# Problem 3 Solution Let's solve each part step by step. --- ## **a) Expression for Horizontal Divergence of Geostrophic Wind** ### **Step 1: Definition of Geostrophic Wind** The geostrophic wind components are: \[ u_g = -\frac{1}{f} \frac{\partial \Phi}{\partial y} \] \[ v_g = \frac{1}{f} \frac{\partial \Phi}{\partial x} \] where: - \( u_g \), \( v_g \): Zonal and meridional geostrophic wind components - \( f \): Coriolis parameter (\( f = 2\Omega \sin\phi \)) - \( \Phi \): Geopotential ### **Step 2: Expression for Horizontal Divergence** By definition, \[ D = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \] For geostrophic wind, \[ D_g = \frac{\partial u_g}{\partial x} + \frac{\partial v_g}{\partial y} \] Substitute the expressions for \( u_g \) and \( v_g \): \[ D_g = \frac{\partial}{\partial x} \left(-\frac{1}{f} \frac{\partial \Phi}{\partial y}\right) + \frac{\partial}{\partial y} \left( \frac{1}{f} \frac{\partial \Phi}{\partial x} \right) \] Assuming \( f \) is constant (valid for small regions): \[ D_g = -\frac{1}{f} \frac{\partial^2 \Phi}{\partial x \partial y} + \frac{1}{f} \frac{\partial^2 \Phi}{\partial y \partial x} \] Since mixed partial derivatives are equal (if \( \Phi \) is smooth): \[ D_g = -\frac{1}{f} \frac{\partial^2 \Phi}{\partial x \partial y} + \frac{1}{f} \frac{\partial^2 \Phi}{\partial x \partial y} = \] ### **Physical Interpretation** **The horizontal divergence of the geostrophic wind is zero** (for constant \( f \)): - This means the geostrophic wind is non-divergent; it neither converges nor diverges horizontally. - Physically, this is because the geostrophic wind flows parallel to isobars (contours of constant pressure) and does not "pile up" or "spread out" air horizontally. --- ## **b) Calculate the Divergence at \( 45^\circ N \) for \( v_g = 10 \) m/s** From part (a), we derived that: \[ D_g = \] So, **the divergence is zero** at any latitude (as long as \( f \) is constant and the wind is geostrophic), including at \( 45^\circ N \). --- ## **Final Answers** ### **a)** **Expression:** \[ D_g = \frac{\partial u_g}{\partial x} + \frac{\partial v_g}{\partial y} = \] **Physical Interpretation:** The geostrophic wind is non-divergent; its flow does not cause horizontal convergence or divergence. --- ### **b)** **The divergence of the geostrophic wind at \( 45^\circ N \) is:** \[ \boxed{} \]