If the extra term found in the Bernoulli equation is considered a pressure correction: Where on the surface of the earth (i.e., at what latitude) will this pressure correction be the largest? What is the absolute size of the maximum pressure correction when changes in \( R \) on a streamline are 1 m, 1 km, and \( 10^3 \) km?
[Sol] \(\because\) At any latitude the local rotation rate at the surface of the earth is \( \Omega_z \cos \theta \) where \(\theta\) is zero at the pole and 90° at the equator
\(\therefore\)The pressure correction term is \( \frac{\rho}{2} \Omega_z^2 R^2 \cos^2 \theta \) where \(R\) represents the distance from the local origin of coordinates
\(\max\left( \frac{\rho}{2} \Omega_z^2 R^2 \cos^2 \theta \right) = \frac{\rho}{2} \Omega_z^2 R^2 \text{ when } \theta = 0\), \(\left( \frac{\rho}{2} \Omega_z^2 R^2 \cos^2 \theta \right)_{\theta = 0} = \frac{\rho}{2} \Omega_z^2 R^2 \propto R^2\)
\(\therefore\) consider two points along a streamline where the first point lies at the origin of coordinates and the second lies a distance \(R\) away. The maximum value of the pressure correction will be \[ \frac{\rho}{2} \Omega_z^2 R^2 = \frac{(1.2 \, \text{kg/m}^3)(2\pi/\text{day})^2 (\text{day}/24\,\text{hr})^2 (\text{hr}/3600\,\text{s})^2 R^2}{2} \] \(= 3.2 \times 10^{-9} \, \text{Pa}\) for \(R = 1 \, \text{m}\)
\(= 3.2 \times 10^{-3} \, \text{Pa}\) for \(R = 1 \, \text{km}\)
\(= 3.2 \, \text{kPa}\) for \(R = 10^3 \, \text{km}\)
\(\because\) 1 atmosphere of pressure \(\approx\) 100 kPa \(\therefore\)a body-force potential for the new term can be found, but its impact might only be felt for relatively large atmospheric- or oceanic-scale flowsa body-force potential for the new term can be found, but its impact might only be felt for relatively large atmospheric- or oceanic-scale flows