(e) Conceptual: Small Rossby Number and Quasi Two-Dimensionality

A small Rossby number ( $\mathrm{Ro} \ll 1$) requires the horizontal flow to be in **Geostrophic Balance** (Coriolis $\approx$ Pressure Gradient). For a steady, inviscid flow that is dominated by rotation ( $\mathrm{Ro} \to 0$), the governing equation reduces to $\mathbf{\Omega} \times \mathbf{u} \approx -\frac{1}{2\rho_0} \nabla p$. Taking the curl ( $\nabla \times$) of this balance for a constant rotation vector $\mathbf{\Omega}$ leads to the **Taylor-Proudman Theorem**:

$$(\mathbf{\Omega} \cdot \nabla)\mathbf{u} \approx 0$$

In a local Cartesian system where $\mathbf{\Omega} \approx \Omega \hat{\mathbf{z}}$, this simplifies to $\partial \mathbf{u}/\partial z \approx 0$. This means the flow velocity must be nearly **independent of the vertical coordinate** ($z$). Consequently, large-scale, low $\mathrm{Ro}$ flows tend to be organized in columns ( $\mathbf{u} \approx \mathbf{u}(x, y)$) parallel to the rotation axis, exhibiting **quasi two-dimensional** behavior. This phenomenon is a direct consequence of the **Geostrophic Balance** identified in part (c).

2 Thermal-wind balance

Assume a steady, hydrostatic, geostrophic flow in the Boussinesq approximation:

$$-fv = -\frac{1}{\rho_0}\frac{\partial p}{\partial x}, \qquad fu =... ...\frac{\partial p}{\partial y}, \qquad \frac{\partial p}{\partial z} = -\rho g.$$

Let the density be decomposed as $\rho = \rho_0 + \rho'(x,y,z)$, with $\vert\rho'\vert \ll \rho_0$.

1. Eliminate $p$ to derive the thermal-wind relation:
   $$f\frac{\partial u}{\partial z} = -\frac{g}{\rho_0} \frac{\partial... ...\partial v}{\partial z} = \frac{g}{\rho_0} \frac{\partial \rho'}{\partial x}.$$

2. Discuss physically what kind of stratification corresponds to vertical shear of the geostrophic flow.
3. Explain what happens to the thermal wind relation near the equator $f=0$.

Solutions