A column of air mass extending from surface to a height of $10 \:\mathrm{km}$ moving eastward along $30^{\circ} \: \mathrm{N}$ strikes a north-south oriented mountain range. While crossing the mountain range, the air mass acquires a relative vorticity of $-3.65 \times 10^{-5} \mathrm{~s}^{-1}$ at the top. If the air mass maintains the same latitude and conserves potential vorticity, the height of the mountain range is $\_\_\_\_\_\_\_$ km . (Round off to the nearest integer.)
[Assume the angular velocity of the Earth is $7.3 \times 10^{-5} \mathrm{~s}^{-1}$ and initial relative vorticity is zero.]
