Question

The lapse rate is the rate at which the temperature in Earth's atmosphere decreases with altitude. For example, a lapse rate of \(6.5^{\circ}\) Celsius/km means the temperature decreases at a rate of \(6.5^{\circ} \mathrm{C}\) per kilometer of altitude. The lapse rate varies with location and with other variables such as humidity. However, at a given time and location, the lapse rate is often nearly constant in the first 10 kilometers of the atmosphere. A radiosonde (weather balloon) is released from Earth's surface, and its altitude (measured in \(\mathrm{km}\) above sea level) at various times (measured in hours) is given in the table below. $$\begin{array}{lllllll} \hline \text { Time (hr) } & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 \\ \text { Altitude (km) } & 0.5 & 1.2 & 1.7 & 2.1 & 2.5 & 2.9 \\ \hline \end{array}$$ a. Assuming a lapse rate of \(6.5^{\circ} \mathrm{C} / \mathrm{km},\) what is the approximate rate of change of the temperature with respect to time as the balloon rises 1.5 hours into the flight? Specify the units of your result, and use a forward difference quotient when estimating the required derivative. b. How does an increase in lapse rate change your answer in part (a)? c. Is it necessary to know the actual temperature to carry out the calculation in part (a)? Explain.

Short answer

Answer: The approximate rate of change of the temperature with respect to time at 1.5 hours is -5.2 °C/hr. If the lapse rate increases, the temperature decreases more rapidly with altitude, resulting in a faster decrease in temperature with respect to time. The actual temperature is not necessary for this calculation, as it only involves the lapse rate and the altitude change rate.

Step-by-step solution

Calculate the rate of altitude change with respect to time at 1.5 hours

We need the altitude change from 1.5 hours to 2 hours to estimate the derivative using a forward difference quotient. From the given table, we have the following data points: \((1.5, 2.1) \text{ and } (2, 2.5)\) The forward difference quotient for altitude with respect to time, which is the estimated rate of change at 1.5 hours, is calculated as follows: \(\frac{d}{dt}(\text{altitude})\approx\frac{2.5-2.1}{2-1.5}= \frac{0.4}{0.5}=0.8 \ \mathrm{km/hr}\)

Calculate the rate of temperature change with respect to altitude

To estimate the derivative in terms of temperature, we multiply the lapse rate with the derivative of altitude with respect to time. The given lapse rate is -6.5 \(^{\circ} \mathrm{C} / \mathrm{km}\) (Note that the negative sign indicates the temperature decreases as altitude increases). The rate of temperature change with respect to time is calculated as follows: \(\frac{d}{dt}(\text{temperature}) = \text{lapse rate} \times \frac{d}{dt}(\text{altitude})\) \(\frac{d}{dt}(\text{temperature}) = -6.5 \frac{^{\circ} \mathrm{C}}{\mathrm{km}} \times 0.8 \ \mathrm{km/hr}\) \(\frac{d}{dt}(\text{temperature}) = -5.2 \frac{^{\circ} \mathrm{C}}{\mathrm{hr}}\) a. The approximate rate of change of the temperature with respect to time as the balloon rises 1.5 hours into the flight is -5.2\(^{\circ} \mathrm{C} / \mathrm{hr}\). b. Increasing the lapse rate If the lapse rate increases, it means that the temperature decreases more rapidly with altitude. Since the lapse rate is a multiplicative factor in the calculation of the rate of temperature change with respect to time, a higher lapse rate will result in a more negative value for the temperature change rate. Therefore, an increase in the lapse rate would make the temperature decrease faster with respect to time compared to the given lapse rate. c. Necessity of the actual temperature To calculate the rate of change of temperature with respect to time at a specific time, we used the lapse rate and the altitude change rate. Knowledge of the actual temperature is not required for this calculation, as it only tells us how much the temperature decreases with each kilometer of altitude increase. The problem asks for the temperature change rate, which doesn't require the actual temperature value as it's not used in our calculations.

Key concepts

Lapse Rate

The concept of a lapse rate is essential when studying how temperature changes with altitude in the Earth's atmosphere. Simply put, the lapse rate refers to the rate at which temperature decreases as you ascend into the sky. For example, a lapse rate of \(6.5^{\circ} \text{C/km}\) means that for each kilometer you go higher, the temperature drops by 6.5 degrees Celsius.

In real-world scenarios, the lapse rate isn't always constant. It can change depending on location and atmospheric conditions such as humidity and wind patterns. But in some situations, like in the first 10 kilometers of the atmosphere, it remains nearly constant. This makes it a helpful tool for meteorologists and scientists when forecasting weather and understanding atmospheric behaviors.

The lapse rate plays a crucial role in understanding thermal dynamics and can influence weather patterns and air movements, impacting everything from local climates to extreme weather conditions. By examining the lapse rate, scientists can gain insights into how quickly temperature changes with altitude, aiding in more accurate weather predictions.

Derivative Estimation

Derivative estimation is a key principle in calculus used to approximate the rate at which one quantity changes with respect to another. In the context of our exercise, it helps us understand how quickly the altitude of the weather balloon changes as time passes. This is necessary to estimate the rate at which the temperature changes.

In mathematical terms, a derivative represents the slope of a function at a particular point. It's a measure of how a function value changes as its input changes. In our explanation, the altitude is a function of time, and we estimate its change using a derivative to understand how quickly the balloon rises.

To find the rate of temperature change with respect to time, we first calculate the rate of altitude change using derivatives. We then apply the known lapse rate to convert this data into temperature change. This approach allows us to estimate derivatives even when dealing with discrete data points, which are commonly encountered in practical applications.

Forward Difference Quotient

The forward difference quotient is a simple method to estimate derivatives when working with discrete data, like measurements or observations. It's particularly useful in situations where continuous data isn't available, which is often the case in real-world applications.

In essence, the forward difference quotient helps calculate the slope between two successive data points. To do this, it takes the difference in the values of a function and divides it by the difference in the input values. In our exercise, we used the forward difference to estimate how rapidly the altitude changes as the weather balloon moves between two time intervals:

• Take two points: the altitude at 1.5 hours and at 2 hours.
• Calculate the change in altitude: \(2.5 - 2.1 = 0.4 \text{ km}\).
• Compute the time difference: \(2 - 1.5 = 0.5 \text{ hr}\).
• The forward difference gives us an estimated derivative: \(\frac{0.4}{0.5} = 0.8 \text{ km/hr}\).

This step-by-step approach provides a practical way to estimate the derivative, allowing us to derive useful insights even from limited or discrete data. Using such techniques is common in science and engineering when precise data expressions are not accessible.