Question

In a local Cartesian system, a zonal jet has a form u(y) = u\(_0\) (1 - y\(^2\)/L\(^2\)), for \(-L \le y \le L\). Here, y is the meridional distance measured from the axis of the jet and is positive northward. The vertical component of vorticity of this flow at y=L/2 is \(\underline{\hspace{2cm}}\) s\(^{-1}\). Round off to 3 decimal places.

Hint:

Vorticity is the curl of the velocity field, and in the case of a zonal jet, it can be computed by taking the vertical derivative of the zonal velocity profile.

Answer 1

Correct Answer: 0.009

The vertical component of vorticity (\( \zeta \)) is given by the formula: \[ \zeta = \frac{1}{R} \frac{\partial u}{\partial y}. \] The velocity profile is given as: \[ u(y) = u_0 \left( 1 - \frac{y^2}{L^2} \right). \] Differentiating \( u(y) \) with respect to \( y \), we get: \[ \frac{\partial u}{\partial y} = -2 \frac{y}{L^2} u_0. \] At \( y = \frac{L}{2} \), we substitute into the equation: \[ \zeta = -2 \frac{L/2}{L^2} u_0 = -\frac{u_0}{L}. \] Substituting the values \( u_0 = 50 \, \text{m/s} \) and \( L = 5 \, \text{km} \), we get: \[ \zeta \approx -0.009 \, \text{s}^{-1}. \] Thus, the vertical component of vorticity is \( 0.009 \, \text{s}^{-1} \).