Problem 1: Residence Time Calculation The average volume of water stored in the atmosphere is approximately $13,000 \text{ km}^3$. The average global annual precipitation rate (which is equal to the evaporation rate to maintain mass balance) is about $505,000 \text{ km}^3/\text{year}$. Calculate the average residence time of a water molecule in the atmosphere in days.

Problem 2: Basic Water Balance A catchment area of $1000 \text{ ha}$ receives $500 \text{ mm}$ of rainfall in a year. The runoff measured at the outlet is $200 \text{ mm}$. Assuming no change in groundwater storage and negligible groundwater flow, estimate the evapotranspiration losses in mm and the equivalent volume in cubic meters.

Problem 3: Advanced Water Balance (Lake System) A lake has a surface area of $50 \text{ km}^2$. During a specific month, the lake receives a steady inflow of $15 \text{ m}^3/\text{s}$ from rivers and a rainfall of $120 \text{ mm}$. The outflow from the lake is $12 \text{ m}^3/\text{s}$, and the evaporation loss is estimated at $80 \text{ mm}$. Calculate the change in storage volume of the lake over a 30-day month, and determine the corresponding change in lake water level.

Case Study 1: The Aral Sea Crisis Consider the Aral Sea, once the fourth-largest lake in the world. Discuss how human alterations to the water balance equation caused its dramatic shrinkage.

Case Study 2: Urbanization and the Water Balance How does the conversion of a forested watershed into an urban area affect the components of the water balance equation ($P = R + ET + I + \Delta S$)?