Sigma Coordinate Pressure Calculation

This problem requires calculating the pressure at a specific sigma ($\sigma$) level within a numerical weather model, given the surface pressure at different locations.

Sigma ($\sigma$) Coordinate Definition

The sigma coordinate system is a vertical coordinate system used in meteorology where the vertical coordinate is normalized by the surface pressure ($p_s$). It is defined as:

\(\sigma = \frac{p}{p_s}\)

Here, '$p$' represents the pressure at a certain level, and '$p_s$' is the surface pressure.

Calculating Pressure ($p$) using $\sigma$

To find the pressure ($p$) at a given sigma level, the formula can be rearranged:

\(p = \sigma \times p_s\)

In this question, we need to calculate the pressure for $\sigma = 0.5$.

Step-by-Step Calculation for Locations

We apply the rearranged formula using the given surface pressures for locations A and B.

Location A ($p_{s} = 1000 \text{ hPa}$):

• Surface pressure ($p_{s,A}$) = $1000 \text{ hPa}$.
• Sigma level ($\sigma$) = $0.5$.
• Pressure calculation ($p_A$):

  \(p_A = 0.5 \times 1000 \text{ hPa} = 500 \text{ hPa}\)

Location B ($p_{s} = 500 \text{ hPa}$):

• Surface pressure ($p_{s,B}$) = $500 \text{ hPa}$.
• Sigma level ($\sigma$) = $0.5$.
• Pressure calculation ($p_B$):

  \(p_B = 0.5 \times 500 \text{ hPa} = 250 \text{ hPa}\)

Conclusion

The calculated pressures at $\sigma = 0.5$ are $500 \text{ hPa}$ for Location A and $250 \text{ hPa}$ for Location B. This corresponds to the values presented in Option 2.