Grid Spacing Calculation

This solution determines the grid spacing at the equator for a General Circulation Model (GCM) based on its resolution in spherical harmonic waves.

Model Resolution and Wavelength

The GCM uses 256 spherical harmonic waves to cover the globe horizontally. This number indicates the highest zonal wavenumber ($m_{max}$) the model resolves. The shortest wave corresponds to this highest wavenumber.

The wavelength ($\lambda$) of a wave is calculated as the Earth's circumference ($C = 2 \pi R_E$) divided by its wavenumber. At the equator, the shortest wavelength ($\lambda_{min}$) is:

$ \lambda_{min} = \frac{2 \pi R_E}{m_{max}} $

Given Earth's radius ($R_E = 6400 \text{ km}$) and $m_{max} = 256$:

$ \lambda_{min} = \frac{2 \pi \times 6400 \text{ km}}{256} $

$ \lambda_{min} = \frac{12800 \pi}{256} \text{ km} $

$ \lambda_{min} = 50 \pi \text{ km} $

Grid Spacing Determination

The condition "exactly resolve the shortest of those waves" implies that the grid spacing ($\Delta x$) at the equator is set to half the shortest wavelength. This ensures two grid points per wavelength, which is a standard for accurate representation in numerical models.

Therefore, the grid spacing is:

$ \Delta x = \frac{\lambda_{min}}{2} $

Substituting the value of $\lambda_{min}$:

$ \Delta x = \frac{50 \pi \text{ km}}{2} $

$ \Delta x = 25 \pi \text{ km} $

Final Calculation and Rounding

To find the numerical value, we use $\pi \approx 3.14159265$:

$ \Delta x = 25 \times 3.14159265... \text{ km} $

$ \Delta x \approx 78.5398 \text{ km} $

Rounding the result to one decimal place yields 78.5 km.