Topographic Retention Basin Drainage (Environmental & Civil Engineering)

An environmental and water resources engineer studies a high-resolution topographic survey contour map of an alluvial valley in the Las Vegas wash basin to design a regional stormwater detention basin. The contour lines represent lines of constant ground surface elevation $E(x, y)$ marked in meters ($\text{m}$) above sea level.

At a prospective retention inlet located at survey coordinate point $P(x_0, y_0)$:

• The contour lines run perfectly horizontal (straight east-to-west, parallel to the $x$-axis).
• Moving vertically northward (in the positive $y$-direction) from point $P$, the contour elevation labels decrease steadily: $105\text{ m} \to 100\text{ m} \to 95\text{ m}$.
• The contour lines representing $5\text{ m}$ elevation drops are spaced tightly at $\Delta y = 10\text{ m}$ intervals on the ground.

1. State the visual rules for determining the sign of partial derivatives $\frac{\partial E}{\partial x}$ and $\frac{\partial E}{\partial y}$ from a contour plot.
2. Determine the signs and instantaneous numerical values (or approximations) of $\frac{\partial E}{\partial x}$ and $\frac{\partial E}{\partial y}$ at point $P$.
3. Explain why the horizontal orientation of the contour line forces $\frac{\partial E}{\partial x} = 0$.
4. In hydrological engineering, stormwater runoff velocity $V$ is proportional to the hydraulic ground gradient $S = \sqrt{(\partial E/\partial x)^2 + (\partial E/\partial y)^2}$. Calculate $S$ at point $P$ and state the direction in which surface water will flow.