(c) Thermal-Wind Relation Near the Equator ($f=0$)

The thermal-wind relation is:

$$f\frac{\partial u}{\partial z} = \frac{g}{\rho_0} \frac{\partial ... ...{\partial v}{\partial z} = -\frac{g}{\rho_0} \frac{\partial \rho'}{\partial x}$$

Near the equator, the Coriolis parameter $f = 2\Omega \sin \phi$ approaches zero ( $\phi \to 0^\circ$).

• If the horizontal density gradients ( $\frac{\partial \rho'}{\partial y}$ and $\frac{\partial \rho'}{\partial x}$) are non-zero, then for the thermal wind relation to hold as $f \to 0$, the vertical shear terms must become infinite:
  $$\frac{\partial u}{\partial z} = \frac{1}{f} \left(\frac{g}{\rho_0} \frac{\partial \rho'}{\partial y}\right) \to \infty$$   as $$f \to 0$$

• Physically, this infinite vertical shear is **impossible**.