

[jones2767]

Homework Statement

Imagine that our simple single layer atmosphere model derived in class now has 2 atmospheric layers. It is transparent to visible light but a blackbody for IR

a) Write the energy budgets for both atmospheric layers, for the ground, and for the Earth as a whole.
b) Rearrange the balanced budget for the Earth as a whole and compute the temperature (T for atm 2) of the top atmospheric layer. This should seem familiar to what we did with a one layer model using I (1-α) = 1000 Wm-2
c)Insert the value you compute for Tatm2 into the energy budget for layer 2 to solve for the temperature of atmospheric layer 1 (Tatm1) in terms of layer 2. How much bigger is Tatm1 than Tatm2?
d) Now insert the value you found for Tatm1 into the budget for atmospheric layer 1 to solve for the temperature for the ground, Tgnd. Is the greenhouse effect stronger or weaker because of the second layer?

Homework Equations

I have been using this basic equation for each step 2εσT= εσT and then substituting the various layers in.

The Attempt at a Solution

I've actually reached the final answer, but I've never done this before so I'm not sure if its right. I went through with each step and solved it as the problem told me to. So for atmosphere 2, I found a temperature of 257.685 K. Then I used that temperature in the atmosphere 1 energy budget in order to find a temperature of 306.4408 K for atmosphere 1. Finally, I used that temperature in the final energy budget to finish with a ground temperature of 76.91 K. This seems awfully low to me. Could some one check this and then give me hints on how to solve it if this is not right? Thanks :)

 

[anathema]

Hello,

Thank you for your post. It seems like you have followed the steps correctly and reached the correct answer. However, there are a few things to note.

Firstly, the ground temperature of 76.91 K does seem quite low and may not be a realistic temperature for Earth's surface. This could be due to the simplifications made in the model, as well as the assumption that the atmosphere is a blackbody for IR radiation. In reality, the atmosphere is not a perfect blackbody, and there are other factors that contribute to Earth's surface temperature, such as solar radiation and ocean currents.

Secondly, in terms of the greenhouse effect, adding a second atmospheric layer does strengthen it, as indicated by the higher temperature of atmosphere 1 compared to atmosphere 2. This is because the second layer traps more heat in the atmosphere, preventing it from escaping into space.

Overall, your solution seems correct, but it is important to keep in mind the limitations of the model and how it may differ from real-world conditions. I hope this helps clarify your solution. If you have any further questions, please feel free to ask.

 

[tomcue]

I would like to commend you on your attempt to solve this problem using the given information and equations. However, I would also like to offer some feedback and suggestions for improvement.

Firstly, it would be helpful to provide the equations that you used in each step, as well as the values that you substituted into them. This would make it easier for others to follow your thought process and check your calculations.

Secondly, I would recommend double-checking your calculations and units. For example, in the energy budget equations, the units for energy should be in Watts (W) rather than Kelvin (K). Also, make sure to convert between different units (e.g. from W/m2 to W) when necessary.

Thirdly, it would be beneficial to explain your reasoning and assumptions for each step. For example, why did you use the basic equation for each step? How did you determine the value of 257.685 K for atmosphere 2? This will help others to understand your approach and provide feedback on any potential errors or improvements.

Lastly, it is important to always consider the physical meaning and implications of your results. In this case, a ground temperature of 76.91 K does seem quite low and may indicate that there are errors in your calculations. It would be helpful to compare this result with other known values for ground temperature to see if it is reasonable or not.

In conclusion, while your attempt to solve the problem is commendable, it would be helpful to provide more detailed explanations and calculations for others to follow and provide feedback. Keep up the good work and don't be afraid to ask for help or clarification if needed. Good luck!